PHYSICAL REVIEW B 92, 014306 (2015)

Quantum Kibble-Zurek physics in the presence of spatially correlated dissipation
P. Nalbach,1,2 Smitha Vishveshwara,3 and Aashish A. Clerk4
1

I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
2
The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
3
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA
4
Department of Physics, McGill University, 3600 rue University, Montreal, QC Canada H3A 2T8(Received 29 March 2015; revised manuscript received 9 July 2015; published 31 July 2015)
We study how the universal properties of quantum quenches across critical points are modified by a weakcoupling to a thermal bath, focusing on the paradigmatic case of the transverse field Ising model. Beyond thestandard quench-induced Kibble-Zurek defect production in the absence of the bath, the bath contributes extra
thermal defects. We show that spatial correlations in the noise produced by the bath can play a crucial role:
one obtains quantitatively different scaling regimes depending on whether the correlation length of the noise issmaller or larger than the Kibble-Zurek length associated with the quench speed, and the thermal length set by thetemperature. For the case of spatially correlated bath noise, additional thermal defect generation is restricted to awindow that is both quantum critical and excluded from the nonequilibrium regime surrounding the critical point.
We map the dissipative quench problem to a set of effectively independent dissipative Landau-Zener problems.
Using this mapping along with both analytic and numerical calculations allows us to find the scaling of the excess
defect density produced in the quench, and it suggests a generic picture for such dissipative quenches.
DOI: 10.1103/PhysRevB.92.014306

PACS number(s): 75.10.Jm, 03.65.Yz, 05.30.Rt, 64.60.Ht

I. INTRODUCTION

Quantum quenches involve the explicit time-dependent
tuning of a Hamiltonian, and they are among the most basic
and generic of phenomena in quantum many-body dynamics.
They have garnered considerable interest recently, particularly
with regard to tuning through a quantum critical point (QCP),
for which the quantum Kibble-Zurek (KZ) mechanism forms
a standard paradigm [1–6], analogous to the correspondingargument for classical, thermal quenches [7–10]. The basic
idea is that for gapped quantum phases separated by agapless critical point, as one approaches the critical point,
the system’s relaxation time (i.e., inverse gap, −1 ) diverges.
When this relaxation time becomes comparable to the timeassociated with the quench speed v [11], the system falls out of
equilibrium. This nonequilibrium regime dictates post-quench
behavior, such as deviations from the final ground state anddefect densities. One obtains the universal scaling behavior ofsuch quantities, ultimately governed by the proximity to the
QCP.
Here we turn our attention to the important yet relatively
unstudied effect of thermal fluctuations on the quantum
Kibble-Zurek scenario. We focus on the prototypical systemof a quench in a transverse-field Ising model (TFIM), wherea dissipative thermal bath produces noise in the transverse
field on each lattice site. Unlike previous studies of thisproblem [12,13], we allow for the noise produced by thethermal bath to be correlated over a finite length scale ξN .
Such correlations are physically well-motivated: in particular,
any noise in the global time-dependent field used to implement
the quench will naturally produce spatially correlated noise.
We show that a crucial role is played by the ratio of ξN to boththe length√scale of the dissipation-free Kibble Zurek problem
ξKZ ∝ 1/ v and to the thermal length scale ξT ∝ 1/kB T .
Patanè et al. [12,13] investigated the limit ξN  ξKZ ,ξT , where
spatial noise correlations are essentially irrelevant, and eachlattice site is effectively coupled to an independent bath. In
1098-0121/2015/92(1)/014306(8)contrast, we analyze the opposite but equally important limit
where the noise correlation length, while finite, is nonethelesslarger than both ξKZ and ξT ; this regime will genericallybe reached at intermediate temperatures and quench speeds
whenever the bath noise is correlated over distances greaterthan the lattice constant. Then, the relatively long-range bathnoise correlations lead to strikingly different behaviors.
In the next section, we discuss thermal defect generation
in a quench by employing a simple scaling ansatz before
we rigorously show our findings in the TFIM. For this,
we introduce the model of a TFIM with a global bath in
the following section. Section IV describes the mapping
to a dissipative Landau-Zener problem. Section V presents
analytical estimates as well as numerical results for the thermal
defect density after a quench. In Sec. VI, we discuss how
to extend our findings to cases with a finite bath correlation
length, and the final section contains our conclusions.

