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+{"text":"\\section{Introduction}\n \\label{sec:introduction}\n\nQuantum chromodynamics (QCD)\nshows a variety of phases, and in passing over a phase\nboundary one would encounter either first (discontinuous) or second\norder (continuous) transition, depending on temperature, density, quark\nmasses, the number of flavors, etc.\nChiral phase transition of QCD with two massless quarks at the\nvanishing chemical potential has been studied with various approaches\nfor a long time, since it provides us with a solid basis in the study\nof $2+1$-flavor QCD in the real world.\nNevertheless, the nature of the transition of this relatively simple\nsystem is yet ambiguous, and is counted as one of the longstanding\nproblems.\n\nBased on the universality argument and the results of the leading\norder $\\epsilon$ expansion, Pisarski and Wilczek analyzed the\nrenormalization group (RG) flow of the three-dimensional scalar field\ntheory, which shares the same internal symmetry with massless QCD around\nthe critical temperature ($T_c$), and pointed out that, in the\ntwo-flavor case, the order of the transition could crucially depend on\nthe presence (or the absence) of the flavor singlet axial ($U_A(1)$)\nsymmetry at $T_c$~\\cite{Pisarski:1983ms}.\nIf $U_A(1)$ symmetry is largely violated at $T_c$, the second order\nphase transition with the $O(4)$ scaling becomes possible although not\nmandatory.\nOn the other hand, when the symmetry is effectively and fully restored at\n$T_c$, the leading order calculation of the $\\epsilon$ expansion\nsuggests no infrared fixed point (IRFP), and hence the second order\nphase transition is excluded.\nBut, later, further studies using different advanced techniques\nfound evidence of IRFP, and the confirmation of the presence of the\nIRFP is under active\ninvestigation~\\cite{Butti:2003nu,Aoki:2012yj,Pelissetto:2013hqa,Nakayama:2014sba}.\nThus, the transition in this case again can be either of first or\nsecond order.\nRecently, a novel possibility is pointed out, following the RG flow\nanalysis: in the presence of small but finite $U_A(1)$ symmetry\nbreaking, the system may undergo the second order transition with the\n$O(4)$ scaling but one of the critical exponents related to the\nscaling dimension of the leading irrelevant operator is different from\nthat of the $O(4)$~\\cite{Sato:2014axa}, although again the second order\ntransition is not mandatory.\n\nNumerical simulations of QCD on the lattice can, in principle,\ndetermine the order of the transition as well as the universality class,\nto which massless two-flavor QCD belongs, by performing the scaling\nstudy~\\cite{Karsch:1993tv,Karsch:1994hm,Iwasaki:1996ya,AliKhan:2000iz,\nD'Elia:2005bv,Bonati:2009yg,RBCBi09,Kanaya:2010qj,Bonati:2014kpa}.\nHowever, it is not easy to keep all the systematic and statistical\nuncertainties under control in the chiral limit due to large\ncomputational costs.\nFurthermore, it appears that, in practice, the standard scaling study\nmay not be efficient enough to distinguish the first and the second\norder transitions because the scaling functions are similar between the\n$Z_2$ and $O(4)$ universality classes~\\cite{Brandt:2013mba}.\nWith lattice QCD simulations, one can also study the presence (or\nabsence) of $U_A(1)$ symmetry through the Dirac spectrum.\nFor recent progress, see, for example,\nRefs.~\\cite{Cossu:2013uua,Buchoff:2013nra,Bhattacharya:2014ara,Dick:2015twa}.\n\nClarifying this point is important not only for understanding the QCD phase\ndiagram but also for the scenario of the axion dark matter.\nThe axion abundance is essentially determined by the temperature\ndependence of the topological susceptibility, $\\chi_t(T)$, which\nvanishes when $U_A(1)$ symmetry is fully and effectively restored.\nIf $\\chi_t$ vanishes very rapidly right above $T_c$, too many axions\nwould be produced, and the axion dark matter scenario becomes hard or is\neven excluded~\\cite{Kitano:2015fla}, depending on how rapidly it\nvanishes.\nThe lattice studies to test the axion dark matter scenario has recently\nbegun in the quenched\napproximation~\\cite{Kitano:2015fla,Berkowitz:2015aua,Borsanyi:2015cka}.\n\nIn this paper, we follow the approach proposed in\nRef.~\\cite{Ejiri:2012rr}, in which the phase transition of two-flavor\nQCD is studied by adding many extra heavy quarks.\nWe call this the many flavor approach.\nThe transition of two-flavor QCD at a finite quark mass is known to be\na crossover, but it can turn to a first order transition when adding\nmany extra flavors.\nThis property is used to explore the nature of the phase transition of\nmassless two-flavor QCD.\nThe extra quarks are incorporated in the form of the hopping parameter\nexpansion (HPE) through the reweighting.\nThen the number of the extra flavors ($N_f$) and their mass parameter\n($\\kappa_h$) appear in a single parameter\n\\begin{eqnarray}\n     h\n &=& 2\\,N_f\\,(2\\kappa_h)^{N_t} \\ ,\n  \\label{eq:hcdef}\n\\end{eqnarray}\nwhere $N_t$ denotes the number of lattice sites in the temporal\ndirection.\nWe determine the critical value of the parameter ($h_c$) at which the\nfirst order and the crossover regions are separated, and examine its\ndependence on the two-flavor mass.\nThe order of the transition for a given $h$ is discriminated by the\nshape of the constraint effective potential at $T_c$, which is\nconstructed from the probability distribution function (PDF) for a\ngeneralized plaquette.\nNamely, it is discriminated by whether the potential at $T_c$ is in\nsingle- or double-well shape.\nIt is important to note that, in the determination of $h_c$ in this\napproach, the convergence of the HPE is not the matter since we can\nconsider arbitrary small $\\kappa_h$ by considering arbitrary large $N_f$ as\nseen from Eq.(\\ref{eq:hcdef}).\n\nWe perform an exploratory study on $N_t=4$ and try to see how $h_c$\ndepends on the two-flavor mass.\nThen, $h_c$ is found to stay constant against the change of the\ntwo-flavor mass in the range we have studied and to remain positive and\nfinite in the chiral limit.\nSince the two-flavor system is equivalent to the $2+N_f$-flavor system\nwith $h=0$, our result suggests that massless two-flavor QCD belongs to\nthe region of second order phase transition.\nThis kind of extension of QCD is useful also for the study of the phase\nstructure in the presence of finite chemical\npotential~\\cite{Ejiri:2012rr}.\nThe similar approach but introducing a finite imaginary chemical\npotential is taken in Ref.~\\cite{Bonati:2014kpa}.\n\nThe paper is organized as follows.\nAfter the central idea of many flavor approach is explained in\nSec.~\\ref{sec:approach}, the method is described in detail in\nSec.~\\ref{sec:method}.\nThe lattice setup and the main part of this paper are given in\nSec.~\\ref{sec:numerical_results}.\nIn Sec.~\\ref{sec:consistency-check}, two independent analyses are\nperformed for the consistency check.\nFinally, the conclusion and perspectives are stated in\nSec.~\\ref{sec:summary}.\nThe preliminary result of this work is available in\nRef.~\\cite{Ejiri:2015pva}.\n\n\n\\section{Many flavor approach}\n\\label{sec:approach}\n\nThe central idea of many flavor approach is outlined.\nFigure~\\ref{fig:columbia-plot} (a) shows the so-called Columbia plot for\n$2+1$-flavor QCD~\\cite{Brown:1990ev}, which summarizes the present\nknowledge on the mass dependent nature of the phase transition of\nQCD as a function of $m_{ud}$ and $m_s$.\nThe physical point is believed to be located in the crossover\nregion~\\cite{RBCBi09,Aoki:2006we}.\nThe plot tells us that there are two distinct first order regions lying\naround the quenched limit ($m_{ud}=m_s=\\infty$) and the chiral limit of\nthree-flavor QCD ($m_{ud}=m_s=0$), respectively.\nIn what follows, we focus on the latter.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/columbia-2+1_nopoint_cap_uds_rev20151029.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/columbia-2+nf_nopoint_cap_udh_rev20151029.eps}\\\\\n (a)\\ 2+1 flavor QCD&\n (b)\\ 2+$N_f$ flavor QCD\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{Basic idea of \"many flavor approach\".}\n\\label{fig:columbia-plot}\n\\end{center}\n\\end{figure}\n\nOur interest is in whether the massless two-flavor QCD point ($m_{ud}=0$\nand $m_s=\\infty$) is inside the first order region or not and hence in\nthe shape of the first order region.\nIf it ends at a finite $m_s$, called the tricritical point, massless\ntwo-flavor QCD is on the line of the second order phase transition (the\nsolid curve on the $m_{ud}=0$ line).\nOn the other hand, if the first order region extends to $m_s=\\infty$,\nmassless two-flavor QCD should undergo the first order transition ({\\it\ni.e.} the dotted curve near the $m_{ud}=0$ line).\nIn either case, if we could resolve which of the solid or dotted curves\nis realized, we would be able to answer the question.\nHowever, it is difficult to trace the critical line for the $2+1$-flavor\ncase, because the critical line is located in the small $m_s$ region\nwhen $m_{ud}$ is moderately\nsmall~\\cite{RBCBi09,Ding:2013lfa,Jin:2014hea}.\n\nThe situation will change in $2+N_f$-flavor QCD as shown in\nFig.~\\ref{fig:columbia-plot} (b).\nThe bottom-right corner of Fig.~\\ref{fig:columbia-plot} (b) is of\nfirst order for $N_f \\geq 3$.\nThe critical line is expected to move upward as $N_f$ increases, and for\nsufficiently large $N_f$ it could enter the region where the hopping\nparameter expansion from the static limit (crosses at\n$m_h=\\infty$) works well (above the dashed line).\nThen, one should be able to easily identify how the critical line runs\nas a function of $m_{ud}$.\nIf the critical heavy mass $m_h^c(m_{ud})$ remains finite in the\n$m_{ud}\\to 0$ limit, it immediately means that massless two-flavor QCD\ncorresponding to the point $(m_{ud},m_h)=(0,\\infty)$ is the outside of\nthe first order region.\nAn important remark is that, in the limit of $m_s\\to\\infty$ or\n$m_h\\to\\infty$, both $2+1$- and $2+N_f$-flavor QCD end up with the same\ntheory, which we want to study.\nThus, the original question is simplified to the one whether the\ncritical heavy mass in the chiral limit stays finite or not.\n\nWhile most of the current knowledge on the shape of the first order\nregion are only qualitative, in a context of the Taylor-expanded\nreweighting method~\\cite{BS02} we can derive a solid statement about the\nslope at $m_{ud}=m_h$ if the same quark action is employed for two- and\n$N_f$-flavors.\nSuppose that $m_{ud}=m + \\Delta m_{ud}$ and $m_{h}=m + \\Delta m_h$ and\nexpand the logarithm of the quark determinants in terms of $\\Delta\nm_{ud}$ and $\\Delta m_h$.\nThen, one will be aware that the partition function does not change as\nlong as $\\Delta m_h=- 2\\, \\Delta m_{ud}\/N_f$ and hence that physics is\nidentical along the line of slope $-2\/N_f$ near $m_{ud} = m_h$ line in\nthe $(m_{ud}, m_h)$ plane~\\cite{ejiri04}.\nThis means that the critical line in $2+N_f$-flavor QCD crosses the line\nof $m_{ud}=m_h$ with a slope milder than the $2+1$ flavor case.\n\nIn Ref~\\cite{Ejiri:2012rr}, it is demonstrated that by adding extra\nflavors the end point (or the critical line) indeed enters the region\nreachable by the hopping parameter expansion.\nIn this paper, we examine the light quark mass dependence of the\nend point.\nAs we will explain below, $m_h^c(m_{ud})$ seems to remain finite in\nthe chiral limit of $m_{ud}$, suggesting the phase transition of\nmassless two-flavor QCD is of second order.\n\n\n\\section{Calculational method}\n\\label{sec:method}\n\nWe first generate two-flavor configurations at finite temperatures\nfollowing the standard hybrid Monte Carlo method.\nUsing the hopping parameter expansion and the reweighting method,\nwe incorporate extra $N_f$ flavors of heavy quarks into those\nconfigurations, and measure the probability distribution function\nfor a generalized plaquette to construct the constraint effective\npotential for $2+N_f$-flavor QCD.\n\nAfter the HPE, the hopping parameter for heavy quarks $\\kappa_h$ and\n$N_f$ appear only in a single parameter, $h$ [see Eq.~(\\ref{eq:h})].\nIf the parameter $h$ is in the crossover region, the effective potential\nshould take a single-well shape at the pseudocritical temperature,\n$T_{pc}$.\nOn the other hand, when the parameter $h$ enters the first order region,\na double-well shape should emerge at $T_c$.\nBy scanning $h$, we determine the critical value $h_c$, at which the\nfirst order and the crossover regions are separated.\nThe critical value $h_c$ is determined at four values of two-flavor\nmass to see the light quark mass dependence of $h_c$.\nIn the following, the calculational procedure is described in detail.\n\nThe PDF was introduced in Refs.~\\cite{Bruce:1981,Binder:1981sa} and has\nbeen extensively used in various fields to study the critical\nproperties of various materials~\\cite{Plascak:2013} or the phase\ndiagrams of QCD~\\cite{whot11}.\nIn our study of 2+$N_f$-flavor QCD, the PDF $w$ for a quantity $\\hat X$\nis defined by\n\\begin{eqnarray}\n    w(X; \\beta, \\kappa_l, \\kappa_h,N_f)\n&=& \\int \\!\\! {\\cal D} U\\, \\delta(X- \\hat{X}) \\ \n    \\big[\\det M(\\kappa_h)\\big]^{N_f}\n    e^{-S_{\\rm gauge}(\\beta) - S_{\\rm light}(\\kappa_l)},\n\\label{eq:pdist}\n\\end{eqnarray}\nwhere $S_{\\rm gauge}(\\beta)$ and $S_{\\rm light}(\\kappa_l)$ are the lattice\nactions for the gauge field and two flavors of light quarks,\nrespectively.\n$\\beta=6\/g_0^2$ is the simulation parameter setting the temperature\nthrough the lattice spacing, and $\\kappa_l$ is the light quark mass\nparameter.\nNote that the method  described below works for any kinds of light quark\naction unless it contains $\\beta$ dependent coefficients.\nThe action for the $N_f$ extra flavors is written in the determinant\nform in Eq.~(\\ref{eq:pdist}), where $M(\\kappa_h)$ is the lattice Dirac\noperator for heavy quarks with a mass parameter $\\kappa_h$.\nA quantity to be constrained, $\\hat X$, is basically arbitrary, but the\norder parameter would be the most natural choice if it is available.\nIn this paper, following previous\nworks~\\cite{Ejiri:2007ga,whot11,Ejiri:2012rr,Ejiri:2013lia}, $\\hat X$ is\nchosen to be the generalized plaquette\n\\begin{eqnarray}\n  \\hat P\n= c_0\\, \\hat W_P + 2 c_1\\, \\hat W_R\\ ,\n\\end{eqnarray}\nwhere $\\hat W_P$ and $\\hat W_R$ denote the averaged plaquette and\nrectangle, respectively, and $c_0$ and $c_1$ satisfying $c_0=1-8 c_1$\nare the improvement coefficients for lattice gauge action.\nIn terms of $\\hat P$, the gauge action is written as\n\\begin{eqnarray}\n    S_{\\rm gauge}(\\beta)\n= - 6\\, N_{\\rm site}\\, \\beta\\, \\hat P\\, ,\n\\end{eqnarray}\nwhere $N_{\\rm site}= N_{\\rm s}^3 \\times N_t$ represents the number of\nsites in four-dimensional lattice volume.\nOur main analysis is carried out with $\\hat X=\\hat P$, but the\ncalculation for $\\hat X=\\hat L$ with $\\hat L$ the real part of the\nPolyakov loop averaged over spatial sites is also performed as a\nconsistency check (see Sec.~\\ref{subsec:polyakov}).\nChoosing $\\hat X=\\hat P$ brings a great simplification in the\nnumerical analysis as explained below.\nIn principle, we could choose other quantities, {\\it e.g.} the\nchiral condensate, to be $\\hat X$.\nBut, whenever a quantity other than $\\hat P$ is chosen, we lose not only\nthe advantage for $\\hat X=\\hat P$ but also the accuracy in the results\nas demonstrated in Sec.~\\ref{subsec:polyakov} for $\\hat X=\\hat L$.\n\nWith the PDF thus obtained, the constraint effective potential $V$ is\ncalculated by\n\\begin{eqnarray}\n       V(X;\\beta,\\kappa_l,\\kappa_h,N_f)\n&=&  - \\ln w(X;\\beta,\\kappa_l,\\kappa_h,N_f)\n \\label{eq:effective-potential}\n\\end{eqnarray}\nIn practice, the PDF is not directly accessible, and hence we instead\ncalculate the histogram defined by\n\\begin{eqnarray}\n    H(X;\\beta,\\kappa_l,\\kappa_h,N_f)\n&=& \\frac{w(X;\\beta,\\kappa_l,\\kappa_h,N_f)}{Z(\\beta,\\kappa_l,\\kappa_h,N_f)}\\ ,\n \\label{eq:histogram}\n\\end{eqnarray}\nwhere $Z(\\beta)$ is the partition function.\nNote that $Z(\\beta)$ is not calculable, but its ratio at two\ndifferent $\\beta$ values is calculable~\\cite{Ejiri:2008xt}.\n\nIn order to see the shape of the potential, we need to calculate the\npotential over a certain range of $X$.\nHowever, a simulation at a single $\\beta$ provides the potential only in\na limited range of $X$.\nThus, the potential calculated at a certain $\\beta$ needs to be\ntranslated to that at other $\\beta$.\nWe call the temperature, at which we want to calculate the potential,\nthe reference temperature (or $\\beta_{\\rm ref}$).\nThen, the potential is calculated as\n\\begin{eqnarray}\n&&      V(X;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f) + \\ln Z(\\beta^*,\\kappa_l',\\kappa_h',N_f')\n\\nonumber\\\\\n&=& - \\ln H(X';\\beta',\\kappa_l',\\kappa_h',N_f')\n    - \\ln \\left(\\frac{w(X'';\\beta'',\\kappa_l'' ,\\kappa_h'' ,N_f'')}\n                     {w(X';\\beta'  ,\\kappa_l',\\kappa_h',N_f')}\n          \\right)\n    - \\ln \\left(\\frac{w(X;\\beta_{\\rm ref},\\kappa_l ,\\kappa_h ,N_f)}\n                     {w(X'';\\beta''       ,\\kappa_l'',\\kappa_h'',N_f'')}\n          \\right)\n\\nonumber\\\\\n& & - \\ln \\frac{Z(\\beta',\\kappa_l',\\kappa_h',N_f')}{Z(\\beta^*,\\kappa_l',\\kappa_h',N_f')}\n\\ .\n \\label{eq:effective-potential-2}\n\\end{eqnarray}\nAll the intermediate quantities such as $\\beta'$, $X'$, $\\kappa_l'$, $\\kappa_h'$\nand $N_f'$ are arbitrary as well as $\\beta^*$.\nFor a practical reason, we take\n\\begin{eqnarray}\n X'=X''=X,\\ \\ \\\n \\beta'' = \\beta_{\\rm ref},\\ \\ \\\n \\kappa_l' = \\kappa_l'' = \\kappa_l,\\ \\ \\\n \\kappa_h' = \\kappa_h'' = 0\\,\\ \\ \\\n N_f' = N_f'' = 0\\ ,\n \\label{eq:setup}\n\\end{eqnarray}\nand $\\beta^*$ is chosen to be the vicinity of $\\beta_{\\rm ref}$.\n\nIn the following, we take $\\hat X=\\hat P$, then the effective potential\nis simplified as\n\\begin{eqnarray}\n    V(P; \\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f) + \\ln Z(\\beta^*,\\kappa_l',\\kappa_h',N_f')\n&=&   V_{\\rm light}(P; \\beta_{\\rm ref},\\kappa_l)\n    - \\ln R(P;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\\ .\n \\nonumber\\\\\n\\label{eq:vefftrans}\n\\end{eqnarray}\nThe first term is defined by\n\\begin{eqnarray}\n    V_{\\rm light}(P; \\beta_{\\rm ref},\\kappa_l)\n= - \\ln H(P;\\beta,\\kappa_l,0,0)  \n  - 6\\,N_{\\rm site}\\,(\\beta_{\\rm ref} - \\beta)\\,P\n  - \\ln \\frac{Z(\\beta,\\kappa_l,0,0)}{Z(\\beta^*,\\kappa_l,0,0)}\n\\label{eq:V-two-flavor}\n\\end{eqnarray}\nand represents the constraint effective potential for two flavors alone.\nThe second term of Eq.~(\\ref{eq:vefftrans}) is defined by\n\\begin{eqnarray}\n    R(P;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\n&=& \\left\\langle\n    \\displaystyle\n    \\left[ \\det M(\\kappa_h) \\right]^{N_f}\n    \\right\\rangle_{P: {\\rm fixed},(\\beta_{\\rm ref},\\kappa_l)}, \\ \n\\label{eq:lnr}\\\\\n    \\langle \\cdots \\rangle_{P: {\\rm fixed}, (\\beta_{\\rm ref},\\kappa_l)}\n&\\equiv&\n    \\frac{\\langle \\delta(P- \\hat{P}) \\cdots \\rangle_{(\\beta_{\\rm ref},\\kappa_l)}}\n         {\\langle \\delta(P- \\hat{P}) \\rangle_{(\\beta_{\\rm ref},\\kappa_l)}}\\ ,\n\\label{eq:exp-fixedP}\n\\end{eqnarray}\nwhere $\\langle \\cdots \\rangle_{(\\beta,\\kappa_l)}$ denotes the ensemble\naverage over two-flavor configurations generated with $\\beta$ and $\\kappa_l$.\nIt is important to note that, separating the effective potential into\nthe two-flavor part and the extra heavy part as in\nEq.~(\\ref{eq:vefftrans}), the latter becomes independent of\n$\\beta_{\\rm ref}$.\nThe reason is as follows.\nDue to the operator $\\delta(P-\\hat P)$, the factor of\n$\\exp(6 N_{\\rm site}\\beta_{\\rm ref} P)$ comes out of the brackets in\nEq.~(\\ref{eq:exp-fixedP}).\nSince this factor cancels between the numerator and the denominator in\nEq.~(\\ref{eq:exp-fixedP}), $\\beta_{\\rm ref}$ dependence disappears.\nThis simplification takes place only for $\\hat X=\\hat P$.\n\nFor the extra heavy quarks, we employ the unimproved Wilson fermion\nbecause it suffices for the present purpose.\nFor sufficiently small $\\kappa_h$, the determinant in Eq.~(\\ref{eq:lnr}) can\nbe approximated by the leading order of the HPE as\n\\begin{eqnarray}\n   \\ln \\left[ \\det M (\\kappa_h) \\right]^{N_f}\n&\\approx&\n   N_f\n    \\left( 288\\, N_{\\rm site}\\, \\kappa_h^4\\, \\hat W_P\n         + 12 N_s^3 (2 \\kappa_h)^4\\, \\hat{L}\n    \\right)\n\\nonumber\\\\\n&=& 6\\,N_s^3\\,h\\,\\hat Y\n\\label{eq:detmw-1}\\\\\n&=&   9\\,N_{\\rm site}\\,\\frac{h}{c_0}\\, \\hat P\n    + 6\\,N_s^3\\,h\\,\\hat Z \\ ,\n\\label{eq:detmw-2}\n\\end{eqnarray}\nwhere the following quantities have been introduced:\n\\begin{eqnarray}\n    h\n&=& 2\\,N_f\\,(2 \\kappa_h)^4\\ ,\n\\label{eq:h}\\\\\n    \\hat Y\n&=& 6\\,\\hat W_P + \\hat{L}\\ ,\n\\label{eq:Y}\\\\\n    \\hat Z\n&=& - \\frac{12\\,c_1}{c_0}\\,\\hat W_R + \\hat{L}\\ .\n\\label{eq:Z}\n\\end{eqnarray}\nHere and hereafter, $N_t=4$ is assumed because we take that value in\nnumerical simulations, but the extension to other values of $N_t$ is\nstraightforward though the expression becomes complicated.\nThen, $\\ln R$ in Eq.~(\\ref{eq:vefftrans}) can be approximated as\n\\begin{eqnarray}\n    \\ln R(P;\\kappa_l,h)\n&\\approx&\n    \\ln \\left\\langle\n    \\displaystyle\n    \\exp\\left( 6\\,N_s^3\\,h\\,\\hat Y \\right)\n    \\right\\rangle_{P: {\\rm fixed},(\\beta,\\kappa_l)}\n\\label{eq:r-1}\\\\\n&=&  9\\,N_{\\rm site}\\,\\frac{h}{c_0}\\, P\n   + \\ln R'(P;\\kappa_l,h)\n\\label{eq:r-2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n    \\ln R'(P;\\kappa_l,h)\n&=& \\ln \\left\\langle\n    \\displaystyle\n    \\exp\\left( 6\\,N_s^3\\,h\\,\\hat Z \\right)\n    \\right\\rangle_{P: {\\rm fixed},(\\beta,\\kappa_l)}\\ .\n\\label{eq:rdash}\n\\end{eqnarray}\nAlthough Eqs.~(\\ref{eq:r-1}) and (\\ref{eq:r-2}) are algebraically\nidentical, the equality is not necessarily trivial in numerical data\nbecause the $\\delta$ function is approximated by\n\\begin{eqnarray}\n        \\delta(x)\n \\approx 1\/(\\Delta \\sqrt{\\pi}) \\exp[-(x\/\\Delta)^2]\\ .\n \\label{eq:delta-func}\n\\end{eqnarray}\nThen, the difference can arise when\n\\begin{eqnarray}\n  \\big\\langle e^{6 N_s^3 h \\hat Y}\n          \\exp\\left[-(P-\\hat P)^2\/\\Delta^2\\right]\n  \\big\\rangle_{(\\beta,\\kappa_l)}\n- e^{9 N_{\\rm site} \\frac{h}{c_0}P}\n  \\big\\langle e^{6 N_s^3 h \\hat Z}\n  \\exp\\left[-(P-\\hat P)^2\/\\Delta^2\\right]\n  \\big\\rangle_{(\\beta,\\kappa_l)}\n  \\neq 0  ,\\nonumber\n\\end{eqnarray}\nwhich should vanish for sufficiently small $\\Delta$, but then the\nstatistical error will enlarge.\nIn the following analysis, both expressions are examined to check\nthe consistency.\n\nIt is important to note that, after the HPE, the number of extra heavy\nflavors ($N_f$) and their mass parameter ($\\kappa_h$) appear only in a single\nparameter $h$, Eq.~(\\ref{eq:h}).\nBecause of this, $\\kappa_h$ and $N_f$ have been replaced by $h$ in the\narguments of $R$ and $R'$, and our purpose turns to finding the critical\nvalue of $h$, $h_c$.\nIt should be also noted that the second derivatives of\nEqs.~(\\ref{eq:r-1}) and (\\ref{eq:rdash}) with regard to $P$ are\nidentical because the difference is proportional to $P$.\n\nOne side remark is below.\nThus far, we have restricted the extra heavy quarks to be degenerate.\nBut the extension to the nondegenerate case is straightforward by\ninterpreting $h$ as $h=2 \\sum_{f=1}^{N_f} (2 \\kappa_h)^{N_t}$.\nIn the following, we only consider the degenerate case for simplicity.\n\nChoosing $\\hat X=\\hat P$ significantly simplifies the procedure to find\nthe critical value of $h$~\\cite{Ejiri:2007ga,Ejiri:2012rr} as follows.\nWe are interested in the shape of the potential at the (pseudo)critical\ntemperature, which requires $\\beta_{\\rm ref}$ to be tuned to its\n(pseudo)critical value, $\\beta_{c}$ (or $\\beta_{pc}$).\nThis tuning can be totally skipped if we look at the curvature, {\\it\ni.e.} the second derivative of the potential with respect to $P$,\nbecause it is independent of $\\beta_{\\rm ref}$.\n$R$ is independent of $\\beta_{\\rm ref}$ as stated above.\n$V_{\\rm light}$ depends on $\\beta_{\\rm ref}$, but its second derivative\ndoes not as explained below.\n\nThe finite temperature transition of two-flavor QCD is always a\ncrossover for the two-flavor masses adopted in this paper.\nThen, at any temperatures, the shape of the PDF (or equivalently\nhistogram) for $\\hat P$ in two-flavor QCD can be well approximated,\naround the peak, by Gaussian form,\n\\begin{eqnarray}\n   w(P;\\beta,\\kappa_l,0,0)\\big|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n\\propto\n   \\exp\\left[ - \\frac{6\\,N_{\\rm site}\\,\n   \t              (\\,P - \\bar P(\\beta,\\kappa_l)\\,)^2}\n                     {2\\,\\chi_P(\\beta,\\kappa_l)}\n       \\right]\\ ,\n\\end{eqnarray}\nwhere $\\bar P(\\beta,\\kappa_l)=\\langle \\hat P \\rangle_{\\beta,\\kappa_l}$ is the\naverage of generalized plaquette at $\\beta$ and $\\kappa_l$, and $\\chi_P$\nis the susceptibility of $P$, given by\n\\begin{eqnarray}\n  \\chi_P(\\beta,\\kappa_l)\n= 6 N_{\\rm site}\n  \\langle\\, (\\,\\hat P - \\bar P(\\beta,\\kappa_l)\\, )^2\\,\n  \\rangle_{\\beta,\\kappa_l}\\ .\n\\end{eqnarray}\nSubstituting this into Eq.~(\\ref{eq:V-two-flavor}) yields, up to a\nconstant shift,\n\\begin{eqnarray}\n      V_{\\rm light}(P; \\beta_{\\rm ref},\\kappa_l)\n      \\big|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n&=&   \\frac{6\\,N_{\\rm site}\\,\n          (\\, P - \\bar P(\\beta,\\kappa_l)\\,)^2}\n\t {2\\,\\chi_P(\\beta,\\kappa_l)}\n    - 6(\\beta_{\\rm ref} - \\beta)\\,N_{\\rm site}\\,P\n    \\ .\n\\end{eqnarray}\nThen, the first and second derivatives are given by\n\\begin{eqnarray}\n      \\frac{d V_{\\rm light}(P;\\beta_{\\rm ref},\\kappa_l)}{dP}\n      \\bigg|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n&=&   \\frac{6\\,N_{\\rm site}\\,\n            (\\, P - \\bar P(\\beta,\\kappa_l)\\,)}\n\t   {\\chi_P(\\beta,\\kappa_l)}\n    - 6(\\beta_{\\rm ref} - \\beta)\\,N_{\\rm site}\\ ,\n\\label{eq:d1V0}\n\\\\\n      \\frac{d^2 V_{\\rm light}(P;\\kappa_l)}{dP^2}\n      \\bigg|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n&=&   \\frac{6\\,N_{\\rm site}}{\\chi_P(\\beta,\\kappa_l)}\\ .\n\\label{eq:d2V0}\n\\end{eqnarray}\nThus, we can calculate the curvature of the two-flavor part\nby collecting $\\chi_P(\\beta,\\kappa_l)$ obtained at various $\\beta$.\nImportantly, Eq.~(\\ref{eq:d2V0}) is independent of\n$\\beta_{\\rm ref}$.\nIn summary, the curvature of the total effective potential\n\\begin{eqnarray}\n  \\frac{d^2 V(P;\\beta_{\\rm ref},\\kappa_l,h)}{dP^2}\n= \\frac{d^2 V_{\\rm light}(P;\\beta_{\\rm ref},\\kappa_l)}{dP^2}\n- \\frac{d^2 \\ln R(P; \\kappa_l, h)}{dP^2}\\ ,\n\\label{eq:curvature}\n\\end{eqnarray}\nis independent of $\\beta_{\\rm ref}$.\n\nThe procedure to identify $h_c$ in the chiral limit of two flavors goes\nas follows.\nAt $h=0$, the contribution of the extra heavy flavors is trivially zero,\nand the system is reduced to two-flavor QCD, where the transition is a\ncrossover.\nTherefore, the second derivative of the potential is always positive.\nAs $h$ is increased from zero, the minimum of the curvature takes zero\nat some point, which gives $h_c$.\nIn this procedure, one needs not tune $\\beta_{\\rm ref}$ to\n$\\beta_{\\rm pc}$ or $\\beta_{\\rm c}$, because the curvature is\nindependent of the temperature or $\\beta_{\\rm ref}$.\nThis simplification does not occur in general, and one such example\nis explicitly shown in Sec.~\\ref{subsec:polyakov}.\nBy looking at the light quark mass dependence of $h_c(\\kappa_l)$, we try to\nextract $h_c$ in the chiral limit.\n\n\n\\section{Numerical results}\n\\label{sec:numerical_results}\n\\subsection{Simulation parameters}\n\\label{subsec:lat-para}\n\nFollowing Ref.~\\cite{whot10}, we take the Iwasaki gauge action\n($c_1=-0.331$) and the $O(a)$-improved Wilson fermion action with the\nperturbatively improved $c_{\\rm sw}$ for two flavors of light quarks.\nSimulations are performed on $N_{\\rm site}=16^3\\times 4$ lattices with\n25 to 32 $\\beta$ values at each of four $\\kappa_l$,\nand 10,000 to 40,000 trajectories have been accumulated at each\nsimulation point.\nFour light quark masses are ranging from $\\kappa_l=0.145$ to 0.1505.\nFigure \\ref{fig:histogram-action} shows the histogram of the generalized\nplaquette.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.1450-csw1.650-nosmr-apbc-sld.eps}& \n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.1475-csw1.677-nosmr-apbc-sld.eps}\\\\\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.150-csw1.707-nosmr-apbc-sld.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.1505-csw1.712-nosmr-apbc-sld.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{Histogram of the generalized plaquette at four values of\n $\\kappa_l$.}\n\\label{fig:histogram-action}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[t]\n \\centering\n \\begin{tabular}{cc|ccccc}\n $\\kappa_l$ & $c_{\\rm sw}$ & $\\beta_{\\rm pc}$ & $a m_\\pi$ & $a m_\\rho$\n& $m_\\pi\/m_\\rho$ & $a m_{\\rm pcac}$\\\\\n  \\hline\n   0.1450 & 1.650 & 1.778 &0.779(1) & 1.172(\\ 2)& 0.665(33)\n          & 0.0535(1)\\\\\n   0.1475 & 1.677 & 1.737 & 0.651(1) & 1.130(\\ 5)& 0.576(28)\n          & 0.0350(2)\\\\\n   0.1500 & 1.707 & 1.691 & 0.514(2) & 1.099(10) & 0.468(24)\n          & 0.0202(2)\\\\\n   0.1505 & 1.712 & 1.681 & 0.495(2) & 1.082(13) & 0.458(23)\n          & 0.0186(2)\\\\\n \\end{tabular}\n \\caption{Simulation parameters ($\\kappa_l$ and $c_{\\rm sw}$) and the\n pseudocritical $\\beta$ ($\\beta_{\\rm pc}$) in two-flavor QCD,\n determined from the peak of the susceptibility for the generalized\n plaquette.