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Large deviations in transport and twist field correlation functions
from hydrodynamics
Benjamin Doyon
Royal Society Leverhulme Trust Senior Research Fellow
Department of Mathematics, King’s College London

with:
M. Joe Bhaseen, King’s College London
Rosemary Harris, Queen Mary University London
Jason Myers, CANES doctoral program
Osaka, 11 September 2019

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What this is about? One goal is to characterise nonequilibrium states, where quantities are beingtransported across a surface.

? Fluctuations of the total amount of a quantity passing through an interface in a time t.
At t → ∞, universal laws, analogues of equilibrium entropy and free energy.

? These are related to two-point functions of twist fields in thermal states / GGEs.
? Theory based on Euler hydrodynamics, widely applicable: quantum and classical field
theories, chains, gases, integrable or not, stochastic or deterministic. In integrable systems, we
use generalised hydrodynamics.

? Extension to interacting models of Lesovik-Levitov free-fermion formula, of Bernard-Doyon
1dCFT formula, of exponential behaviour of dynamical correlation functions as in XX chain.

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Examples of transport fluctuation problems

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Particles have length ` and jump by one site to the right, if there is space, with probability p.
A steady state is a distribution of positions invariant under this stochastic dynamics.
Homogeneous, and fully characterised by the density of particles ρ ∈ [0, 1/`].
random variable = J&(t)= # particles that crossed the interface in time t

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Examples of transport fluctuation problems

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The hard rod gas [Lebowitz, Spohn1970’s; Boldrighini,
Dobrushin,` Sukhov 1983]<latexit sha1_base64="NiCNFQINCFG5T4iOY+JPxVaf5XA=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NJFmyu3fs7gnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelAhurO9/e6W19Y3NrfJ2ZWd3b/+genjUNnGqGbZYLGLdiahBwRW2LLcCO4lGKiOBj9HkNvcfn1AbHqsHO00wlHSk+JAzanOph0L0qzW/7s9BVklQkBoUaParX71BzFKJyjJBjekGfmLDjGrLmcBZpZcaTCib0BF2HVVUogmz+a0zcuaUARnG2pWyZK7+nsioNGYqI9cpqR2bZS8X//O6qR1ehxlXSWpRscWiYSqIjUn+OBlwjcyKqSOUae5uJWxMNWXWxVNxIQTLL6+S9kU98OvB/WWtcVPEUYYTOIVzCOAKGnAHTWgBgzE8wyu8edJ78d69j0VryStmjuEPvM8fDWaOOw==</latexit><latexit<latexit sha1_base64="NiCNFQINCFG5T4iOY+JPxVaf5XA=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NJFmyu3fs7gnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelAhurO9/e6W19Y3NrfJ2ZWd3b/+genjUNnGqGbZYLGLdiahBwRW2LLcCO4lGKiOBj9HkNvcfn1AbHqsHO00wlHSk+JAzanOph0L0qzW/7s9BVklQkBoUaParX71BzFKJyjJBjekGfmLDjGrLmcBZpZcaTCib0BF2HVVUogmz+a0zcuaUARnG2pWyZK7+nsioNGYqI9cpqR2bZS8X//O6qR1ehxlXSWpRscWiYSqIjUn+OBlwjcyKqSOUae5uJWxMNWXWxVNxIQTLL6+S9kU98OvB/WWtcVPEUYYTOIVzCOAKGnAHTWgBgzE8wyu8edJ78d69j0VryStmjuEPvM8fDWaOOw==</latexit><latexit<latexit sha1_base64="NiCNFQINCFG5T4iOY+JPxVaf5XA=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NJFmyu3fs7gnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelAhurO9/e6W19Y3NrfJ2ZWd3b/+genjUNnGqGbZYLGLdiahBwRW2LLcCO4lGKiOBj9HkNvcfn1AbHqsHO00wlHSk+JAzanOph0L0qzW/7s9BVklQkBoUaParX71BzFKJyjJBjekGfmLDjGrLmcBZpZcaTCib0BF2HVVUogmz+a0zcuaUARnG2pWyZK7+nsioNGYqI9cpqR2bZS8X//O6qR1ehxlXSWpRscWiYSqIjUn+OBlwjcyKqSOUae5uJWxMNWXWxVNxIQTLL6+S9kU98OvB/WWtcVPEUYYTOIVzCOAKGnAHTWgBgzE8wyu8edJ78d69j0VryStmjuEPvM8fDWaOOw==</latexit><latexit<latexit sha1_base64="NiCNFQINCFG5T4iOY+JPxVaf5XA=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NJFmyu3fs7gnhyF+wsVDE1j9k579xL7lCEx8MPN6bYWZelAhurO9/e6W19Y3NrfJ2ZWd3b/+genjUNnGqGbZYLGLdiahBwRW2LLcCO4lGKiOBj9HkNvcfn1AbHqsHO00wlHSk+JAzanOph0L0qzW/7s9BVklQkBoUaParX71BzFKJyjJBjekGfmLDjGrLmcBZpZcaTCib0BF2HVVUogmz+a0zcuaUARnG2pWyZK7+nsioNGYqI9cpqR2bZS8X//O6qR1ehxlXSWpRscWiYSqIjUn+OBlwjcyKqSOUae5uJWxMNWXWxVNxIQTLL6+S9kU98OvB/WWtcVPEUYYTOIVzCOAKGnAHTWgBgzE8wyu8edJ78d69j0VryStmjuEPvM8fDWaOOw==</latexit><latexit