II. SCALING ANSATZ

Our main result, applicable to this correlated noise regime
and generic form of system-bath coupling, reflects the following general picture, as graphically depicted in Fig. 1. Thermal
fluctuations affect a quench only under certain conditions.
First, at a given time t, the bath temperature needs to be large
enough to be able to produce defects, implying that one must
be in the quantum critical regime kB T > (t) [14]. However,this is not enough: even if the temperature is large enough, thecoherent system dynamics needs sufficient time for the weakdissipation to also play a role. Thus, thermal defect generation
is suppressed in the “nonequilibrium” Kibble-Zurek regime
close to the critical point. Thermal fluctuations only give riseto additional defects during (at most) a limited portion of thequench protocol.
Focusing on temperatures small enough that only longwavelength excitations can be produced, the above arguments

014306-1©2015 American Physical Society

P. NALBACH, SMITHA VISHVESHWARA, AND AASHISH A. CLERKPHYSICAL REVIEW B 92, 014306 (2015)

couples to the system in the same way as this time-dependent
control field (i.e., it acts as a noisy transverse field). As aconsequence, dissipation only plays a role in the vicinity ofthe quantum critical point, but not far from it; this ensures that
defects produced during the quench protocol can be measured
once the quench is over. We also mainly focus on the limit
in which the bath noise is spatially uniform, like the averagetransverse field itself. The net Hamiltonian takes the form
Ĥ = ĤS + ĤB + ĤSB , whereĤS + ĤSB = −Jσ̂jx σ̂jx+1 − (h + X̂)
σ̂jz
(2)
j

FIG. 1. (Color online) Quench trajectories for two temperatures
T1 and T2 at a fixed quench rate. At T1 , far from the QCP, the spectral
gap  is too large for thermal excitations to be created. Thermal defectgeneration thus turns on only once the trajectory enters the quantum
critical regime kB T > . It is, however, suppressed again once thesystem enters the nonequilibrium regime near the QCP, where theintrinsic relaxational rate is slower than the quench rate. The thickyellow dashed line indicates the portion of the quench where thermal
defects are produced. For low temperature (T2 ), the entire quantum
critical regime is subsumed within the nonequilibrium regime near
the QCP, and hence there is no appreciable thermal defect generation.

lead to a general prediction that a weakly coupled bath only
gives rise to additional defect generation if the temperaturesatisfies kB T > kB Tmin . One can make a physically motivatedscaling ansatz, based on the Kibble-Zurek criterion for the
crossover scale into the nonequilibrium regime, generalizing
what has been done in the dissipation-free case [1,5,15].
Technical details are given in Appendix A. This leads to thefollowing prediction:
kB Tmin ∝ v νz/(1+νz) .

(1)

Here ν is the critical exponent describing the divergence ofthe system correlation length near the transition, and z is thecorresponding dynamical critical exponent. For the specificcase of the transverse-field Ising model (where ν = z = 1),3
the excess√ defect density is predicted to scale as (kB T ) /v aslong as v  kB T , but it is strongly suppressed for lower
temperatures.
To explicitly show that this scaling ansatz holds, werigorously treat the dissipative TFIM by using a mapping to an
ensemble of dissipative Landau-Zener (LZ) systems [16–19].
Such a mapping has been extremely powerful for the study of
nondissipative problems; its application to the dissipative caseis, however, nontrivial. We show that a mapping can still be
made, but only if one focuses on our limit of large ξN , and also
on a restricted class of observables. Using this mapping, we
calculate the defect production by both analytic and numericalapproaches, explicitly verifying Eq. (1).
III. MODEL

We consider a one-dimensional TFIM where a quench
occurs by tuning the transverse field as a function of time.
Similar to Refs. [12,13], we consider the case in which the bathjrepresents the TFIM
 system and system-bath coupling Hamiltonian, HB = ν ων b̂ν† b̂ν is the bath Hamiltonian, and X̂ =x,z†denote Pauli matrices for the spin
ν λν (b̂ν + b̂ ). Here, σ̂jat site j , J is the exchange coupling, and h is a Zeeman field in
the z direction. We have set  = 1. Without loss of generality,
we take J,h  0. The system is in an Ising ferromagneticallyordered phase when h < J , while for h > J the system is in aparamagnetic phase. The two phases are separated by a QCP
at hc = J. The bath is characterized by the spectral density
J (ω) = ν λ2ν δ(ω − ων ) = γ ωs ωc1−s e−ω/ωc , where γ is thedimensionless coupling strength and ωc is a cutoff frequency.
We focus on the standard case of an Ohmic bath, where s = 1.
Employing the standard Jordan-Wigner transformation [20], we reexpress
the Hamiltonian in terms of spinless
fermions, ĉi = ( j <i σ̂jz )σ̂i+ . Working in momentum spaceand setting the lattice constant a = 1, we have
 