\n $m_\\pi\/m_\\rho$ and other quantities are determined on a $16^3\\times 32$\n lattice at $\\beta_{\\rm pc}$ for each $\\kappa_l$.}\n \\label{tab:spectrum}\n\\end{table}\n\nAt the pseudocritical point $\\beta_{\\rm pc}$ for each $\\kappa_l$, we carried\nout zero temperature simulations on $16^3\\times 32$ lattices to find the\nmass ratio of pseudoscalar and vector mesons, $m_\\pi\/m_\\rho$, and the\nquark mass defined through the partially conserved axialvector current\n(PCAC),\n\\begin{eqnarray}\n    am_{\\rm pcac}\n&=& \\frac{\\langle \\sum_{\\vec x}\n          (A_4(N_t\/2+1,\\vec x)-A_4(N_t\/2-1,\\vec x)) P(0)\\rangle}\n         {4\\, \\langle \\sum_{\\vec x}P(N_t\/2,\\vec x) P(0) \\rangle}\\ ,\n\\end{eqnarray}\nwhere $A_4(x)$ and $P(x)$ are the flavor nonsinglet, local axialvector\nand pseudoscalar operator, respectively, and $N_t=32$.\nThen, the four values of $\\kappa_l$ in this paper turn out to cover\n$0.46 < m_\\pi\/m_\\rho < 0.66$, or $0.019 < am_{\\rm pcac} < 0.054$.\nThese results are tabulated in Table~\\ref{tab:spectrum}.\n\n\n\\subsection{Main results}\n\\label{subsec:main-result}\n\nWe present the numerical results for the two terms in the right-hand\nside of Eq.~(\\ref{eq:curvature}) separately, and focus on the second\nterm first.\nWith the approximated $\\delta$ function (\\ref{eq:delta-func}),\n$\\ln R(P;\\kappa_l,h)$ [Eq.~(\\ref{eq:r-1})] and $\\ln R'(P;\\kappa_l,h)$\n[Eq.~(\\ref{eq:rdash})] are calculated.\nWe take two values of $\\Delta$, 0.0001, 0.00025, to see the\nstability of the results, and the discrepancy arising from a different\nchoice of $\\Delta$ is taken as the systematic uncertainties.\nIn the following plots, the results with $\\Delta=0.0001$ are shown\nunless otherwise stated.\n\nThe $P$ dependence of $\\ln R(P;\\kappa_l,h)$ and $\\ln R'(P;\\kappa_l,h)$ are shown\nin Fig.~\\ref{fig:lnr}.\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.145-d0.0005-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.1505-d0.0005-Xsp1.eps}\\\\[-2ex]\n (a)&(b)\\\\[2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.145-d0.0005-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.1505-d0.0005-Xsp2.eps}\\\\[-2ex]\n (c)&(d)\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{$P$ dependence of $\\ln R$ [Eq.~(\\ref{eq:r-1})]\n at $\\kappa_l=0.145$ (a) and 0.1505 (b),\n and $\\ln R'$ [Eq.~(\\ref{eq:rdash})] at $\\kappa_l=0.145$ (c) and 0.1505 (d).\n The results at $h$=0.2 to 0.4 are shown.\n }\n\\label{fig:lnr}\n\\vspace{-1ex}\n\\end{center}\n\\end{figure}\nThe statistical errors are invisible on this scale.\nThe sizes of $\\ln R$ and $\\ln R'$ differ by an order of magnitude, which\nindicates that the difference proportional to $P$ is large and explains\nwhy the curvature in $\\ln R$ is less clear than that in $\\ln R'$.\nThe curvature in $\\ln R$ and $\\ln R'$ originates from the rapid increase\nof the Polyakov loop ($\\hat L$) contained in Eqs.~(\\ref{eq:Y}) or\n(\\ref{eq:Z}) around $\\beta_{\\rm pc}$.\n\nWe then calculate the slope and curvature by fitting the data\nof $\\ln R$ and $\\ln R'$.\nHere let us mention the fits and the selection of the results.\nEach data point shown in Fig.~\\ref{fig:lnr} is obtained in a separate\nsimulation, and is totally independent of other points.\nThe data points are fit to polynomial functions of $P$ as\n\\begin{eqnarray}\n    \\ln {R^{(')}}^{\\rm fit}(P)\n&=& \\sum_{i=0}^{N_{\\rm poly}} c_i\\, P^i \\ ,\n\\label{eq:fit-func}\n\\end{eqnarray}\nover three fit ranges (or three different numbers of the data points),\ntwo different polynomial orders, and two values of $\\Delta$.\nSince not all the fits are successful, we only keep the fit results,\nwhich satisfy $\\chi^2\/{\\rm dof} < 3$, in the following analysis.\n\nFigure~\\ref{fig:chi2} shows the values of $\\chi^2\/{\\rm dof}$ obtained\nfrom the fit of $\\ln R$ and $\\ln R'$ as a function of the number of the\ndata points used, where only the results with $\\chi^2\/{\\rm dof}<3$ are\nplotted.\nIt is seen that $\\chi^2\/{\\rm dof}$ with $\\Delta=0.00025$ is always\nlarger than that with $\\Delta=0.0001$.\nOnce the fit parameters are determined, it is straightforward to\ncalculate the curvature of the potential.\n\\begin{figure}[t]\n\\begin{center}\n\\vspace*{-3ex}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.145-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1475-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.150-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1505-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.145-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1475-dall-Xsp2.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.150-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1505-dall-Xsp2.eps}\\\\[-2ex]\n\\end{tabular}\n\\vspace{-3ex}\n\\caption{The resulting $\\chi^2$\/dof as a function of the number of data\n points.\n The results for the $n$-th order polynomial fit are shown.}\n\\label{fig:chi2}\n\\end{center}\n\\end{figure}\n\nAlthough we analyze the second derivative to derive the main result,\nwe discuss the first derivative because it is instructive.\nThe first derivative of Eq.~(\\ref{eq:vefftrans}) is obtained as follows.\nFirst, the numerical values of $\\bar P(\\beta,\\kappa_l)$ and\n$\\chi_P(\\beta,\\kappa_l)$ are calculated and substituted in\nEq.~(\\ref{eq:d1V0}) to obtain the two-flavor contribution.\nThen, the $N_f$ flavor's contribution, the first derivative of\n$\\ln R^{(')}$, is determined using the fit results of\nEq.~(\\ref{eq:fit-func}).\nBy adding up, we obtain the first derivative of the full effective\npotential.\nFigure~\\ref{fig:dvdp} shows the typical behavior of the first derivative\nof the potential, where the five curves in each plot represent the\nresults for $h=0.0$, 0.1, $\\dots$, 0.4 from top to bottom and the fit\nresults with $\\Delta=0.00010$ and $n=5$ are used.\nIt is clear that for $h=0$ the curve is monotonically increasing for all\n$\\kappa_l$ while the ``S'' shape is seen for $h=0.4$.\nIn principle, it may be possible to determine the critical value of $h$\nusing these plots, however it is not easy to clearly distinguish an\n``S'' shape from a monotonic increase.\nThus, we use the second derivative to determine $h_c$ as described\nbelow.\n\nThe results for the curvature are plotted in Fig.~\\ref{fig:d2lnr} for\n$\\ln R$ (solid curves) and $\\ln R'$ (dashed curves), where the results\nfor $h=0.2$ and 0.4 are shown as examples.\nAgain, the fit results with $\\Delta=0.00010$ and $n=5$ are used.\nThe difference in the curvature between $\\ln R$ and $\\ln R'$ turns out\nto be reasonably small at all $\\kappa_l$.\n\\begin{figure}[t]\n\\vspace*{-1ex}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.145-Xsp1.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.1505-Xsp1.eps}\\\\\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.145-Xsp2.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.1505-Xsp2.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{The first derivative of the full effective potential is shown\n as a function of $P$.\n $h$ = 0.0, 0.1, 0.2, 0.3, 0.4 from top to bottom.}\n\\label{fig:dvdp}\n \\vspace{-1ex}\n\\end{center}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.145-d0.0001.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.1475-d0.0001.eps}\\\\\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.150-d0.0001.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.1505-d0.0001.eps}\\\\\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{The second derivative of the first term and the second term of\n Eq.~(\\ref{eq:curvature}) are shown as a function of $P$.\n The second term contribution exceeds that of the first term in a range\n of $P$ when $h=0.4$, which indicates the occurrence of the first order\n transition at such a value of $h$.}\n\\label{fig:d2lnr}\n \\vspace{-2ex}\n\\end{center}\n\\end{figure}\n\nNext, the curvature of the first term in Eq.~(\\ref{eq:curvature}) is\npresented, which can be easily calculated using the averaged value and\nthe susceptibility of $\\hat P$ at each $\\beta$ as in\nEq.~(\\ref{eq:d2V0}).\nThe curvatures thus obtained are shown in Fig.~\\ref{fig:d2lnr} together\nwith the statistical error, where the fit results obtained with a fifth\norder of polynomial are shown by dotted curves.\nIt is seen that, independently of $\\kappa_l$, $d^2V_{\\rm light}\/dP^2$ is\nalways positive as expected.\n\nFigure~\\ref{fig:d2lnr} shows that $d^2 \\ln R\/dP^2$ and $d^2 \\ln R'\/dP^2$\nhave a peak slightly below the $P$ value at which\n$d^2V_{\\rm light}\/dP^2$ takes the minimum.\nThis indicates that, in the many flavor system, the phase transition or\nrapid crossover occurs at $P$ smaller than the two-flavor case.\nFor all $\\kappa_l$, it is observed that the peak of $d^2 \\ln R\/dP^2$ or\n$d^2 \\ln R'\/dP^2$ is almost touching the curve of\n$d^2V_{\\rm light}\/dP^2$ at $h=0.2$, and exceeds $d^2V_{\\rm light}\/dP^2$\nat $h=0.4$ in a certain region of $P$.\n\nThe resulting curvature of the full effective potentials\nEq.~(\\ref{eq:curvature}) for $\\kappa_l=0.1450$ and 0.1505 are shown in\nFig.~\\ref{fig:d2Veff-action}.\nIt is seen that the minimum of the curvature with $h=0.0$ (solid curve)\napproaches to zero towards the chiral limit of two light flavors.\nThus, with this observation alone, one might expect that the potential\nwith $\\kappa_l=0.1505$ requires only a small $h$ to bring it into the double\nwell shape.\nHowever, at $\\kappa_l=0.1505$, it is also true that adding the heavy quarks\ndoes not reduce the minimum of the curvature by much.\nAs a consequence, $h_c$ takes a similar value at $\\kappa_l=0.1450$ and\n0.1505.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.6 \\textwidth,clip=true]\n{figs\/d2veff-n6-rng3-d0.0001-Xsp2.eps}\\\\\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{The curvature of the effective potential at $\\kappa_l=0.1450$ and\n 0.1505 and for $h=0.0$ (solid), 0.2 (dashed) and 0.4 (dotted).}\n\\label{fig:d2Veff-action}\n\\vspace{-2ex}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\vspace*{-2ex}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.145-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1475-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.150-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1505-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.145-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1475-dall-Xsp2.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.150-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1505-dall-Xsp2.eps}\\\\[-2ex]\n\\end{tabular}\n\\vspace{-2ex}\n \\caption{The critical value of $h$ as a function of the number of the\n data points used.\n }\n\\label{fig:scan-hc}\n\\end{center}\n\\end{figure}\n\nTo determine $h_c$, we iterate the calculation with $h$ varying in\nsteps of 0.02.\nFigure~\\ref{fig:scan-hc} shows the critical values of $h$\nas a function of the number of data points used in the fit,\ncorresponding to Fig.~\\ref{fig:chi2}.\nSince no reason exists to select the best result from them,\nwe take all the results satisfying $\\chi^2\/$dof $<$ 3 as the final\nresults, and the systematic uncertainty is chosen to cover the whole\naccepted results.\n\n\\begin{figure}[tb]\n\\vspace*{-1ex}\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics*[width=0.6 \\textwidth,clip=true]\n{figs\/scan-hc-P-k0.145-0.1505-Xspall.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{$h_c$ and $P$ at $h_c$: The results from $\\ln R$ (open symbols)\n and $\\ln R'$ (filled symbols).}\n\\label{fig:hc-P}\n\\end{center}\n\\vspace{-2ex}\n\\end{figure}\n\nWe can also determine the value of $P$ at $h_c$, denoted by $P_c$\n(for the numerical values, see Table~\\ref{tab:hc-result}).\nIn Fig.~\\ref{fig:hc-P}, all the results with $\\chi^2\/{\\rm dof} < 3$ are\nplotted together on the $P_c$-$h_c$ plane.\nWhile $h_c$ is insensitive to the two-flavor mass, $P_c$ is found\nto decrease towards the chiral limit of the two-flavor mass.\nThis qualitative feature is tested in the direct simulations of\n$2+N_f$-flavor QCD in Sec.~\\ref{subsec:direct-sim}.\n\n\\begin{table}[tb]\n \\centering\n \\begin{tabular}{c|cc}\n $\\kappa_l$ & $h_c$ & $P$ at $h_c$\\\\\n  \\hline\n   0.1450 & 0.23( 6) & 1.627(34) \\\\\n   0.1475 & 0.27( 8) & 1.595(27) \\\\\n   0.1500 & 0.26( 5) & 1.561(41) \\\\\n   0.1505 & 0.27(10) & 1.572(24) \\\\\n \\end{tabular}\n \\caption{Numerical results of $h_c$ and $P$ at $h_c$.\n }\n \\label{tab:hc-result}\n\\vspace{-1ex}\n\\end{table}\n\n\\begin{figure}[t]\n\\vspace*{-1ex}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-mpmr2-0.eps} &\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-pcac-0.eps} \\\\\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-mpmr2.eps} &\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-pcac.eps} \\\\\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{The light quark mass dependence of $h_c$.\n Top: A constant fit.\n Bottom: A linear fit is performed with the horizontal axis\n proportional to $m_l^{2\/5}$, testing the mean field prediction\n Eq.~(\\ref{eq:meanfield-scaling}).\n The band appearing in the top right corner is $h_c$ determined in the\n direct simulation with $\\kappa_l=0$ and $N_f=50$ (for details, see\n Sec.~\\ref{subsec:direct-sim}).\n }\n\\label{fig:lq-dep}\n\\vspace{-2ex}\n\\end{center}\n\\end{figure}\n\nThe light quark mass dependence of $h_c$ is plotted in\nFig.~\\ref{fig:lq-dep} as a function of the $m_\\pi\/m_\\rho$ ratio (left)\nand the PCAC quark mass (right).\nThe error is dominated by the systematic uncertainty associated with\nthe fitting procedure.\nIn the top-right corner of each plot, $h_c$ determined from the direct\n$2+N_f$ flavor simulation with $\\kappa_l=0$ ({\\it i.e.} $m_l=\\infty$) and\n$N_f=50$ is shown with an uncertainty (for the direct simulations, see\nSec.~\\ref{subsec:direct-sim}).\n$h_c$ at $\\kappa_l=0$ is clearly larger than those around\n$0.145 \\le \\kappa_l \\le 0.1505$, which indicates that $h_c$ gradually\ndecreases towards the chiral limit as a global behavior.\nIn the range of $0.145 \\le \\kappa_l \\le 0.1505$, $h_c$ does not show\nsignificant dependence on $m_l$ within the error, and a constant fit\nyields $h_c=0.23(1)$ in the chiral limit.\n\nIf this mild dependence is continued down to the chiral limit and hence\n$h_c$ in the chiral limit remains positive and finite, $h_c$ in the\nchiral limit corresponds to the tricritical point in\nFig.~\\ref{fig:columbia-plot}.\nMean field analysis of an effective theory predicts the tricritical\nscaling~\\cite{Wilczek:1992sf,Rajagopal:1992qz,Ukawa:1995tc,Ejiri:2008nv},\n\\begin{eqnarray}\n h_c \\sim (\\mbox{const.})\\times m_l^{2\/5} + \\mbox{const.}\n \\label{eq:meanfield-scaling}\n\\end{eqnarray}\nin the vicinity of the tricritical point, where the power $2\/5$ is\nindependent of $N_f$.\nIn addition to the constant fit (top), we also fit the data to a linear\nfunction of $m_l^{2\/5}$ in each plot, yielding $h_c=0.35(26)$ and\n$0.32(17)$ in the chiral limit of the two-flavor mass, respectively.\nNote that the slope is undetermined and consistent with zero.\nIn either case, a positive value of $h_c$ is favored in the chiral\nlimit, which suggests the second order transition of massless two-flavor\nQCD.\nFurther checks require more extensive lattice calculations and are\npostponed to future papers.\n\n\n\\section{Consistency check}\n\\label{sec:consistency-check}\n\n\\subsection{Effective potential constraining Polyakov loop}\n\\label{subsec:polyakov}\n\nAs an independent check of the results obtained in\nSec.~\\ref{sec:numerical_results}, we try to estimate $h_c$ with a\ndifferent method.\nThe quantity to be constrained to obtain the PDF is arbitrary as long as\nit has an overlap with the order parameter.\nIn this section, we take the real part of the Polyakov loop, $\\hat L$.\nRecalling Eq.~(\\ref{eq:effective-potential-2}), the constraint effective\npotential for $\\hat L$ is given by\n\\begin{eqnarray}\n      V_L(L;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\n&=& - \\ln w_L(L;\\beta_{\\rm ref},\\kappa_l,0,0)\n    - \\ln \\left(\\frac{w_L(L;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)}\n                     {w_L(L;\\beta_{\\rm ref},\\kappa_l,0  ,0)}\n\t\t   \\right)\\\\\n&=&   V_{L,\\,\\rm light}(L; \\beta_{\\rm ref},\\kappa_l)\n    - \\ln R_L(L;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\\ ,\n \\label{eq:effective-potential-poly}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n      V_{L,\\,\\rm light}(L; \\beta_{\\rm ref},\\kappa_l)\n&=& - \\ln w_L(L;\\beta_{\\rm ref},\\kappa_l,0,0) \\\\\n&=& - \\ln \\langle\\,\n             \\delta(L-\\hat L)\\,\n             e^{6\\,(\\beta_{\\rm ref}-\\beta)\\,N_{\\rm site}\\, \\hat P}\\,\n          \\rangle_{(\\beta,\\kappa_l)}\\, ,\\\\\n    R_L(L; \\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\n&=& e^{6\\,N_s^3\\,h\\,L}\n    \\left\\langle \\displaystyle\n       \\exp\\left[\n            36\\,N_s^3\\,h\\,\\hat W_P\n\t  +  6\\,(\\beta_{\\rm ref}-\\beta)\\,N_{\\rm site}\\, \\hat P\n\t   \\right]\n    \\right\\rangle_{L: {\\rm fixed},(\\beta,\\kappa_l)}\\ .\n\\end{eqnarray}\nUnlike the case with $\\hat X=\\hat P$, $\\beta_{\\rm ref}$ dependence\nremains in the second derivative of the potential with respect to $L$,\nwhich means $\\beta_{\\rm ref}$ has to be explicitly tuned to the\n(pseudo)critical temperature for each value of $h$.\nIn the crossover region, the minimum of the potential and the minimum of\nits second derivative are realized at the same value of $L$.\nOn the other hand, in the first order region, $\\beta_{\\rm ref}$ has to\nbe tuned until the two minima in the potential take the same depth.\n\nThe potential, $V_L(L)$, is shown in Fig.~\\ref{fig:Veff-RePoly}\nfor $h=0.0$, 0.2 and 0.4, where $V_L(L)$ at each $h$ is shifted in a\nvertical direction for comparison.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/poly-case-k0.145.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/poly-case-k0.1505.eps}\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{The effective potential constructed by constraining $\\hat L$\n near the critical temperatures.\n The absolute values of the potential are shifted for comparison.\n The results at $h=0.0$, 0.2 and 0.4 are shown for $\\kappa_l=0.145$ (left)\n and 0.1505 (right).\n }\n\\label{fig:Veff-RePoly}\n\\end{center}\n\\end{figure}\nFirst of all, the effective potential for the Polyakov loop is not as\nclean as that for the generalized plaquette, especially at nonzero $h$,\nwhich only allows us to extract the qualitative feature.\nIt is seen that the potential has a positive curvature at $h=0$ in our\nlightest and heaviest light quarks.\nWhen $h=0.2$, the potential around the minimum becomes almost flat,\nindicating that it is close to the end point.\nIf $h$ is further increased to 0.4, the double well shape appears to emerge\nthough the statistical error makes it ambiguous.\nThese qualitative features are consistent with the findings in the\nprevious section.\n\n\n\\subsection{Direct simulations of $2+N_f$ flavor QCD}\n\\label{subsec:direct-sim}\n\nAlthough the convergence of hopping parameter expansion is not the\nmatter for the discussion in the previous section, it is interesting to\ninvestigate the convergence of the HPE for future applications.\nWe study this by explicitly performing simulations of $2+N_f$ flavor QCD\nand comparing the results with those based on the HPE.\nHowever, thoroughly precise calculations of the many flavor system\nrequire an extensive scan of simulation parameters ($\\kappa_h$, $\\beta$ and\n$N_f$) even after fixing $\\kappa_l$.\nFurthermore, in general, it is not easy to locate the end point of the\nthe first order transition accurately, because the statistical noise\ngrows as one approaches the end point.\nInstead, we draw a thermal cycle on the $\\beta$-$P$ plane, which is\nobtained by increasing (or decreasing) $\\beta$ until passing its\n(pseudo) critical value and then reversing the direction.\nFor representative values of $\\kappa_h$, $\\kappa_l$ and $N_f$, we accumulated\n400 trajectories at each point of the thermal cycle.\n\nWe take $\\kappa_l=0, 0.145$ and 0.1505, and choose $N_f$ ranging from 4 to\n50, depending on $\\kappa_l$, and monitor each thermal cycle whether the\nhysteresis curve occurs or not.\nIf it occurs, the parameters chosen turn out to be in the first order\nregion.\nBy repeating this, we try to find the critical value of $\\kappa_h$, $\\kappa_{h_c}$,\nfor fixed values of $\\kappa_l$ and $N_f$.\nThe thermal cycles at several simulation parameters are shown in\nFig.~\\ref{fig:hysteresis}.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+16_rg-KL0.0-cswL0.0-nosmr-apbc.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+50_rg-KL0.0-cswL0.0-nosmr-apbc.eps}\\\\[-2ex]\n (a)&(b)\\\\[2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+32_rg-KL0.1505-cswL1.712-nosmr-apbc.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+10_rg-KL0.1505-cswL1.712-nosmr-apbc.eps}\\\\[-2ex]\n (c)&(d)\\\\[2ex]\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{The thermal cycle on the ($P$, $\\beta$)-plane in $2+N_f$ flavor\n QCD. The vicinity of the end point, {\\it i.e.} $\\kappa_h\\approx\\kappa_{h_c}$, is\n shown.}\n\\label{fig:hysteresis}\n\\end{center}\n\\end{figure}\n\nFigures~\\ref{fig:hysteresis} (a) and \\ref{fig:hysteresis} (b) show the\nthermal cycle at $\\kappa_l=0$ and $N_f=10$, 16, 32, 50, which tells us that\n$\\kappa_{h_c}$ decreases with $N_f$.\nFigure~\\ref{fig:hysteresis} (c) shows the same but with nonzero $\\kappa_l$.\nComparing the $N_f=16$ data in Figs.~\\ref{fig:hysteresis} (a) and\n\\ref{fig:hysteresis} (c) or the $N_f=32$ data in\nFigs.~\\ref{fig:hysteresis} (b) and \\ref{fig:hysteresis} (c), it is found\nthat $\\kappa_{h_c}$ is clearly different between $\\kappa_l=0$ and 0.1505.\nIn Fig.~\\ref{fig:hc-P} of Sec.~\\ref{subsec:lat-para}, we have discussed\nthat the HPE predicts that $P$ at $h_c$ decreases with $\\kappa_l$.\nFigure~\\ref{fig:hysteresis} (d) seems to show that it is the case, at\nleast, qualitatively.\n\nThe critical values, $\\kappa_{h_c}$, obtained at each $\\kappa_l$ and $N_f$\nare translated into $h_c$ using Eq.~(\\ref{eq:h}), and plotted as a\nfunction of $N_f$ in Fig.~\\ref{fig:direct-hc}.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.7 \\textwidth,clip=true]\n{figs\/nf_vs_h_c.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{$N_f$ dependence of $h_c$ through direct simulations of $2+N_f$\n flavor QCD.}\n\\label{fig:direct-hc}\n\\end{center}\n\\end{figure}\nIt is expected that, for a sufficiently large $N_f$,\n$h_c$ approaches an asymptotic value, but the value of $N_f$ we have\nstudied seems not to reach such a region yet although the increasing\nrate looks slowing down.\nImportantly, it is seen that $\\kappa_{h_c}$ does not differ by much between\n$\\kappa_l=0.145$ and 0.1505, which is consistent with the observation in the\nHPE analysis.\nIt is interesting to include the next-to-leading order contribution of\nthe HPE analysis.\n\n\n\\section{Summary and discussion}\n\\label{sec:summary}\n\nWe have studied the finite temperature phase transition of QCD with two\nlight and many heavy quarks at zero chemical potential, where the heavy\nquarks are introduced in the form of the hopping parameter expansion via\nthe reweighting method.\nThe phase structure was scanned on the $\\kappa_l$-$h$ plane to identify the\ncritical line separating the continuous crossover and the first order\nregions.\n\nThe nature of the transition is identified by the shape of the\nconstraint effective potential constructed from the probability\ndistribution function of the generalized plaquette.\nFor $h=0$, the system reduces to two-flavor QCD, which always shows\ncontinuous crossover and hence the potential has a single well.\nAs one increases $h$, at some point the potential takes a double-well\nshape, which defines a critical value, $h_c$.\nWe have determined $h_c$ at four light quark masses, and observed that\n$h_c$ is independent of two-flavor mass in the range we have studied\n($0.46 \\le m_\\pi\/m_\\rho\\le 0.66$).\nThis result indicates that the critical heavy mass remains finite in the\nchiral limit of the two flavors, suggesting the phase\ntransition of massless two-flavor QCD is of second order.\nSome of the qualitative features observed in the main analysis were\nchecked by two independent analyses.\n\nThe approach in this study can be said as follows.\nTwo-flavor QCD with a finite mass is enforced to undergo a first order\nphase transition by adding extra quarks.\nIt is then likely that those extra quarks are necessary to keep the\nfirst order transition down to the chiral limit of two-flavor QCD.\nThis method is applicable for any kinds of lattice fermions unless they\ncontain $\\beta$ dependent coefficients.\nAccording to the definition of $h$, Eq.~(\\ref{eq:h}), $\\kappa_{h_c}$ can be\nconsidered to be arbitrarily small for a given $h_c$ by assuming\narbitrarily large $N_f$, and thus we do not have to care about the\nconvergence of the hopping parameter expansion.\nNevertheless, it is interesting to see the limitation of the HPE for a\nfixed $N_f$ for further applications.\n\nIn order to establish our finding, possible systematic uncertainties,\nwhich are not investigated in the present paper, need to be\nunderstood.\nSince, at the end point, the second order phase transition occurs, a\nsizable finite volume effect is possible in the vicinity of the point.\nHowever, we do not expect it to be significant in the many flavor\napproach.\nIn this approach, the end point is determined through the extrapolation\nof the effective potential with regard to $h$, and the extrapolation is\nperformed in a region free from finite size effects since the two-flavor\nconfigurations are all generated at a parameter region away from the\nsecond order end point.\nNamely, at the price of the uncertainty due to the extrapolation, we\ncould have avoided the finite volume effect associated with the second\norder phase transition.\nNevertheless, it is clearly important to explicitly check that the\neffect is under good control.\nSuch work is ongoing.\n\nAlthough the behavior observed in Fig.~\\ref{fig:lq-dep} seems to suggest\nthat the chiral limit of $h_c$ is finite and positive, it then has to\nshow the tricritical scaling [Eq.~(\\ref{eq:meanfield-scaling})].\nAt the present, our results allow us to fit to any smooth function\nin $m_l$.\nIn order to improve the situation, we need to explore lighter quark\nmasses and reduce the systematic uncertainty associated with the\nfitting procedure.\nHowever, during the preliminary study, we realized that the lightest\nquark mass presented in this paper is the lower limit in our lattice\nsetup.\nTo go beyond the limit, the setup has to be changed.\nTowards the ultimate goal, the discretization effects also have to be\nexamined.\nA systematic study of these uncertainties requires large scale\nsimulations, and we postpone them to future works.\n\nWe can extend the many flavor approach to explore QCD at finite chemical\npotential as initiated in Refs.~\\cite{Ejiri:2012rr,Iwami:2015mqa}.\nIn this case, mean field analysis predicts that the critical line\nruns like ${m_l}^c \\sim |\\mu|^5$~\\cite{Ejiri:2008nv}.\nWe believe that such a study brings valuable information to\nunderstand the rich QCD phase diagram.\n\n\n\\section*{Acknowledgments}\n\nWe would like to thank members of the WHOT-QCD Collaboration for\nuseful discussions.\nWe also thank Ken-Ichi Ishikawa\nfor providing us his simulation codes.\nThis work is in part supported by\nJSPS KAKENHI Grant-in-Aid for Scientific Research (B)\n(No.\\ 15H03669 [N.~Y.],\n       26287040 [S.~E.]\n)\nand (C)\n(No.\\ 26400244 [S.~E.]), and by the Large Scale Simulation Program of High\nEnergy Accelerator Research Organization (KEK) No.\\ 14\/15-23.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1} Introduction}\nCircular dichroism (CD) spectroscopy is a widely used experimental technique which enables the characterization of chiral molecular samples. This technique, which measures the differential absorption of molecules to left- and right-handed circularly polarized light (CPL) excitation, permits, for instance, to characterize the secondary structure of biomolecules and to determine the enantiopurity of pharmaceutical drugs ~\\cite{Barron}. Despite its wide applicability, the sensitivity of CD measurements is quite limited due to the weak nature of chiral light-matter interactions. Nanostructured materials have been instrumental enhancing the sensitivity of other molecular spectroscopic techniques such as Surface-Enhanced Raman Scattering (SERS) and Surface-Enhanced Infrared Absorption (SEIRA) spectroscopy, increasing their performance several orders of magnitude \\cite{SEIRA1, SEIRA2, SERS1, SERS2, SERS3}. Similarly, increasing the sensitivity of CD spectroscopy with the aid of nanostructures would be highly desirable and it is a research objective which has been pursued in the recent past.