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Rods have length ` and move at velocities vi ∈ R, with elastic collisions.
A steady state is a distribution of positions and velocities invariant under thisdeterministic evolution. Homogeneous, and fully characterised by the distribution ofvelocities ρ(v) ∈ R+

R

dv ρ(v) ∈ [0, 1/`]).
hYi
measure on initial condition = χ{|xi − xj | ≥ `}dxi dvi ρ(vi )(constrained toirandom variable = J(t)= total amount of the quantity h(v) transferred from left to rightXX=h(vi (t)) −h(vi )
i:xi >0i:xi (t)>0

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Examples of transport fluctuation problems

The classical sinh-Gordon field (or any other field theory)1
H = H[Φ, Π] =
2

Z


 2
m2
2
dx Π + (∂x Φ) + 2 (cosh(gΦ) − 1)g

A steady state is a distribution of field and canonical momentum invariant underdynamics {H, ·}. Homogeneous, and characterised (at least formally) by the Lagrange
parameters βi for all conserved quantities: boosted / generalised Gibbs ensemble.
measure on initial condition = DΦDΠ erandom variable = J

(t)

−

P

j βj Qj= total amount of the quantity Qi transferred from left to right
Z ∞
Z ∞
=
dx qi (x, t) −dx qi (x, 0)
0

0

where

Z
Qi = Qi [Φ, Π] =&5dx qi (x)%'$Nonequilibrium steady states

We can evaluate the above in nonequilibrium steady states. For instance, those generated by
the partitioning protocol, or Riemann problem, where nontrivial currents emerge in
steady-state region if there is ballistic transport.
Nonequilibrium steady state

eiHt<latexit sha1_base64="nBa9ky8BlAAlzINkvvk/webX8ZI=">AAAB9HicbVBNS8NAEJ34WetX1aOXxSJ4KokIeix66bGC/YA2ls120y7dbOLupFBCf4cXD4p49cd489+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKwecJJwP6IDJULBKFrJ549Z1tUREdMaTnulsltx5yCrxMtJGXLUe6Wvbj9macQVMkmN6Xhugn5GNQom+bTYTQ1PKBvRAe9YqmjEjZ/Nj56Sc6v0SRhrWwrJXP09kdHImEkU2M6I4tAsezPxP6+TYnjjZ0IlKXLFFovCVBKMySwB0heaM5QTSyjTwt5K2JBqytDmVLQheMsvr5LmZcVzK979Vbl6m8dRgFM4gwvw4BqqUIM6NIDBEzzDK7w5Y+fFeXc+Fq1rTj5zAn/gfP4A7JKSLw==</latexit>

e

l Hl

<latexit sha1_base64="8rdIsBAA9gkJPcOgGkGkiLBSrSw=">AAACCHicbVDLSsNAFJ3UV42vqks3g0VwVRIRdFl047KCfWATymQyaYdOJmHmRgihP+DarX6DO3HrX/gJ/oXTNgvbeuDC4Zx7ufeeIBVcg+N8W5W19Y3Nreq2vbO7t39QOzzq6CRTlLVpIhLVC4hmgkvWBg6C9VLFSBwI1g3Gt1O/+8SU5ol8gDxlfkyGkkecEjDSI3iQeFxGkA9qdafhzIBXiVuSOirRGtR+vDChWcwkUEG07rtOCn5BFHAq2MT2Ms1SQsdkyPqGShIz7Reziyf4zCghjhJlSgKeqX8nChJrnceB6YwJjPSyNxX/8/oZRNd+wWWaAZN0vijKBIYET9/HIVeMgsgNIVRxcyumI6IIBRPSwpYwAzqiRExs22TjLiexSjoXDddpuPeX9eZNmVIVnaBTdI5cdIWa6A61UBtRJNELekVv1rP1bn1Yn/PWilXOHKMFWF+/ZAWaag==</latexit>