ĉk
†
ĤS + ĤSB =[ĉk ĉ−k ](Hk,S + Hk,SB ) †
ĉ−k
0kπ
 (3)



k
ξ
X̂
0
,
, Hk,SB =with Hk,S = k∗k −ξk0 −X̂
where ξk = 2h − 2J cos(k) and √k = 2J sin(k). The energy
dispersion of ĤS is given by k = ξk2 +|k |2 . Critical exponents
ν = z = 1 can be extracted from critical gap behavior, namely
k=0 ∼ |h − hc | (near h = hc = J ) and k ∼ |k| (for k → 0
when h = hc ) [21].
IV. MAPPING TO THE LANDAU-ZENER PROBLEM

We briefly recall the standard treatment of quench dynamics
in the absence of dissipation [1]. Each term in ĤS [cf. Eq. (3)]describes an effective two-level system, where the two statescorrespond to having the orbitals (k,−k) either both empty or
both occupied. The most common quench protocol involves alinear ramp of the formh(t) = −vt.

(4)

For concreteness, we consider a quench that starts in aparamagnetic phase at t = −∞ and ends in the ferromagnetic
phase at t = 0. Thus, each term in ĤS describes a LandauZener problem [16–19]: a two-level system subject to a
constant x magnetic field (k ) and a linearly time-varying
z magnetic field (ξk = −2vt − 2J cos k), which runs throughthe avoided crossing at ξk = 0.
Evaluating the density of defects nKZ produced by thequench now amounts to calculating the final population of

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PHYSICAL REVIEW B 92, 014306 (2015)

quasiparticles at the end of the quench. For slow quenches(with velocity v  J 2 ), the dominant contribution to thispopulation comes from low-k fermion modes, where the gap
k at the avoided crossing is the smallest. The excitation
probability of such modes is well-described by the asymptotic,
infinite-time LZ probability [2] that the two-level system(TLS) transitions to the final excited state [16–19]: Pk ∼exp (−π 2k /v). Using this expression and integrating over all
momentummodes, one obtains the power-law form nKZ ∼√v, consistent with the Kibble-Zurek form [2].

2π J (2k )nB (2k )], multiplied by the effective time spentat the avoided crossing (tcross ∼ k /v).
Hence, as sketched in Fig. 1, the dissipation only influencesmodes where k is sufficiently large to yield a near-adiabatic
transition, but also small enough that the Bose-Einstein factor
in Eq. (5) is appreciable. Thus, the only modes affected bydissipation satisfy√(6)(KZ = v)  k  kB T .

Mapping to the dissipative Landau-Zener problem

Although a general dissipative TFIM quench cannot bemapped exactly to a set of independent dissipative LandauZener problems, this is possible for a restricted class ofobservables if one takes the large ξN limit, where thetranslational invariance of the fermionic system is maintained[cf. Eq. (3)]. For each k > 0, we have an effective two-levelsystem (TLS) whose detuning has a noisy part (due to the bath)and an average part that is linearly ramped in time. As the bathnoise X̂ couples to every k mode, it will correlate the different
effective TLS. These correlations are severely constrained byfermionic momentum conservation: in particular, the singleparticle fermion Green functions for each momentum k arecompletely decoupled from one another. Technical details are
given in Appendix B. As a result, the defect density can be
rigorously calculated by assuming that each effective TLS
is independent, making our mapping to the dissipative LZ
problem exact for the quantity of interest. Such dissipative LZ
problems with diagonal noise have been well-studied in theliterature [22–31].
V. RESULTS
A. Analytic estimates