\n\nSince the seminal contributions of Y. Tang and A. E. Cohen~\\cite{tang_cohen}, several nanophotonic platforms have been proposed to augment the sensitivity of CD experiments through the local and global enhancement of $C(\\mathbf{r})$, the local density of electromagnetic chirality. To mention a few, high refractive index nanoparticles~\\cite{PRB, CDEnh2, OldACSAitzol, Curto, Vladimiro, IvanACS}, plasmonic chiral~\\cite{PRXDesignPples, fCD3000} and nonchiral nanoantennas~\\cite{CDEnh1, MicOrCD, fCD3000}, optical waveguides~\\cite{Vazquez} and metasurfaces~\\cite{ACSDionne, NanophotonicPlatforms, Quidant} have been explored as enablers of surface-enhanced circular dichroism and other chiral spectroscopy techniques. \n\nGlobally enhancing CD spectroscopy around a nanostructure necessitates duality symmetry in the design of the device so that the helicity of the scattered fields is conserved ~\\cite{PRLMolina}. Unfortunately, common nanophotonic platforms can only guarantee this symmetry at particular, non-resonant, excitation wavelengths. This limits the efficiency of light-matter interactions between the scattered fields and the molecules under study, limiting the magnitude of the achievable enhancements in CD spectroscopy. \n\nIn this work, we present a novel approach to design helicity preserving and optically resonant nanostructured devices to enhance the CD signal exploiting the lattice modes of periodic arrays of dual high refractive index nanoparticles. Starting from the well-known example of a silicon nanosphere capable of enhancing CD spectroscopy by a factor of 10~\\cite{PRB}, we first characterize the resonances of an infinite 1D array of such nanoparticles. We observe that these systems support two different types of emerging lattice resonances: far-field diffractive modes and a non-diffractive lattice resonance mediated by mid- and near-field interactions. Secondly, we find that in finite particle arrays finite chain modes can also be observed and exploited. \n\nWe show that all of these helicity preserving resonant modes can globally enhance the sensitivity of CD spectroscopy over two orders of magnitude, providing one additional order of magnitude enhancement over the response of a single, non-resonant nanoparticle ~\\cite{PRB}. Even if the maximum enhancement is given for the far-field diffractive modes, we find that enhancements obtained for the rest of the modes are comparable.\n\n\\section{\\label{sec:level2} Enhanced chiral light-matter interactions and helicity conservation}\n\nFor molecules illuminated by a left ($+$) or right ($-$) handed circularly polarized plane wave in vacuum, the CD signal of a molecule, $\\text{CD}_{cpl}$, can be expressed as \\cite{tang_cohen, PRB} \n\\begin{equation}\n\\text{CD}_{cpl}=-\\frac{4}{\\varepsilon_0}\\text{Im}(G)|C_{cpl}|,\n\\end{equation}\nwhere $G$ is the molecular chiral polarizability, $C_{cpl}=\\pm \\frac{\\varepsilon_0\\omega}{2c}E_0^2$ the local density of chirality of a plane wave in vacuum, $c$ the speed of light, $\\omega$ the angular frequency, $E_0$ the amplitude of the incoming electric field and $\\varepsilon_0$ the permittivity of free space. Ref.~8 concluded that in the presence of nonchiral antennas, the CD signal could be locally expressed as\n\\begin{equation}\n\\text{CD} = -\\frac{4}{\\varepsilon_0}\\text{Im}(G)C(\\mathbf{r}),\n\\label{CD_C}\n\\end{equation}\nwith $C(\\mathbf{r}) = -\\frac{{\\omega}}{2c^2}\\text{Im}\\Big( \\mathbf{E}(\\mathbf{r})^*\\cdot \\mathbf{H}(\\mathbf{r}) \\Big)$  being the local density of optical chirality\\cite{tang_cohen}, where $\\mathbf{E}(\\mathbf{r})$ and $\\mathbf{H}(\\mathbf{r})$ are the local electric and magnetic fields, respectively. Although $G$ is a fixed molecular parameter, $C(\\mathbf{r})$, can be in principle engineered and enhanced in the presence of optical antennas. Importantly, in the presence of nonchiral antennas one can define a local enhancement factor such that $\\text{CD}=f_\\text{CD}\\text{CD}_{cpl}$ where\n\n\\begin{equation} \\label{fCD}\nf_\\text{CD}=\\frac{\\text{CD}}{\\text{CD}_{cpl}}=\\frac{C(\\mathbf{r})}{|C_{cpl}|}=-\\frac{Z_0}{E_0^2}\\text{Im}\\Big(\\mathbf{E}(\\mathbf{r})^*\\cdot \\mathbf{H}(\\mathbf{r})\\Big),\n\\end{equation}\n$Z_0$ being the impedance of vacuum.\n\nContrary to what could be intuitively expected, the local density of \nchirality, $C(\\mathbf{r})$, is not necessarily related to the local handedness of the field. Nevertheless, as shown in Eq. (\\ref{CD_C}), the CD signal is directly proportional to $C(\\mathbf{r})$ in the presence of nonchiral environments. Therefore, an essential element to enhance the CD signal is to engineer the local fields, so that the overall sign of $C(\\mathbf{r})$ (and, thus, also the sign of $f_{\\text{CD}}$) is preserved, while its magnitude is maximized. Our work is focused on finding a nanostructure which, when illuminated by CPL of a given handedness, can enhance the absolute value of $C(\\mathbf{r})$ while locally preserving its sign in space. In such a scenario, molecules in the vicinity of our optical resonator will interact with the local field in a way that always contributes positively to CD signal. This allows for a strict spatial control on the molecular absorption rates and facilitates the practical measurement of the field-enhanced CD signal.\n\nIn the case of monochromatic fields, the electromagnetic density of chirality  is  intimately related to the local density of helicity~\\cite{barnett, cameron, nieto2017chiral, Lisa}. \nThe helicity operator can be expressed as\n\\begin{equation}\n\\Lambda = \\frac{\\mathbf{J} \\cdot \\mathbf{P}}{|\\mathbf{P}|} = \\frac{1}{k}\\nabla \\times\n\\label{helicity}\n\\end{equation}\nwhere $\\mathbf{J}$ is the total angular momentum of light, $\\mathbf{P}$ is the linear momentum and the second equality in Eq. \\eqref{helicity} is only held for monochromatic fields \\cite{Messiah}. The helicity operator is the generator of dual transformations, i.e. rotations of the electric and magnetic fields~\\cite{PRLMolina}. Hence, the helicity is a conserved magnitude in the interaction of light with systems symmetric under the exchange of electric and magnetic fields, the so-called dual systems.\n\nIn single particles with a dominant dipolar response, duality symmetry is fulfilled for wavelengths which meet the first Kerker's condition, in other words, for wavelengths where the electric and magnetic polarizabilities of the nanostructure are equal ($\\alpha_e=\\alpha_m$) \\cite{nieto2011angle,Mole_k,Zambrana, Aitzol_moebius,olmos2019enhanced}. This condition can be achieved for particular wavelengths of light and materials~\\cite{strong_magnetic}. Similarly, in a collection of nanoparticles, duality is a symmetry of the whole system if every individual particle satisfies the first Kerker's condition~\\cite{PRLMolina,MikolajPRL} \n\nHelicity conservation in such scenarios allows us to consider the following expressions for the local electric and magnetic fields\\cite{MikolajPRL}\n\\begin{align}\n\\label{DualE}\n\\Lambda \\mathbf{E}(\\mathbf{r}) &=  \\frac{1}{k}\\nabla \\times \\mathbf{E}(\\mathbf{r}) = p\\mathbf{E}(\\mathbf{r})\\\\\n\\label{DualH}\n\\Lambda \\mathbf{H}(\\mathbf{r}) &=  \\frac{1}{k}\\nabla \\times \\mathbf{H}(\\mathbf{r}) = p\\mathbf{H}(\\mathbf{r}),\n\\end{align}\nwhere the helicity eigenvalue, $p = \\pm 1$. From Eqs. \\eqref{DualE}-\\eqref{DualH} and Maxwell's equations one obtains $\\mathbf{H}(\\mathbf{r}) = \\frac{-i p k}{\\omega\\mu_0}\\mathbf{E}(\\mathbf{r})$. Applying this expression to Eq. \\eqref{fCD}, the local CD enhancement factor for dual systems can be reformulated as:\n\n\\begin{equation}\n\\label{fCDdualE}\nf^{dual}_{\\text{CD}} = \\frac{1}{E_0^2}|\\mathbf{E}(\\mathbf{r})|^2.\n\\end{equation}\n\nOn the one hand, Eq. \\eqref{fCDdualE} confirms that under duality symmetry, $C(\\mathbf{r})$ does not change sign locally upon scattering. Most importantly, this result predicts that we can enhance the local CD experimental signal on a dual structure as far as we are able to enhance the fields close to our optical system. Finally, this expression comes into agreement with other results in the literature, which express the chirality in terms of the Riemann-Silberstein vectors~\\cite{RS, IvanACS}.\n\n\\section{\\label{sec:level3}Lattice resonances}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth, scale=0.6]{jon_spheres_v55.png}\n\t\\caption{Sketch of the setup under study: a group of nanoparticles arranged in a 1D periodic arrangement. In the bottom right part, the extinction cross section of a single isolated silicon nanoparticle in our system.}\n\t\\label{Array}\n\\end{figure}\n\nAs we determined in the previous section, dual systems allow for the enhancement of $f_\\text{CD}$ while keeping its local sign unaltered.  Nevertheless, duality symmetry constriction restricts the chirality enhancement to specific wavelengths which fulfill Kerker's first condition. Typically, this wavelength does not coincide with resonant modes of optical cavities and resonators. As a consequence, the electromagnetic fields used in typical surface-enhanced CD devices are not very intense on isolated nanoparticles. Previous results for single high refractive index nanoparticles \\cite{PRB} predict a modest enhancement of one order of magnitude in the molecular CD signal. In order to overcome this limitation we propose to exploit lattice resonances arising in ordered collections of similar dual, non-resonant, individual dielectric nanoparticles. For the sake of clarity, we restrict our study to 1D periodic arrangements of silicon nanoparticles. Nevertheless, ideas presented in this article can be easily generalized for 2D or 3D particle arrays.\n\nLattice resonances are high-quality factor collective excitations of periodically arranged groups of nanoparticles. Equation \\eqref{fCDdualE} indicates that on any dual system $f_\\text{CD}$ scales with the square of the field enhancement that is produced. Thus, our hypothesis is that although the individual nanoparticles are non-resonant at the first Kerker's condition, their collective behaviour at particular separation distances can give rise to resonant, very intense circularly polarized fields capable of uniformly enhancing CD spectroscopic signal of molecules over the entire extension of the array. Thus, in order to guarantee duality symmetry, we will operate at a particular frequency at which Kerker's first condition is fulfilled by every particle in the lattice, while we modify the separation distance between them to induce a resonant behaviour of the periodic system.\n\nUnder certain circumstances, the optical response of each nanoparticle can be described as a combination of a point electric dipole and a point magnetic dipole. Thus, in such situations, the collective behaviour of the system can be analyzed using an approach based on the Coupled Dipole Approximation (CDA). For a finite collection of $N$ particles, the optical response of the coupled electric and magnetic dipoles has a well-known algebraic solution:\n\n\\begin{equation}\n\\vec{\\Psi} = \\left[\\bm{I} - k^2 \\bm{G}_F \\bm{\\alpha}_F\\right]^{-1} \\vec{\\Psi}_{0}.\n\\end{equation}\n$\\vec{\\Psi}$ is a $6N$ dimensional vector which contains the self-consistent solution for each electric and magnetic field components in every site. $\\bm{I}$ is the $6N \\times 6N$ identity matrix and $\\bm{\\alpha}_F$ is a matrix of the same size containing the values of the electric and magnetic polarizabilities. $\\bm{G}_F$ is a matrix that contains the coupling coefficients (built as lattice sums of Green's functions) between all the different dipoles in the system. Finally, $\\vec{\\Psi}_{0}$ is a $6N$ dimensional vector containing the values of the incident electric and magnetic fields at each of the positions of the particles (more detailed information can be found in the Supplementary Material).\n\nCollective resonances emerge when $\\text{det}\\left( \\bm{I} - k^2 \\bm{G}_F \\bm{\\alpha}_F \\right) \\approx 0$ (notice that real poles would signal the existence of bound states). In these situations the scattered fields around the nanoparticles are strongly enhanced and, thus, $f_\\text{CD}$ increases substantially. Note that since the incident wavelength and, therefore, the polarizabilities of the particles are fixed to fulfill duality symmetry, the emergence of this collective modes depends only on geometrical values, namely, the spatial distribution of the nanoparticles and the direction of incidence of the probing CPL.\n\nIt is important to note that there are different interaction mechanisms through which nanoparticles can interact in a periodic structure to give rise to collective resonances. On the one hand, if the photonic elements are placed at relatively long separation distances, they interact through a $r^{-1}$ dependent far-field dipolar interaction giving rise to diffractive modes ~\\cite{hicks}. On the other hand, if the particles are more closely spaced, $r^{-3}$ and $r^{-2}$ dependent near-field interactions can also give rise to hybridized chain modes~\\cite{ross,halas_review}.  \n\nIn what follows, we show that lattice resonances based on both types of interaction mechanisms are, in practice, useful to enhance the local density of optical chirality $C(\\mathbf{r})$ and, thus, enhance the CD signal, setting a useful photonic paradigm for experimental testing. We have chosen our proof-of-concept system to be a finite 1D array of silicon dual nanoparticles showing that both types of lattice resonances give rise to an important enhancement of $f_\\text{CD}$.\n\n\\section{\\label{sec:level4}A finite chain of dual nanospheres}\n\nIn the following, we study a 1D array of $N=2000$ silicon nanospheres of radius $a = 150$ nm located along the $OY$ axis such that the nanospheres are placed at $\\mathbf{r}_n = (n-1)D \\mathbf{\\hat{u}_y}$ where $n = 1,2,..., N$ and $D$ is the distance between two adjacent nanoparticles from center to center. In addition, the wavevector ($\\mathbf{k}$) of the incoming circularly polarized plane wave is contained in the $YZ$ plane and forms an angle $\\theta$ with the $OZ$ axis perpendicular to the chain. A sketch of the parameters of the system can be seen in Fig. \\ref{Array}.\n\nTo characterize the resonant features of the system, we first study the extinction cross section, $C_{ext}$ of the particle chain. In this particular system, Kerker's first condition is exclusively fulfilled at $\\lambda = \\lambda_k = 1208$ nm, so we explore the extinction cross section of the system as a function of two parameters: the incidence angle, $\\theta$, and the interparticle distance, $D$. The results of this calculation are shown in Fig. \\ref{Diego_Extinction}a, where the normalized extinction cross section, $Q_{ext} = C_{ext}\/(N\\cdot C_{ext,1})$, is shown and $C_{ext,1}$ is the extinction cross section of a single silicon particle at $\\lambda_k$. In Fig. \\ref{Diego_Extinction}b, a comparison of our results with the extinction cross section of an infinite chain of particles is shown (details of the theoretical formalism describing the infinite case can also be found in the Supplementary Material). As expected, both calculations give very similar results since collective interactions are known to converge for a given number of elements in a periodic structure~\\cite{plasmon_pol, size_matters}. Interestingly, the number of nanospheres required in our system to reproduce the behaviour of the infinite case is very big comparing to what is found in the literature.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{AfterDiego_ExtMap_a-b_changed_HOT_back.png}\n\t\\caption{Normalized extinction cross section map under Kerker's first condition as a function of interparticle distance for: {\\bf a)} a finite periodic 1D array of 2000 silicon nanospheres, {\\bf b)} an infinitely periodic 1D array of silicon nanospheres.}\n\t\\label{Diego_Extinction}\n\\end{figure}\n\nThe analytic calculations for the infinite chain allow us to identify the narrow peaks at low angles ($\\theta \\sim 0$) as diffractive modes emerging at $D\/\\lambda = m\/(1 \\pm \\sin \\theta)$ where $m$ takes positive integer values. As stated above, these peaks emerge as a consequence of the constructive interference of the far-field interactions. This is also observable in Fig. \\ref{LambdaSweep}b, where, together with the single-particle spectrum, narrow diffraction lines emerge when nanoparticles are placed at $\\lambda=5\\lambda_k$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{LambdaSweep2.png}\n\t\\caption{Normalized extinction cross section map as a function of the illuminating wavelength for: {\\bf a)} a single silicon nanosphere, {\\bf b)} a chain of 2000 nanospheres with $D = 5\\lambda_k$, {\\bf c)} a chain of 2000 nanospheres with $D = 1.5\\lambda_k$ and {\\bf d)} a chain of 2000 nanospheres with $D = 0.5\\lambda_k$.}\n\t\\label{LambdaSweep}\n\\end{figure}\n\nHowever, close to grazing incidence ($\\theta \\rightarrow \\pi\/2$) a much wider resonance emerges. This mode does not fit with diffractive characteristics and, thus, has to be interpreted in terms of a different interaction mechanism. As we show in Fig. \\ref{LambdaSweep}, this resonance emerges when particles in the chain are placed closer than $D = 1.5\\lambda_k$ and, therefore, near- and mid-field interparticle contributions become important. As particles are placed closer together, the magnetic single-particle resonance of silicon ($\\lambda_{mag} \\sim 1110$ nm) redshifts and it can approach the wavelength of the Kerker condition, $\\lambda_k$. All these considerations point out that this resonance is related to a \\emph{bonding} transversal mode of the chain, a configuration which has been extensively studied in nanoparticle dimers and metamaterials~\\cite{halas_review}.\n\nMoreover, at $\\theta = \\pi\/2$, the complete physical configuration (incident wave included) is rotationally symmetric. Due to the simultaneous duality and axial symmetry, helicity can be expressed in this situation as:\n\n\\begin{equation}\n\\Lambda = \\frac{J_yP_y}{|P_y|},\n\\end{equation}\nwhere both $J_y$ (y-component of the total angular momentum of light) and $\\Lambda$ are conserved quantities. Therefore, $P_y$ (y-component of the linear momentum of light) cannot change sign under scattering and backward scattering is not allowed~\\cite{kerker}. As a consequence, this mode in an exclusively forward propagating one.\n\nOnce we have thoroughly analyzed the optical resonances that can emerge in a dual 1D array of nanospheres, we proceed to show that, as predicted, these resonances give rise to important enhancements of $f_\\text{CD}$. In order to model realistic environments of practical interest, we define two averaged enhancement factors: the average CD value around a particle in position $i$ ($f_{\\text{CD}}^{i}$) and the array-averaged CD enhancement factor, $f_{\\text{CD}}^{avg}$, defined as\n\n\\begin{eqnarray}\n\\label{fCDi}\nf_{\\text{CD}}^{i} &=& \\int_{S_i}\\frac{1}{4\\pi} f_{\\text{CD}}(\\xi,\\eta)\\sin\\xi d\\xi d\\eta, \\\\\n\\label{fCDavg}\nf_{\\text{CD}}^{avg} &=& \\frac{1}{N}\\sum_{i = 1}^N f_{\\text{CD}}^{i}.\n\\end{eqnarray}\nThe integrals are carried out around a spherical surface $S_i$ ($\\xi$ and $\\eta$ are, respectively, the polar and azimuthal spherical coordinates respect to the center of the $i$-th nanoparticle) separated $1~\\text{nm}$ apart from the surface of the $i$-th silicon nanoparticle.  \n\nFig. \\ref{Extinction_fCD}a presents the normalized extinction cross section of the silicon array as a function of $D\/\\lambda$, for $\\theta = 0$. Under this incidence, the extinction shows a resonant behaviour at the diffraction condition $D = m\\lambda_k$, for $m \\in \\mathbb{N}$. However, the strength of the resonances decreases as $m$ is increased, something which is already noticeable in Fig. \\ref{Extinction_fCD}a. In Fig. \\ref{Extinction_fCD}b we can compare the extinction resonances with the CD enhancement averaged over the whole structure, $f_{\\text{CD}}^{avg}$, for $\\theta=0$. We obtain two orders of magnitude CD enhancement (a factor of $\\sim 110$) averaged over the entire structure for the diffractive mode peaking at $D\/\\lambda=1$. Note that previous results for induvidual nanoparticles \\cite{PRB} concluded that individual silicon nanoparticles of similar radius, were capable of inducing an enhancement of one order of magnitude in the molecular CD signal. Thus, this dual lattice design strategy provides an additional order of magnitude enhancement in $f_{\\text{CD}}^{avg}$. \n\nAlso, the CD enhancement variation among the individual spheres of the chain is shown in Fig. \\ref{Extinction_fCD}c. In this figure, we reflect the $f_{\\text{CD}}^i$ enhancement value around every sphere, for the two resonant distances appearing in Fig. \\ref{Extinction_fCD}b. The average values of the resonant enhancement for the whole structure (coinciding with the peak values of Fig. \\ref{Extinction_fCD}b) are presented as black dotted line. We can observe that both resonances have similar symmetric field distributions, whose maximum is located right at the center of the array.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.92\\textwidth]{Image_inkscape_theta0_and_thetapi41.png}\n\t\\caption{ {\\bf a)} Normalized extinction for an incidence of $\\theta = 0$. {\\bf b)} $f_\\text{CD}^{avg}$ for $\\theta = 0$. {\\bf c)} Distribution of the local CD enhancement factor for the two resonant conditions in b). {\\bf d)} Normalized extinction for an incidence of $\\theta = \\pi\/2$. {\\bf e)} $f_\\text{CD}^{avg}$ for $\\theta = \\pi\/2$. {\\bf f)} Distribution of the local CD enhancement factor for the two resonant conditions in e). Black dotted line: average enhancement for the whole structure ($f_\\text{CD}^{avg}$) at the resonant conditions in Figs. {\\bf b)} and {\\bf c)}.}\n\t\\label{Extinction_fCD}\n\\end{figure}\n\n\nOn the other hand, under incidence parallel to the array axis, at $\\theta = \\pi\/2$, the normalized extinction cross section in Fig. \\ref{Extinction_fCD}d shows a single broad peak around $D\/\\lambda \\sim 1.1$. Analyzing the array-averaged CD enhancement factor, though, two peaks are evident in Fig. \\ref{Extinction_fCD}e, one at $D\/\\lambda \\sim 1.1$ and another one around $D\/\\lambda \\sim 0.5$, both showing maximum values of $f_\\text{CD}^{avg} \\sim 60$. As pointed out above, the $f_\\text{CD}^{avg}$ peak centered at $D\/\\lambda \\sim 1.1$ is related to the redshifted single-particle magnetic resonance and, thus, it is directly related to the the peak in Fig. \\ref{Extinction_fCD}d.\n\nHowever, the second $f_\\text{CD}^{avg}$ peak at $D\/\\lambda \\sim 0.5$ is not related to an extinction peak, but to the small oscillations that the normalized extinction shows in Fig. \\ref{Extinction_fCD}d for $D\/\\lambda \\le 0.5$. This oscillations do not appear in the infinite case, which allows us to understand it as a finite chain effect. Finite chain modes have been widely studied over the literature and are known to appear for the geometric condition $D\/\\lambda < 0.5$ (concretely for $D\/\\lambda = \\frac{(N-2)n + 1}{2N(N-1)}$, with $n \\in [1,2,..., N]$)~\\cite{finite_chain}, which very well fits the range in which both the extinction oscillations and the $f_\\text{CD}^{avg}$ peak are found. The nature of these modes makes them have nearly $0$ dipole moment, which implies that they can barely contribute to the extinction but can still present very strong near-fields~\\cite{DarkPlasmon}, giving rise to the second peak in Fig. \\ref{Extinction_fCD}e. This happens because finite chain modes are typically characterized by sign changing dipole distributions~\\cite{finite_chain}, whose contributions interfere destructively in the far-field. Figure \\ref{Extinction_fCD}f shows the spatial distribution of $f_\\text{CD}^{avg}$ in the chain for the two resonant conditions in Fig. \\ref{Extinction_fCD}e. Finally, note that the $f_\\text{CD}$ distribution for the finite chain mode in Fig. \\ref{Extinction_fCD}f represents a guided mode of the finite chain, with a considerable propagation length.\n\nIn conclusion, we have presented a systematic procedure to design periodic nanophotonic platforms capable of enhancing molecular CD spectroscopy resonantly. As an example, we applied the method to design a dual 1D periodic nanostructure made of high refractive index nanoparticles. By analyzing the emergent lattice resonances in the system we have found that they can be classified in three different types: diffractive modes, non-diffractive modes and finite chain modes. We have shown that far-field diffractive resonances can increase the array-averaged CD enhancement up to a factor $\\sim 110$, providing an additional order of magnitude enhancement compared to the optical response of a single particle. Moreover, we find that non-diffractive and finite chain modes, which have not been previously considered in the field-enhanced CD literature, also give rise to comparable enhancement magnitudes (over a factor of $\\sim 60$). This method can be easily extended to more complicated platforms such as 2D metasurfaces or 3D photonic crystals, opening venues to new designs and research directions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGalaxy formation is a complex process.\nDespite continuing efforts incorporating higher resolution numerical simulations with more and more sophisticated subgrid stellar physics, it has hitherto proven impossible to satisfy observational constraints simultaneously at both dwarf and massive galaxy scales.\nIntroduction of feedback from massive black holes has helped alleviate the problem of excessive production of  massive galaxies~\\citep{2014MNRAS.445..175G}, but  nearby dwarf galaxies provide a well studied and more challenging environment.\nSupernova feedback seems incapable of resolving their  paucity~\\citep{2015ApJ...807..154B}, the too-big-to-fail problem~\\citep{2013MNRAS.433.3539G}, and the ``missing'' baryon fraction issue~\\citep{2015JATIS...1d5003B}.\n\nThis failure has motivated many attempts at modifying the nature of Dark Matter (DM), for example into warm~\\citep{Dodelson:1993je,Dolgov:2000ew,2017MNRAS.464.4520B}, fuzzy~\\citep{Hu:2000ke,2014MNRAS.437.2652M} and strongly self-interacting~\\citep{Spergel:1999mh,2015MNRAS.453...29E} variants.\nAll of these attempts seem to create as many problems as they try to\nresolve~\\citep{2016arXiv161109362S}.\n\n\\smallskip\n\nHere we take a different tack via the mirror DM~\\citep{Berezhiani:2005ek,Foot:2004pa,1983SvA....27..371B,1991SvA....35...21K}.\nWe demonstrate that a subdominant component of dissipative dark matter, containing dark baryons and dark photons identical to ordinary sector particles, naturally produces Intermediate Mass Black Holes (IMBHs), in the mass range $(10^4-10^5) \\, \\rm M_\\odot$, at the epoch of first structure formation.\nThis behavior derives from the suppression of mirror molecular hydrogen, due to a much lower fraction of free mirror electrons, which act as catalyzers.\n\nBy accretion, a few of these black holes can transform into the Supermassive Black Holes (SMBHs) observed at $z \\sim 7$ (see~\\cite{2017arXiv170303808V} and references therein), whose existence is still an unexplained issue in astrophysics.\nThis can happen because we have massive seeds and they can accrete two non-interacting dissipative matter sectors (ordinary and mirror).\n\nThe paper is organized as follows. In Sec.~\\ref{Sec:Mirror} we describe our dark matter model, whose thermal history we study in Sec.~\\ref{Sec:Thermal}.\nSection~\\ref{Sec:Structures} is devoted to the description of the structure formation in the mirror sector and the estimate of the IMBH number density.\nWe discuss the accretion of the BH seeds in~\\ref{Sec:Accretion}, and summarize our results in~\\ref{Sec:Conclusions}.\n\n\n\\section{Mirror World}\n\\label{Sec:Mirror}\nWe assume the existence of a parallel sector of mirror particles which is completely identical, in terms of particle physics properties, to the Standard Model (SM) particle sector.\nMirror particles interact with the SM only via gravitational interactions and all the portals (e.g. photon and Higgs portals) are chosen to be very small.\nWe further assume the existence of a cold DM component, which does not interact appreciably with the baryons (ordinary and mirror).\n\nWe leave the particle physics details to future work.\nFor the purpose of this article, it will suffice to note that, in the simplest scenario, the whole theory is invariant with respect to an unbroken discrete mirror parity that exchanges the fields in the two sectors, although there needs to be a breaking in the very early universe to allow different initial conditions in the two sectors~\\cite{Berezhiani:2000gw}.\nThe DM component can simply be an axion, i.e. the Goldstone of an anomalous, spontaneously broken $U(1)_{\\rm PQ}$ Peccei-Quinn (PQ) symmetry, with the $U(1)_{\\rm PQ}$ charges carried by both the ordinary and mirror Higgses (for an example, see~\\citep{Berezhiani:2000gh}).\nIn summary, from a cosmological point of view, we have:\n\\begin{itemize}\n    \\item[$\\diamond$]\n    A duplicate of the SM matter.\n    The relativistic degrees of freedom of this sector are mirror photons and neutrinos, contributing an energy density $\\Omega_{\\rm r}'$.\n    The non-relativistic degrees of freedom are mirror baryons with energy density $\\Omega_{\\rm b}'$.\nHere and in the following, the symbol ($'$) denotes the physical quantities of the mirror world.\n    \\item[$\\diamond$]\n    A Cold Dark Matter (CDM) candidate, whose energy density is denoted $\\Omega_{\\rm c}$, such that the total matter energy fraction is $\\Omega_{\\rm m} = \\Omega_{\\rm c} + \\Omega_{\\rm b} + \\Omega_{\\rm b}'$.\n\\end{itemize}\n\nAll the differences between the two sectors can be described in terms of two macroscopic parameters which are the only free parameters of the model:\n\\begin{equation}\n    x = T'_\\gamma\/T_\\gamma \\, , \\qquad \\beta = \\Omega_{\\rm b}'\/\\Omega_{\\rm b} \\, ,\n\\end{equation}\n$T_\\gamma$ being the photon temperature.\nFor simplicity, the results showed in the next sections are derived by taking  $\\beta = 1$, i.e.~$\\Omega_{\\rm b}'=\\Omega_{\\rm b}$.