e

r Hr

<latexit sha1_base64="UBcE/cho9THJh8Cktdlvp+uz1yI=">AAAB+XicbVDLSgNBEJz1GeNr1aOXwSB4MeyKoMeglxwjmAck6zI76U2GzD6Y6Q2EJX/ixYMiXv0Tb/6Nk2QPmljQUFR1090VpFJodJxva219Y3Nru7RT3t3bPzi0j45bOskUhyZPZKI6AdMgRQxNFCihkypgUSChHYzuZ357DEqLJH7ESQpexAaxCAVnaCTftuEpv+wFgMxXtO6rqW9XnKozB10lbkEqpEDDt796/YRnEcTIJdO66zopejlTKLiEabmXaUgZH7EBdA2NWQTay+eXT+m5Ufo0TJSpGOlc/T2Rs0jrSRSYzojhUC97M/E/r5theOvlIk4zhJgvFoWZpJjQWQy0LxRwlBNDGFfC3Er5kCnG0YRVNiG4yy+vktZV1XWq7sN1pXZXxFEip+SMXBCX3JAaqZMGaRJOxuSZvJI3K7derHfrY9G6ZhUzJ+QPrM8f9aqTNQ==</latexit><latexit sha1_base64="/7olCdOHjqTCaYRqVAFuZqHyCU0=">AAAB+XicbVDLSgNBEJz1GeNr1aOXwSB4MeyKoMeglxwjmAck6zI76U2GzD6Y6Q2EJX/ixYMiXv0Tb/6Nk2QPmljQUFR1090VpFJodJxva219Y3Nru7RT3t3bPzi0j45bOskUhyZPZKI6AdMgRQxNFCihkypgUSChHYzuZ357DEqLJH7ESQpexAaxCAVnaCTftuEpv+wFgMyXtO7LqW9XnKozB10lbkEqpEDDt796/YRnEcTIJdO66zopejlTKLiEabmXaUgZH7EBdA2NWQTay+eXT+m5Ufo0TJSpGOlc/T2Rs0jrSRSYzojhUC97M/E/r5theOvlIk4zhJgvFoWZpJjQWQy0LxRwlBNDGFfC3Er5kCnG0YRVNiG4yy+vktZV1XWq7sN1pXZXxFEip+SMXBCX3JAaqZMGaRJOxuSZvJI3K7derHfrY9G6ZhUzJ+QPrM8f41aTKQ==</latexit>

&

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Nonequilibrium steady states

In quantum chains there is the Lieb-Robinson bound. Phenomenology from conventionalhydrodynamics involves shocks and rarefaction waves.

trarefaction wave

shock<latexit sha1_base64="fInOqGTCCrWkRGJFOZWK1l6FLBY=">AAAB6HicbVBNS8NAEJ34WetX1aOXYBE8lUQEPRa9eGzBfkAbymY7adduNmF3IpTSX+DFgyJe/Une/Ddu2xy09cHA470ZZuaFqRSGPO/bWVvf2NzaLuwUd/f2Dw5LR8dNk2SaY4MnMtHtkBmUQmGDBElspxpZHEpshaO7md96Qm1Eoh5onGIQs4ESkeCMrFSnXqnsVbw53FXi56QMOWq90le3n/AsRkVcMmM6vpdSMGGaBJc4LXYzgynjIzbAjqWKxWiCyfzQqXtulb4bJdqWIneu/p6YsNiYcRzazpjR0Cx7M/E/r5NRdBNMhEozQsUXi6JMupS4s6/dvtDISY4tYVwLe6vLh0wzTjabog3BX355lTQvK75X8etX5eptHkcBTuEMLsCHa6jCPdSgARwQnuEV3pxH58V5dz4WrWtOPnMCf+B8/gDgKYz4</latexit><latexit

constant

constant

constantx<latexit sha1_base64="f2yzimwbR/Dgjzp6tZ360fHRqNI=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N2IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB5jmM/A==</latexit><latexit

Our theory will work generically in steady state regions away from shocks and rarefaction
waves. In rarefaction waves: nonlinear fluctuating hydrodynamics instead (e.g. TASEP fromstep initial condition).

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Nonequilibrium steady states

In integrable systems, including e.g. the hard rods, the phenomenology is different:
generically smooth profiles, because of the continuum of normal mode’s velocities [Castro
Alvaredo, BD, Yohimura 2016; Bertini, Collura, De Nardis, Fagotti 2016]. In fluid dynamic terms, solution is acontinuum of contact singularities [Castro Alvaredo, BD, Yohimura 2016].

Our theory will work in integrable models.

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Dynamical two-point functions of twist fields

In all the examples mentioned, the variable can be re-expressed as an integral over the flux,

J (t) =Z t
ds ji (0, s)
0which is related to the charge density by the continuity equation∂t qi + ∂x ji = 0
Consider the generating function for the cumulants of J (t) :F (λ; 0, t) = loghe

λJ (t)

i = logX

P(J (t) = J)eλJ

J

Solve the continuity equation by defining ϕi such thatqi = ∂x ϕi , ji = −∂t ϕiThen

F (λ; 0, t) = heλϕi (0,0) e−λϕi (0,t) i
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Dynamical two-point functions of twist fields

We can consider the more general problem of evaluating
λϕi (0,0) −λϕi (x,t)
F (λ; x, t) = logheei = loghexp λ(x, t)