The above mapping enables us to invoke results [22,23]
for the dissipative LZ problem to evaluate the contributionof thermal fluctuations toward defect production during thequench. For modes having 2k  v, the transition throughthe avoided crossing is not adiabatic, and there is a large
probability for the effective TLS to be excited even without
dissipation. For such modes, dissipation (to leading order) does
not yield any additional probability of excitation [22,23].
In contrast, for modes having 2k > v, the transition
through the avoided crossing is very nearly adiabatic; without
dissipation, they remain close to the ground state. For such“slow” modes, the system spends enough time near the avoided
crossing for dissipation to give rise to transitions to the excitedstate. This leads to an additional dissipation-induced excitation
probability δPk that for weak dissipation simply adds to the
LZ expression. The probability δPk was explicitly calculated
in Refs. [22,23]:
δPk =

2π k
J (2k ) nB [2k ,T ],v(5)where nB [E,T ] is a Bose-Einstein distribution evaluated atenergy E and at temperature T of the dissipative bath. This is
Fermi’s Golden Rule rate for excitation of the TLS at the
avoided crossing via absorption of a bath phonon [ exc =For sufficiently slow quench velocities, all the relevant modes
satisfying this condition correspond to small k, where k ∼J k. This justifies the general picture provided in Fig. 1. Outside
the quantum critical regime,  ≡ k=0 > T , and there are
no modes that satisfy Eq. (6); dissipation thus has no effecthere (e.g., the blue region in Fig. 1). In the nonequilibrium
regime (orange in Fig. 1), k=0 < KZ and v is too large fordissipation to affect any of the modes. Finally, in the portion of
the quantum-critical region that lies outside the nonequilibrium
regime, we have instead KZ < k=0 < kB T ; we thus havemodes satisfying Eq. (6), and dissipation can create excitations.
Here, we observe dynamics different from the KZ scenario.
Integrating δPk of Eq.
√ (5) over the range defined by Eq. (6),v, we find that the total dissipationand assuming kB T
induced excitation density obeys
nth ∼ γ kB T [(kB T )2 /v],

(7)

in agreement with the scaling ansatz of Eq. (1). Like thestandard KZ result, it is based on analyzing long-wavelength
modes, and hence it is restricted to energies J .
B. Numerics

To confirm the above picture, we perform a rigorous
numerical analysis of a TFIM quench within the dissipative LZ
framework. We use a weak system-bath coupling Markovian
approach [32], which has been shown to be reliable [33]in comparison to numerically exact approaches [27,28].
Technical details about the methods are given in Appendix C.
The quench is explicitly started at h(t0 ) = −8J and endedat h(0) = 0, implying v = −8J /t0 . We have ensured that
results do not depend on this initialization. Solving theadiabatic-Markovian master equation for each k, we obtain the
total probability P (k,v; γ ,T ) for a given effective TLS to end
in the excited state. The thermal contribution Pth (k,v; γ ,T )is determined by subtracting the dissipation-free probability
P (k,v; γ = 0,T ) from the total one. Integrating over all k, wefind the total excitation density produced during the quench,
 π
dk P (k,v; γ ,T ) ≡ nKZ (v) + nth (v; γ ,T ),ntot (v; γ ,T ) =0

where nKZ is the defect density without dissipation, and nth is
the additional one due to thermal dissipation.
In Fig. 2 we plot the total defect density produced by
the quench as a function of quench velocity v; different
curves correspond to different bath temperatures. The defectdensity for the dissipation-free case γ = 0 is√also plotted (redsolid line); this curve exhibits the standard v Kibble-Zurek
scaling. For finite γ and T , we observe with decreasing v an
increasing defect probability due to additional thermal defects.
Thus, at a fixed temperature a minimal total defect density is014306-3

P. NALBACH, SMITHA VISHVESHWARA, AND AASHISH A. CLERK

Markovian master equation is least reliable [33], and thus theobserved behavior in the low-temperature regime in Fig. 3
cannot be taken to be conclusive.