\n\nIn order to avoid the CMB (Cosmic Microwave Background) and BBN (Big Bang Nucleosynthesis) bounds on dark radiation, one needs the condition $x\\lesssim 0.3$~\\citep{Berezhiani:2000gw,Foot:2014uba}.\nIf this is the case, we will see in the next section that mirror matter behaves like CDM at the time of CMB last scattering  (mirror baryons are bounded in neutral mirror hydrogen atoms).\n\n\n\\section{A brief thermal history of the mirror universe}\n\\label{Sec:Thermal}\nAs discussed, in our setup the Friedmann equation reads:\n\\begin{multline}\n    H(z) = H_0 \\Bigl[ \\Omega_{\\rm r}(1+x^4)(1+z)^4 + \\\\\n    \\left(\\Omega_{\\rm b}(1+\\beta)\n    + \\Omega_{\\rm c}\\right)(1+z)^3\n    +\\Omega_\\Lambda \\Bigr]^{1\/2} \\ ,\n\\end{multline}\nwhere $H_0$ is today's Hubble constant.\n\nAn important stage for structure formation is the matter-radiation equality epoch, which occurs at the redshift\n\\begin{equation}\n\\begin{split}\n    1+z_{\\rm eq} =& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm r}^{\\rm tot}}\n    =\\frac{\\Omega_{\\rm b}(1+\\beta) + \\Omega_{\\rm c}}{\\Omega_{\\rm r}(1+x^4)} \\\\\n    =& \\frac{\\rho_{\\rm c}^0}{T_{\\gamma,0}^4} \\, \\frac{\\Omega_{\\rm m}}{ \\pi^2\/30 \\,  g_*(T_{\\gamma, 0})\\,  (1+x^4)} \\, \\ ,\n\\end{split}\n\\end{equation}\nwhere $g_*$ is the number of relativistic degrees of freedom and $T_{\\gamma,0}$ is the CMB temperature today.\nUsing the best-fit Planck parameters~\\cite{Ade:2015xua}, one gets $1+z_{\\rm eq} \\simeq 3396\/(1+x^4)$.\nSince $x \\ll 1$, the matter-radiation equality is untouched in presence of a colder mirror sector.\n\n\\medskip\nThe evolution of the free electron number fraction $X_e$ and gas temperature $T_g$ as a function of the redshift $z$ for the ordinary and mirror sectors are ruled by the following coupled differential equations~\\cite{Giesen:2012rp}:\n\\begin{align}\n    \\begin{split}\n        \\frac{{\\rm d}X_e}{{\\rm d}z} ={}& \\frac{\\mathcal P_2}{(1+z)H(z)}  \\Big(\\mathcal \\alpha_H(T_g)  n_{\\rm H} X_e^2 \\\\\n        &- \\mathcal \\beta_H(T_g) e^{-E_\\alpha\/T_g} (1-X_e) \\Big) \\, ,\n    \\end{split} \\\\\n    \\frac{{\\rm d}T_{\\rm g}}{{\\rm d}z} ={}& \\frac{1}{1+z} \\[ 2 T_g - \\gamma_{\\rm C} \\(T_\\gamma(z) - T_g \\) \\] \\, ,\n\\label{eq:evolXe&Tg}\n\\end{align}\nwhere $E_{\\alpha}$ is the Ly-$\\alpha$ energy, $\\beta_H$ is the effective photoionization rate from $n = 2$ (per atom in the $2s$ state), and $\\alpha_H$ is the case-B recombination coefficient.\nWe have defined the dimensionless coefficient\n\\begin{equation}\n    \\gamma_{\\rm C} \\equiv \\frac{8 \\sigma_{\\rm T} a_r T_\\gamma^4}{3 H m_e c} \\frac{X_e}{1 +X_{\\rm He} + X_e} \\, ,\n    \\label{eq:compton}\n\\end{equation}\n(and analogous for the mirror sector) with $\\sigma_{\\rm T}$ the Thomson cross-section, $a_r$ the radiation constant, $m_e$ the electron mass and $X_{\\rm He}$ the number fraction of helium.\nThe coefficient $\\mathcal P_2$ represents the probability for an electron in the $n = 2$ state to get to the ground state before being ionized, given by~\\cite{Giesen:2012rp}\n\\begin{equation}\n    \\mathcal{P}_2 = \\frac{1 + K_H \\Lambda_H n_H (1-X_e)}{1 + K_H (\\Lambda_H + \\beta_H) n_H (1-X_e)} \\, ,\n\\end{equation}\n(and analogous expression for the mirror sector) where $\\Lambda_H = 8.22458 \\, {\\rm s^{-1}}$ is the decay rate of the $2s$ level, and $K_H = \\lambda^3_{\\rm Ly\\alpha}\/(8 \\pi H(z))$ accounts for the cosmological redshifting of the Ly-$\\alpha$ photons.\n\nFor the ordinary sector, the boundary conditions are $X_e(z_M)=1$ and $T_g(z_M)=T_{\\gamma, 0}(1+z_M)$~\\footnote{Here $z_M=2500$. We have checked that for $z>z_M$ the solutions are stable.}.\nFor the mirror sector, the equations take the same form, with the substitutions $X_e \\to X_e'$, $T_g \\to T_g'$, $n_H \\to n_{H'}$, $\\gamma_C \\to \\gamma'_C$, $T_{\\gamma}(z) \\to x\\, T_{\\gamma}(z)$.\nThe boundary conditions in the mirror sector are $X'_e(z_M\/x)=1$ and $T'_g(z_M\/x)=T_{\\gamma, 0}(1+z_M\/x)$.\n\n\\medskip\nFrom eq.~\\eqref{eq:compton} in the mirror sector, we notice that the Compton rate is a factor $x^{4}$ smaller than that in the ordinary sector.\nAs a consequence, the recombination in the speculative sector is much faster.\nWe solve numerically Eqs.~\\eqref{eq:evolXe&Tg}, showing the results in fig.~\\ref{fig:recombination}.\nThe left panel shows the electron fraction $X_e$ for the ordinary and mirror sectors as a function of redshift. For the mirror sector we show the results for three benchmark values of $x$: $x=0.3$ (blue dot-dashed line), $x=0.1$ (magenta dot-dashed line) and $x=0.01$ (red dot-dashed line).\n\nAt the time of the ordinary recombination ($z\\simeq 1100$) the mirror hydrogen is fully recombined.\nIndeed, the residual mirror ionization fraction $X_e'$ at $z=1100$ is always less than $10^{-5}$ for the three benchmark models we consider.\nAs a consequence, mirror hydrogen behaves like CDM with respect to the ordinary plasma evolution before the CMB last scattering.\nHence, by $z \\simeq 1100$, the total amount of CDM is exactly the one measured by the Planck satellite: i.e.~$\\Omega_{\\rm DM}=\\Omega_{\\rm m}-\\Omega_{\\rm b} = \\Omega_{\\rm b}'+\\Omega_{\\rm c}$.\nThe right panel of fig.~\\ref{fig:recombination} shows instead the evolution of the gas temperatures $T_g$, $T'_g$ as a function of redshift.\nSince the Compton heating process is not efficient in keeping the mirror baryons and mirror photons in thermal equilibrium, the temperature of the mirror gas at redshifts relevant for the structure formation is much smaller than the ordinary one (not simply by a factor $x$).\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.4925\\textwidth]{Xe} \\\n\\includegraphics[width=0.4925\\textwidth]{TIGM}\n\\caption{{\\bf Left panel:} Free electron number abundance $X_e$ as a function of redshift $z$ for the ordinary (black solid), and mirror sector for different values of the $x$ parameter (dot-dashed lines). {\\bf Right panel:} Background gas temperature $T_g$ as a function of redshift $z$.}\n\\label{fig:recombination}\n\\end{figure*}\n\n\\section{Structure formation}\n\\label{Sec:Structures}\n\\subsection{Qualitative picture}\nThe different chemical initial conditions of the ordinary and mirror gas are crucial for understanding the differences in the formation of the first structures in the two sectors.\n\nLet us follow a spherical overdensity of mirror baryons.\nWhile $\\delta \\equiv \\delta \\rho\/\\rho \\ll 1$, it will expand with the rest of the universe, but lagging behind as $\\delta \\rho \\sim a^{-2}$.\nAt some point, when $\\delta \\sim 4.55$, the overdensity turns around and starts to collapse~\\citep{2010gfe..book.....M}.\nIf the matter interacts only gravitationally (as the CDM component), the final result of the collapse will be a halo of particles supported by velocity dispersion, corresponding to an effective virial temperature $T_{\\rm vir} = \\mu m_p G_N M \/ (5 k_B R)$, where $\\mu$ is the mean molecular weight, $m_p$ the proton mass, and $M$, $R$ the mass and radius of the overdensity, respectively.\nHowever, unlike CDM, the mirror gas does not undergo shell-crossing, being instead heated by shocks.\nThe end result of the collapse of the  gas is approximately a mirror cloud heated to a temperature $\\sim (\\gamma-1) T_{\\rm vir}$, where $\\gamma \\simeq 5\/3$ is the adiabatic index, at the virial density $\\rho_{\\rm vir} \\simeq 178 \\, \\rho_{cr} \\Omega_{\\rm b}'(1+z)^3$~\\cite{2010gfe..book.....M}.\n\nAt this point, the chemistry of the gas needs to be considered.\nIn particular, we have to ask whether the mirror cloud can cool, losing pressure support and contracting further, or it will remain as a hot dilute halo (resembling a CDM component).\nAt low temperatures, in the absence of heavy elements, there is not enough energy to excite mirror atomic transitions.\nTherefore, as in the ordinary sector, the main cooling mechanism for a hydrogen-helium gas is through $\\mathrm{H}_2$, which at low densities is produced mainly through the reactions $\\mathrm{H} + e^{-} \\to \\mathrm{H}^{-} + \\gamma$, $\\mathrm{H}^{-} + \\mathrm{H} \\to \\mathrm{H}_2 + e^{-}$, in which free electrons act as catalyzers.\n\nSince molecular cooling can bring the temperature down to $T_g \\sim 200 \\, \\mathrm{K}$, ordinary baryons are able to form small structures.\nHowever, in the mirror sector the initial abundance of free electrons, as shown in the left panel of Fig.~\\ref{fig:recombination}, is very suppressed, and therefore the production of mirror molecular hydrogen is slower.\nThis allows some mirror clouds to not undergo catastrophic cooling, as we will show.\n\nFirst of all, it is clear that mirror clouds with very high virial temperatures, $T_{\\rm vir} \\gtrsim 10^4 \\, \\rm K$, will behave essentially as ordinary clouds, as the mirror hydrogen quickly undergoes full ionization, independently on the initial conditions.\nThe same fate happens to massive clouds which are adiabatically heated to high temperatures at the beginning of their evolution.\nOn the other hand, at very low virial temperatures, $T_{\\rm vir} \\lesssim 1000 \\, \\rm K$, a mirror cloud is a dilute cloud of neutral atomic hydrogen, which does not cool efficiently (at least in the absence of metals) and just behaves as cold DM, as shown in Sec.~\\ref{sec:timecol}.\nWe thus expect a range of masses in between these two extrema in which the behavior of mirror clouds can be different from ordinary baryons.\n\n\n\n\n\\subsection{Semi-analytical model}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.93\\textwidth]{Plottatutto_z40}\n\n\\caption{Evolution of several physical quantities as a function of the gas number density $n_g$ at $z_{\\rm vir}=40$.\nThe panels in the first column describe the properties of the ordinary sector while the second and third columns refer to the mirror sectors with $x=0.1$ and $x=0.01$ respectively.\nIn all plots, the light gray area on the left denotes the region of the parameter space where the gas is not yet virialized.\n{\\bf First row:} Evolution of the free fall $t_{\\rm ff}$ (grey lines), sound-crossing $t_{\\rm s}$ (red lines) and cooling $t_{\\rm c}$ (blue lines) timescales for three virial temperatures ($T_{\\rm vir}=1000\\, \\rm K$ (solid lines), $T_{\\rm vir}=3500\\, \\rm K$ (dot-dashed lines) and $T_{\\rm vir}=12000\\, \\rm K$ (dotted lines)).\n{\\bf Second row:} Evolution of the gas temperature $T_g$. Three scenarios are possible: efficient gas cooling at $\\sim 200\\, \\rm K$ (red lines); quasi-isothermal collapse in the range $\\sim (500 - 900) \\, \\rm K$ (orange lines, depending on the value of $x$); quasi-isothermal collapse at $\\sim 9000 \\, \\rm K$ (blue lines).\n{\\bf Third row:} Evolution of the number fraction of molecular hydrogen $X_{\\rm H_2}$ for the scenarios discussed before.\n{\\bf Fourth row:} Evolution of the mass accretion rate estimated as the Jeans mass to free fall time $\\dot M = M_{\\rm J}\/t_{\\rm ff}$. The cyan region gives a threshold above which a {\\sc DCBH}\\ is  possible.}\n\\label{fig:All_Results}\n\\end{figure*}\n\nWe can try to understand more in detail the dynamics of a mirror cloud by considering the evolution of averaged temperature, total number density and species abundances, as done for instance in~\\cite{2000ApJ...534..809O,2001ApJ...546..635O}.\nOf course, this is only a first approximation, but it is useful to check the existence of halos in which very massive black holes may be formed by direct collapse.\n\nThe chemistry evolution of the cloud is solved using the \\textsc{krome} package~\\cite{2014MNRAS.439.2386G}.\nFor the details about the model, we refer to the built-in one-zone collapse released with the package.\nWe follow the abundance of 9 mirror species ($\\mathrm{H}$, $\\mathrm{H}^{-}$, $p$, $e$, $\\mathrm{H}_2$, $\\mathrm{H}_2^{+}$, $\\mathrm{He}$, $\\mathrm{He}^{+}$, $\\mathrm{He}^{++}$), tracking 21 reactions.\nFor the ordinary sector we use $\\mu = 1.22$, corresponding to a gas of hydrogen and helium in their standard BBN abundances.\nFor the mirror sector, we study two models, $x=0.1$ and $x=0.01$, for which \\cite{Berezhiani:2000gw} showed that the mirror helium abundance is negligible, therefore we have $\\mu'=1$.\nThe thermodynamic evolution is given by the equation\n\\begin{equation}\\label{eq:evolution}\n\\frac{\\dot{T_g}}{T_g} - (\\gamma-1 ) \\frac{\\dot{n}_g}{n_g} = \\frac{(\\gamma-1)}{k_{\\rm B} T_g n_g} (\\mathcal{H-C}) \\, ,\n\\end{equation}\nwhere $\\mathcal{C}$ and $\\mathcal{H}$ are, respectively, the cooling and heating rates per unit volume.\nThe inverse of the right hand side of Eq.~\\eqref{eq:evolution} is defined as (minus) the cooling time scale $t_{\\rm c}$.\nThe density evolution is approximated by a free-fall or isobaric evolution, depending on the shorter timescale between the sound-crossing time $t_{\\rm s}$ and the free-fall time $t_{\\rm ff}$~\\footnote{The sound-crossing time is $t_{\\rm s} = R\/c_s$, where $R$ is the radius of the cloud and $c_s$ the sound speed. The free-fall time is $t_{\\rm ff} = (3 \\pi \/ (32 G \\rho_{\\rm tot}))^{1\/2}$, $\\rho_{\\rm tot}$ being the total mass density.}.\nIf $t_{\\rm s} > t_{\\rm ff}$, we take the number density evolution to be $\\dot{n}_g = n_g\/t_{\\rm ff}$~\\cite{2000ApJ...534..809O}; if instead $t_{\\rm s} < t_{\\rm ff}$, the number density is inversely proportional to the temperature, $n_g \\propto T_g^{-1}$.\n\nOur results are shown in fig.~\\ref{fig:All_Results}, which illustrates the evolution of several physical quantities as a function of the gas density $n_g$ at a virialization redshift $z_{\\rm vir}=40$.\nFrom top to bottom, the rows show the evolution of: free-fall, sound-crossing and cooling timescales; gas temperature $T_g$; number fraction of molecular hydrogen $X_{\\rm H_2}$; and finally the accretion rate, estimated as  $\\dot{M} = M_{\\rm J}\/t_{\\rm ff}$, where $M_{\\rm J}$ is the Jeans mass.\nFrom left to right, we show the ordinary sector, the mirror with $x=0.1$ and the mirror with $x=0.01$.\nFrom the second to fourth row, the different curves correspond to different halo virial masses.\n\nFocusing on the phase diagram $(T_g-n_g)$, we can see that all the halos of the ordinary sector, after an initial phase of adiabatic contraction (in some cases followed by a short isobaric evolution) produce enough molecular hydrogen to quickly cool down to $\\sim 200 \\, \\rm K$ (red lines), according to the results reported in~\\cite{2000ApJ...534..809O,2001ApJ...546..635O}.\nThe mirror sectors, instead, shows markedly different behavior for moderate virial temperatures.\nFirst, as shown by the red curves, the more massive halos which manage to attain a temperature above the Ly$-\\alpha$ line will ionize the mirror neutral hydrogen, thus behaving as ordinary halos.\nHowever, at lower virial temperatures, the orange curves show that the evolution settles down to a quasi-isothermal collapse at a temperature in the range $\\sim (500 - 900)\\, \\rm K$ (depending on the value of $x$), at least until the density reaches $10^{10}\\,  \\mathrm{cm^{-3}}$, when 3-body reactions become important.\nAs apparent from the evolution of the timescales and of the $\\mathrm{H_2}$ abundance, this behavior is due to a balance between the cooling induced by trace amounts of molecular hydrogen and the compressional heating.\nWe also observe a third qualitative behavior in a narrow range of virial masses, illustrated by the blue curves.\nThese halos follow a trajectory which brings them close to the Ly-$\\alpha$ line, and they start to produce molecular hydrogen.\nHowever, before cooling occurs, the density reaches a critical value for which the cooling function changes behavior in $n_g$~\\cite{1999MNRAS.305..802L}.\nAfter that, there is a collapse in which the temperature decreases very slowly, with negligible amounts of $\\mathrm{H}_2$.\nThe end result of this scenario looks somewhat similar to the case in which molecular hydrogen is destroyed by Lyman-Werner photons~\\cite{2003ApJ...596...34B}, but in our case the time evolution of the halos is very slow ($\\sim \\rm Gyr$).\nTherefore, as we discuss below, these halos cannot collapse fast enough before undergoing merger events.\n\nAt this point, we would like to discuss the endpoint of the collapse of ordinary and mirror halos, whether we produce Population III ({\\sc POPIII}) stars or direct collapse black holes ({\\sc DCBH}).\nWe rely on the results of~\\cite{2014MNRAS.443.2410F,2016PASA...33...51L}, which give a threshold $\\dot{M} \\sim 10^{-2} \\, \\rm M_{\\odot}\/\\mathrm{yr}$ above which the result of the halo collapse is a {\\sc DCBH}.\nThis is shown as the cyan region in the last row of fig.~\\ref{fig:All_Results}.\nIf we choose to evaluate $\\dot{M}$ when the halo reaches a minimum temperature, which presumably means it fragments into Jeans-supported structures, we find that the ordinary sector can only form {\\sc POPIII}\\ stars, while the mirror sector is able to form {\\sc DCBH} s in the cases represented by the orange and blue curves (which however evolve too slowly).\nFrom the dashed black lines depicted in the second row of fig.~\\ref{fig:All_Results} we estimate the mass of such black holes, {\\it at their formation time}, as $(10^4-10^5) \\rm \\, M_{\\odot}$, the Jeans mass at (presumed) fragmentation (for the orange curves).\nThe number of black holes we expect to form can be as low as a few and as high as $(\\Omega'_{\\rm b}\/{\\Omega_{\\rm c}}) \\, M_{\\rm vir}\/M_{\\rm J,min}$, where $M_{\\rm vir}$ is the virial mass of the original halo and $M_{\\rm J,min}$ the Jeans mass at the temperature minimum.\nFor illustration, in the mirror sector with $x=0.1$, the maximum number of DCBHs with a mass $\\simeq 10^4\\, \\rm M_\\odot$, in a single halo, is in the range $(100-500)$ at $z_{\\rm vir}=40$.\n\n\\subsection{Time of collapse}\n\\label{sec:timecol}\nBefore concluding that each halo produces black holes, we need to check that the single-halo evolution discussed above is a reasonable approximation.\nIn particular, we have neglected the fact that structures in a $\\Lambda$CDM universe form by continuous merging of smaller objects into larger halos.\nTherefore, to be consistent, we have to require that the collapse time $t_{\\rm Coll}$ of any halo, defined as the time when the density grows super-exponentially, is shorter that a typical ``merger'' timescale, which we take as the Hubble time at the virialization epoch $t_{\\rm H}$.\nWe show this in fig.~\\ref{fig:Collasso}.\nFrom left to right, we have the ordinary sector, mirror with $x=0.1$, and mirror with $x=0.01$.\nIn each plot, we show the estimated age of the universe when we consider the halo collapsed (and a BH or a star formed), as a function of the virial temperature, for different virialization redshifts (at fixed $T_{\\rm vir}$, the redshift increases from top to bottom).\nThe gray region represents times larger than the age of the Universe today.\nThe stars denote the halos which satisfy our criterion that $t_{\\rm Coll} < t_{\\rm H}$, while the dots denote the ones which do not.\n\nThe evolution of the ordinary sector is always very fast, and we conclude that each halo we consider ends up producing {\\sc POPIII}\\ stars.\nThe evolution in both mirror sectors is instead very different.\nThere are 2 different ranges in virial temperatures which we can describe within the single-halo approximation, which correspond to all of the red curves (higher $T_{\\rm vir}$), which end up forming mirror {\\sc POPIII}\\ stars, and some of the orange curves (lower $T_{\\rm vir}$) in fig.~\\ref{fig:All_Results}, which end up forming IMBH seeds.\nAs anticipated, all the blue curves in fig.~\\ref{fig:All_Results} evolve so slowly we cannot really trust our conclusions.\nWe can say the same about the small halos with low $T_{\\rm vir}$ (dots on the left in the mirror panels of fig.~\\ref{fig:Collasso}).\nThese will evolve very slowly, thus resembling a cold dark matter component before undergoing mergers with other halos.\n\\begin{figure*}\n\\includegraphics[width=0.31\\textwidth]{Collasso_Ordinary} \\\n\\includegraphics[width=0.31\\textwidth]{Collasso_Mirrorp1} \\\n\\includegraphics[width=0.31\\textwidth]{Collasso_Mirrorp01}\n\\caption{Typical collapse timescales as a function of the virial temperature $T_{\\rm vir}$. From the left to the right we show the ordinary and the mirror sectors with $x=0.1$ and $x=0.01$ respectively. For a given $T_{\\rm vir}$, the different points are obtained by assuming several virialization redshifts. From bottom ($z_{\\rm vir} =60$) to top ($z_{\\rm vir} =10$) the step in redshift is $\\Delta z = 5$. The stars satisfy the criterium $t_{\\rm Coll}<t_H$ while the dots do not.}\n\\label{fig:Collasso}\n\\end{figure*}\n\\begin{table*}\n\\centering\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|}\n\\cline{2-10}\n& \\multicolumn{3}{c|}{Ordinary}\n& \\multicolumn{3}{c|}{$x=0.1$}\n& \\multicolumn{3}{c|}{$x=0.01$} \\\\\n\\cline{2-10}\n& $z_{\\rm vir}=50$       & $z_{\\rm vir}=40$ &  $z_{\\rm vir}=30$\n& $z_{\\rm vir}=50$      & $z_{\\rm vir}=40$ & $z_{\\rm vir}=30$\n& $z_{\\rm vir}=50$       & $z_{\\rm vir}=40$ & $z_{\\rm vir}=30$  \\\\\n\\hline\n\\hline\n\\hline\n\\multicolumn{1}{|l|}{DCBH seeds}  & $0$      & $0$     & $0$        & $403$        & $1.8\\times 10^5$     & $2.7 \\times 10^7$  & $0.01$ & $0$ &   $0$ \\\\\n\\hline\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm min}$ [$M_\\odot$]}  & $\/$      & $\/$   & $\/$   & $3.4 \\times 10^5$  & $6.2 \\times 10^5$    & $1.4 \\times 10^6$  & $1.7 \\times 10^6$  & $\/$ & $\/$ \\\\\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm max}$ [$M_\\odot$]}  & $\/$       & $\/$   & $\/$   & $4.7 \\times 10^6$  & $5.0 \\times 10^6$    & $7.7 \\times 10^6$  & $2.4 \\times 10^6$  & $\/$ & $\/$ \\\\ \\hline\n\\hline\n\\multicolumn{1}{|l|}{Star sites} & $1.1\\times 10^5$     & $3.6 \\times 10^7$     & $4.3\\times 10^9$                    & $1.2 \\times 10^{-8}$        & $0.01$  & $505$  & $1.2 \\times 10^{-8}$  & $1.4 \\times 10^{-2}$  &  $258$ \\\\\n\\hline\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm min}$ [$M_\\odot$]}  & $9.0 \\times 10^4$      & $1.2 \\times 10^5$  & $1.9 \\times 10^5$  & $8.0 \\times 10^6$  & $1.1 \\times 10^7$    & $1.9 \\times 10^7$  & $8.0 \\times 10^6$  & $1.3 \\times 10^7$ & $1.9 \\times 10^7$ \\\\\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm max}$ [$M_\\odot$]}  & $\\infty$     & $\\infty$  & $\\infty$  & $\\infty$  & $\\infty$    & $\\infty$  & $\\infty$  & $\\infty$ & $\\infty$ \\\\    \\hline\n\\end{tabular}\n\\caption{Number density of candidate halos, in $\\mathrm{Gpc}^{-3}$. For each virialization redshift we show the number density of halos able to produce {\\sc DCBH} s seed or {\\sc POPIII}\\ star forming sites,\nas well as the minimum and maximum virial masses, which are the extremes of integration of the Press-Schechter mass function (inferred from Fig.~\\ref{fig:Collasso}).\nFrom left to right, we show results for the ordinary and mirror sectors with $x=0.1$ and $x=0.01$.}\n\\label{tab:Candidates}\n\\end{table*}\n\\subsection{Number of candidate halos}\nFinally, we would like to estimate the number of black hole seeds produced by the mirror sector.\nWe can say that each halo which undergoes quasi-isothermal collapse at $T_g\\simeq (500-900)\\, \\rm K$ (orange lines in the second panel of Fig.~\\ref{fig:All_Results}) could hosts at least one, at most $(\\Omega'_{\\rm b}\/{\\Omega_{\\rm c}}) \\, M_{\\rm vir}\/M_{\\rm J}$ black holes.\nAn approximate answer to our question is the estimate of the number of such halos, which we will do using the Press-Schechter mass function.\nIn detail, we integrate the mass function over the ranges of halo virial masses able to produce {\\sc DCBH} s or stars, as identified in fig.~\\ref{fig:Collasso}.\nOur results, in number of halos per comoving $\\mathrm{Gpc}^3$, are shown in table~\\ref{tab:Candidates}, for 3 representative redshifts of virialization ($z_{\\rm vir}=30, 40, 50$).\n\nAs one can see, the amount of halos containing stars in the mirror sector is much smaller with respect to the ordinary sector, especially at high redshift.\nWe then expect that the mirror sector is, in general, less contaminated by metals, which cause halo cooling and fragmentation.\nThis can be understood by noticing that, in the case of the lowest virialization redshift we use ($z_{\\rm vir} = 30$), in the ordinary sector there is already one star formation site per halo with a mass above $\\simeq 2 \\times 10^5 \\, \\rm M_{\\odot}$ (corresponding to the lowest virial temperature we consider).\nIndeed, our estimate is of the same order ($\\sim 10^{9}\/\\rm Gpc^3$ when the Universe was $t_{\\rm vir}(z=30)+t_{\\rm coll} \\simeq 200$ Myr old) of the (extrapolated) galaxy number density of the Universe~\\cite{2016ApJ...830...83C,2015ApJ...803...34B,2013ApJ...763..129S}, while in the mirror sector we expect the star formation sites to be in negligible number ($\\sim 500$).\nFor this reason, we do not fully trust our calculations at a lower redshift, as we expect that at least the ordinary sector is polluted by metals, thus changing the collapse dynamics of both the ordinary and the mirror sector.\n\nThe star formation history in the mirror sector is different than in the ordinary sector, and we expect that mirror stars are less abundant and bigger than the ordinary ones, at the same epoch.\nFurthermore, for the mirror sector with $x=0.1$ we estimate that there are many {\\sc DCBH}\\ seeds ($\\sim 10^{7}\/\\rm Gpc^3$), more than the mirror star formation sites.\nFrom the numbers we get, we estimate that {\\sc DCBH}\\ seeds are about a factor $10^{-2}$ less abundant than the sites of star formation in the ordinary sector, in other words we expect that about one percent of galaxies will host a black hole.\n\nAs for the sector with $x=0.01$, we see that it forms very few {\\sc DCBH}\\ seeds, formed only above $z=40$, and a few mirror stars.\nIndeed, this sector is very cold and remains mostly dilute during the cosmic evolution.\n\n\n\\section{Accretion of BH seeds}\n\\label{Sec:Accretion}\nOur results so far give an interesting indication that many halos can produce very massive black hole seeds, by direct collapse of mirror baryon clouds.\nThe next question is to understand whether we can reach BH masses of up to $(10^9 - 10^{10}) \\, \\rm M_{\\odot}$ by redshift $\\sim 7$, as observed~\\cite{2017arXiv170303808V}.\n\nAfter formation, we expect that the seed black hole will grow by accretion, of both ordinary and mirror baryons.\nIn first approximation, we assume~\\cite{2004ApJ...614L..25Y} that a BH formed at time $t_0$ with mass $M_0$ grows continuously to reach a mass at time $t$\n\\begin{equation}\\label{eq:accretion}\nM(t) = M_0 \\, e^{(t-t_0)\/t_{\\rm Sal}} \\, .\n\\end{equation}\nHere, $t_{\\rm Sal}$ is the Salpeter time~\\cite{1964ApJ...140..796S}\n\\begin{equation}\nt_{\\rm Sal} = \\frac{\\epsilon M c^2}{(1-\\epsilon) L}\n\\simeq 400 \\, \\mathrm{ Myr} \\frac{\\epsilon}{1-\\epsilon} \\frac{L_{\\rm Edd}}{L} \\, ,\n\\end{equation}\nwhere $\\epsilon$ is the radiative efficiency, $L$ the BH luminosity and $L_{\\rm Edd}$ the Eddington luminosity.\nIn the presence of an ordinary and a mirror sector, which only interact gravitationally with each other, we expect that the Salpeter time roughly halves, assuming that the accretion happens in both sectors at similar luminosity and efficiency.\nTherefore, in a universe with both ordinary and mirror matter, even small seed black holes grow very fast, with an e-folding timescale of $\\sim 22 \\, \\mathrm{Myr}$ if the accretion happens at Eddington luminosity and $\\epsilon \\simeq 0.1$.\nNow, focusing on the mirror sector with $x=0.1$, by $z_{\\rm vir}=30$ we have a large amount of {\\sc DCBH}\\ seeds ($\\sim 10^{7}\/\\rm Gpc^3$) with initial masses $M_0 \\simeq 10^4 \\, \\rm M_\\odot$ formed when the Universe was $t_{\\rm vir}(z=30)+t_{\\rm Coll} \\equiv t_0 \\simeq 200 \\, \\rm Myr$.\nBy substituting these numbers in Eq.~\\eqref{eq:accretion}, we get that, at the time $t(z=7) \\simeq 800 \\, \\rm Myr$ when SMBHs with mass bigger than $10^9\\, \\rm M_\\odot$ are observed~\\cite{2011Natur.474..616M}, in our framework we are able to saturate such high masses even if the {\\sc DCBH}\\ seeds continuously accrete mass with a high radiative efficiency and luminosity less than Eddington in either sector.\nAs far as we know, ours is the only model~\\footnote{More generally, this happens in models with dissipative matter which can form dense structures.} which can accelerate the black hole growth so quickly.\nEven significant super-Eddington growth of SMBH in the ordinary sector requires seed black holes of at least $\\sim  100 \\rm \\, M_\\odot$ present at $z\\sim 20$~\\cite{2016MNRAS.458.3047P}, and in any case observations of the SMBH in quasars point out to accretion at a fraction of the Eddington luminosity~\\cite{2017ApJ...836L...1T}.\n\n\\medskip\nNevertheless, we cannot reliably estimate the final mass reached by redshift $\\sim 7$ because the exponential accretion is too simple a scenario. There are indeed two major issues to be addressed:\n\\begin{itemize}\n\\item\nFirst, there might be radiative feedback effects which shut off the mass accretion above a limiting mass~\\cite{2014MNRAS.443.2410F}.\n\n\\item\nThen, perhaps more importantly, our seed BHs are born in relatively small halos, of up to $\\sim 10^7 \\, \\rm M_{\\odot}$.\nAs a consequence, we should quantify whether the BH, once it has swallowed most of its host environment, can grow further.\nThis will be possible, because of halo accretion.