Z (x,t)(0,0)(x, t)<latexit sha1_base64="zuoL9EgHnayPYkWQ4aMVRoLe6dU=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiF48VTFtoQ9lsN+3SzSbsTsRS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MJXCoOt+Oyura+sbm4Wt4vbO7t5+6eCwYZJMM+6zRCa6FVLDpVDcR4GSt1LNaRxK3gyHt1O/+ci1EYl6wFHKg5j2lYgEo2glv/J0jmfdUtmtujOQZeLlpAw56t3SV6eXsCzmCpmkxrQ9N8VgTDUKJvmk2MkMTykb0j5vW6pozE0wnh07IadW6ZEo0bYUkpn6e2JMY2NGcWg7Y4oDs+hNxf+8dobRdTAWKs2QKzZfFGWSYEKmn5Oe0JyhHFlCmRb2VsIGVFOGNp+iDcFbfHmZNC6qnlv17i/LtZs8jgIcwwlUwIMrqMEd1MEHBgKe4RXeHOW8OO/Ox7x1xclnjuAPnM8f6guOFQ==</latexit><latexit sha1_base64="qSkLmVT2BPdVI3IIZ6IqkiC7RWY=">AAAB/3icbVDLSgNBEOz1GddX1KOXwSB4MeyKoMegF48JmAckS5idnU2GzD6Y6RXCkoNnr/oN3sSrn+In+BdOkj2YxIKGoqqb7i4/lUKj43xba+sbm1vbpR17d2//4LB8dNzSSaYYb7JEJqrjU82liHkTBUreSRWnkS952x/dT/32E1daJPEjjlPuRXQQi1AwikZqXPbLFafqzEBWiVuQChSo98s/vSBhWcRjZJJq3XWdFL2cKhRM8ondyzRPKRvRAe8aGtOIay+fHToh50YJSJgoUzGSmfp3IqeR1uPIN50RxaFe9qbif143w/DWy0WcZshjNl8UZpJgQqZfk0AozlCODaFMCXMrYUOqKEOTzcKWIEM2ZFRObNtk4y4nsUpaV1XXqbqN60rtrkipBKdwBhfgwg3U4AHq0AQGHF7gFd6sZ+vd+rA+561rVjFzAguwvn4BhHqWBA==</latexit>

(ji dt − qi dx) i'i (x, t)<latexit sha1_base64="L6P+bv14gz2jDxjDPnAPqQsyiiE=">AAACB3icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF2yptKJPJpBk6Mwkzk0IJ/QDXbvUb3IlbP8NP8C+ctlnY1gMXDufcy733BCln2rjut1NaW9/Y3CpvV3Z29/YPqodHbZ1kitAWSXiiHgOsKWeStgwznD6mimIRcNoJhrdTvzOiSrNEPphxSn2BB5JFjGBjpafeCKs0Zn3Wr9bcujsDWiVeQWpQoNmv/vTChGSCSkM41rrruanxc6wMI5xOKr1M0xSTIR7QrqUSC6r9fHbwBJ1ZJURRomxJg2bq34kcC63HIrCdAptYL3tT8T+vm5no2s+ZTDNDJZkvijKOTIKm36OQKUoMH1uCiWL2VkRirDAxNqOFLWFmSEwwn1QqNhtvOYlV0r6oe27du7+sNW6KlMpwAqdwDh5cQQPuoAktICDgBV7hzXl23p0P53PeWnKKmWNYgPP1C1vRmdU=</latexit><latexit sha1_base64="zuoL9EgHnayPYkWQ4aMVRoLe6dU=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiF48VTFtoQ9lsN+3SzSbsTsRS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MJXCoOt+Oyura+sbm4Wt4vbO7t5+6eCwYZJMM+6zRCa6FVLDpVDcR4GSt1LNaRxK3gyHt1O/+ci1EYl6wFHKg5j2lYgEo2glv/J0jmfdUtmtujOQZeLlpAw56t3SV6eXsCzmCpmkxrQ9N8VgTDUKJvmk2MkMTykb0j5vW6pozE0wnh07IadW6ZEo0bYUkpn6e2JMY2NGcWg7Y4oDs+hNxf+8dobRdTAWKs2QKzZfFGWSYEKmn5Oe0JyhHFlCmRb2VsIGVFOGNp+iDcFbfHmZNC6qnlv17i/LtZs8jgIcwwlUwIMrqMEd1MEHBgKe4RXeHOW8OO/Ox7x1xclnjuAPnM8f6guOFQ==</latexit><latexit sha1_base64="zuoL9EgHnayPYkWQ4aMVRoLe6dU=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiF48VTFtoQ9lsN+3SzSbsTsRS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MJXCoOt+Oyura+sbm4Wt4vbO7t5+6eCwYZJMM+6zRCa6FVLDpVDcR4GSt1LNaRxK3gyHt1O/+ci1EYl6wFHKg5j2lYgEo2glv/J0jmfdUtmtujOQZeLlpAw56t3SV6eXsCzmCpmkxrQ9N8VgTDUKJvmk2MkMTykb0j5vW6pozE0wnh07IadW6ZEo0bYUkpn6e2JMY2NGcWg7Y4oDs+hNxf+8dobRdTAWKs2QKzZfFGWSYEKmn5Oe0JyhHFlCmRb2VsIGVFOGNp+iDcFbfHmZNC6qnlv17i/LtZs8jgIcwwlUwIMrqMEd1MEHBgKe4RXeHOW8OO/Ox7x1xclnjuAPnM8f6guOFQ==</latexit><latexit sha1_base64="aRf/TvuyoZmqDiVkm7GeNAFThME=">AAACLXicbZDLSsNAFIYnXmu8RV26GSyCm5ZEBF0W3bisYC/QljCZTNuxk4szJ2IJeQofxLVbfQYXgrgV38JJW9S2/jDw851zOGd+LxZcgW2/GQuLS8srq4U1c31jc2vb2tmtqyiRlNVoJCLZ9IhigoesBhwEa8aSkcATrOENLvJ6445JxaPwGoYx6wSkF/IupwQ0cq1SOyDQp0TgG5fjtC0D7GeAS/iH3/7ye9cq2mV7JDxvnIkpoomqrvXV9iOaBCwEKohSLceOoZMSCZwKlpntRLGY0AHpsZa2IQmY6qSjb2X4UBMfdyOpXwh4RP9OpCRQahh4ujM/Vs3WcvhfrZVA96yT8jBOgIV0vKibCAwRzjPCPpeMghhqQ6jk+lZM+0QSCjrJqS1+AjRPKTNNnY0zm8S8qR+XHbvsXJ0UK+eTlApoHx2gI+SgU1RBl6iKaoiiB/SEntGL8Wi8Gu/Gx7h1wZjM7KEpGZ/fYWCn3g==</latexit>(0, 0)<latexit sha1_base64="J+Nb7fD6d44Iorb+ri9Qi+oMQd8=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBahgpSNCHosevFYwbSFNpTNdtMu3WzC7kYIob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMCxLBtcH42ymtrW9sbpW3Kzu7e/sH1cOjto5TRZlHYxGrbkA0E1wyz3AjWDdRjESBYJ1gcjfzO09MaR7LR5MlzI/ISPKQU2Ks5NXxBT4fVGu4gedAq8QtSA0KtAbVr/4wpmnEpKGCaN1zcWL8nCjDqWDTSj/VLCF0QkasZ6kkEdN+Pj92is6sMkRhrGxJg+bq74mcRFpnUWA7I2LGetmbif95vdSEN37OZZIaJuliUZgKZGI0+xwNuWLUiMwSQhW3tyI6JopQY/Op2BDc5ZdXSfuy4eKG+3BVa94WcZThBE6hDi5cQxPuoQUeUODwDK/w5kjnxXl3PhatJaeYOYY/cD5/ABS/jYk=</latexit>