−6

γ = 1.3∗10
T = 0.21 J
T = 0.46 J
T = 0.69 J
T = 1.04 J-1

VI. FINITE BATH CORRELATION LENGTH

ntot

10

10

T = 1.5 J
T = 2.25 J
T = 3.38 J
T = 5.06 J

-2

γ=0
-4

-3

10

-2

10
2
v [J ]

10

FIG. 2. (Color online) Total excitation density ntot (v; γ ,T ) vsJ , and a system-bathquench speed v for various temperatures, ωccoupling γ = 1.3 × 10−6 . The red full line shows nKZ (v), i.e., theexcitation density for γ = 0.

observed for an optimal quench speed. The thermal defectdensity also increases with increasing temperature.
To focus on the bath contribution to the defect production
during the quench, in Fig. 3 we now plot results for the excess
thermal defect density as a function of scaled temperature,
and we observe data collapse.√The analytic estimate predictsa scaling nth ∼ (kB T )3 /v for v  kB T  J [see Eq. (7)],which is indeed observed (see Fig. 3). For smaller T , we
see that the thermal defect density is suppressed, also inaccordance with our prediction that thermal defect production
is suppressed in the “nonequilibrium” regime 1/(t) > |t|.
The inset of Fig. 3 plots the momentum-resolved thermal
defect density Pth (k,v; γ ,T ) exhibiting a clear peak. With
increasing temperature, the momentum√ of the peak maximumv, the main contribution
increases. For temperatures
T→√
is from modes at kv. Note that here the adiabatic-

10-1

nth / T [J ]

10

-5 2

v = 7.5*10 J-4 2
v = 1.9*10 J-3 2
v = 1.4*10 J2
v = 0.01 J-6 3
nth = 2*10 T /v

-2

10

10

-1

-3

-6

γ = 1.3*10

-4

10
10

0.01

k

0.1

1

-3 2

v = 1.4*10 J
Pth(k,v,γ,T)

-5

T/v
T/v
T/v

-6

1

10

= 100
= 20
=5

*10
*10

*10

1/2

T/v

4

3

2

PHYSICAL REVIEW B 92, 014306 (2015)

2
1.5
1
0.5
0

While the above analysis was for noise that is correlated
over the entire lattice, it remains valid for situations in which
the bath noise has a finite spatial correlation length ξN , aslong as this length is much longer than all other relevant
length scales. In particular, this requires that ξ√
N be larger
than both the Kibble-Zurek length ξKZ = J a/ v and thethermal length ξT = J a/kB T . It does not, however, requirethat ξN approach the size of the system. The fact that thesystem correlation length diverges at the QCP is not relevant;heuristically, the bath only plays a role outside the nonequilibrium regime, and thus it does not see the full divergence of
this length. A more rigorous justification for why our results
apply whenever ξN
ξKZ ,ξT (based on a Keldysh analysis)
including technical details is provided in Appendix D. In the
opposite limit of a very small correlation length ξN  ξT ,one approaches the limit of spatially uncorrelated dissipation,
as studied by Patanè et al. [12,13], with a very different
temperature dependence of the thermal defect density from
Eq. (7). Thus, the ratios ξN /ξKZ and ξN /ξT will in general playa crucial role in determining the influence of dissipation on aquench.
VII. CONCLUSIONS

We have studied the general problem of how thermal dissipation and noise give rise to defect generation in a quantum
quench beyond the amount predicted for zero dissipation by the
Kibble-Zurek scenario. We have explicitly analyzed quenches
in the TFIM coupled to a global thermal bath and shown3
that the excess√ thermal defect density scales as (kB T ) /v
as long as v  kB T  J , in clear contrast to previous
results obtained for spatially uncorrelated dissipation. These
first steps toward understanding the interplay between quench
and dissipation elicit a host of further studies, for instancethose regarding more complex systems, different crossover
regimes, general scaling laws, and the effect of an alternative
bath or coupling forms. They also serve as a starting point
for a rigorous scaling analysis that requires taking recent
renormalization-group analyses of nonequilibrium quantum
critical dynamics to a new regime that couples in the effect ofnoise. As with nondissipative quantum quenches, certain cold
atomic and magnetic systems would form the experimental
testbeds for these studies.

100

ACKNOWLEDGMENTS

FIG. 3. (Color online) Excess thermal defect density associatedwith the quench, as a function of √scaled temperature, for threedifferent quench velocities v. For v  T  J , the data follow
nth ∝ T 3 /v in line with the scaling form given in Eq. (7) (orange
solid line). Inset: plot of momentum-resolved thermal defect density
Pth (k,v; γ ,T ) for quench velocity v = 1.4 × 10−3 J2 and threetemperatures.