\nIn fact, using the (extrapolation of the) median mass accretion rate derived in~\\cite{2010MNRAS.406.2267F}, we can determine the evolution of a typical halo by solving $\\mathrm{d} M \/ \\mathrm{d} z = 25.3 (1+1.65 z)\/(1+z) \\left[ M \/ (10^{12} {\\rm M_{\\odot}}) \\right]^{1.1} H_0^{-1} {\\rm M_{\\odot}} \\,\\mathrm{yr}^{-1}$.\nFocusing again on the mirror with $x=0.1$, we estimate that the final halo mass at $z = 7$ is $\\sim 10^8 \\, \\rm M_{\\odot}$ and $\\sim 1.6 \\times 10^9 \\, \\rm M_{\\odot}$ for, respectively, initial virial temperatures $T_{\\rm vir} = 1200$ and $T_{\\rm vir} = 3900$, corresponding to the limits of the range of BH-producing halos virialized at $z_{\\rm vir}=30$ (as shown in Fig.~\\ref{fig:Collasso}).\nThe mass in both baryonic components is a fraction $\\Omega_{\\rm b} (1+\\beta)\/\\Omega_{\\rm m}$.\nThe accretion history of a single halo has a large variability in its final mass, and the numbers we quote correspond to a median rate.\nGiven that the largest halos we consider accrete, in the median, a baryonic mass of the order of the few observed SMBHs, we find it likely that these can be produced by our mechanism.\n\n\\end{itemize}\n\n\n\n\\section{Conclusions and Outlook}\n\\label{Sec:Conclusions}\nThe presence of supermassive black holes at redshift as high as 7 is still an open issue in astrophysics.\nMoreover, baryonic feedback by supernovae has difficulty in resolving outstanding problems in dwarf galaxy formation.\nA new ingredient may be needed  that might involve IMBH feedback in dwarfs, but it is far from obvious how such IMBH might form in the early universe.\n\nWe provide a DM solution to this problem by appealing to a subdominant dissipative component of the dark sector, which for simplicity we consider to be a mirror copy of the Standard Model, while the rest of the DM sector is assumed to be a standard cold candidate.\nConcerning the particle physics details, we have in mind a model consisting of a mirror copy of the Standard Model with the same coupling constants and masses, with an axion-like particle shared by both ordinary and mirror sector.\nThe axion can potentially solve the strong-CP problem~\\cite{Berezhiani:2000gh}, but we leave the details to future work.\n\nOf course, this is not the only model which can give the phenomenology we discussed.\nWe expect that similar conclusions can be reached by changing couplings and masses (or even the particle content).\nIn particular, models with larger binding energy of the dark hydrogen atom will have a larger atomic line cooling temperature, and the cooling by molecular hydrogen is hindered because the formation rate is less efficient.\nThe cooling due to a dark hydrogen-like atom has been recently analyzed in~\\cite{Rosenberg:2017qia}.\n\n\\smallskip\n\nComing back to our mirror scenario, using a zero-dimensional approach, we showed that, if the mirror CMB temperature is much lower than the ordinary one, the thermodynamics and chemistry of collapsing mirror clouds follow a different path, leading to the production of IMBH seeds by direct collapse.\nWe have estimated the frequency, mass range and growth by accretion of the seeds, finding that they might in fact be helpful to resolve the many issues that perplex astrophycisists.\nMore in detail, we expect that by $z \\simeq 7$, in the case $x=0.1$, the bulk of the population of black holes has masses around $10^{7} - 10^{8} \\, \\rm M_{\\odot}$, with a tail that extends above $\\sim 10^{9}\\, \\rm M_{\\odot}$, and power the quasars we observe.\nThe mirror sector IMBHs are sufficiently numerous to be at the centers of the dwarf galaxies, such that their feedback could plausibly resolve many of the dwarf galaxy issues in the standard sector.\nOf course, detailed numerical simulations need to be performed in order to transform our estimates into more precise values, and analyze cosmological data in terms of our scenario.\n\nIn addition to the black holes, we will form mirror stars, more massive and much less abundant than in the ordinary sector, as shown in table~\\ref{tab:Candidates}.\nIn star-forming sites, we expect that part of the dissipative dark matter will organize itself in disks, rather than halos.\nThis has been already pointed out in similar models, for instance in~\\cite{Fan:2013yva,Fan:2013tia,Agrawal:2017rvu,Agrawal:2017pnb}.\nThe mirror substructures can be detected by large-scale structure observations, such as lensing, and by small-scale (galactic) probes, such as stellar dynamics.\n\nToday we have a new window to look at the sky: gravitational waves.\nWe expect to have IMBH mergers, and a few mergers of smaller black holes born from the death of mirror stars, at redshifts all the way up to $\\sim 30$.\nWhile IMBH mergers belong to the LISA range of frequencies, smaller black holes can be observed by LIGO.\nIt is therefore very interesting to evaluate the expected rate and mass range of these merger events.\n\n\\smallskip\n\n\n\n\n\\bigskip\n\n\\section*{acknowledgments}\nWe thank Stephane Charlot, Pratika Dayal, Anna Feltre, Andrea Macci\\`o, Mallory Roberts and Dominik Schleicher for useful discussions.\nP.P., J.S. and A.L. acknowledge support by the European Research Council ({\\sc Erc}) under the EU Seventh Framework Programme (FP7\/2007-2013)\/{\\sc Erc} Advanced Grant 267117 (`{\\sc Dark}') hosted by Universit\\'e Pierre \\& Marie Curie - Paris 6.\nA.L. aknowledges support by the European Research Council (ERC), Project no. 614199 (`BLACK').\nS.B. acknowledges funding through the DFG priority program `The Physics of the Interstellar Medium' (projects BO 4113\/1-2).\nP.P. is grateful to the Institute d'Astrophysique de Paris where this work was initiated.\n\n\n\\bibliographystyle{apsrev4-1}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet  $G$ be a group and  $n\\geqslant 1$. Recall that an \\emph{elementary Nielsen transformation} on a $n$-tuple  $T=(g_1,\\ldots, g_n)$ of elements of  $G$   is one of the following three types of moves:\n\\begin{enumerate}\n\\item[(T1)] For some $i\\in \\{1,\\ldots, n\\}$ replace $g_i$ by $g_i^{-1}$ in $T$.\n\\item[(T2)] For some $i\\neq j$, $i,j\\in \\{1, \\ldots, n\\}$ interchange $g_i$ and $g_j$ in $T$. \n\\item[(T3)] For some $i\\neq j$, $i,j\\in \\{1, \\ldots, n\\}$ replace $g_i$ by $g_ig_j$ in $T$.\n\\end{enumerate}\nWe  say that two $n$-tuples $T$ and $T'$   are \\emph{Nielsen equivalent}  if there exists a finite sequence of $n$-tuples \n$$T=T_0, T_1, \\ldots, T_{k-1}, T_k=T'$$ \nsuch that $T_i$ is obtained from $T_{i-1}$ by an elementary Nielsen transformation for $1\\leqslant i\\leqslant k$. Nielsen equivalence clearly defines an equivalence relation for tuples of elements of $G$.\n\n\\smallskip\n \nNielsen equivalence in Fuchsian groups  have been extensively studied by many authors, see  \\cite{Fine, Lustig1, Lustig2, Lustig3, Lustig4, P,Rosenberger1,  R, Zieschang} for example. However the techniques deployed so far are mainly algebraic such as normal form and K-theoretic arguments.\tIn \\cite{Dutra} we give a new geometric insight to the problem by showing that  any generating tuple of the fundamental group of a sufficiently large $2$-orbifold is represented by a special almost orbifold covering.  We still do not know if the same holds  for non sufficiently large orbifolds. The interesting case are orbifolds  with three cone points whose underlying surface is a sphere. The purpose of  this note  is to show that   generating pairs of the fundamental group of such orbifolds also have this geometric description.\n\\begin{theorem}{\\label{mainthm}}\nLet  $\\mathcal{O}=S^2(p_1, p_2, p_3)$   and  \n$$G=\\pi_1^o(\\mathcal{O})\\cong \\Delta(p_1, p_2, p_3)=\\langle s_1, s_2, s_3 \\ | \\ s_1^{p_1}=s_2^{p_2}=s_3^{p_3}=s_1s_2s_3=1\\rangle .$$ \n\nThen for any generating pair   $T$  of  $G$  there is a special almost orbifold covering  $\\eta':\\mathcal{O}'\\rightarrow \\mathcal{O}$ and a generating pair  $T' $ of $\\pi_1^o(\\mathcal{O}')$ such that $\\eta_{\\ast}'(T') $ and $T$ are Nielsen   equivalent.   \n\\end{theorem}\n\n\n\nA natural question that arises  is  whether  this holds  for arbitrary generating tuples, that is: \n\\begin{question}\nAre all  generating tuples of a triangle group also  represented by a (special)  almost orbifold coverings?\n\\end{question}\n\n  \n  \nThe proof of Theorem~\\ref{mainthm} relies heavily on the  description of all generating pairs of two-generated Fuchsian groups given  by Fine-Knapp-Matelsky-Purzitsky-Rosenberger \\cite{Fine, Knapp,Matelski, P,R}.  \n  \n  \n  \n\\section{Orbifolds and almost orbifold coverings}{\\label{secorbifold}}\n\nIn this section we give a quick review  about  orbifolds, orbifold fundamental groups and orbifold coverings.  More details about orbifolds can be found in the beautiful  article of P. Scott~\\cite{Scott}.\n\n \nA  $2$-\\emph{orbifold}  $\\mathcal{O}$ is a pair $(F, p)$ where $F$ is a  connected  surface, called the \\emph{underlying surface} of $\\mathcal{O}$, and $p:F\\rightarrow \\mathbb{N}$ is a function such that \n$$\\Sigma(\\mathcal{O}):=\\{x\\in F \\ | \\ p(x)\\geqslant 2\\}$$ \nis finite and contained in the interior   of $F$. The number $p(x)$ is called the \\emph{order} of   $x$ and the points of $\\Sigma(\\mathcal{O})$   are called \\emph{cone points}.  An  orbifold is said to be \\emph{compact} (resp. \\emph{closed}) if its underlying surface is  compact (resp. closed).  \n\nA compact orbifold   $\\mathcal{O}=(F, p)$  with $\\Sigma(\\mathcal{O})=\\{x_1, \\ldots, x_r\\}$  will also be denoted by   $F(p_1, \\ldots, p_r)$   where   $p(x_i)=p_i$ for $i=1,\\ldots, r$. \n\n\n\\smallskip \n\nThe \\emph{fundamental group} $\\pi_1^{o}(\\mathcal{O})$ of  $\\mathcal{O}=(F, p)$  is defined in terms of the fundamental group of the underlying surface $F$  as follows.  Let $x_1, x_2, \\ldots$ be the cone points of $\\mathcal{O}$ and for each $i=1, 2 , \\ldots$ let  $D_i\\subseteq \\text{Int} (F)$ be a disk centered at $x_i$ such that $ {D}_i\\cap {D}_j= \\emptyset$  for $i\\neq j$.   Set   \n$$ S_{\\mathcal{O}} = {F -  \\underset{i=1,2,\\ldots}\\cup \\text{Int}(D_i )}.$$  \nThen $\\pi_1^{o}(\\mathcal{O})$ is the  group obtained from $\\pi_1 (S_{\\mathcal{O}})$ by adding the relations $s_i^{p_i}=1$ for all $i=1,2,\\ldots$,   where $s_i$ is the homotopy class represented by $\\partial D_i$ and $p_i$ is the order of  $x_i$. For example the fundamental group of   $\\mathcal{O}=S^2(p_1, p_2, p_3)$  has presentation   \n$$\\pi_1^o(\\mathcal{O})=\\langle s_1, s_2, s_3 \\ | \\  s_1^{p_1}=s_2^{p_2}=s_3^{p_3}=s_1s_2s_3=1 \\rangle.$$\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_1.pdf}\n\\end{center}\n\\caption{The generator $s_i$ is represented by a simple closed curve that goes once around the cone point $x_i$.}\\label{fig:almost1}\n\\end{figure} \n \n \n\nIn order to define almost orbifold coverings we first  recall the definition of an orbifold covering. Let $\\mathcal{O}=(F,p)$ and $\\mathcal{O}'=(F', p')$ be orbifolds. An \\emph{orbifold covering} $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ is a continuous surjective map $\\eta:F'\\rightarrow F$ \nwith the following properties:\n\\begin{enumerate}\n\\item For each point $y\\in F'$, the order of $y$ divides the order of $\\eta(y)$.\n\n\\item For each point $x\\in \\text{Int}(F)$ the set $\\eta^{-1}(x)\\subseteq F'$ is discrete and, over a small disk in $F$ centered at $x$,  the map $\\eta$ is equivalent to the map \n$$(z, y)\\in \\mathbb{D}^2 \\times \\eta^{-1}(x) \\mapsto e^{( \\frac{2\\pi p'(y)}{p(x)}i ) } z\\in \\mathbb{D}^2$$ \nand $x$ corresponds to $0$ in $\\mathbb{D}^2$.\n\n\n\\end{enumerate} \nNote that the map $\\eta|_{F'- \\eta^{-1}(\\Sigma(\\mathcal{O}))}:F'- \\eta^{-1}(\\Sigma(\\mathcal{O}))\\rightarrow F-\\Sigma(\\mathcal{O})$ is a genuine covering.  The \\emph{degree}  of  $\\eta:\\mathcal{O}'\\rightarrow\\mathcal{O}$ is defined as  the degree of $\\eta|_{F'- \\eta^{-1}(\\Sigma(\\mathcal{O}))}$. It is not hard to see that an orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ induces a monomorphism $\\eta_{\\ast}:\\pi_1^o(\\mathcal{O}')\\rightarrow \\pi_1^o(\\mathcal{O}).$ \nConversely, for any subgroup $H\\leq \\pi_1^o(\\mathcal{O})$ there is an orbifold covering $\\eta:\\mathcal{O}_{H}\\rightarrow  \\mathcal{O}$ such that $\\eta_{\\ast}(\\pi_1^o(\\mathcal{O}_{H}))=H$.  \n\n\n  \n\\begin{definition*}[Almost orbifold covering]{\\label{def:almost}}\nLet $\\mathcal{O}'=(F', p')$ and $\\mathcal{O}=(F, p)$ be compact  orbifolds.  An \\emph{almost orbifold covering} $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ is a (non necessarily surjective) continuous map $\\eta:F'\\rightarrow F$ having the following properties:\n\\begin{enumerate}\n\\item[(C1)] For each $y\\in F'$ the order of $y$ divides the order  of $\\eta(y)$. \n\n\\item[(C2)] There is a point $x\\in\\text{Int}(F)$ of order $p\\geqslant 1$,  called the \\emph{exceptional point}, and a disk ${D}\\subseteq\\text{Int}(F)$ centered at $x\\in F$, called the \\emph{exceptional disk}, with \n$$(D-\\{x\\})\\cap \\Sigma(\\mathcal{O})=\\emptyset$$  \nsuch that $\\eta$ restricted to $F'-\\eta^{-1}(\\text{Int}(D))$ defines  an orbifold covering of finite degree between the compact $2$-orbifolds  \n$$\\mathcal{Q}':=(F'-\\eta^{-1}(\\text{Int}(D)), p'|_{F'-\\eta^{-1}(\\text{Int}(D))})\\subseteq \\mathcal{O}'$$ and $$\\mathcal{Q}:=(F-\\text{Int}(D), p|_{F-\\text{Int}(D)})\\subseteq\\mathcal{O}.$$\n\n\\item[(C3)] $\\eta^{-1}(D)=D_1\\sqcup D_2\\sqcup\\ldots\\sqcup D_t\\sqcup C_1\\sqcup \\ldots \\sqcup C_u$, where $t\\geqslant 0$, $u\\geqslant 1$  and  \n\\begin{enumerate}\n\\item[(C3.a)]  $C_1, \\ldots, C_u\\subseteq \\partial F'$ are boundary components of $\\mathcal{O}'$, called the \\emph{exceptional boundary components} of $\\mathcal{O}'$.\n\n\\item[(C3.b)] Each $D_j\\subseteq \\text{Int}(F')$ ($1\\leqslant j\\leqslant t$) is a disk and $\\eta|_{D_j}:D_j\\rightarrow D$ is equivalent to the map\n$$z\\in \\mathbb{D}^2\\longmapsto e^{\\frac{2\\pi p}{q}i}  z\\in \\mathbb{D}^2$$\nand  $x\\in D$ corresponds to $0$ in $\\mathbb{D}^2$, where  $q$ is the order of the  point $\\eta^{-1}(x)\\cap D_j$.  \n\\end{enumerate}\n\n\\end{enumerate}\nThe degree of $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ is defined as the degree of the orbifold covering  $$\\eta|_{\\mathcal{Q}'}:\\mathcal{Q}'\\rightarrow \\mathcal{Q}.$$  \nWe further call an almost orbifold covering   $\\eta :\\mathcal{O}'\\rightarrow \\mathcal{O}$   \\emph{special} if $u=1$ (i.e.   $\\mathcal{Q}'$  has a single exceptional boundary component) and  the degree of the map  $\\eta|_{C_1}:C_1\\rightarrow \\partial D$ is at most  the order $p$ of the exceptional point $x$.\n\\end{definition*} \n\n  \n\n\\begin{example}{\\label{ex:almost}}\nLet  $\\mathcal{O}=(F,p)$ be a closed orbifold and  $x$ an arbitrary  point of $F$. Let further   $D\\subseteq F$ be a small disk centered at $x$  such that $D$ contains no cone points except possibly  $x$. Denote by $\\mathcal{O}_0$ the orbifold $( F-\\text{Int}(D), p|_{F-\\text{Int}(D)})$ and by $j$ the inclusion map $ F-\\text{Int}(D) \\hookrightarrow F$. \n\nAny orbifold covering  $\\eta':\\mathcal{O}'\\rightarrow \\mathcal{O}_0$    of finite degree induces an almost orbifold covering  of $\\mathcal{O}$,  namely the map  $\\eta:= j\\circ \\eta':F'\\rightarrow F.$\n Note that the  exceptional point is   $x$,  the exceptional disk is $D$ and \n$$\\eta^{-1}(D)=\\partial \\mathcal{O}'= C_1\\sqcup \\ldots \\sqcup C_u$$  are  the exceptional boundary components of $\\eta$.  When   the degree of $\\eta'$ is one, an hence the degree of $\\eta$ is also   one,  then  $\\mathcal{O}'=\\mathcal{O}_0$.  In this case   we say that $\\eta$   is   trivial.\t\n\\end{example}\n\n \n \n\n\\section{Proof of Theorem~\\ref{mainthm}}\nThroughout  this section $\\mathcal{O}$ will denote the orbifold $S^2(p_1,p_2,p_3)$   and $G$ its fundamental  group. In \\cite{Matelski} (see also Knapp \\cite{Knapp} for the particular case of  generating pairs consisting  of elliptic elements and \\cite{Fine} for a more general discussion)  Matelski  implicitly shows that   if  $T$ is  a generating pair of $G$ (the arguments in \\cite{Matelski} work also  for non-hyperbolic triangle groups)  then  $T$ is equivalent to $T'$ where one of the following holds:\n\\begin{enumerate}\n\\item $T'=(s_{i_1}^{\\nu_1},s_{i_2}^{\\nu_2})$ with  $1\\leqslant i_1<i_2\\leqslant 3$ and $(p_{i_j}, \\nu_j)=1$ for $j=1,2$.\n\n\n\\item $p_1=2$,  $p_3\\geqslant 3$ odd and  $T'=( s_2^{\\nu}, s_1 s_2^{\\nu'}  s_1^{-1})= ( s_2^{\\nu}, s_1  s_2^{\\nu'} s_1 )$  with  $\\nu $ and $\\nu'$ coprime with $p_2$. \n \n\\item  $p_1=2$, $p_2=3$  and $p_3\\geqslant 3$ is odd,   and   $T'=(s_1, s_2^{-1}   \\cdot  s_3^{\\nu}  \\cdot  s_2)$ with $(\\nu, p_3)=1$.\n\n \n\\item $p_1=2$, $p_2=3$ and $p_3\\geqslant 4$  coprime with  $3$, and   $$T'=( s_1s_2s_1 \\cdot  s_3^{\\nu} \\cdot (s_1s_2s_1)^{-1}, s_2) $$   with $\\nu$ coprime with  $p_3$.\n\n\\item $p_1=2$, $p_2=3$ and $p_3\\geqslant 5$  coprime with  $4$, and   \n$$T'=( (s_2s_1)^2 \\cdot  s_3^{\\nu}\\cdot    (s_2s_1)^{-2}, s_3^{\\nu'})$$ \nwith $ \\nu$ and $\\nu'$ coprime with  $p_3$.\n\n\n\\item  $(p_1,p_2,p_3)=(2,3,5)$, and one of the following holds:\n\\begin{enumerate}\n\\item[(6.a)]  $T'=( s_1, (s_2^2s_1)^2\\cdot  s_2 \\cdot  (s_2^2s_1)^{-2})$. \n \n\\item[(6.b)]  $T'=(s_2\\cdot s_1 \\cdot s_2^{-1}, (s_1s_2)^2 s_1\\cdot   s_3^{\\nu}\\cdot   s_1^{-1} (s_1s_2)^{-1})$ with  $\\nu \\in \\{1, 2\\}$.\n \n \n \\item[(6.c)]  $T'=( s_2s_1 \\cdot s_2 \\cdot  (s_2s_1)^{-1}, s_3^3  s_1\\cdot    s_3^{\\nu} \\cdot   (s_3^3s_1)^{-1}) $ with $\\nu\\in \\{1, 2\\}$.\n \n \n \\item[(6.d)] $T'=(s_2  s_1 \\cdot  s_2  \\cdot (s_2s_1)^{-1}, (s_1s_2)^2 s_1\\cdot   s_3^{\\nu}\\cdot    s_1^{-1} (s_1s_2)^{-1})$. \n\\end{enumerate}\n \n\n\\item $(p_1,p_2,p_3)=(2,3,7)$, and  one of the following holds:   \n\\begin{enumerate}\n\\item[(7.a)] $T'=(s_2 ,  s_3^{-3} s_1 \\cdot    s_3^{\\nu} \\cdot  s_1    s_3^3)$ with $ \\nu$ coprime with $7$\n\n\\item[(7.b)] $T'= ((s_1s_2^2s_1s_2)^2s_2s_1s_2, (s_2(s_2s_1)^2)^2 s_2s_1) $.\n\\end{enumerate}\n\n \n\\item $p_1=p_2=3$ and $p_3\\geqslant 4$  coprime with $3$,  and   $$T'= (s_1s_2^2, s_2^2s_1).$$  \n\n \n\n\\item $p_1=2$, $p_2=4$ and $p_3\\geqslant 7$ odd, and \n$$T'=( s_1s_2^2,  s_2^3  s_1  s_2^3).$$ \n\n\n\\item $p_1=2$, $p_2=3$ and $p_3\\geqslant 7$ coprime with  $6$, and  $$T'=(s_1s_2s_1s_2^2, s_2^2s_1s_2s_1).$$\n\\end{enumerate}\nNote that the list (1)-(10) above does not give a classification of generating pairs, that is, pairs from distinct items might be Nielsen equivalent.  For example the   generating  pair $T'=(s_2^{\\nu}, s_1s_2^{\\nu'}s_1^{-1})$ given in (2) with $\\nu=\\nu'=1$   is Nielsen equivalent to the standard pair  $(s_1, s_3^2)$.  For a complete classification of generating pairs of triangle groups see \\cite{Fine}.  \n \nFor  each   generating pair given in  (1)-(10) we will  describe a  special almost orbifold covering that represents $T$.  First note that standard generating tuples, that is, those given in   (1),  are represented by trivial almost orbifold coverings, see Example~\\ref{ex:almost} and  \\cite[Example 1.9]{Dutra}.  Thus  we are left with the generating pairs given in  items  (2)-(10). \n\n\\smallskip \n\n\nLet $D\\subseteq S^2$ be a disk centered at the   cone point  $x_3$  that does not contain $x_1$ and $x_2$ as shown in Fig.~\\ref{fig:case0}. Denote the orbifold obtained from $\\mathcal{O}$ by removing the interior of  $D$  by $\\mathcal{O}_0$. Then $\\mathcal{O}_0$  is a disk  $D_0:=S^2-\\text{Int}(D)$ with two cone points $x_1$ and $x_2$ of order $p_1$ and $p_2$ respectively. Consequently  $\\pi_1^o(\\mathcal{O}_0)$ has the following presentation\n$$ \\langle S_1 , S_2 \\ | \\ S_1^{p_1}=S_2^{p_2}=1\\rangle.$$  \nThe  inclusion map $i:D_0\\rightarrow S^2$  induces a surjective  homomorphism   $$i_{\\ast} :\\pi_1^{o}(\\mathcal{O}_0)\\twoheadrightarrow \\pi_1^o(\\mathcal{O})$$\n that sends $S_i$ onto $s_i$ ($i=1,2$)  and $S_1S_2$ onto $s_3^{-1}$. \n\nLet further  $\\alpha$ and $\\beta$    be arcs in $S^2$  with $\\partial \\alpha =\\{x_1, x_3\\} $ and  $\\partial \\beta= \\{x_2, x_3\\}$  as shown in   Fig.~\\ref{fig:case0}.   Let further $\\alpha_0$ and  $\\beta_0$ be the arcs $D_0\\cap \\alpha $ and $ D_0\\cap \\beta$  in $D_0$. We also  label the cone point   $x_1$ (of order $p_1$) by a small square an the cone point $x_2$ (of order $p_2$) by a small disk. The  arcs $\\alpha$, $\\alpha_0$, $\\beta$ and $\\beta_0$   and  the labels of the points  $x_1$ and $x_2$  will help us to visualize the defined orbifold maps. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{graph_gamma.pdf}\n\\end{center}\n\\caption{ The  arcs  $\\alpha $ and $ \\beta$ in $S^2$  and the orbifold $\\mathcal{O}_0=D_0(p_1,p_2)\\subseteq \\mathcal{O}$.}\\label{fig:case0}\n\\end{figure} \n\n \n\nIn all cases we will describe an almost orbifold covering $\\mathcal{O}'\\rightarrow \\mathcal{O}$  where the   orbifold $\\mathcal{O}'$  is either a disk with two cone points   or a torus with no cone points that has an open disk removed. Moreover, in all cases   $x_3$ will be  the exceptional point and $D$  the exceptional disk.\n\n \\smallskip \n\n\\noindent \\textbf{Case (2).} Recall that $p_1=2$ and that  $p_3\\geqslant 3$ is odd. The pair $T'$ is equal to $(s_2^{\\nu} , s_1s_2^{\\nu'}s_1^{-1})$ with $(\\nu, p_2)=(\\nu', p_2)=1$. \n\nConsider the surjecitve  map $\\eta':D^2\\twoheadrightarrow D_0$ described in Fig.~\\ref{fig:case2}, that is, $\\eta'(\\alpha_1)=\\eta'(\\alpha_2)=\\alpha_0$, $\\eta'(\\beta_1)=\\eta'(\\beta_2)=\\beta_0$ and $\\eta'$  maps each component of $$D^2-\\alpha_1\\cup \\alpha_2\\cup \\beta_1\\cup \\beta_2$$ homeomorphically onto $D_0-\\alpha_0\\cup   \\beta_0$.\n \n Denote by  $\\mathcal{O}'$   the orbifold with underlying surface $ D^2$   and with cone points $y_1$ and $y_2$ both of order $p_2$. Thus $\\mathcal{O}'=D^2(p_2,p_2)$.  Then  it is not hard to see that $\\eta'$ defines an orbifold covering of degree $2$  from $\\mathcal{O}'$ onto $\\mathcal{O}_0\\subseteq \\mathcal{O}$  that corresponds to the subgroup $\\langle S_2 , S_1S_2S_1^{-1}\\rangle$ of $\\pi_1^o(\\mathcal{O}_0)$. As $\\eta_{\\ast}':\\pi_1^o(\\mathcal{O}')\\rightarrow \\pi_1^o(\\mathcal{O}_0)$ is 1-1 we  may assume that $$\\pi_1^o(\\mathcal{O}')=\\langle S_2 , S_1S_2S_1^{-1}\\rangle.$$ \n\nIt follows from Example~\\ref{ex:almost} that the map $\\eta:=i\\circ \\eta':D^2\\rightarrow S^2$ defines an almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$. As the degree of $\\eta|_C:C\\rightarrow \\partial D$ is equal to two, which is strictly smaller than $p_3$, we conclude that $\\eta$ is   special. Moreover, \n$$ \\eta_{\\ast}(S_2^{\\nu})=i_{\\ast}(S_2^{\\nu})= s_2^{\\nu} \\ \\  \\text{ and } \\ \\   \\eta_{\\ast}(S_1S_2^{\\nu'}S_1^{-1})=i_{\\ast}(S_1S_2^{\\nu'}S_1^{-1})=s_1s_2^{\\nu'}s_1^{-1}$$ \nwhich means that $\\eta$ represents the generating pair $T'$.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_2.pdf}\n\\end{center}\n\\caption{ $\\eta=\\eta'\\circ i:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $2$. }\\label{fig:case2}\n\\end{figure} \n  \n\\smallskip \n\n \n\\noindent\\textbf{Case  (3).} Recall that in this case we have $p_1=2$, $p_2=3$ and $p_3\\geqslant 3$ odd, and  $T'=(s_1, s_2^{-1}    s_3^{\\nu}   s_2)$ with $(\\nu, p_3)=1$.  \nLet $\\eta:D^2\\rightarrow S^2$ be the map  described in Fig.~\\ref{fig:case3}, that is,   $\\eta(\\alpha_1)=\\eta(\\alpha_2)=\\alpha_0$, $\\eta(\\alpha_3)=\\alpha$, $\\eta (\\beta_1)=\\eta (\\beta_2)=\\beta_0$,  $\\eta (\\beta_3)=\\beta$ and $\\eta $ maps each component of \n$$D^2-\\text{Int}(D_1)\\cup \\alpha_1\\cup\\alpha_2\\alpha_3\\cup\\beta_1\\cup\\beta_2\\cup \\beta_3$$\nhomeomorphically onto $D_0-\\alpha_0\\cup \\beta_0$ and the disk  $D_1$ homeomorphically onto $D$.  \n\n\nDenote by $\\mathcal{O}'$   the orifold  with underlying surface $D^2$ and with two cone pints $y_1$  and $y_2$ of order $p_1=2$ and $p_3$ respectively, i.e. $\\mathcal{O}'=D^2(p_1, p_3)$.   Then  $\\eta$  defines an almost orbifold covering of degree $3$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n\n\nBy the description of $\\eta$ we see that   $\\eta^{-1}(D)=D_1 \\sqcup  C$.  The restriction of $\\eta$ to the disk $D_1$ defines an orbifold covering of degree $1$ from the orbifold $\\mathcal{Q}'=D_1(p_2)$ (a disk with a single cone point, $y_2$ ,  order $p_3$)   onto  $\\mathcal{Q}=D(p_3)$ (a disk with a single cone point, $x_3$ , of order $p_3$).  Moreover,   the exceptional boundary component $C$ is mapped  by $\\eta$ onto $\\partial D$ with degree $2$ which implies that   $\\eta$ is a special almost orbifold covering. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_3.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(3)}.  $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $3$.  }\\label{fig:case3}\n\\end{figure} \n\nPut $F:= D^2-\\text{Int}(D_1)$. It is easy to see that  the map \n$$\\eta':=\\eta|_{F}:F:=D^2-\\text{Int}(D_1)\\twoheadrightarrow D_0$$  defines an orbifold covering of degree $3$ from the orbifold $\\mathcal{O}_0'=F(p_1)$ (an annulus  with a single cone point $y_1$ of order $p_1=2$) onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$ and that   $\\eta'$ corresponds to the subgroup  $\\langle  S_1 , S_2^{-1}\\cdot  S_1S_2\\cdot  S_2\\rangle$ of $\\pi_1^o(\\mathcal{O}_0)$. Thus we may assume that   \n$$\\pi_1^o(\\mathcal{O}_0')=\\langle S_1 , S_2^{-1}\\cdot S_1S_2\\cdot S_2\\rangle.$$\nThe inclusion map $j:F\\hookrightarrow D^2$  induces a surjective homomorphism $$j_{\\ast}:\\pi_1^o(\\mathcal{O}_0')\\rightarrow \\pi_1^o(\\mathcal{O}').$$ Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  \n$$t_1:=j_{\\ast}(S_1) \\ \\ \\text{ and } \\ \\ t_2:=j_{\\ast}(S_2^{-1}\\cdot S_1S_2 \\cdot S_2).$$     \nAs  $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ and   $i_{\\ast}(S_1S_2)=s_3$,  we see that $\\eta_{\\ast}$ sends the generating pair $(t_1, t_2^{-\\nu})$  of $\\pi_1^o(\\mathcal{O}')$   onto the pair $T'$, and hence  the special almost orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ represents the generating pair $T'$. \n\n\n\\smallskip \n\n\\noindent \\textbf{Case (4).}  In this case we have $p_1=2$, $p_2=3$ and $p_3\\geqslant 4$ with $(p_3, 3)=1$, and \n$$T'=( s_1s_2s_1 \\cdot   s_3^{\\nu} \\cdot   (s_1s_2s_1)^{-1}, s_2) $$ \nwith $(\\nu, p_3)=1$. Consider the map   $\\eta:D^2\\rightarrow S^2$   described in Fig.~\\ref{fig:case4} (its precise description is given as in   case (3)) and let $\\mathcal{O}'$ be the orifold  with underlying surface $D^2$ and with two cone points $y_1$  and $y_2$ of order $p_2=3$ and $p_3$ respectively, i.e. $\\mathcal{O}'=D^2(p_2, p_3)$.   Then $\\eta$  defines an almost orbifold covering of degree $4$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n \\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_4.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(4)}.  $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $4$. }\\label{fig:case4}\n\\end{figure}\n\nThe description of $\\eta$  implies  that   $\\eta^{-1}(D)=D_1  \\sqcup  C$.  The restriction of $\\eta$ to the disk $D_1$ defines an orbifold covering of degree $1$ from the orbifold $\\mathcal{Q}'=D_1(p_2)$ (a disk with a single cone point $y_2$  order $p_3$)   onto  $\\mathcal{Q}=D(p_3)$ (a disk with a single cone point $x_3$  of order $p_3$).    Moreover,   the exceptional boundary component $C$ is mapped  by $\\eta$ onto $\\partial D$ with degree $3$,  and so $\\eta$ is a special almost orbifold covering. \n \n \nThe restriction $\\eta'$  of $\\eta$ to the  surface  $F:=D^2-\\text{Int}(D_1) \\subseteq D^2$ defines an orbifold covering of degree $4$ from the orbifold $\\mathcal{O}_0'=F(p_2)$ (an annulus  with a single cone point $y_1$ of order $p_2=3$) onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$ such that \n$$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle S_1S_2S_1   \\cdot S_1S_2  \\cdot S_1S_2^2S_1\n, S_2\\rangle. $$ \nThus we may assume that  $\\pi_1^o(\\mathcal{O}_0')=\\langle S_1S_2S_1   \\cdot S_1S_2  \\cdot S_1S_2^2S_1 , S_2\\rangle.$ \n\nNow the inclusion map $j:F\\hookrightarrow D^2$  induces a surjective homomorphism $j_{\\ast}$ from $\\pi_1^o(\\mathcal{O}_0')$ onto $\\pi_1^o(\\mathcal{O}').$ Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  $$t_1:=j_{\\ast}(S_1S_2S_1   \\cdot S_1S_2  \\cdot S_1S_2^2S_1) \\ \\ \\text{ and } \\ \\  t_2:=j_{\\ast}(S_2). $$ \nSince $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ and since $i_{\\ast}(S_1S_2)=s_3^{-1}$,  we see that $\\eta_{\\ast}$ sends the generating pair $(t_1^{-\\nu}, t_2)$  of $\\pi_1^o(\\mathcal{O}')$   onto the pair $T'=( s_1s_2s_1   s_3^{\\nu}  (s_1s_2s_1)^{-1}, s_2)$. \n \n \n  \n\\smallskip \n\n \n \n\\noindent \\textbf{Case (5).} Recall that $p_1=2$, $p_2=3$ and $p_3\\geqslant 5$ with $(p_3, 4)=1$, and that   $T'$ is equal to $( (s_2s_1)^2\\cdot   s_3^{\\nu} \\cdot    (s_2s_1)^{-2}, s_3^{\\nu'})$ with $\\nu$ and $\\nu'$ coprime with $p_3$. \n\nConsider the  map $\\eta:D^2\\rightarrow S^2$  described in Fig.~\\ref{fig:case5} (again, its precise definition is given as in case (3))  and let $\\mathcal{O}'$ be the orifold  with underlying surface $D^2$ and with two cone pints $y_1$  and $y_2$  both of  order $p_3$, i.e. $\\mathcal{O}'=D^2(p_3, p_3)$.   Then it is easy to see that  $\\eta$  defines an almost orbifold covering of degree $6$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_5.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(5)} $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $6$.}\\label{fig:case5}\n\\end{figure} \n  \n  \nPut $F:=D^2-\\text{Int}(D_1)\\cup \\text{Int}(D_2)$.   It is not hard to see that  the map   $$\\eta':=\\eta|_{F}:F  \\rightarrow D_0$$ defines an orbifold covering of degree $6$ from the orbifold $\\mathcal{O}_0'=F$ (a pair of pants without cone points)  onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$.  