ji dt

qi dx

=

=
(0, 0)'i(0, 0)<latexit sha1_base64="L6P+bv14gz2jDxjDPnAPqQsyiiE=">AAACB3icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF2yptKJPJpBk6Mwkzk0IJ/QDXbvUb3IlbP8NP8C+ctlnY1gMXDufcy733BCln2rjut1NaW9/Y3CpvV3Z29/YPqodHbZ1kitAWSXiiHgOsKWeStgwznD6mimIRcNoJhrdTvzOiSrNEPphxSn2BB5JFjGBjpafeCKs0Zn3Wr9bcujsDWiVeQWpQoNmv/vTChGSCSkM41rrruanxc6wMI5xOKr1M0xSTIR7QrqUSC6r9fHbwBJ1ZJURRomxJg2bq34kcC63HIrCdAptYL3tT8T+vm5no2s+ZTDNDJZkvijKOTIKm36OQKUoMH1uCiWL2VkRirDAxNqOFLWFmSEwwn1QqNhtvOYlV0r6oe27du7+sNW6KlMpwAqdwDh5cQQPuoAktICDgBV7hzXl23p0P53PeWnKKmWNYgPP1C1vRmdU=</latexit><latexit sha1_base64="J+Nb7fD6d44Iorb+ri9Qi+oMQd8=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBahgpSNCHosevFYwbSFNpTNdtMu3WzC7kYIob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMCxLBtcH42ymtrW9sbpW3Kzu7e/sH1cOjto5TRZlHYxGrbkA0E1wyz3AjWDdRjESBYJ1gcjfzO09MaR7LR5MlzI/ISPKQU2Ks5NXxBT4fVGu4gedAq8QtSA0KtAbVr/4wpmnEpKGCaN1zcWL8nCjDqWDTSj/VLCF0QkasZ6kkEdN+Pj92is6sMkRhrGxJg+bq74mcRFpnUWA7I2LGetmbif95vdSEN37OZZIaJuliUZgKZGI0+xwNuWLUiMwSQhW3tyI6JopQY/Op2BDc5ZdXSfuy4eKG+3BVa94WcZThBE6hDi5cQxPuoQUeUODwDK/w5kjnxXl3PhatJaeYOYY/cD5/ABS/jYk=</latexit><latexit sha1_base64="J+Nb7fD6d44Iorb+ri9Qi+oMQd8=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBahgpSNCHosevFYwbSFNpTNdtMu3WzC7kYIob/BiwdFvPqDvPlv3LY5aOuDgcd7M8zMCxLBtcH42ymtrW9sbpW3Kzu7e/sH1cOjto5TRZlHYxGrbkA0E1wyz3AjWDdRjESBYJ1gcjfzO09MaR7LR5MlzI/ISPKQU2Ks5NXxBT4fVGu4gedAq8QtSA0KtAbVr/4wpmnEpKGCaN1zcWL8nCjDqWDTSj/VLCF0QkasZ6kkEdN+Pj92is6sMkRhrGxJg+bq74mcRFpnUWA7I2LGetmbif95vdSEN37OZZIaJuliUZgKZGI0+xwNuWLUiMwSQhW3tyI6JopQY/Op2BDc5ZdXSfuy4eKG+3BVa94WcZThBE6hDi5cQxPuoQUeUODwDK/w5kjnxXl3PhatJaeYOYY/cD5/ABS/jYk=</latexit>