We thank Alessandro Silva and Anatoli Polovnikov foruseful conversations. P.N. acknowledges financial support by
the excellence cluster “The Hamburg Centre for Ultrafast
Imaging—Structure, Dynamics and Control of Matter at theAtomic Scale” and by the project “Nonequilibrium Quantum
Energy Transport through Nanostructures” (NA 394/2-1) of
the Deutsche Forschungsgemeinschaft. S.V. acknowledges the
National Science Foundation Grant No. DMR 0644022-CAR.

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QUANTUM KIBBLE-ZUREK PHYSICS IN THE PRESENCE . . .

PHYSICAL REVIEW B 92, 014306 (2015)

A.A.C. thanks NSERC. A.A.C. and S.V. also thank the Kavli
Institute for Theoretical Physics for their hospitality.

With regard to the regime of interest here, namely that ofthermal excitations, the two constraints mentioned above need
to be satisfied. First, as discussed previously in Refs. [12,13],the bath temperature must be greater than the gap, i.e., kB T >, yielding the scaling relationship kB T ∼ δ νz . Second, thequench must fall short of entering the nonequilibrium regime,or equivalently, t(δ) > (kB T )−1 . This condition, combinedwith the scaling relationship obtained from the first constraint
and with Eq. (A5), provides the following temperature lower
bound for a given quench rate:

APPENDIX A: SCALING ARGUMENTS

We generalize the scaling arguments presented in previous
works [1,5,15] with regard to the universal nonequilibrium
dynamics associated with Kibble-Zurek-type physics. We
focus on the influence of dissipation from a weakly coupled
thermal bath that produces spatially correlated noise. We
also focus on a system-bath coupling that is analogous totransverse-field noise in the transverse-field Ising model, i.e.,the system-bath coupling Hamiltonian commutes with thecoherent system Hamiltonian away from the critical point.
As the quench process evolves through the quantum criticalpoint, we expect excitations produced by coupling to thethermal bath, nth , to become important in the vicinity of
the critical point where the system’s energy gap  is small
compared to the bath temperature scale kB T . However, very
close to the critical point, the system’s relaxational time scale,τ , diverges and the quench process becomes too rapid for thesystem to form the coherences needed to allow bath-induced
transitions. These constraints provide the following bounds
for the regime in which thermal excitations are significant
and the following scaling forms for their behavior in terms ofcharacteristic energy/time scales.
Consider a quantum phase transition characterized by aparameter α such that the quantum critical point occurs at avalue αc [14]. Close to the critical point, the system’s typicalcorrelation length scale diverges asξ ∼ δ −ν ,(A1)where ν is the associated critical exponent and δ = |α − αc |measures the deviation from the critical point. The relaxationaltime diverges asτ ∼ −1 ∼ δ −νz ,(A2)where z is the dynamic critical exponent.
With regard to Kibble-Zurek scaling behavior [1,5,15],
consider a linear quench at a characteristic rate v −1 ,α(t) = αc + vt.

(A3)

A given deviation δ thus occurs at a time t(δ) = δ/v; |t(δ)|
represents the time remaining until the system reaches thecritical point. Now the quench enters the nonequilibrium
regime under the conditiont(δ) < τ,(A4)or, from Eq. (A2), δ/v < δ −νz . This crossover criterion
establishes a scaling relationship between the quench rate anddistance to criticality, namely
δ 1+νz ∼ v.

(A5)

From this relationship, one can derive the scaling of standard
Kibble-Zurek variables. For instance, the density of defects
produced in the nonequilibrium region scales as nD ∼ ξ −d ∼v νd/(1+νz) . In the case of the one-dimensional transverse Isingsystem, where ν = z =√d = 1, one obtains the well-known
scaling behavior nD ∼ v.

(kB T )(1+νz)/(νz) > v,

(A6)

as presented in Eq. (1) in our main text.
In Ref. [15], similar to equilibrium quantum critical scaling,
the effect of finite temperature has been discussed in thecontext of nonequilibrium quantum critical scaling. This isdone by introducing a dimensionless parameter that is naturally
defined by the scaling relationship between temperature andquench rate in Eq. (A6). It is interesting to note that the excess
thermal density that we predict in Eq. (8) of the main text,
given that it has dimensions of inverse volume, is consistent
with the scaling form hypothesized in Ref. [15] for genericsituations:nth ∼ (kB T )d/z F[(kB T )(1+νz) v −νz ],(A7)