Moreover,  \n$$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle    (S_2S_1)^2 \\cdot S_2^2S_1 \\cdot     (S_2S_1)^{-2}, S_2^2S_1 )  \\rangle.  $$ \nThus  we can assume that    $\\pi_1^o(\\mathcal{O}_0')=\\langle    (S_2S_1)^2 \\cdot S_2^2S_1 \\cdot     (S_2S_1)^{-2}, S_2^2S_1 )  \\rangle$. \n \n \nThe inclusion map $ j:F\\hookrightarrow D^2$  induces  a  surjective homomorphism $j_{\\ast}$ from $\\pi_1^o(\\mathcal{O}_0')$ onto $ \\pi_1^o(\\mathcal{O}')$.  Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  \n$$t_1:=j_{\\ast}((S_2S_1)^2 \\cdot S_2^2S_1 \\cdot     (S_2S_1)^{-2})  \\ \\ \\text{ and } \\ \\  t_2:=j_{\\ast}(S_2^2S_1).$$ \nAs $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ and since $i_{\\ast}$ maps the element $(S_2^2S_1)$ onto $ s_3$,  we see that  the   pair $(t_1^{-\\nu}, t_2)$  is mapped by $\\eta_{\\ast}$    onto the pair $T'$. \n\n \n\\smallskip\n  \n  \n  \n\\noindent \\textbf{Case (6).}  $(p_1,p_2,p_3)=(2,3,5)$. It is not hard to see that    all generating pairs given in (6.a)-(6.d) are equivalent to those given in (1)-(2).  For example, suppose that    (6.b) holds, that is,     $T'=(s_2s_1s_2^{-1}, (s_1s_2)^2s_1 \\cdot s_3^{\\nu} \\cdot s_1(s_1s_2)^{-2})$ with $\\nu \\in \\{1,2\\}$. If  $\\nu =1$  then we have \n$$ T'\\sim_{NE} (s_1 , s_3^2)$$ \n which is a standard generating pair,  and if $ \\nu=2$ then  we have\n$$T'\\sim_{NE} (s_3^2 , s_1 s_3^2  s_1^{-1})$$ \nwhich is case (2.b) with $p_2=3$ and $p_3=5$. We point out that for all cases (6.a)-(6.d)  it is also possible to construct a special almost orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$   and find a   generating pair $T_{\\mathcal{O'}}$    of $\\pi_1^o(\\mathcal{O}')$ that is mapped by $\\eta_{\\ast}$ onto $T'$.  If we take  case (6.b)   for example,  then  the almost orbifold covering  $\\eta$ described in Figure~\\ref{fig:case_6} represents $\\tau'$. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=1]{sphere_case_6.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(6.b)}.  $\\eta$ has degree $6$ and $\\mathcal{O}'$ is equal to $D^2(p_1, p_3)$.}\\label{fig:case_6}\n\\end{figure} \n \n\n\\smallskip  \n \n\\noindent \\textbf{Case (7).} $(p_1, p_2, p_3)=(2,3,7)$.  We start with the generating par  given in (7.a), that is, we define  a special  almost orbifold covering  that represents  \n$$T'=(s_2 ,  s_3^{-3} s_1 \\cdot    s_3^{\\nu} \\cdot  s_1    s_3^3)$$ \nwhere $(\\nu, 7)=1$. \n\n\nLet $\\eta:D^2\\rightarrow S^2$ be the map described in Fig.~\\ref{fig:case7a} and let $\\mathcal{O}'$ be the orbifold  with underlying surface $D^2$ and with two cone pints $y_1$  and $y_2$ of order $p_2=3$ and $p_3=7$ respectively, that is, $\\mathcal{O}'=D^2(3,  7)$.   Then $\\eta$  defines an almost orbifold covering of degree $10$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n\nThe description of $\\eta$ implies that  $\\eta^{-1}(D)=D_1\\sqcup D_2 \\sqcup  C$ and that the restriction of $\\eta$ to the disk $D_1$    defines an orbifold covering of degree $7$ from the orbifold $\\mathcal{Q}'=D_1$ (a disk without cone points) onto  $\\mathcal{Q}=D(p_3)$ (a disk with a single cone point $x_3$  of order $p_3=7$). Moreover,  $\\eta|_{ D_2}:D_2\\rightarrow D$  defines a  genuine   covering (of spaces) of degree one.   The exceptional boundary component $C$ is mapped  by $\\eta$ onto $\\partial D$ with degree $3$ which is smaller than $p_e=7$. Thus   $\\eta$ is a special almost orbifold covering. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_7_a.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(7.a)}. $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold  covering of degree $10$.}\\label{fig:case7a}\n\\end{figure} \n\n\nIt is not hard to see that    the restriction $\\eta'$  of $\\eta$ to the  surface  $$P:=D^2-\\text{Int}(D_1)\\cup \\text{Int}(D_2)\\subseteq D^2$$  \ndefines an orbifold covering of degree $10$ from the orbifold $\\mathcal{O}_0'=P(p_2)$ (a pair of pants with a single cone point $y_1$ of order $p_2=3$) onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$.   Moreover,  we have  $\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle  T_1, T_2, T_1T_2\\rangle $ \nwhere $T_1= S_2$, \n$$T_2= S_1S_2S_1S_2S_1S_2^2 \\cdot S_1S_2 \\cdot S_2S_1 S_2^2S_1S_2^2S_1$$ \nand $T_3=(S_1S_2)^7$. By the injectivity of $\\eta_{\\ast}'$   we can assume that  $\\pi_1^o(\\mathcal{O}_0')=\\langle T_1 , T_2 ,T_3\\rangle.$\nUsing the relations in $G$  we see that  $i_{\\ast} \\circ \\eta_{\\ast}' (T_1)=s_2$, that \n\\begin{eqnarray}\ni_{\\ast} \\circ \\eta_{\\ast}' (T_2) & = &  s_1s_2s_1s_2s_1s_2^2 \\cdot s_1s_2 \\cdot s_2s_1 s_2^2s_1s_2^2s_1 \\nonumber \\\\ \n          & = & s_3^{-3} s_1 \\cdot  s_3^{-1} \\cdot s_1  s_3^3 \\nonumber\n\\end{eqnarray} \nand that  $i_{\\ast} \\circ \\eta_{\\ast}'(T_3)=(s_1s_2)^7=s_3^{-7}=1.$ The inclusion map $j:P\\hookrightarrow D^2$ induces a surjective homomorphism $j_{\\ast}:\\pi_1^o(\\mathcal{O}_0')\\rightarrow \\pi_1^o(\\mathcal{O}')$   whose kernel is normally generated by $T_3$.   Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  \n$$t_1:=j_{\\ast}(T_1) \\ \\ \\text{ and } \\ \\ t_2:=j_{\\ast}(T_2).$$   \n As we clearly have $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ we see that $\\eta_{\\ast}$ sends the generating pair $(t_1, t_2^{\\nu})$  of $\\pi_1^o(\\mathcal{O}')$   onto the pair $T'$.\n\n\\smallskip \n\n \nFor the reaming of this paper  $F$  will denote  the surface  obtained from a torus by removing an open  open disk   and   $\\mathcal{O}'$  will denote  the orbifold with underlying surface $F$ and without cone points, that is, $\\mathcal{O}'$ is isomorphic to the surface $F$.\n\n\n\\smallskip\n\nWe now turn our attention to case (7.b). Thus we must find a special almost orbifold covering that represents  the pair \n$$T'= ((s_1s_2^2s_1s_2)^2s_2s_1s_2, (s_2(s_2s_1)^2)^2 s_2s_1).$$ \nThe map $\\eta:F\\rightarrow S^2$  described  in Fig.~\\ref{fig:case7b}  defines an almost orbifold covering of degree $17$ from the orbifold  $\\mathcal{O}' $ to $\\mathcal{O}$. The description of $\\eta$ implies that\n$$\\eta^{-1}(D)=D_1\\sqcup D_2 \\sqcup  C$$  and that  \n$$\\eta|_{D_i}:D_i\\rightarrow D$$  defines an orbifold covering of degree $7$ from the orbifold $\\mathcal{Q}_i'=D_i$ (a disk without cone points) onto $\\mathcal{Q}=D(p_3)$ (a disk with a single point of order $p_3=7$) for $i=1,2$. As   $\\eta|_{C}:C\\rightarrow \\partial D$ has degree $4$ we conclude that   $\\eta$ is a special almost  orbifold covering.\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_7_b.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(7.b)}. $\\eta:F\\rightarrow S^2$ defines a special  almost orbifold covering of degree $17$.}\\label{fig:case7b}\n\\end{figure} \n \nPut $F':=F-\\text{Int}(D_1)\\cup \\text{Int}(D_2)$.  It is not hard to see that the  map   \n$$\\eta':=\\eta|_{F'}: F'  \\rightarrow D_0$$  \n defines an orbifold covering of degree $17$ from the  orbifold $\\mathcal{O}_0'=F'$  onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$.  Moreover, we have \n $$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle  T_1, T_2, T_3 , T_4, T_5\\rangle \\leq \\pi_1^o(\\mathcal{O}_0)$$ \n where \n\\begin{eqnarray}\nT_1 & = & (S_1S_2^2S_1S_2)^2S_2S_1S_2  \\nonumber \\\\\nT_2 & = &  (S_2(S_2S_1)^2)^2 S_2S_1) \\nonumber \\\\\nT_3 & = &  S_1S_2^2\\cdot (S_1S_2)^7\\cdot S_1S_2^{2} \\nonumber \\\\\nT_4 & = & (S_1S_2)^{3}S_2S_1\\cdot (S_1S_2)^7 S_1S_2 (S_1S_2)^3 \\nonumber \\\\\nT_5 & = &  (S_1S_2)^{7} \\nonumber \n\\end{eqnarray}\nBy the injectivity of $\\eta_{\\ast}'$   we can assume that   $\\pi_1^o(\\mathcal{O}_0')=\\langle T_1 , T_2 ,T_3, T_4, T_5\\rangle.$\nUsing the relations in $G$  we see that  $i_{\\ast} \\circ \\eta_{\\ast}' (T_1)= (s_1s_2^2s_1s_2)^2s_2s_1s_2$, that  $i_{\\ast} \\circ \\eta_{\\ast}' (T_2)= (s_2(s_2s_1)^2)^2 s_2s_1)$ and that $$i_{\\ast} \\circ \\eta_{\\ast}' (T_3)=i_{\\ast} \\circ \\eta_{\\ast}' (T_4)=i_{\\ast} \\circ \\eta_{\\ast}' (T_5)=1.$$\n\n Now the inclusion map    $j: F'\\hookrightarrow F$  induces a surjective homomorphism  $$j_{\\ast}:\\pi_1^o(\\mathcal{O}_0')\\rightarrow \\pi_1^o(\\mathcal{O}')$$    whose kernel is normally generated by $\\{T_3, T_4, T_5\\}$.   Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  $$t_1:=j_{\\ast}(T_1) \\ \\ \\text{ and } \\ \\  t_2:=j_{\\ast}(T_2).$$   \nSince $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ we see that $\\eta_{\\ast}$ sends the  pair $(t_1, t_2)$   onto the pair $T$, and hence  the special almost orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ represents  $T'$. \n \n\\smallskip\n\n\\noindent{\\textbf{Case (8).}}  Recall that  $p_1=p_2=3$ and $p_3\\geqslant 4$ with $(p_3, 3)=1$. We must find a special almost orbifold covering that represents the generating pair $T'= (s_1s_2^2, s_2^2s_1).$  \n\nLet $\\eta':F\\twoheadrightarrow D_0\\subseteq S^2$ be the surjective map described in Fig.~\\ref{fig:case9}, that is, $\\eta'(\\alpha_i)=\\alpha_0$, $\\eta'(\\beta_i)=\\beta$ for $1\\leqslant i\\leqslant 3$ and $\\eta'$ maps each component of $$F- \\alpha_1\\cup\\alpha_2\\cup\\alpha_3\\cup \\beta_1\\cup \\beta_2\\cup\\beta_3$$ \nhomeomorphically onto $D_0-\\alpha_0\\cup\\beta_0$. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_9.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(8)}.   $\\eta =i\\circ \\eta' :F\\rightarrow S^2$ defines an almost orbifold covering of degree $3$.}\\label{fig:case9}\n\\end{figure} \n\n\nIt is not hard to see that  $\\eta'$  defines an  orbifold covering   of degree $3$ from $\\mathcal{O}'$ onto  $\\mathcal{O}_0$. Note further  that  the orbifold covering $\\eta'$ corresponds  to the subgroup \n $\\langle  t_1, t_2\\rangle$ of $\\pi_1^o(\\mathcal{O}_0)$ where $t_1= S_1S_2^2$ and $t_2= S_2^2S_1$. Thus   we  can  assume  that $\\pi_1^o(\\mathcal{O}')= \\langle t_1 , t_2\\rangle$.  Moreover,  $\\eta'$  maps the boundary $C$ of $\\mathcal{O}'$ onto  $\\partial  D$  with degree $3$. \n\nIt follows from Example~\\ref{ex:almost} that $\\eta:=i\\circ \\eta':F\\rightarrow S^2$  defines an almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$. Since $\\eta|_C$ has degree $3$, which is smaller than $p_3$, we conclude that $\\eta$ is a special almost orbifold covering. Now   the generating pair $(t_1, t_2)$ is clearly  mapped by $\\eta_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ onto the pair $T'$. \n\n\n\\smallskip\n\n\n\\noindent{\\textbf{Case (9).}} In this case   $p_1=2$, $p_2=4$ and $p_3\\geqslant 7$ odd, and  $T'=( s_1s_2^2,  s_2^3  s_1  s_2^3)$.\n\n\nLet $\\eta':F\\twoheadrightarrow D_0\\subseteq S^2$ be the surjective map described in Fig.~\\ref{fig:case10}. Thus  $\\eta'$ defines an  orbifold covering of degree $4$ from $\\mathcal{O}'$ onto  $\\mathcal{O}_0$.  Moreover,  \n$$(\\eta_n')_{\\ast}\\pi_1^o(\\mathcal{O}')=\\langle T_1, T_2\\rangle$$ \nwhere   $t_1= S_1S_2^2$  and $t_2= S_2^3  S_1  S_2^3$, consequently we may assume that $\\pi_1^o(\\mathcal{O}')$ is generated by the pair  $(t_1, t_2)$.  \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_10.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(9)}.  $\\eta =i\\circ \\eta ' :F\\rightarrow S^2$ defines a special almost orbifold covering of degree $4$.}\\label{fig:case10}\n\\end{figure}\n\n\n\nIt follows from Example~\\ref{ex:almost} that $\\eta:=i\\circ \\eta':F\\rightarrow S^2$  defines an almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$ which is special as  the degree of the map  $\\eta|_C$  is equal to $4$ which  is smaller than $p_3$. The generating pair $(t_1, t_2)$  of $\\pi_1^o(\\mathcal{O}'$)   clearly gets mapped by $\\eta_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ onto the pair $\\tau'$. \n \n\n\\smallskip \n\n\\noindent{\\textbf{Case (10).}} In this case we have $p_1=2$, $p_2=3$ and $p_3\\geqslant 7$ with $(p_3,6)=1$, and  $T'$ is equal to  $(s_1s_2s_1s_2^2, s_2^2s_1s_2s_1).$\n\n\nThe map $\\eta':F\\twoheadrightarrow D_0$  described   in Fig.~\\ref{fig:case8} defines an orbifold covering of degree $6$ from $\\mathcal{O}'$ onto $\\mathcal{O}_0$ such that \n$$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}'))  = \\langle S_1S_2S_1S_2^2 ,  S_2^2S_1S_2S_1  \\rangle.  $$\nBy applying  Example~\\ref{ex:almost} we conclude that $\\eta:=i\\circ \\eta'$ defines a almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$. Moreover it is easy to see that \n$$ \\eta_{\\ast}(S_1S_2S_1S_2^2 ,  S_2^2S_1S_2S_1 )=T'$$  \nwhich concludes the proof. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_8.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(10)}.   $\\eta =i\\circ \\eta ' :F\\rightarrow S^2$ defines a special almost orbifold covering of degree $6$.}\\label{fig:case8}\n\\end{figure}\n \n  \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nQuasi-probability distributions such as the Wigner function have been an important tool in quantum mechanics since the early days of the theory \\cite{wigner:1932}. Indeed, the Wigner function lies at the heart of the phase-space representation of quantum mechanics which is tightly connected to classical statistical mechanics and provides an alternative route to quantization \\cite{qm_phase:book}. In classical statistical mechanics, averages of observables can be described as averages taken with respect to some underlying probability distribution. This probability distribution fully describes the system under consideration and reflects the lack of knowledge of the observer. If one has perfect knowledge of the behavior of all degrees of freedom, there would be no need for probability distributions and measurement outcomes could be predicted in a deterministic manner. {This is not possible in quantum mechanics due to the probabilistic nature of the theory.} Similarly to classical statistical mechanics, the fluctuations in quantum mechanics can be encoded in \\textit{quasi}-probability distributions. These distributions can take on negative values which is ultimately a consequence of the non-commutativity of operators, reflecting the fact that a measurement inevitably disturbs all subsequent measurements of observables which do not commute with the first one. Negative quasi-probability distributions therefore reflect the impossibility of accessing observables without disturbing them, indicating a non-classical behavior of the system which requires to take into account the action of the measurement apparatus to predict its outputs.\nThe concept of negative quasi-probability distributions, or \\textit{negativity} in short, is therefore closely related to the concept of contextuality which deals with the dependence of measurement outcomes on the experimental setup \\cite{spekkens:2008,ferrie:2008,ferrie:2009,ferrie:2011}.\n\nSo far, most quasi-probability distributions describe the quantum state of a system. This holds true for the Wigner function and its generalizations to finite dimensional Hilbert spaces \\cite{wootters:1987,gibbons:2004,ferrie:2011}, as well as a number of other distributions used in quantum optics \\cite{cahill:1969,walls:book}. Negativity in these distributions indicates non-classicality of the quantum state. However, a state that exhibits negativity in some distribution might not do so in another one \\cite{lutkenhaus:1995}, implying that non-classicality is not necessarily a general feature but depends on the way a state is interrogated (see also Refs.~\\cite{revzen:2005,spekkens:2008,ferrie:2008,ferrie:2009}). A well studied quasi-probability distribution which does not describe the quantum state of a system but rather its dynamics is the full counting statistics (FCS) \\cite{levitov:1996,nazarov:book2,nazarov:2003,bednorz:2010,clerk:2011,bednorz:2012,hofer:2016}. Negativity in the FCS has recently been connected to a peculiar interference effect and it was shown that systems which do not show any negativity in their Wigner function can still show negativity in the FCS \\cite{hofer:2016}. This further stresses the fact that systems might exhibit completely classical behavior in some aspects, e.g. their instantaneous states, while exhibiting non-classicality in others, e.g. their dynamics which determine the temporal fluctuations within the system. It would therefore be highly desirable to have access to a quasi-probability distribution which encodes only the aspects of interest of a system.\nThis is in marked contrast to Refs.~\\cite{spekkens:2008,ferrie:2008,ferrie:2009,zhu:2016} which consider a full description of states and measurements in terms of quasi-probabilities.\n\nThe goal of the present paper is to introduce a general quasi-probability distribution that can be tailored to the problem at hand and thus provides a general framework for investigating fluctuations of non-commuting observables. Because this distribution is based on the Keldysh formalism, we call it the Keldysh quasi-probability distribution (KQPD). When the observables of interest are the position and momentum operators, the KQPD reduces to the Wigner function. Similarly, when one is interested in the time-integral of an observable, the FCS of the corresponding operator is obtained. Furthermore, the outcome of von Neumann type measurements \\cite{vonneumann:book} can be predicted by convolving the corresponding KQPD with the Wigner function of the detectors [cf.~Sec.~\\ref{sec:vonneumann}]. This convolution takes into account the imprecision as well as the back-action of the performed measurement. An example is provided by the Husimi $Q$-function which is obtained by convolving the Wigner function with Gaussians and predicts the outcomes of simultaneous but imprecise measurements of position and momentum \\cite{stenholm:1992}.\n\nThe rest of the article is structured as follows. In Sec.~\\ref{sec:keldysh}, we introduce the KQPD, discuss the Wigner function, the FCS, and two subsequent spin measurements as examples, and discuss negativity of the KQPD  as an indicator for non-classical behavior. In Sec.~\\ref{sec:vonneumann}, we provide a more physical motivation for investigating the KQPD by connecting it to measurement outcomes which requires the introduction of measurement imprecision and back-action. Before concluding in Sec.~\\ref{sec:conclusions}, we discuss applications of the KQPD in Sec.~\\ref{sec:applications}, including the distribution of thermodynamic work, (anomalous) weak values, and Leggett-Garg inequalities. Throughout this paper, we set $\\hbar=1$.\n\n\n\\section{The Keldysh quasi-probability distribution}\n\\label{sec:keldysh}\nIn this section we briefly introduce the Keldysh formalism, a powerful framework for tackling out-of-equilibrium problems, where the KQPD arises quite naturally through its moment generating function. Here we only touch upon the ideas in the Keldysh formalism which are relevant for our discussion. For a detailed introduction to the subject we refer the reader to Ref.~\\cite{kamenev:book}. We will start by considering the somewhat trivial example of the statistics of a single observable before we generalize the results to the highly non-trivial scenario of multiple, non-commuting observables at different times.\n\n\\subsection{Single observable}\n\nThe main idea that we will be concerned with is the evolution along the closed time contour illustrated in Fig.~\\ref{fig:ctc}, an idea that goes back to Schwinger \\cite{schwinger:1961}. In this section, we are interested in the statistics of a single observable $\\hat{A}$ at time $\\tau$. Its average value can be written as\n\\begin{equation}\n\\label{eq:avga}\n\\begin{aligned}\n\\langle A \\rangle_\\tau &= {\\rm Tr}\\left\\{\\hat{U}(0,\\tau)\\hat{A}\\hat{U}(\\tau,0)\\hat{\\rho}_0\\right\\} \\\\&=\\frac{1}{2}{\\rm Tr}\\left\\{\\hat{U}(0,\\mathcal{T})\\hat{U}(\\mathcal{T},\\tau)\\hat{A}\\hat{U}(\\tau,0)\\hat{\\rho}_0\\right\\} +\\frac{1}{2}{\\rm Tr}\\left\\{\\hat{U}(0,\\tau)\\hat{A}\\hat{U}(\\tau,\\mathcal{T})\\hat{U}(\\mathcal{T},0)\\hat{\\rho}_0\\right\\},\n\\end{aligned}\n\\end{equation}\nwhere $\\hat{U}(t',t)=\\hat{U}^\\dag(t,t')$ is the time-evolution operator from time $t$ to time $t'$ and $\\hat{\\rho}_0$ denotes the density matrix at time $t=0$. Throughout this article, we consider unitary time evolution. Generalizing the results to non-unitary time evolution is in principle straightforward and discussed in App.~\\ref{app:nonunitary}. Writing the density matrix as a mixture of pure states, the first line of the last expression can be interpreted as time evolving the states up to time $\\tau$, evaluating the operator $\\hat{A}$, and evolving the states backward to the initial time $t=0$ [cf.~the left-hand side of Fig.~\\ref{fig:ctc}]. On the second line, we inserted identities $\\mathbb{1}=\\hat{U}(\\tau,\\mathcal{T})\\hat{U}(\\mathcal{T},\\tau)$, extending the contour along which the state is time evolved. We call the resulting contour, from $t=0$ to $t=\\mathcal{T}$ and back again, the Keldysh contour. We can evaluate the operator $\\hat{A}$ either along the forward-in-time or along the backward-in-time branch of the contour. As discussed below, the most natural choice is to treat both contours on an equal footing as illustrated graphically in Fig.~\\ref{fig:ctc}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{ctc}\n\\caption{Illustration of the closed time contour. The average $\\langle\\hat{A}\\rangle_\\tau$ can be obtained by evolving the state (or a mixture of states) up to time $\\tau$, where the operator $\\hat{A}$ is evaluated, and back again. Alternatively, one can extend this closed-time contour to the Keldysh contour (from $t=0$ to $t= \\mathcal{T}$ and back again) and evaluate the operator either on the forward-in-time or on the backward-in-time branch.}\n  \\label{fig:ctc}\n\\end{figure}\n\nMore information on the observable $\\hat{A}$ can be obtained by considering its moment generating function given by\n\\begin{equation}\n\\label{eq:momgena1}\n\\Lambda(\\lambda;\\tau)=\\langle e^{-i\\lambda\\hat{A}}\\rangle_\\tau = {\\rm Tr}\\left\\{\\hat{U}(0,\\tau)e^{-i\\frac{\\lambda}{2}\\hat{A}}\\hat{U}(\\tau,\\mathcal{T})\\hat{U}(\\mathcal{T},\\tau)e^{-i\\frac{\\lambda}{2}\\hat{A}}\\hat{U}(\\tau,0)\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nAll higher moments can be obtained from this generating function by taking derivatives with respect to $\\lambda$\n\\begin{equation}\n\\label{eq:momentsa1}\n\\langle \\hat{A}^k\\rangle_\\tau=\\left.i^k\\partial_\\lambda^k\\Lambda(\\lambda;\\tau)\\right|_{\\lambda=0}.\n\\end{equation}\nIn anticipation of the discussion below, we wrote Eq.~\\eqref{eq:momgena1} symmetrically in terms of the Keldysh contour. Taking the derivative with respect to $\\lambda$ at $\\lambda=0$, we recover Eq.~\\eqref{eq:avga}. Since the moment generating function is the average of the exponential of $\\hat{A}$, it corresponds to the Fourier transform of a probability distribution associated to the fluctuations of $\\hat{A}$\n\\begin{equation}\n\\label{eq:probaa1}\n\\Lambda(\\lambda;\\tau)=\\int_{-\\infty}^\\infty dA \\mathcal{P}(A;\\tau)e^{-i\\lambda A},\\hspace{1cm}\\mathcal{P}(A;\\tau)=\\sum_{A'}\\delta(A-A')\\langle A'|\\hat{U}(\\tau,0)\\hat{\\rho}_0\\hat{U}^\\dag(\\tau,0)|A'\\rangle,\n\\end{equation}\nwhere the sum over $A'$ goes over all eigenstates $\\hat{A}|A'\\rangle=A'|A'\\rangle$ and has to be replaced by an integral if $\\hat{A}$ has a continuous spectrum. In the case of a single observable, the KQPD $\\mathcal{P}(A;\\tau)$ is a positive probability distribution which predicts the outcome of projectively measuring $\\hat{A}$ at time $\\tau$.\n\nThe symmetric treatment with respect to the forward-in-time and the backward-in-time branches of the Keldysh contour might seem like an unnecessary complication at this point. Below, we show how such a treatment for multiple observables leads to a time-ordering prescription which is relevant for von Neumann type measurements. Before discussing multiple observables in detail, we express the moment generating function in terms of the classical and quantum fields obtained by a Keldysh rotation [cf.~Eq.~\\eqref{eq:keldyshrot}]. To this end, we assume that the observable $\\hat{A}$ has a discrete spectrum and that its eigenstates span the whole Hilbert space, i.e. $\\mathbb{1}=\\sum_{A}|A\\rangle\\langle A|.$\nWe then divide the time evolution along the Keldysh contour in Eq.~\\eqref{eq:momgena1} into $2N$ steps ($N$ for the forward-, $N$ for the backward-in-time branch) of infinitesimal length $\\delta t$ and insert identities in between the steps. Writing the trace in terms of the eigenstates of $\\hat{A}$, we then find\n\\begin{equation}\n\\label{eq:momgensuma1}\n\\Lambda(\\lambda;\\tau)=\\sum_{\\boldsymbol{A}_+,\\boldsymbol{A}_-}\\delta_{A_\\mathcal{T}^+,A_\\mathcal{T}^-}e^{-i\\frac{\\lambda}{2}(A_\\tau^++A_\\tau^-)}\\mathcal{A}(\\boldsymbol{A}_+)\\mathcal{A}^*(\\boldsymbol{A}_-)\\langle A_0^+|\\hat{\\rho}|A_0^-\\rangle.\n\\end{equation}\nHere, $A_t^{+(-)}$ denotes the state that was inserted at time $t$ on the forward-in-time (backward-in-time) branch, the vectors $\\boldsymbol{A}_\\pm$ gather all $A_t^\\pm$ and the amplitudes along the branches read\n\\begin{equation}\n\\label{eq:amplitudesa1}\n\\mathcal{A}(\\boldsymbol{A}_\\pm)=\\langle A_\\mathcal{T}^\\pm|\\hat{U}(\\mathcal{T},\\mathcal{T}-\\delta t)|A_{\\mathcal{T}-\\delta t}^\\pm\\rangle\\langle A_{\\mathcal{T}-\\delta t}^\\pm|\\cdots |A^\\pm_{\\delta t}\\rangle\\langle A_{\\delta t}^\\pm|\\hat{U}(\\delta t, 0)|A_0^\\pm\\rangle.\n\\end{equation}\nThe Kronecker delta in Eq.~\\eqref{eq:momgensuma1} is due to the fact that the point $t=\\mathcal{T}$ connects the two branches (cf.~Fig.~\\ref{fig:ctc}). We can write the moment generating function in the form of a path integral\n\\begin{equation}\n\\label{eq:pathintegral}\n\\Lambda(\\lambda;\\tau)=\\int \\mathcal{D}[A_t^+,A_t^-]e^{-i\\frac{\\lambda}{2}(A_\\tau^++A_\\tau^-)}e^{iS[A_t^+,A_t^-]},\n\\end{equation}\nwhere the action $S$ is determined by $\\exp(iS[A_t^+,A_t^-])=\\mathcal{A}(\\boldsymbol{A}_+)\\mathcal{A}^*(\\boldsymbol{A}_-)\\langle A_0^+|\\hat{\\rho}|A_0^-\\rangle$\nand the integration measure reads\n$\\int \\mathcal{D}[A_t^+,A_t^-]=\\sum_{\\boldsymbol{A}_+,\\boldsymbol{A}_-}\\delta_{A_\\mathcal{T}^+,A_\\mathcal{T}^-}$.\nFinally, we perform a Keldysh rotation by introducing the new fields \\cite{kamenev:book}\n\\begin{equation}\n\\label{eq:keldyshrot}\nA^{cl}_t=\\frac{1}{2}(A_t^++A_t^-),\\hspace{2cm} A^q_t = A_t^+-A_t^-,\n\\end{equation}\nwhere the superscript stands for classical and quantum respectively. This nomenclature derives from the fact that the forward- and backward-in-time evolutions are equal for classical processes. Using these new integration variables, the path integral in Eq.~\\eqref{eq:pathintegral} reduces to\n\\begin{equation}\n\\label{eq:pathintegral2}\n\\Lambda(\\lambda;\\tau)=\\int \\mathcal{D}[A_t^{cl},A_t^q]e^{-i\\lambda A_\\tau^{cl}}e^{iS[A_t^{cl},A_t^q]}.\n\\end{equation}\nThe corresponding probability distribution then reads\n\\begin{equation}\n\\label{eq:probapathint}\n\\mathcal{P}(A;\\tau)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{i\\lambda A}d\\lambda\\Lambda(\\lambda;\\tau)=\\int \\mathcal{D}[A_t^{cl},A_t^q]\\delta(A-A_\\tau^{cl})e^{iS[A_t^{cl},A_t^q]}.\n\\end{equation}\nThe moment generating function thus describes the fluctuations of $A_\\tau^{cl}$ and the probability density $\\mathcal{P}(A;\\tau)$ is given by the sum over all paths for which $A_\\tau^{cl}=A$, weighted by the complex exponential of the action in complete analogy to Feynman path integrals \\cite{altland:book}. In the case of a single observable, the quantum field is zero (i.e. $A_\\tau^+=A_\\tau^-$), making its introduction seem like an unnecessary complication. However, when multiple observables are involved, the quantum fields are no longer bound to be zero. Considering the fluctuations of the classical fields then corresponds to a given time-ordering prescription. We now generalize the results of this section to an arbitrary number of (possibly non-commuting) observables at arbitrary times.\n\n\\subsection{Multiple observables}\n\nThe KQPD, denoted by $\\mathcal{P}(\\boldsymbol{A})$, can be defined via its moment generating function which is obtained by generalizing Eq.~\\eqref{eq:momgena1} and reads\n\\begin{equation}\n\\label{eq:kmomgen}\n\\begin{aligned}\n\\Lambda(\\boldsymbol{\\lambda})&\\equiv\\int d\\boldsymbol{\\lambda}e^{-i\\boldsymbol{\\lambda}\\cdot\\boldsymbol{A}}\\mathcal{P}(\\boldsymbol{A})\\\\\n&\\equiv{\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_+\\exp\\left[-\\frac{i}{2}\\int_0^\\mathcal{T} d\\tau\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)\\right]\\hat{\\rho}_0\\hat{\\mathcal{T}}_-\\exp\\left[-\\frac{i}{2}\\int_0^\\mathcal{T} d\\tau\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)\\right]\\right\\}\\\\&\n={\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_K\\exp\\left[-\\frac{i}{2}\\int_K d\\tau\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)\\right]\\hat{\\rho}_0\\right\\}.\n\\end{aligned}\n\\end{equation}\nHere $\\hat{\\mathcal{T}}_{+(-)}$ is the (anti) time-ordering operator, the variables $\\lambda_l$ and $A_l$ are grouped in vectors $\\boldsymbol{\\lambda}$ and  $\\boldsymbol{A}$, and the operators are given in the Heisenberg picture, i.e.\n\\begin{equation}\n\\label{eq:aheisen}\n\\hat{A}_l(t)=\\hat{U}(0,t)\\hat{A}_l\\hat{U}(t,0),\n\\end{equation}\nwhere $\\hat{A}$ denotes the operator in the Schr\u00f6dinger picture (where all considered operators are time-independent). In the last line of Eq.~\\eqref{eq:kmomgen}, $\\hat{\\mathcal{T}}_K$ denotes time-ordering along the Keldysh contour (from $t=0$ to $t=\\mathcal{T}$ and back again). The functions $f_l(\\tau)$ take into account a possibly time-dependent coupling to the observables $\\hat{A}_l$. When using the Keldysh contour, the functions $f_l$ take on the same values on the forward- and the backward-in-time branch.