The variables eϕi (x,t) are twist fields associated to the symmetry generated by Qi . By
the continuity relation the integral is independent of the shape of the path. A typical example isthe vertex operator in the sine-Gordon model, the twist field associated to its topologicalcharge (equivalently, the U (1)-twist field in the massive Thirring model); or thehσ + (0, 0)σ − (x, t)i correlation function in the XX chain.
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Quantum

Subtleties in quantum models:• The observable J

(t)

=

Rt
0

ds ji (0, s) does not represent well the transport fluctuations,

as it is not a natural observable on which we can make measurements. Instead, could useR∞dx qi (x):hi
Q+ −iHt Q+
Q+ iHt Q+
Tr PJ2 e
PJ1 ρ0 PJ1 e PJ2

two projective measurement protocol for the half-charge Q+ = 0

P(J

(t)

X

= J) =

J2 −J1 =J

where ρ0 = e−Pj βj Qj. For instance: quantum field theories (UV regularisationneeded), quantum chains, etc.

• By non-commutativity,
λϕi (0,0) −λϕi (x,t)
loghe
e
i=
6 loghexp λZ (x,t)(0,0)


(ji dt − qi dx) i

But in the limit of large time / space-time, these problems are expected to disappear...

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Universal behaviours: large deviations / large space-time distances

Consider the large-time behaviour. J (t) is extensive, and it concentrates on some value t̄.
More precisely, we expect a large deviation principle to hold [e.g. Touchette, 2009]:

P(J (t) = tj)  e−tI(j) , t → ∞I(j) is the large-deviation rate function. It describes rare but significant fluctuations of theextensive state characteristics J (t) . It is the analogue of an entropy. It satisfies
I(̄) = 0,&

I(j) > 0 (j 6= ̄)

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Universal behaviours: large deviations / large space-time distances

We will argue that I(j) is universal:• I(j) is determined by the Euler hydrodynamics of the system, independently from its
microscopic details.

• In the quantum case, I(j) appears to be largely independent of the measurementRt(t)protocol, and we can use the distribution of the operator J= 0 ds ji (0, s).
• In quantum field theory, I(j) appears to be free from UV divergencies.

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Universal behaviours: large deviations / large space-time separations

More useful to work with is the Legendre-Frenchel transform of I(j): the scaled cumulant
generating function(t)

F (λ) = lim t−1 logheλJ i =
t→∞

∞
X
λn
n=1

n!

cn .

F (λ) is (up to sign) a nonequilibrium analogue of the specific free energy. We have the scaling
λJ (t)

hei  etF (λ)

cn are the cumulants of J (t) scaled by 1/t. c1 = ̄ is the average current. In mesoscopic
physics, c2 is the zero-frequency noise.

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Universal behaviours: large deviations / large space-time separations

On more general rays, the large deviation principle implies the exponential behaviour oftwist field correlation functions at large space-time separations

heλϕi (0,0) e−λϕi (ξt,t) i  etF (λ;ξ) , t → ∞This large-t behaviour holds also in the quantum case, despite the lack of commutativity.

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Nonequilibrium steady states and fluctuation relations
Nonequilibrium steady state

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In nonequilibrium steady states, nonequilibrium fluctuation relations relate probabilities forrare events where stuff goes in the “wrong” direction to those where it goes in the “right”
direction

Pness (J (t) = tj)t(βr −βl )jePness (J (t) = −tj)⇔

Fness(βr ,βl ) (βl − βr − λ) = Fness(βr ,βl ) (λ)

In 1dCFT and in free models, the stronger “extended fluctuation relation” holds [Bernard, Doyon
2012, 2013]dFness(βr ,βl ) (λ) = hji iness(βr +λ/2,βl −λ/2)
dλ
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Main result: ballistic fluctuation framework[Myers, Bhaseen, Harris, Doyon 2019; Doyon, Myers 2019]

A general framework giving an exact result for F (λ) (scaled cumulant generating function)and more generally F (λ; ξ) (exponential behaviour of twist field two-point function), in any
homogeneous, stationary maximal-entropy state, such as (G)GEs, NESSs and all states
discussed in the examples.

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Main result: ballistic fluctuation framework

Maximal entropy states (MES): a Lagrange parameter β j for every conserved quantity Qj ,

e

−

P

j β

j

Qj

↔

h· · ·iββ is a coordinate on the MES manifold.
But there’s no need for this explicit distribution. Can define β j as coordinates associated toconserved charges Qj , which form vector in the tangent space of the MES manifold. The
tangent space is the Hilbert space of pseudolocal charges [Doyon 2017]∂hO(0, 0)iβ = (qj , O)βj∂β
Z ∞(O, O 0 )β =dx hO(x, 0)O 0 (0, 0)icβ
−

where

−∞

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Main results: ballistic fluctuation framework

Consider set of averages of all local or quasi-local densities and associated fluxes

qj = qj (β) = hqj (0, 0)iβ ,jj = jj (β) = hjj (0, 0)iβ

Seeing fluxes as functions of densities jj = jj (q), calculate the so-called flux Jacobian