\nThe case of $N$ observables probed instantaneously at different times $\\tau_l$ is obtained by taking $f_l(\\tau_l)=\\delta(t-\\tau_l)$ (for $l\\leq N$) resulting in the moment generating function\n\\begin{equation}\n\\label{eq:momgenmultsub}\n\\Lambda(\\boldsymbol{\\lambda})={\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_N}{2}\\hat{A}_N(\\tau_N)}\\cdots e^{-i\\frac{\\lambda_1}{2}\\hat{A}_1(\\tau_1)}\\hat{\\rho}_0e^{-i\\frac{\\lambda_1}{2}\\hat{A}_1(\\tau_1)}\\cdots e^{-i\\frac{\\lambda_N}{2}\\hat{A}_N(\\tau_N)} \\right\\}.\n\\end{equation}\nThe apparent symmetry between terms on the left and on the right of the density matrix is equivalent to treating the forward- and the backward-in-time branch of the time evolution symmetrically and determines the time-ordering of operators when evaluating higher moments (see below).\n\n\n\\subsection{Examples}\nWe now introduce two well-known examples of the KQPD: the Wigner function and the Full counting statistics (FCS). To recover the Wigner function from the moment generating function in Eq.~\\eqref{eq:kmomgen}, we set\n\\begin{equation}\n\\label{eq:lamsetwig}\n\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)=\\delta(\\tau)\\left[\\lambda_x\\hat{x}(\\tau)+\\lambda_p\\hat{p}(\\tau)\\right],\n\\end{equation}\nwhich results in\n\\begin{equation}\n\\label{eq:wigmom}\n\\Lambda(\\lambda_x,\\lambda_p)={\\rm Tr}\\left\\{e^{-\\frac{i}{2}(\\lambda_x\\hat{x}+\\lambda_p\\hat{p})}\\hat{\\rho}_0e^{-\\frac{i}{2}(\\lambda_x\\hat{x}+\\lambda_p\\hat{p})}\\right\\},\n\\end{equation}\nwhere we identified $\\hat{x}=\\hat{x}(0)$ and similarly for $\\hat{p}$. It is straightforward to verify that the Wigner function is indeed given by\n\\begin{equation}\n\\label{eq:wigner}\n\\mathcal{W}(x,p)=\\frac{1}{(2\\pi)^2}\\int d\\lambda_xd\\lambda_p e^{i\\lambda_x x+i\\lambda_p p} \\Lambda(\\lambda_x,\\lambda_p)=\\frac{1}{2\\pi}\\int d\\lambda_pe^{-i\\lambda_p p}\\left\\langle x+\\frac{\\lambda_p}{2}\\right|\\hat{\\rho}_0\\left|x-\\frac{\\lambda_p}{2}\\right\\rangle.\n\\end{equation}\nThe Wigner function is thus recovered as the KQPD corresponding to a simultaneous observation of $\\hat{x}$ and $\\hat{p}$. In the next section, we discuss how the Wigner function can be used to predict a simultaneous (but imprecise) measurement of position and momentum \\cite{arthurs:1964}. The moments obtained from the Wigner function\n\\begin{equation}\n\\label{eq:momwig}\n\\langle x^n p^m\\rangle_W=\\int dx dp \\,x^np^m \\mathcal{W}(x,p),\n\\end{equation}\nare determined by the Weyl order \\cite{carmichael:book} obtained by complete symmetrization, i.e. every possible order is given the same weight, e.g.\n\\begin{equation}\n\\label{eq:weylord}\n\\langle x p\\rangle_W=\\frac{1}{2}{\\rm Tr}\\left\\{(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})\\hat{\\rho}_0\\right\\},\\hspace{1cm}\n\\langle x^2p\\rangle_W=\\frac{1}{3}{\\rm Tr}\\left\\{(\\hat{x}^2\\hat{p}+\\hat{x}\\hat{p}\\hat{x}+\\hat{p}\\hat{x}^2)\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nThis is a general result for the KQPD; when considering multiple operators at equal times, the moments generated by the KQPD are determined by the quantum averages of the completely symmetrized operators.\n\nThe next example we consider in this section is the FCS which is obtained from the moment generating function in Eq.~\\eqref{eq:kmomgen} upon setting\n\\begin{equation}\n\\label{eq:lamsetfcs}\n\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)=\\lambda \\hat{A}(\\tau),\n\\end{equation}\nwhich results in\n\\begin{equation}\n\\label{eq:momgenfcs}\n\t\\Lambda(\\lambda)={\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_+e^{-i\\frac{\\lambda}{2}\\int_{0}^{\\mathcal{T}}d\\tau\\hat{A}(\\tau)}\\hat{\\rho}_0 \\hat{\\mathcal{T}}_-e^{-i\\frac{\\lambda}{2}\\int_{0}^{\\mathcal{T}}d\\tau\\hat{A}(\\tau)}\\right\\}={\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_Ke^{-i\\frac{\\lambda}{2}\\int_Kd\\tau\\hat{A}(\\tau)}\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nThe FCS is given by the Fourier transform of the last expression\n\\begin{equation}\n\\label{eq:fcs}\n\\mathcal{F}(m)\\equiv\\frac{1}{2\\pi}\\int d\\lambda e^{i\\lambda m}\\Lambda(\\lambda),\n\\end{equation}\nand gives information on the fluctuations of the time integral of $\\hat{A}$. The FCS has been investigated intensively in the context of electron transport \\cite{nazarov:book2}, where $\\hat{A}$ corresponds to the current operator and one is interested in the fluctuations of the charge that is transferred through a conductor during a time interval $[0,\\mathcal{T}]$.\n\nThe moments of the FCS can conveniently be expressed using the Keldysh contour\n\\begin{equation}\n\\label{eq:momentsfcs}\n\\int dm\\, m^k\\mathcal{F}(m)=i^k\\partial_\\lambda^k\\Lambda(\\lambda)\\big|_{\\lambda=0}=\\int_Kd\\tau_1\\cdots d\\tau_k{\\rm Tr}\\left\\{ \\hat{\\mathcal{T}}_K\\hat{A}(\\tau_1)\\cdots \\hat{A}(\\tau_k) \\hat{\\rho}_0\\right\\}.\n\\end{equation}\nWe thus find that the time-ordering corresponding to the KQPD is determined by the \\textit{Keldysh time-ordering}, implying that operators are ordered along the Keldysh contour.\nThe Wigner function as well as the FCS have been discussed extensively in the literature. Explicit evaluations exhibiting negativity can be found in, e.g., Refs.~\\cite{kenfack:2004,hofer:2016}.\n\nTo further illustrate the KQPD, we consider the very simple example of two subsequent spin measurements. The operators of interest are given by the rotated Pauli matrices $\\hat{\\sigma}_1$ and $\\hat{\\sigma}_2$. For simplicity, we consider a qubit in a pure state $|+_0\\rangle$ which is an eigenstate of $\\hat{\\sigma}_0$ with eigenvalue $+1$. The KQPD then reads\n\\begin{equation}\n\\label{eq:kqpdspins}\n\\mathcal{P}(\\Sigma_1,\\Sigma_2)=\\frac{1}{(2\\pi)^2}\\int d\\lambda_1d\\lambda_2e^{i\\lambda_1\\Sigma_1+i\\lambda_2\\Sigma_2}{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_2}{2}\\hat{\\sigma}_2}e^{-i\\frac{\\lambda_1}{2}\\hat{\\sigma}_1}|+_0\\rangle \\langle +_0|e^{-i\\frac{\\lambda_1}{2}\\hat{\\sigma}_1}e^{-i\\frac{\\lambda_2}{2}\\hat{\\sigma}_2}\\right\\}.\n\\end{equation}\nWriting the eigenstates of $\\hat{\\sigma}_j$ in the eigenbasis of $\\hat{\\sigma}_1$\n\\begin{equation}\n\\label{eq:basis}\n|+_0\\rangle = \\alpha|+_1\\rangle +\\beta|-_1\\rangle\\hspace{2cm}|+_2\\rangle = \\gamma|+_1\\rangle +\\delta|-_1\\rangle,\n\\end{equation}\nwe can evaluate Eq.~\\eqref{eq:kqpdspins}, which can be written as\n\\begin{equation}\n\\mathcal{P}(\\Sigma_1,\\Sigma_2)=\\sum_{\\sigma_1=0,\\pm 1}\\sum_{\\sigma_2=\\pm 1}\\tilde{\\mathcal{P}}(\\sigma_1,\\sigma_2)\\delta(\\Sigma_1-\\sigma_1)\\delta(\\Sigma_2-\\sigma_2),\n\\end{equation}\nwith the discrete probability distribution\n\\begin{equation}\n\\label{eq:kqpdspinsdisc}\n\\begin{aligned}\n&\\tilde{\\mathcal{P}}(+1,+1)=|\\alpha|^2|\\gamma|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(-1,+1)=|\\beta|^2|\\delta|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(0,+1)=2{\\rm Re}\\left\\{\\alpha\\beta^*\\gamma^*\\delta\\right\\},\\\\&\n\\tilde{\\mathcal{P}}(+1,-1)=|\\alpha|^2|\\delta|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(-1,-1)=|\\beta|^2|\\gamma|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(0,-1)=-2{\\rm Re}\\left\\{\\alpha\\beta^*\\gamma^*\\delta\\right\\}.\n\\end{aligned}\n\\end{equation}\nNote that although the operators $\\hat{\\sigma}_j$ have eigenvalues $\\pm 1$, the KQPD is non-zero also for $\\sigma_1=0$. Furthermore, these terms can become negative. We will discuss the origin of these terms and their negativity in more detail in the next section. For now, we only mention that the negativity reflects the fact that a measurement of $\\hat{\\sigma}_1$ disturbs a subsequent measurement of $\\hat{\\sigma}_2$. The negative terms thus vanish if these two operators are simultaneously measurable, $[\\hat{\\sigma}_1,\\hat{\\sigma}_2]=0$, if the first measurement does not influence the initial state, $[\\hat{\\sigma}_0,\\hat{\\sigma}_1]=0$, or if a measurement of $\\hat{\\sigma}_2$ has a completely random outcome no matter if $\\hat{\\sigma}_1$ is measured or not. The latter is the case if $\\hat{\\sigma}_0=\\hat{\\sigma}_x$, $\\hat{\\sigma}_1=\\hat{\\sigma}_y$, and $\\hat{\\sigma}_2=\\sigma_z$. In all these cases, a measurement of $\\hat{\\sigma}_2$ yields outcomes which are independent of the presence or absence of a previous measurement of $\\hat{\\sigma}_1$. The KQPD for two subsequent spin measurements is illustrated in Fig.~\\ref{fig:barchart}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.6\\textwidth]{barchart}\n\\caption{Keldysh quasi-probability distribution for two subsequent spin measurements on a pure qubit state. While the second argument is restricted to the eigenvalues of $\\hat{\\sigma}_2$, the first argument can also be zero. (a) $\\hat{\\sigma}_0=\\hat{\\sigma}_2=\\hat{\\sigma}_x$ and $\\hat{\\sigma}_1=\\hat{\\sigma}_z$. In this case, a measurement of $\\hat{\\sigma}_1$ influences a subsequent measurement of $\\hat{\\sigma}_2$ which results in a negative value for $\\tilde{\\mathcal{P}}(0,-1)$. (b) $\\hat{\\sigma}_0=\\hat{\\sigma}_x$, $\\hat{\\sigma}_1=\\hat{\\sigma}_y$, and $\\hat{\\sigma}_2=\\sigma_z$. In this case, the measurement outcome of $\\hat{\\sigma}_2$ is completely random, no matter if the first measurement is performed or not. A measurement of $\\hat{\\sigma}_1$ does therefore not influence a subsequent measurement of $\\hat{\\sigma}_2$ and the KQPD is purely positive.}\n  \\label{fig:barchart}\n\\end{figure}\n\n\n\\subsection{Negativity}\n\\label{sec:negativity}\n{As mentioned above, negativity in the KQPD indicates that measuring the considered observables necessarily influences the outcomes (i.e. measuring all of the operators results in different statistics than measuring each of them in a different realization). Following Ref.~\\cite{hofer:2016}, we corroborate this claim by connecting negative values in the KQPD to coherent superpositions of different eigenstates of the observables. In the presence of such superpositions, any measurement (partly) collapses the state, possibly influencing subsequent measurement outcomes.} For clarity, we consider the case of $N$ observables at subsequent times as described by the moment generating function in Eq.~\\eqref{eq:momgenmultsub}.\nAssuming that the eigenstates of each operator $\\hat{A}_l$ span the whole Hilbert space, we replace\n\\begin{equation}\n\\label{eq:exprepl}\ne^{-i\\frac{\\lambda_l}{2}\\hat{A}_l(\\tau_l)}=\\sum_{A_l^\\alpha}e^{-i\\frac{\\lambda_l}{2}A_{l}^\\alpha}\\hat{U}(0,\\tau_l)|A_{l}^{\\alpha}\\rangle\\langle A_{l}^{\\alpha}|\\hat{U}(\\tau_l,0),\n\\end{equation} \nwhere $\\hat{A}_l|A_{l}^\\alpha\\rangle=A_{l}^\\alpha|A_{l}^\\alpha\\rangle$ and the superscript $\\alpha=L,R$ labels if the replacement happened on the left ($L$) or on the right ($R$) of the density matrix in Eq.~\\eqref{eq:momgenmultsub}, i.e. $\\alpha$ labels the branch (forward- or backward-in-time).\nThis allows us to take the $\\lambda$-dependent factors out of the trace and obtain the KQPD by a Fourier transformation\n\\begin{equation}\n\\label{eq:kqpdtraj}\n\\mathcal{P}(\\boldsymbol{A})=\\sum_{\\boldsymbol{A}_L,\\boldsymbol{A}_R}\\delta_{A_N^L,A_N^R}\\delta\\left(\\boldsymbol{A}-\\frac{\\boldsymbol{A}_L+\\boldsymbol{A}_R}{2}\\right)\\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R)\\langle A^L_1|\\hat{\\rho}_{\\tau_1}|A^R_1\\rangle.\n\\end{equation}\nHere the vectors $\\boldsymbol{A}_{L\/R}$ have as their entries $A_l^{L\/R}$ and thus denote a particular manifestation of eigenvalues that we call a trajectory. The amplitudes associated with such a trajectory read\n\\begin{equation}\n\\label{eq:aampl}\n\\mathcal{A}(\\boldsymbol{A}_\\alpha)=\\langle A_N^\\alpha|\\hat{U}(\\tau_N,\\tau_{N-1})|A_{N-1}^\\alpha\\rangle\\langle A_{N-1}^\\alpha|\\cdots|A_{2}^\\alpha\\rangle\\langle A_{2}^\\alpha| \\hat{U}(\\tau_2,\\tau_1)|A_1^\\alpha\\rangle,\n\\end{equation}\nthe Dirac delta for vectors is defined as\n\\begin{equation}\n\\delta\\left(\\boldsymbol{A}-\\frac{\\boldsymbol{A}_L+\\boldsymbol{A}_R}{2}\\right)=\\prod_l \\delta\\left(A_l-\\frac{A_l^R+A_l^L}{2}\\right),\n\\end{equation}\nand $\\hat{\\rho}_{\\tau_1}=\\hat{U}(\\tau_1,0)\\hat{\\rho}_0\\hat{U}(0,\\tau_1)$.\n\nEquation \\eqref{eq:kqpdtraj} expresses the KQPD as a sum over pairs of trajectories. These pairs, here denoted left ($L$) and right ($R$) trajectory, are equivalent to a single trajectory along the Keldysh contour. A given pair of trajectories contributes with a weight that is given by the interference term $\\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R)$ times a density matrix element $\\langle A^L_1|\\hat{\\rho}_{\\tau_1}|A^R_1\\rangle$ corresponding to the ``starting point'' of the trajectories. Each pair contributes to the KQPD at the argument which corresponds to the \\textit{mean} value of the observable on the two trajectories, again underlining the fact that the KQPD considers the fluctuations of the \\textit{classical} field in the Keldysh language (see below). Finally, the Kronecker delta in Eq.~\\eqref{eq:kqpdtraj} results from the trace and reflects the fact that the disturbance of the last measurement is irrelevant.\n\nFrom Eq.~\\eqref{eq:kqpdtraj}, we see that as long as $\\boldsymbol{A}_L=\\boldsymbol{A}_R$, the KQPD is strictly greater or equal to zero. It is therefore the interference terms of trajectories that differ from each other which allow the KQPD to become negative. This is a genuine quantum effect since these interference terms are a direct result of a coherent superposition between different eigenstates of the observables $\\hat{A}_l$. {Although superposition can be encountered in classical (wave) theories, a coherent superposition between two states which correspond to two different values for an observable does not exist in classical theories (indeed, the presence of such superpositions violates the assumptions made by Leggett and Garg to obtain their inequalities \\cite{leggett:1985}, see Sec.~\\ref{eq:seclg})}.\n\nIn analogy to Eqs.~\\eqref{eq:keldyshrot} and \\eqref{eq:probapathint}, we can introduce a Keldysh rotation\n\\begin{equation}\n\\label{eq:keldyshrot2}\n\\boldsymbol{A}_{cl}=\\frac{1}{2}(\\boldsymbol{A}_L+\\boldsymbol{A}_R),\\hspace{2cm} \\boldsymbol{A}_q = \\boldsymbol{A}_L-\\boldsymbol{A}_R,\n\\end{equation}\nand write the KQPD in Eq.~\\eqref{eq:kqpdtraj} as a path integral\n\\begin{equation}\n\\label{eq:probapathint2}\n\\mathcal{P}(\\boldsymbol{A})=\\int \\mathcal{D}[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]\\delta(\\boldsymbol{A}-\\boldsymbol{A}_{cl})e^{iS[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]},\n\\end{equation}\nwith the action\n\\begin{equation}\n\\label{eq:action}\ne^{iS[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]}=\\mathcal{A}(\\boldsymbol{A}_{cl}+\\boldsymbol{A}_q\/2)\\mathcal{A}^*(\\boldsymbol{A}_{cl}-\\boldsymbol{A}_q\/2)\\langle A^{cl}_1+A^q_1\/2|\\hat{\\rho}_{\\tau_1}| A^{cl}_1-A^q_1\/2\\rangle,\n\\end{equation}\nand the integration measure\n$\\int \\mathcal{D}[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]=\\sum_{\\boldsymbol{A}_{cl},\\boldsymbol{A}_q}\\delta_{A_N^q,0}$. In this language, the KQPD is given by the sum over all paths for which $\\boldsymbol{A}=\\boldsymbol{A}_{cl}$, weighted by the complex exponential of the action. From Eqs.~\\eqref{eq:probapathint2} and \\eqref{eq:action}, we see that any negativity in the KQPD necessarily implies the presence of a non-zero \\textit{quantum} field $\\boldsymbol{A}_q\\neq0$.\nIn contrast to Eq.~\\eqref{eq:probapathint}, which describes a single observable, the quantum field in Eq.~\\eqref{eq:probapathint2} is not necessarily zero implying that the statistics of $\\boldsymbol{A}_{cl}$, determined by the Keldysh time-ordering, are not necessarily equal to the statistics of $\\boldsymbol{A}_L$ or $\\boldsymbol{A}_R$. \n\nWe now illustrate these ideas with the KQPD for two subsequent qubit measurements introduced in the last section. In terms of trajectories, the discrete KQPD in Eq.~\\eqref{eq:kqpdspinsdisc} can be expressed as\n\\begin{equation}\n\\label{eq:kqpdtwospintraj}\n\\tilde{\\mathcal{P}}(\\sigma_1,\\sigma_2)=\\sum_{\\sigma_1^{L\/R}=\\pm }\\delta_{\\sigma_1,(\\sigma_1^L+\\sigma_1^R)\/2}\\langle\\sigma_2|\\sigma_1^L\\rangle\\langle \\sigma_1^L|+_0\\rangle\\langle +_0|\\sigma_1^R\\rangle\\langle\\sigma_1^R|\\sigma_2\\rangle.\n\\end{equation}\nWe thus see that the terms where $\\sigma_1^L\\neq\\sigma_1^R$ are responsible for the terms $\\tilde{\\mathcal{P}}(0,\\sigma_2)$. Note that these terms are only non-zero if the initial state is in a coherent superposition of the two eigenstates of $\\hat{\\sigma}_1$. One can further show that\n\\begin{equation}\n\\sum_{\\sigma_1,\\sigma_2=\\pm 1}\\tilde{\\mathcal{P}}(\\sigma_1,\\sigma_2)=1,\n\\end{equation}\nwhich directly implies that $\\tilde{\\mathcal{P}}(0,+1)=-\\tilde{\\mathcal{P}}(0,-1)$. Thus any contribution to Eq.~\\eqref{eq:kqpdtwospintraj} from pairs of trajectories with $\\sigma_1^L\\neq\\sigma_1^R$ results in negativity in the KQPD as illustrated in Fig.~\\ref{fig:barchart}.\n\n{We close this section with a brief comparison to other non-classicality indicators based on negative quasi-probabilities. As mentioned above, the Wigner function (at time $t$) is the KQPD for the observables $\\hat{x}(t)$ and $\\hat{p}(t)$. A negative Wigner function implies an impossibility of describing the corresponding state classically in the canonical position-momentum phase space (i.e. as a mixture of states with well defined positions and momenta). The KQPD generalizes this concept to arbitrary observables. Negativity in the KQPD implies that it is impossible to describe the system and its time evolution as a mixture of states with well defined values for the observable outcomes. We note that negative values in a discrete analogue of the Wigner function have recently been connected to the computational speed-up of a quantum computer \\cite{veitch:2012,veitch:2013,howard:2014}.\n\nIn quantum optics, one often considers negativity, or strong singularity, in the Glauber-Sudarshan $P$ function as an indicator of non-classicality \\cite{titulaer:1965,mandel:1986}. This criterion captures all states with non-positive Wigner functions as well as squeezed states and entangled states \\cite{sperling:2009}. A non-classical $P$ function implies that the state can not be interpreted as a mixture of coherent states which indicates that the state can not be fully described by classical optics \\cite{mandel:1986}. We note that the $P$ function has recently been extended to multiple times \\cite{vogel:2008,krumm:2016,krumm:2017}. In contrast to the KQPD, this distribution is relevant for photo-detection and its moments correspond to normal ordering and regular time ordering.\n\nWhile these QPDs only contain information on the states and possibly their time-evolution, quasi-probability \\textit{representations} also describe measurements with QPDs \\cite{spekkens:2008,ferrie:2008,ferrie:2009,ferrie:2011}. It has been shown that a positive representation (i.e. a representation that only uses positive probability distributions) of all states and measurements is impossible. Restricting the states and measurements to a given experimental procedure, a procedure is then said to require non-classical resources if no positive representation can be found \\cite{ferrie:2008}. It is an interesting question if negativity in the KQPD implies the absence of a positive representation for a corresponding experimental procedure.}\n\n\\section{von Neumann measurements}\n\\label{sec:vonneumann}\nIn this section, we consider von Neumann type measurements and we show that the corresponding probability distributions can be obtained by convolving the KQPD with the back-action and imprecision associated with the measurement. We start by considering the instantaneous measurement of a single observable before moving on to subsequent instantaneous measurements, simultaneous measurements, and measurements of finite duration.\n\n\\subsection{Instantaneous measurements}\nFollowing von Neumann \\cite{vonneumann:book}, we model the measurement of an observable $\\hat{A}$ at time $\\tau$ by coupling it to a detector described by the canonically conjugate observables $\\hat{r}$ and $\\hat{\\pi}$ via the interaction Hamiltonian\n\\begin{equation}\n\\label{eq:hvn}\n\\hat{H}_m = \\delta(t-\\tau)\\chi\\hat{A}\\hat{\\pi},\n\\end{equation}\nwhere $\\chi$ denotes the coupling strength. This interaction induces the time-evolution\n$\\hat{U}_m=\\exp(-i\\chi\\hat{A}\\hat{\\pi})$,\nwhich displaces the detector coordinate $\\hat{r}$ by an amount that depends on the state of the system. Projectively measuring the position of the detector then yields the measurement of $\\hat{A}$. Measuring a single observable in this way yields the probability distribution (see App.~\\ref{app:vonneumann})\n\\begin{equation}\n\\label{eq:probaone}\nP(A)=\\int \\chi dA'\\langle  \\chi(A-A')|\\hat{\\rho}_{\\rm det}|\\chi(A-A')\\rangle\\mathcal{P}(A';\\tau),\n\\end{equation}\nwhere $\\hat{\\rho}_{\\rm det}$ is the density matrix of the detector, $|\\chi(A-A')\\rangle$ is an eigenstate of the operator $\\hat{r}$, and $\\mathcal{P}(A';\\tau)$ is the KQPD for a single observable given in Eq.~\\eqref{eq:probaa1}. Note that the position uncertainty in the initial detector state directly translates into a measurement uncertainty. A projective measurement can be obtained by choosing a position eigenstate as the initial state for the detector. In this case, the measured distribution coincides with the KQPD. This is due to the fact that the back-action of the measurement is irrelevant for a single observable since we are not interested in the post-measured state.\n\nFor multiple subsequent (instantaneous) measurements, we couple each observable to a different detector. The measurement Hamiltonian then reads \n\\begin{equation}\n\\label{eq:hvnm}\n\\hat{H}_m = \\sum_{l=1}^N\\delta(t-\\tau_l)\\chi_l\\hat{A}_l\\hat{\\pi}_l.\n\\end{equation}\nIn contrast to a single observable, each measurement can now influence all subsequent measurements. To capture this back-action effect, we evaluate the KQPD with the additional term $\\sum_l\\delta(t-\\tau_l)\\gamma_l\\hat{A}_l$ in the Hamiltonian,\nwhere the $\\gamma_j$ are random variables which are distributed according to the detector states (see below). This yields the modified moment generating function\n\\begin{equation}\n\\label{eq:momgenmultsubba}\n\\Lambda(\\boldsymbol{\\lambda};\\boldsymbol{\\gamma})={\\rm Tr}\\left\\{e^{-i\\left(\\frac{\\lambda_N}{2}+\\gamma_N\\right)\\hat{A}_N(\\tau_N)}\\cdots e^{-i\\left(\\frac{\\lambda_1}{2}+\\gamma_1\\right)\\hat{A}_1(\\tau_1)}\\hat{\\rho}_0e^{-i\\left(\\frac{\\lambda_1}{2}-\\gamma_1\\right)\\hat{A}_1(\\tau_1)}\\cdots e^{-i\\left(\\frac{\\lambda_N}{2}-\\gamma_N\\right)\\hat{A}_N(\\tau_N)} \\right\\},\n\\end{equation}\nwhich can be obtained by replacing $\\lambda_l\\rightarrow \\lambda_l\\pm 2\\gamma_l$ in Eq.~\\eqref{eq:momgenmultsub}, where the $+$ ($-$) sign corresponds to terms on the left (right) side of the density matrix.\n\nThe probability distribution which describes subsequent von Neumann measurements can then be shown to read (for a derivation, see App.~\\ref{app:vonneumann})\n\\begin{equation}\n\\label{eq:probasub}\nP(\\boldsymbol{A})=\\int d\\boldsymbol{A}'d\\boldsymbol{\\gamma}\\left[\\prod_l \\mathcal{W}_l(\\chi_l[A_l-A_l'],\\gamma_l\/\\chi_l)\\right]\\mathcal{P}(\\boldsymbol{A}';\\boldsymbol{\\gamma}),\n\\end{equation}\nwhere $\\mathcal{W}_l(r,\\gamma)$ denotes the Wigner function of detector $l$ and $\\mathcal{P}(\\boldsymbol{A}';\\boldsymbol{\\gamma})$ is the KQPD defined as the Fourier transform of the moment generating function in Eq.~\\eqref{eq:momgenmultsubba}. The measurement outcomes are determined by a convolution of the KQPD and the Wigner functions of the detectors. We can thus interpret the KQPD as describing the \\textit{intrinsic} fluctuations of the observables of interest which is being influenced by the effects of the measurement \\cite{clerk:2011}. The position distributions of the detectors determine the precision of the measurement while the momentum distributions determine the back-action via the $\\gamma_l$-dependent terms in the KQPD. The Heisenberg uncertainty relation implies a trade-off between these two effects, indicating that we can never eliminate both influences.\n\nAs an example, we consider a measurement of position $\\hat{x}$ at $\\tau=0$ followed immediately by a momentum $\\hat{p}$ measurement. The probability distribution describing such a measurement reads\n\\begin{equation}\n\\label{eq:probaxp}\nW_{x\\rightarrow p}(x,p)=\\int dx'dp'd\\gamma_xd\\gamma_p\\mathcal{W}_p(\\chi_p[p-p'],\\gamma_p\/\\chi_p)\\mathcal{W}_x(\\chi_x[x-x'],\\gamma_x\/\\chi_x)\\mathcal{W}(x',p'+\\gamma_x).\n\\end{equation} \nHere $\\mathcal{W}_{x\/p}$ denotes the Wigner function of the detector measuring $x\/p$ and $\\mathcal{W}$ without a subscript denotes the KQPD which in this case is given by the Wigner function of the system. The position measurement disturbs the momentum of the system as reflected by the second argument of the KQPD. Note that it is the momentum distribution of the $x$-detector which determines the effect of this disturbance. This reflects the trade-off between the precision of the position measurement (which requires a peaked position distribution for the $x$-detector) and the disturbance of the subsequent momentum distribution (which is minimized for a peaked momentum distribution for the $x$-detector). Note that the last expression is independent of the momentum distribution of the $p$-detector (this can be seen by performing the integral over $\\gamma_p$), reflecting the fact that the back-action of the momentum measurement is irrelevant since it is the last measurement. Inverting the order of the measurements yields a probability distribution which is given by the last expression upon exchanging $\\mathcal{W}(x',p'+\\gamma_x)\\rightarrow \\mathcal{W}(x'-\\gamma_p,p')$.\n\nReturning to arbitrary observables and choosing Gaussian Wigner functions for the detectors\n\\begin{equation}\n\\mathcal{W}_l(r,\\gamma)=\\frac{1}{2\\pi\\sigma_{r_l}\\sigma_{\\gamma_l}}e^{-\\frac{r^2}{2\\sigma_{r_l}^2}}e^{-\\frac{\\gamma^2}{2\\sigma_{\\gamma_l}^2}},\n\\end{equation}\nwe can write the probability distribution in Eq.~\\eqref{eq:probasub} in terms of trajectories [cf.~Eq.~\\eqref{eq:kqpdtraj}] yielding\n\\begin{equation}\n\\label{eq:probasubtraj}\n\\begin{aligned}\nP(\\boldsymbol{A})=\\sum_{\\boldsymbol{A}_L,\\boldsymbol{A}_R}\\delta_{A_N^L,A_N^R}&\\left[\\prod_l\\frac{1}{\\sqrt{2\\pi}\\sigma_{{\\rm imp},l}}e^{-\\frac{1}{2\\sigma_{{\\rm imp},l}^2}\\left(A_l-\\frac{A_l^L+A_l^R}{2}\\right)^2}e^{-\\frac{\\sigma_{{\\rm BA},l}^2}{2}\\left(A_l^L-A_l^R\\right)^2}\\right]\\\\&\\times\\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R)\\langle A^L_1|\\hat{\\rho}_{\\tau_1}|A^R_1\\rangle.\n\\end{aligned},\n\\end{equation}\nwhere $\\sigma_{{\\rm imp},l}=\\sigma_{r_l}\/\\chi_l$ and $\\sigma_{{\\rm BA},l}=\\sigma_{\\gamma_l}\\chi_l$. Comparing the last expression with the KQPD in Eq.~\\eqref{eq:kqpdtraj}, we can clearly identify the effects of the measurement imprecision and back-action. The first Gaussian in the last expression replaces the Dirac delta in Eq.~\\eqref{eq:kqpdtraj}. It implies that even terms with $A_l\\neq (A_l^L+A_l^R)\/2$ contribute to the probability of measuring $A_l$. This effect captures the imprecision of the measurement which becomes more pronounced as the uncertainty in the position of detector $l$ grows. The second Gaussian in the last expression reduces the weight of terms which have $A_l^L\\neq A_l^R$, i.e. a finite quantum field in the Keldysh language. {This effect is due to the measurement back-action which (partly) collapses the coherent superpositions between eigenstates of $\\hat{A}_l$}. The coupling Hamiltonian in Eq.~\\eqref{eq:hvnm} entangles the system with the detector leading to dephasing in the reduced state of the system. As the uncertainty in the detector momentum grows, this effect becomes more pronounced. From the Heisenberg uncertainty relation, we find\n\\begin{equation}\n\\label{eq:heisenberg}\n\\sigma_{{\\rm imp},l}\\sigma_{{\\rm BA},l}=\\sigma_{r_l}\\sigma_{\\gamma_l}\\geq\\frac{1}{2},\n\\end{equation}\nreflecting the trade-off between measurement imprecision and back-action {(see also Ref.~\\cite{clerk:2010})} which ensures that the measured distribution is positive even if the KQPD exhibits negativities. We note that coupling the system to additional degrees of freedom, such as a thermal reservoir, introduces dephasing analogously to the measurement back-action \\cite{hofer:2016}.\n\nTo illustrate the effect of the measurement, we return to our example of two subsequent spin measurements [cf.~Eq.~\\eqref{eq:kqpdspins}]. Since the back-action of the second measurement is irrelevant, we consider a measurement of intermediate strength of $\\hat{\\sigma}_1$ followed by a projective measurement of $\\hat{\\sigma}_2$.\nIn terms of the discrete KQPD given in Eq.~\\eqref{eq:kqpdspinsdisc}, we find for the distribution describing a von Neumann measurement (with Gaussian detectors)\n\\begin{equation}\n\\label{eq:kqpdspinsmeas}\nP(\\sigma_1,\\sigma_2)=\\frac{1}{\\sqrt{2\\pi}\\sigma_{\\rm imp}}\\sum_{\\sigma'_1=0,\\pm 1}e^{-\\frac{1}{2\\sigma_{\\rm imp}^2}(\\sigma_1-\\sigma_1')^2}e^{-2\\sigma_{\\rm BA}^2\\delta_{\\sigma'_1,0}}\\tilde{\\mathcal{P}}(\\sigma_1',\\sigma_2).