Ajk = Ajk (β) =X ∂jj ∂βl∂jj=∂qk
∂βl ∂qk
l

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Main results: ballistic fluctuation framework

This is a (model-specific) matrix that is a function of the MES, hence a function of β . It enters
into the Euler equations of the model, and controls propagation of perturbations andEuler-scale two-point functions:∂t hqj (x, t)O(0, 0)i + Ajk ∂x hqk (x, t)O(0, 0)i = 0
The fluid’s normal modes and their velocities are the obtained by diagonalising A:

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exponential

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qi (x, t)

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Main results: ballistic fluctuation framework

Define a flow λ 7→ β(λ) on the manifold of MES by the differential equation


j
d j
β (λ) = − sgn A(β(λ)) i
dλ
where

 
sgn A = M sgn(v eff )M−1

Main result: We identify the flow parameter λ with the conjugate parameter generating the
cumulants. Recall

F (λ) = lim t
t→∞

−1

loghe

λJ (t)iβ

We set the initial condition of the flow as β(0) = β . Then we haveZ λdµ ji (β(µ))

F (λ) =
0

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Main results: ballistic fluctuation framework

Putting all together, the main formula is

h Z t
i
h Z λ
i
exp λ
ds ji (0, s) β  exp tdµ hji (0, 0)iβ(µ)
0

0

with


j
d j
β (λ) = − sgn A(β(λ)) i ,
dλ

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β(0) = β%'$Main results: ballistic fluctuation framework

This can be extended to other rays. Written in terms of twist fields:h Z λiλϕi (0,0) −λϕi (ξt,t)eiβ  exp thedµ hji (0, 0) − ξqi (0, 0)iβ(µ;ξ)
0

with


j
d j
β (λ; ξ) = − sgn A(β(λ; ξ)) − ξ1 i ,
dλ

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β(0; ξ) = β

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Application to `-TASEP

A single local conserved density: the density of particles ρ. The current can be obtained interms of this [Schoenherr, Schuetz 2004]:j=pρ(1 − `ρ)
1 − (` − 1)ρ

The Lagrange parameter β can also be related to the density:(1 − `ρ)`e =ρ(1 − (` − 1)ρ)`−1β

We calculate A = ∂j/∂ρ and then solve the flow equation. It is trivial here as sgn(A) is
constant, except at the value of ρ where A changes sign (the normal-mode velocity is zero),which separates phases [Erdmann-Pham, Duc, Song 2018].
At ` = 1 the obtained F (λ) agrees with previous calculations [de Gier, Essler 2011; Lazarescu,
Mallick 2011]. For ` > 1, new resuts, for instance, second cumulant is

c2 = p&ρ(`ρ − 1)(`ρ((` − 1)ρ − 2) + 1)(` − 1)ρ − 1
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“Free” systems and constant flux Jacobian

If A is independent of the state, then it is a simple matter to solve for the flow:

β j (λ) = β j − λ sgn(A)ij
That is, the flow corresponds to a shift of the Lagrange parameters proportional to λ. In
particularZ λ
F (λ) =
0dµ hji i{β • −µ sgn(A)i• }When applied to NESSs, this leads to the extended fluctuation relationsdFness(βr ,βl ) (λ) = hji iness(βr +λ/2,βl −λ/2)
dλ

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Application to integrable systems

The exact Euler hydrodynamics of the hard rod system was obtained in [Boldrighini, Dobrushin,
Sukhov 1983], and for integrable systems in [Castro-Alvaredo, Doyon, Yoshimura 2016; Bertini, Collura, De
Nardis, Fagotti 2016]. The flux Jacobian was first evaluated in [Doyon, Spohn 2017]. There is acommon formalism, TBA-like, which works for quantum and classical systems.

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Application to integrable systems

In this formalism, we know how to diagonalise the flux Jacobian. For any “spanning set” offunctions hk (θ) of the “rapidity” θ ,eff
drAjk hdrk (θ) = v (θ)hj (θ)

where v eff (θ) = (E 0 )dr (θ)/(p0 )dr (θ), where p(θ) and E(θ) are momentum and energy of
dr

R dα

(θ) = h(θ) + 2π ϕ(θ, α)n(α)hdr (α), where n = dF()/d is theoccupation, where F() is the non-interacting free energy of a mode of energy , andR dα
P j(θ) = j β hj (θ) + 2π ϕ(θ, α)F ((α)) where ϕ(θ, α) is the differential scattering
a quasiparticle hphase.
For the hard rods: set θ = v the velocity, and ϕ(θ, α) = −` and F() = −e− andvelocity distribution ρ(v) = (1 − `ρ)/2π n(v) where ρ is the total density per unit length.

For the quantum sinh-Gordon model (e.g. at the self-dual point): setϕ(θ, α) = 1/ cosh(θ − α) and F() = − log(1 + e− ), and hj (θ) are one-particle
eigenvalues of a spanning set of conserved charges.

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Application to integrable systems

Consider transport problem (ξ = 0) for simplicity. We get the flow equation for thepseudoenergy∂λ (θ; λ) = − sgn(v eff (θ; λ))hdr
i (θ; λ).
An expression for F (λ) can be obtained:dθ 0 
eff
F (λ) = −E (θ) sgn v (θ; λ) F((θ; λ)) − F((θ; 0))
2π
X
X

−
σ F((θ; λ̃)) − F((θ; 0) ,Zσ∈{±} λ̃∈λσ? (θ)∩[0,λ]where the sets λ±? (θ) are the turning points of the sign of the effective velocity:
eff
eff
λ±
? (θ) = {λ : v (θ; λ) = 0, ∂λ v (θ; λ) ≷ 0}.

• This reduces to the Levitov-Lesovik formula in free-particle models, where ϕ(θ, α) = 0.
• This reproduces the known exponential behaviour, both in time-like and space-like regimes,of hσ+ (0, 0)σ− (x, t) in XX chain (again ϕ(θ, α) = 0) [Itz, Izergin, Korepin, Slavnov 1993].

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Application to integrable systems

From this the cumulants can be obtained, for instance

Z
c2 =

2
dθ ρp |v eff |(hdri ) f,

where f = −d log n/d is the statistical factor, in agreement with [Doyon, Spohn 2017], and

Z
c3 =

i

dr
2˜
dr 2
dθ ρp f |v eff |hdr
(hdrii ) f s + 3 (hi ) f s
hwhere f˜ = −(d log f /d + 2f ) and s = sgn(v eff )

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Application to integrable systems

We have performed numerical checks by simulating the hard rods

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Application to integrable systems

We have checked numerically that the nonequilibrium fluctuation relations are satisfied in theNESSs found in [Castro-Alvaredo, Doyon, Yoshimura 2016; Bertini, Collura, De Nardis, Fagotti 2016]

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Derivation: biasing the measure

Main assumption: clustering at large time differences. This is natural, but its breaking is asignal of a dynamical phase transition.
Modify state h· · ·iβ by a time-integral of the current ji (0, t) ≡ j(0, t):λ(λ)

hO(x, t)i

=

he

R∞
−∞

he

λ

dt j(0,t)R∞−∞

O(x, t)iβ

dt j(0,t)iβBiasing the dynamics, changing the weights of trajectories in order to make rare events“typical” and access their probabilities.
Related to generalised (classical or quantum) Doob transformation [Chetrite, Touchette 2015; Carollo,
Garrahan, Lesanovsky, Pérez-Espigares, 2018]

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Derivation: biasing the measureThis state is useful to calculate F (λ)

dF (λ)
1
= lim
t→∞ t
dλ

Z t/2
ds

he

λ

R t/2
−t/2

he

−t/2

λ

dr j(0,r)

R t/2
−t/2

j(0, s)iβ

dr j(0,r).

iβ

giving, neglecting boundaries in time,dF (λ)= hj(0, 0)i(λ)
dλ

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Derivation: biasing the measure

Crucial observation: The state h· · ·i(λ) is a MES. Therefore, there exists β(λ) such thathOi(λ) = hOiβ(λ) .
With this observation,dhOiβ(λ) =
dλ

Z ∞
−∞

dt hj(0, t)Oicβ(λ)

Choose conserved densities qj (0, 0), get a flow for the qj -coordinates:dqj (λ) =
dλ

Z ∞
−∞

dt hj(0, t)qj (0, 0)icβ(λ)

The right-hand side is a function of the state, hence of the state coordinates q(λ).

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Derivation: biasing the measure

A result from linear fluctuating hydrodynamics is[Spohn 1991, 2014; Mendl, Spohn 2015; Doyon, Spohn 2017]hji (x, t)qj (0, 0)icβ ∼ (Aδ(x − At)C)ij
where the covariance matrix is

Z
Cij =

dx hqi (x, 0)qj (0, 0)iβ = −

∂qj
∂β i

and δ(x − At) = Mδ(x − v eff t)M−1 . Integrating over time, the result is independent of x:

Z ∞

−∞

dt hji (0, t)qj (0, 0)icβ = (sgn[A] C)ij

Also

X dβ k dqj
X dβ k
dqj
=
=−
Cjk
dλ
dλ dβkdλkkThis gives the result.

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Dynamical phase transitions

Recalljd j
β (λ) = − sgn A(β(λ)) i
dλ

where

 
sgn A = M sgn(v eff )M−1
Cumulants aredd2
ji (λ)
c1 = ji (0), c2 =, c3 =ji (λ)
2
dλ
dλ
λ=0
λ=0
Around points where some sgn(vjeff ) changes sign: discontinuities of cn , n ≥ 3, andpossible divergencies exactly at these points.

• At vieff = 0, “stationary” normal modes create strong time-correlations which break thetime-clustering hypothesis.

• Divergence of higher cumulants suggest that the leading Gaussianity (law of large

numbers) is broken. This is where nonlinear fluctuating hydrodynamics [Spohn 2014],
should be applied instead.

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Conclusion

We have obtained a framework for evaluating large-deviation properties of current fluctuations /twist-field correlation functions along arbitrary rays in space-time in a large family of states,
including GGEs and NESSs.

• More analysis near to / at phase transitions and relations to NLFHD• Studying consequences on correlation functions of vertex operators in sine-Gordon modelin GGEs and NESSs• Can we get a full fluctuation theory for ballistic transport akin to macroscopic fluctuation
theory?

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