\n\\end{equation}\nThe effect of the measurement is illustrated in Fig.~\\ref{fig:measured}. We note that the distribution of measurement outcomes for the first measurement depends very strongly on the outcome of the second measurement. Specifically, we see from Fig.~\\ref{fig:measured} that only considering events where the second (projective) measurement yields $-1$ allows one to distinguish the outcomes of the first measurement much better (for a given measurement strength). This effect, which arises from the negativity of the KQPD, is responsible for the occurrence of anomalous weak values as discussed in Sec.~\\ref{sec:weakvalues}. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.8\\textwidth]{measured}\n\\caption{Effect of measurement imprecision and back-action on the KQPD. Here $\\hat{\\sigma}_0=\\hat{\\sigma}_2=\\hat{\\sigma}_x$ and $\\hat{\\sigma}_1=\\hat{\\sigma}_z$ to maximize the negativity in the KQPD. The measurement imprecision and back-action modify the KQPD resulting in a positive probability distribution. Here $\\sigma_{\\rm imp}\\sigma_{\\rm BA}=1\/2$, minimizing the influence of the measurement. Depending on the measurement outcome of the second (projective) measurement, the distribution for the first measurement results in two clearly distinguishable outcomes or not. This is directly related to the terms $\\tilde{\\mathcal{P}}(0,\\sigma_2)$ which are responsible for the negative values in the KQPD. Note that the peaks are shifted with respect to the eigenvalues of $\\hat{\\sigma}_1$ (dashed lines). This effect allows for the occurrence of anomalous weak values as discussed in Sec.~\\ref{sec:weakvalues}.}\n  \\label{fig:measured}\n\\end{figure}\n\n{Note that subsequent von Neumann measurements can also be used to model continuous measurements, a subject which has received a lot of attention recently, see e.g. \\cite{korotkov:1999,korotkov:2001,chantasri:2013,weber:2014,chantasri:2015}.}\n\n\n\\subsection{Simultaneous measurements of position and momentum}\n\nWe now turn to the simultaneous measurement of position and momentum. In analogy to Eq.~\\eqref{eq:hvn} (and following Arthurs and Kelly \\cite{arthurs:1964}), we model the measurement by the simultaneous coupling of two detectors to the position and momentum of the system respectively\n\\begin{equation}\n\\label{eq:hvnmsim}\n\\hat{H}_m = \\delta(t)\\left(\\chi_x\\hat{x}\\hat{\\pi}_x+\\chi_p\\hat{p}\\hat{\\pi}_p\\right).\n\\end{equation}\nIt can be shown that this measurement is described by the probability distribution \\cite{di_lorenzo:2011}\n\\begin{equation}\n\\label{eq:probaxpsim}\nW(x,p)=\\int dx'dp'd\\gamma_xd\\gamma_p\\mathcal{W}_p(\\chi_p[p-p'],\\gamma_p\/\\chi_p)\\mathcal{W}_x(\\chi_x[x-x'],\\gamma_x\/\\chi_x)\\mathcal{W}(x'-\\gamma_p\/2,p'+\\gamma_x\/2).\n\\end{equation}\nThe arguments of the last Wigner function reflect the fact that in a simultaneous measurement, both position and momentum are being disturbed. Note the close similarity between the last expression and the probability distribution describing subsequent measurements of position and momentum [cf.~Eq.~\\eqref{eq:probaxp}]. Notably, the KQPD is in both cases given by the Wigner function. This is a consequence of the commutation relation $[\\hat{x},\\hat{p}]=i$ and does not hold in general for non-commuting observables.\n\nFor a detailed discussion on the von Neumann type simultaneous measurement of position and momentum, we refer the reader to Ref.~\\cite{stenholm:1992}. Here we only discuss the case of Gaussian states of minimal uncertainty for the detectors ($k=x,p$)\n\\begin{equation}\n\\mathcal{W}_{k}(r,\\gamma)=\\frac{1}{\\pi}e^{-r^2\/\\sigma_{k}^2}e^{-\\gamma^2\\sigma_{k}^2},\n\\end{equation}\nwith the additional constraint $\\sigma_x\\sigma_p\/(\\chi_x\\chi_p)=1\/2$. This constrained minimizes the disturbance of the measurement and yields the Husimi $Q$-function as the measured probability distribution \\cite{husimi:1940,stenholm:1992}\n\\begin{equation}\n\\label{eq:husimiq}\nW(x,p)=\\frac{1}{\\pi}\\int dx' dp' e^{-\\frac{(x-x')^2}{2\\sigma^2}}e^{-2\\sigma^2(p-p')^2}\\mathcal{W}(x',p'),\n\\end{equation}\nwhere $\\sigma=\\sigma_x\/\\chi_x$. In addition to this von Neumann type measurement, the Husimi $Q$-function also describes heterodyne measurements of two field quadratures \\cite{leonhardt:1995}.\n\n\n\\subsection{Measuring the FCS}\n\n\nIn this section, we consider a von Neumann type measurement of the time-integral of an observable \\cite{nazarov:2003}. To this end, we couple a detector to the observable of interest for a finite amount of time\n\\begin{equation}\n\\label{eq:hvnc}\n\\hat{H}_m = \\Theta(t)\\Theta(\\mathcal{T}-t)\\chi\\hat{A}\\hat{\\pi}.\n\\end{equation}\nThis measurement can be shown to be described by the probability distribution \\cite{nazarov:2003}\n\\begin{equation}\n\\label{eq:measfcs}\nF(m) = \\int d\\gamma \\mathcal{W}(\\chi[m-m'],\\gamma\/\\chi)\\mathcal{F}(m';\\gamma),\n\\end{equation}\nwhere $\\mathcal{F}(m;\\gamma)$ is obtained through Eqs.~\\eqref{eq:momgenfcs} and \\eqref{eq:fcs} upon adding the term $\\Theta(t)\\Theta(\\mathcal{T}-t)\\gamma\\hat{A}$ to the Hamiltonian.\nAgain, we find a trade-off between measurement imprecision, depending on the spread in the detector position, and back-action which depends on the spread in the detector momentum. As discussed in Ref.~\\cite{hofer:2016}, the last equation can be expressed in terms of trajectories yielding (for a Gaussian detector)\n\\begin{equation}\n\\label{eq:fcsmeastraj}\nF(m)=\\frac{1}{\\sqrt{2\\pi}\\sigma_{\\rm imp}}\\sum_{\\boldsymbol{A}_L,\\boldsymbol{A}_R}\\delta_{A_\\mathcal{T}^L,A_\\mathcal{T}^R}e^{-\\frac{\\left(m-\\frac{1}{2}m_L-\\frac{1}{2}m_R\\right)^2}{2\\sigma^2_{\\rm imp}}} e^{-\\frac{\\sigma_{\\rm BA}^2}{2}\\left(m_L-m_R\\right)^2}\\langle A_0^L|\\hat{\\rho}|A_0^R\\rangle \\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R),\n\\end{equation}\nwhere the trajectory amplitudes are obtained by dividing the time evolution into $N$ time-slices of infinitesimal duration $\\delta t$ and inserting identities resolved in the eigenstates of $\\hat{A}$ after each slice\n\\begin{equation}\n\\label{eq:aamplfcs}\n\\mathcal{A}(\\boldsymbol{A}_\\alpha)=\\langle A_\\mathcal{T}^\\alpha|\\hat{U}(\\mathcal{T},\\mathcal{T}-\\delta t)|A_{\\mathcal{T}-\\delta t}^\\alpha\\rangle\\langle A_{\\tau-\\delta t}^\\alpha|\\cdots|A_{\\delta t}^\\alpha\\rangle\\langle A_{\\delta t}^\\alpha| \\hat{U}(\\delta t,0)|A_0^\\alpha\\rangle.\n\\end{equation}\nThe discrete time-integral over such a trajectory is denoted by $m_\\alpha =\\sum_l A^\\alpha_l \\delta t$. The Heisenberg uncertainty relation applied to the detector variables again enforces $\\sigma_{\\rm imp}\\sigma_{\\rm BA}\\geq 1\/2$, reflecting the trade-off between smearing out the negativities in the FCS by measurement imprecision and reducing the weight of negative terms (which necessarily have $m_L\\neq m_R$) by measurement back-action in complete analogy to Eq.~\\eqref{eq:probasubtraj}.\n\nThis concludes our treatment on von Neumann measurements. Expressing the measured probability distributions in terms of the KQPD suggests that the latter should be interpreted as describing the intrinsic fluctuations of the observables of interest. In the next section, we will build on this interpretation and demonstrate the utility of studying the KQPD to understand the quantum behavior of dynamic systems.\n\n\n\\section{Applications of the KQPD}\n\\label{sec:applications}\nWe now turn to situations where it is beneficial to study the KQPD in the absence of any measurement device. We are especially interested in situations where the KQPD fails to be positive. As discussed in Sec.~\\ref{sec:negativity}, this indicates non-classical behavior. The most prominent KQPD is by far the Wigner function, which is a representation of the density matrix and is used extensively to illustrate the state of a quantum system and to capture its non-classicality \\cite{deleglise:2008,kenfack:2004}. An example for a KQPD which goes beyond instantaneous states is provided by the FCS which describes the fluctuations of an observable over a certain time-interval. While the FCS has mostly been employed to describe electronic transport in systems where it is a purely positive function, its utility for detecting non-classical behavior has been recognized \\cite{hofer:2016,clerk:2011,bednorz:2010,bednorz:2012,belzig:2001,clerk:2010}. Here we discuss three additional applications of the KQPD, stressing its versatility and its utility in capturing non-classical behavior in the dynamics of quantum systems.\n\n\\subsection{Work fluctuations}\n\nSince work is not a state function, it is in general not sufficient to interrogate a system at a single time in order to determine the work performed by the system. As we have seen above, performing measurements at multiple times generally results in a probability distribution which is influenced by the measurement itself. As a consequence, the observed amount of work will in general depend on the way it is measured \\cite{perarnau:2017}. Here we use the KQPD to describe work as a measurement-independent, fluctuating quantity. We note that the KQPD has been used before to describe work fluctuations \\cite{esposito:2009,solinas:2015,solinas:2016}. \n\nIn classical statistical mechanics, work is defined as the energy change in the macroscopic, accessible degrees of freedom, while heat is the energy change in the microscopic, inaccessible degrees of freedom. In quantum systems, all degrees of freedom are in principle microscopic. We can however still define work as being the energy change mediated by the accessible degrees of freedom. Here we consider two different scenarios: In the first, one has access to the energy stored in a part of the system that we call the work storage device. The energy increase of this device is then interpreted as the work performed by the system. {This is the analog of determining the work performed on a suspended weight by measuring its height.} In the second scenario, one has access to the power produced by the system and the work is defined as the time-integral of the power. This captures the situation in thermoelectric heat engines, where one can access the power through a measurement of the electrical current. {In terms of the suspended weight, this corresponds to the analog of measuring the power by measuring the velocity of the weight.}\n\nFor the first scenario, we write the total Hamiltonian as $\\hat{H}=\\hat{H}_w+\\hat{H}_{\\rm rest}$, where $\\hat{H}_w$ denotes the Hamiltonian of the work storage device. Work fluctuations can then be described by the KQPD corresponding to the difference of two subsequent energy measurements on the work storage device\n\\begin{equation}\n\\label{eq:work}\n\\begin{aligned}\n\\mathcal{P}(w;\\tau)=&\\frac{1}{2\\pi}\\int d\\lambda e^{i\\lambda w}{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{U}(\\tau,0)e^{i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{\\rho}_0e^{i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{U}(0,\\tau)e^{-i\\frac{\\lambda}{2}\\hat{H}_w}\\right\\}\\\\\n=&\\sum_{E_\\tau,E_L,E_R}\\delta\\left(w-\\left[E_\\tau-\\frac{E_L+E_R}{2}\\right]\\right){\\rm Tr}\\left\\{\\langle E_\\tau|\\hat{U}(\\tau,0)|E_L\\rangle\\langle E_L|\\hat{\\rho}_0|E_R\\rangle \\langle E_R|\\hat{U}(0,\\tau)|E_\\tau\\rangle\\right\\}.\n\\end{aligned}\n\\end{equation}\nHere $\\hat{H}_w|E_j\\rangle=E_j|E_j\\rangle$ and $\\tau$ denotes the time interval between the two measurements. For initial states which commute with $\\hat{H}_w$, we find $E_L=E_R$ and we recover the probability for obtaining $w$ as the difference of two projective energy measurements \\cite{talkner:2007}. Any negativity in the above distribution is thus due to the coherence in the initial state. We note that the probability distribution obtained by performing two Gaussian energy measurements, as discussed in Ref.~\\cite{talkner:2016}, is obtained from Eq.~\\eqref{eq:work} by including the imprecision and back-action of the measurement as discussed in Sec.~\\ref{sec:vonneumann}.\nThe moments generated by the KQPD are given by\n\\begin{equation}\n\\label{eq:momentswork}\n\\langle w^k\\rangle = \\sum_{l=0}^{k}\\frac{1}{2^k}{k \\choose l}{\\rm Tr}\\left\\{ \\hat{\\mathcal{T}}_-\\left(\\hat{H}_w(\\tau)-\\hat{H}_w\\right)^l\\hat{\\mathcal{T}}_+\\left(\\hat{H}_w(\\tau)-\\hat{H}_w\\right)^{(k-l)}\\hat{\\rho}_0\\right\\},\n\\end{equation}\nwhere $\\hat{H}_w(\\tau)=\\hat{U}(0,\\tau)\\hat{H}_w\\hat{U}(\\tau,0)$. For the first two moments, i.e. $k=1,2$ in the last expression, this reduces to \n\\begin{equation}\n\\label{eq:momentswork12}\n\\langle w^k\\rangle={\\rm Tr}\\left\\{ \\left(\\hat{H}_w(\\tau)-\\hat{H}_w\\right)^k\\hat{\\rho}_0\\right\\},\\hspace{1cm}{\\rm for}\\hspace{.2cm} k=1,2.\n\\end{equation}\nThe mean work obtained through the KQPD is thus just given by the difference of the average energy in the work storage device at $t=\\tau$ and $t=0$. This is in contrast to the probability distribution obtained by performing two projective energy measurements, where the mean value only coincides with Eq.~\\eqref{eq:momentswork12} for initial states that commute with $\\hat{H}_w$. For an explicit evaluation of Eq.~\\eqref{eq:work}, we refer to Ref.~\\cite{solinas:2016}. See Ref.~\\cite{batalhao:2014} for an experimental reconstruction of the work distribution by measuring its moment generating function (for an initial state that commutes with the Hamiltonian).\n\nIn case one has access to a power operator $\\hat{P}$, the KQPD describing work fluctuations is given by the FCS of the power operator. The KQPD and its moments are then given by Eqs.~(\\ref{eq:momgenfcs}-\\ref{eq:momentsfcs}) upon replacing $\\hat{A}$ with $\\hat{P}$. {This scenario is relevant for thermoelectric devices, where the power operator is given by the electrical current operator times the voltage across the device $\\hat{P}=\\hat{I}V$. In this case, the work fluctuations reduce to the problem of charge FCS \\cite{nazarov:book2}. We now return to systems described by $\\hat{H}=\\hat{H}_w+\\hat{H}_{\\rm rest}$, where we take the Hamiltonians to be time-independent for simplicity (see Ref.~\\cite{venkatesh:2015} for a discussion on the power operator for a time-dependent Hamiltonian). In this case, the power operator is given by the energy change of the work storage device per unit time, i.e. $\\hat{P}=i[\\hat{H},\\hat{H}_w]$.}\nIf the power operator commutes with the total Hamiltonian, i.e. $[\\hat{P},\\hat{H}]=0$, we find\n\\begin{equation}\n\\label{eq:powerop}\ne^{-i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{U}(\\tau,0)e^{i\\frac{\\lambda}{2}\\hat{H}_w}=\\hat{\\mathcal{T}}_+e^{-i\\frac{\\lambda}{2}\\int_0^\\tau dt \\hat{P}(t)}.\n\\end{equation}\nThis implies that the FCS of the power operator is equal to the KQPD for two energy measurements given in Eq.~\\eqref{eq:work} \\cite{esposito:2009}. The difference of the two scenarios considered here is then only relevant when taking into account the disturbance from an actual measurement. {For a thermoelectric device, the energy in the work storage device corresponds to the electric potential energy stored in an electronic contact, i.e. $\\hat{H}_w=V\\hat{Q}$, where $\\hat{Q}$ denotes the charge operator in the contact with $\\hat{I}=i[\\hat{H},\\hat{Q}]$. The work provided by a thermoelectric device can thus either be determined by measuring the electrical current (determining the power) or by measuring the number of electrons in an electric contact (determining the energy in the work storage device). We note that both approaches have been considered in the theory of charge FCS \\cite{nazarov:book2}.}\n\n\n\\subsection{Weak values}\n\\label{sec:weakvalues}\nThe weak value of an observable $\\hat{A}$ is obtained by preparing the system in some state $|I\\rangle$, performing a weak measurement of $\\hat{A}$, and post-selecting the measurement outcomes on the final state $|F\\rangle$ \\cite{aharonov:1988}. Describing the weak measurement as a von Neumann measurement, the post-measured state of the \\textit{detector} is determined by the weak value\n\\begin{equation}\n\\label{eq:weakvalue}\nA_w=\\frac{\\langle F|\\hat{A}|I\\rangle}{\\langle F|I\\rangle},\n\\end{equation}\nand the average of the measurement outcome is given by the real part of the last expression \\cite{aharonov:1988,dressel:2015}. Any possible time-evolution between pre-selection, measurement and post-selection is absorbed in the definition of the initial and final states. In the KQPD framework, we can describe the weak value measurement using the distribution describing two subsequent measurements, one for the weak measurement and one for the post-selection. For simplicity, we do not treat the pre-selection as an additional measurement but impose it by setting $\\hat{\\rho}_0=|I\\rangle\\langle I|$. The relevant KQPD then reads\n\\begin{equation}\n\\label{eq:kqpdweak1}\n\\begin{aligned}\n\\mathcal{P}(A,B)=&\\frac{1}{(2\\pi)^2}\\int d\\lambda_Ad\\lambda_Be^{i\\lambda_A A}e^{i\\lambda_B B}{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_B}{2}\\hat{B}}e^{-i\\frac{\\lambda_A}{2}\\hat{A}}|I\\rangle\\langle I|e^{-i\\frac{\\lambda_A}{2}\\hat{A}}e^{-i\\frac{\\lambda_B}{2}\\hat{B}}\\right\\}\\\\\n=&\\sum_{B_F,A_L,A_R}\\delta\\left(B-B_F\\right)\\delta\\left(A-\\frac{A_L+A_R}{2}\\right) \\langle B_F|A_L\\rangle\\langle A_L|I\\rangle\\langle I|A_R\\rangle\\langle A_R|B_F\\rangle,\n\\end{aligned}\n\\end{equation}\nwhere $|B_F\\rangle$ are eigenstates of $\\hat{B}$. The KQPD corresponding to the measurement of the weak value is then obtained by taking into account the post-selection, i.e. fixing the argument corresponding to the second measurement\n\\begin{equation}\n\\label{eq:kqpdweak2}\n\\mathcal{P}_w(A)=\\frac{\\mathcal{P}(A,F)}{\\int dA\\mathcal{P}(A,F)}=\\frac{1}{|\\langle F|I\\rangle|^2}\\sum_{A_L,A_R}\\delta\\left(A-\\frac{A_L+A_R}{2}\\right)\\langle F|A_L\\rangle\\langle A_L|I\\rangle\\langle I|A_R\\rangle\\langle A_R|F\\rangle.\n\\end{equation}\nThe statistics of weak value measurements (with post-selection performed by a projective measurement) can be described using the last expression by including measurement imprecision and back-action as discussed in Sec.~\\ref{sec:vonneumann}. In the weak measurement limit, the disturbance of the measurement does not alter the average value of the KQPD, which is equal to the real part of Eq.~\\eqref{eq:weakvalue}\n\\begin{equation}\n\\label{eq:evgweakvalue}\n\\langle A\\rangle_w =\\int dA\\,A\\mathcal{P}_w(A)={\\rm Re}\\left\\{\\frac{\\langle F|\\hat{A}|I\\rangle}{\\langle F|I\\rangle}\\right\\}.\n\\end{equation}\n\nFrom Eq.~\\eqref{eq:kqpdweak2}, we see that the KQPD vanishes for arguments that lie outside the spectrum of $\\hat{A}$. However Eq.~\\eqref{eq:evgweakvalue} can take on values which lie outside the spectrum of $\\hat{A}$. In such cases, the weak value is called \\textit{anomalous}.\nWe now show that an anomalous weak value implies that the distribution $\\mathcal{P}_w(A)$ takes on negative values. To see this, assume that $\\mathcal{P}_w(A)\\geq 0$. In that case, we find\n\\begin{equation}\n\\label{eq:wvnegkqpd}\n\\langle A\\rangle_w =\\int_{A_{\\rm min}}^{A_{\\rm max}} dA\\,A\\mathcal{P}_w(A)\\leq A_{\\rm max} \\int_{A_{\\rm min}}^{A_{\\rm max}} dA\\mathcal{P}_w(A)=A_{\\rm max},\n\\end{equation}\nwhere $A_{\\rm min \\,(max)}$ is the smallest (largest) eigenvalue of $\\hat{A}$ and a similar inequality restricts $\\langle A\\rangle_w\\geq A_{\\rm min}$. Note that the inequality in Eq.~\\eqref{eq:wvnegkqpd} is only valid for a strictly positive KQPD.\nAn anomalous weak value thus necessarily implies that the KQPD $\\mathcal{P}_w(A)$ [and also $\\mathcal{P}(A,B)$, cf.~Eq.~\\eqref{eq:kqpdweak2}] exhibits negativity indicating non-classical behavior. This is an agreement with Ref.~\\cite{pusey:2014}, where anomalous weak values have been shown to imply contextuality. We note that weak values have been connected to quasi-probability distributions before \\cite{dressel:2015}. In particular, the average momentum at a given position that is obtained from the Wigner function has been shown to be determined by the weak value of the momentum\n\\begin{equation}\n\\label{eq:weakvaluep}\n\\frac{\\int dp\\, p\\mathcal{W}(x,p)}{\\int dp \\mathcal{W}(x,p)}={\\rm Re}\\left\\{\\frac{\\langle x|\\hat{p}|I\\rangle}{\\langle x|I\\rangle}\\right\\},\n\\end{equation}\nwhich follows from our approach when taking $\\hat{A}=\\hat{p}$ and $\\hat{B}=\\hat{x}$.\n\n{Finally, we note that the distribution in Eq.~\\eqref{eq:kqpdweak2} is relevant for a measurement of $\\hat{A}$ with an arbitrary strength. It has the variance}\n\\begin{equation}\n\\label{eq:weakvariance}\n\\langle \\hat{A}^2\\rangle_w-\\left(\\langle \\hat{A}\\rangle_w\\right)^2 = \\frac{1}{2}{\\rm Re}\\left\\{\\frac{\\langle F|\\hat{A}^2|I\\rangle}{\\langle F|I\\rangle}\\right\\}-\\frac{1}{2}\\left({\\rm Re}\\left\\{\\frac{\\langle F|\\hat{A}|I\\rangle}{\\langle F|I\\rangle}\\right\\}\\right)^2.\n\\end{equation}\nFor $\\hat{A}=\\hat{p}$ and $|F\\rangle =|x\\rangle$, the last expression is proportional to the \\textit{quantum potential energy} \\cite{dressel:2015}, a central object in Bohmian mechanics \\cite{bohm:1952}. {See also Ref.~\\cite{dressel:2012} for a connection between weak values and von Neumann measurements of arbitrary strength and post-selection.}\n\nFor our example of two subsequent spin measurements, we find [using Eqs.~\\eqref{eq:kqpdspins}, \\eqref{eq:kqpdspinsdisc}, and \\eqref{eq:kqpdweak2}]\n\\begin{equation}\n\\label{eq:kqpdwvspins}\n\\mathcal{P}_w(\\sigma_1)=\\frac{|\\alpha|^2|\\delta|^2\\delta(\\sigma_1-1)+|\\beta|^2|\\gamma|^2\\delta(\\sigma_1+1)-2{\\rm Re}\\{\\alpha\\beta^*\\gamma^*\\delta\\}\\delta(\\sigma_1)}{|\\alpha\\delta-\\beta\\gamma|^2},\n\\end{equation}\nwhere we chose $|-_2\\rangle$ as the final state $|F\\rangle$.\nWe see that for a sufficiently small denominator, the probability density associated with $\\sigma_1=\\pm 1$ can become very large as long as the presence of the negative term ensures the normalization of $\\mathcal{P}_w(\\sigma_1)$. In this way, the weak value, determined by $\\mathcal{P}_w(1)-\\mathcal{P}_w(-1)$ can be amplified resulting in values far outside the range $[-1,+1]$. This can however only occur for a sufficiently large negative term in the KQPD as can be seen from Eq.~\\eqref{eq:kqpdwvspins} which implies\n\\begin{equation}\n\\label{eq:weakvaluespins}\n\\langle \\sigma_1\\rangle_w>1\\hspace{.5cm}\\Leftrightarrow\\hspace{.5cm} {\\rm Re}\\{\\alpha\\beta^*\\gamma^*\\delta\\}>|\\beta|^2|\\gamma|^2.\n\\end{equation}\n\n\n\\subsection{Leggett-Garg inequalities}\n\\label{eq:seclg}\n\nIn Ref.~\\cite{leggett:1985}, Leggett and Garg derived inequalities which hold under the assumption of \\textit{macroscopic realism} [i.e., a (macroscopic) system will at all times be in one (and only one) of the states available to it] and \\textit{non-invasive measurability}. Clearly quantum systems do not fulfill either of these assumptions and can thus violate Leggett-Garg inequalities. In contrast to Bell inequalities, where multiple spatially separated parties apply local operations, Leggett-Garg inequalities are obtained by probing the same system at multiple times.\nWe note that even if a Leggett-Garg inequality is violated, one can always argue that this happened because the measurement did influence the system, and therefore all subsequent measurements, in an unexpected (but in principle avoidable) way. This is known as the \\textit{clumsiness} loophole. Since it is not possible to certify the non-invasiveness of a measurement, the clumsiness loophole can never be closed \\cite{emary:2014}.\n\nWe will now show that the violation of a particular Leggett-Garg inequality directly implies the occurence of negative values in the corresponding KQPD. The considered inequality reads\n\\begin{equation}\n\\label{eq:leggettgarg}\nK=C_{21}+C_{32}-C_{31}\\leq 1,\n\\end{equation}\nwhere the correlators are given by\n\\begin{equation}\n\\label{eq:lgcorr}\nC_{ij} = \\sum_{Q_i,Q_j=\\pm 1}Q_iQ_jP_{ij}(Q_i,Q_j).\n\\end{equation}\nHere $P_{ij}(Q_i,Q_j)$ is the probability of obtaining the outcome $Q_i$ at time $t_i$ and $Q_j$ at $t_j$ if the system is measured at these two times only. The measurement outcomes are restricted to $Q_i=\\pm 1$. Violation of the inequality in Eq.~\\eqref{eq:leggettgarg} implies that the correlators $C_{ij}$ can not be obtained from a positive probability distribution which describes all three measurements and is independent of the choice of measurements that are performed \\cite{leggett:1985,emary:2014}, i.e. \n\\begin{equation}\n\\label{eq:corrneq}\nK>1\\hspace{.5cm}\\Rightarrow \\hspace{.5cm}C_{ij}\\neq \\sum_{Q_1,Q_2,Q_3=\\pm 1}Q_iQ_jP(Q_3,Q_2,Q_1),\n\\end{equation}\nwith $P(Q_3,Q_2,Q_1)\\geq 0$.\n\nFor projective measurements of a dichotomic observable $\\hat{Q}$, the correlator to be used in Eq.~\\eqref{eq:leggettgarg} is given by \\cite{fritz:2010}\n\\begin{equation}\n\\label{eq:corrproj}\nC_{ij}={\\rm Tr}\\left\\{\\frac{1}{2}\\left\\{\\hat{Q}(t_i),\\hat{Q}(t_j)\\right\\}\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nWe now introduce the KQPD for three subsequent measurements of $\\hat{Q}$\n\\begin{equation}\n\\label{eq:kqpdlg}\n\\begin{aligned}\n\\mathcal{P}(Q_3,Q_2,Q_1)=&\\frac{1}{(2\\pi)^3}\\int d\\lambda_3d\\lambda_2\\lambda_1e^{i\\lambda_3 Q_3+i\\lambda_2 Q_2+i\\lambda_1 Q_1}\\\\&\\times{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_3}{2}\\hat{Q}(t_3)}e^{-i\\frac{\\lambda_2}{2}\\hat{Q}(t_2)}e^{-i\\frac{\\lambda_1}{2}\\hat{Q}(t_1)}\\hat{\\rho}_0e^{-i\\frac{\\lambda_1}{2}\\hat{Q}(t_1)}e^{-i\\frac{\\lambda_2}{2}\\hat{Q}(t_2)}e^{-i\\frac{\\lambda_3}{2}\\hat{Q}(t_3)}\\right\\}.\n\\end{aligned}\n\\end{equation}\nIt is straightforward to show that the correlators in Eq.~\\eqref{eq:corrproj} can directly be obtained from the KQPD\n\\begin{equation}\n\\label{eq:lgcorrkqpd}\nC_{ij} = \\int dQ_1dQ_2dQ_3\\,Q_iQ_j\\mathcal{P}(Q_3,Q_2,Q_1).\n\\end{equation}\nIf the KQPD was strictly positive, it would thus provide a positive probability distribution which describes all three measurements and is independent of the choice of measurements that are performed. The existence of such a probability distribution prevents the Leggett-Garg inequality to be violated \\cite{emary:2014} (see also App.~\\ref{app:lg}).\nInverting the argument, we conclude that a violation of the Leggett-Garg inequality in Eq.~\\eqref{eq:leggettgarg} implies negative values for the KQPD in Eq.~\\eqref{eq:kqpdlg} and thus non-classical behavior in the time-evolution of $\\hat{Q}$.\n\n{While we discussed projective measurements, the KQPD could also find applications when considering Leggett-Garg inequalities obtained from weak continuous measurements \\cite{ruskov:2006,jordan:2006}.}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nThe KQPD provides a generalization of the Wigner function and the FCS to describe arbitrary observables within a quasi-probability formalism. This provides a unified framework for the description of fluctuations in dynamic quantum systems. Negative values of the KQPD can be directly linked to an interference between paths through Hilbert space which visit different eigenstates of the observables of interest. This requires coherent superpositions {of states that correspond to different measurement outcomes}, implying that negativity in the KQPD indicates non-classical behavior. As we have shown, this negativity is witnessed by anomalous weak values and violated Leggett-Garg inequalities (see also Refs.~\\cite{williams:2008,groen:2013} for a connection between anomalous weak values and Leggett-Garg inequalities). The versatility of the KQPD allows to describe the problem at hand with a tailor-made quasi-probability distribution which focuses on the observables of interest.\nWhen measuring these observables in von Neumann measurements, the outcome is determined by the KQPD, corrupted by measurement imprecision and back-action. \nHowever, the underlying KQPD can in principle be reconstructed tomographically by coupling the observables of interest to qubits and performing state tomography on the qubits. This has been discussed extensively for the FCS \\cite{levitov:1996,clerk:2011,hofer:2016,dasenbrook:2016,lebedev:2016}.\n\n{While we presented the point of view of the KQPD being a measurement-independent quantity that captures non-classicality of the system, a complementary viewpoint is provided by only considering von Neumann measurement setups. The appearance of negative values in the KQPD then reflects the presence of non-trivial correlations between the system and the detectors during the measurement.}\n\nHere we only discussed a small number of applications for the KQPD. Other scenarios where the KQPD might be of interest include: measurement based quantum computing \\cite{briegel:2009}; could the KQPD in this case capture the non-classical features that allow for outperforming a classical computer? Quantum-to-classical transition; how much noise is necessary for ensuring positivity of the KQPD and in which cases can the back-action in a measurement be neglected? Bell non-locality, entanglement, and steering; are these notions of non-classicality captured by a negative KQPD?\nIn addition to these open questions, we believe that there are many useful applications for the KQPD yet to be discovered.\n\n\\section*{Acknowledgments}\nI acknowledge stimulating discussions with A. Clerk, N. Brunner, C. Flindt, and M. Perarnau-Llobet. I gratefully acknowledge the hospitality of McGill University where part of this work has been done. This work was funded by the Swiss National Science foundation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}