where F is a scaling function. Specifically, in the onedimensional transverse Ising case, our arguments show that
thermal excitations are important in the regime √(kB T )2 > v
and that they respect the form nth ∼ kB T F(kB T / v).
We emphasize, however, that care needs to be taken
in applying the above scaling. The situation presented by
Ref. [15] presents a closed system having an initial temperature
kB T , and in general, unlike our case, thermal effects need
not give rise to separate additional contributions above thezero-temperature Kibble-Zurek contributions. We believe thatadherence to the expected form is tied to the weak nature of
the bath coupling, as well as the particular choice of an Ohmicbath spectral function. Considering the effects of more generalbath spectral functions and stronger couplings would make for
an interesting and challenging study.
APPENDIX B: MAPPING TO INDEPENDENT DISSIPATIVELANDAU-ZENER TRANSITIONSIn the limiting case in which each site of our TFIM
couples to the same dissipative bath (a “global” system-bathcoupling), we can map our dissipative quench problem onto
a set of independent dissipative Landau-Zener problems.
One might worry that this mapping is only approximate,as it ignores correlations between different fermionic modes
induced by the globally coupled bath. While such correlations
will play a role for some physical observables, they play
no role in determining single-particle properties, such as thequasiparticle occupancies upon which we focus. This is a directconsequence of the fact that the system-bath Hamiltonianof Eq. (3) in the main text conserves the momentum of thefermionic system.

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P. NALBACH, SMITHA VISHVESHWARA, AND AASHISH A. CLERK

More formally, the occupancy of a given quasiparticle modewith momentum k can be written†

Pk (t) ≡ ĉk (t)ĉk (t) ≡ −iG<k (t,t),(B1)

where we have introduced the standard lesser Keldysh Green’s
function associated with this mode [34]. If one treats the coupling to the bath as a perturbation of the coherent (dissipationfree) system, the lesser Green’s function appearing above iscompletely determined by the Keldysh self-energies kα (t,t  )associated with the system-bath interaction; here, the index
α can take the values R,A,K, corresponding to retarded,advanced, and Keldysh self-energies [34]. Note the self-energy
must be diagonal in momentum, as electronic momentum
is conserved. Consider an arbitrary self-energy diagram forkα (t,t  ). As fermion momentum is conserved at each systembath interaction vertex, all internal fermion propagators in thisdiagram involve the same momentum k. Heuristically, this
implies that at least for single-particle Green’s functions, agiven fermion mode with momentum k does not know aboutother modes having a different momentum k  = k.
It follows that we would obtain exactly the same diagrammatic expansion [and hence result for G<
k (t,t)] if we
had coupled each fermion mode to its own independentbath. Formally, this means modifying the system-bath (SB)
Hamiltonian in Eq. (3) of the main text as follows:
 


ĉk
0
X̂k
†
.
(B2)
ĤSB →[ĉk ĉ−k ]†0−X̂k ĉ−k0kπ

We now have an independent bath foreach k mode, witha corresponding noise operator X̂k = ν λk,ν (b̂k,ν + H.c.).
Each of these baths (labeled by k) has identical properties
to the bath appearing in our starting Hamiltonian. They all
have the same temperature T and identical spectral densities:
Jk (ω) = J (ω), where J (ω) is the spectral density of ouroriginal bath (noise operator X̂), as given after Eq. (2) in themain text.
Thus, for computing quasiparticle occupancies, we canexactly treat each fermion k mode as being effectively coupled
to its own independent bath. This then rigorously justifies our
mapping to an ensemble of uncoupled dissipative LandauZener problems.
APPENDIX C: ADIABATIC MARKOVIANMASTER EQUATION

Letting |j (t) (j = 1,2) denote the eigenstates of theinstantaneous coherent Hamiltonian Hk,S (t) √in Eq. (3) (in themain text) (with eigenenergies ±Ek (t) = ± [ξk (t)]2 +2k ), thematrix elements of the effective statistical operator ρ̂k (t) of
the Landau-Zener system after tracing out the bath degrees
of freedom are parametrized in terms of a 3-vector rk (t) as
j (t)|ρ̂k (t)|j  (t) = 12 (1 − rk · τ)jj  , where τ is the vector ofPauli matrices. Suppressing the k index for clarity, the adiabatic
Markovian master equation takes the form [33]⎛⎞−γ1 (t)0θ̇ (t)−γ2 (t) −2E(t)⎠r + γ1 (t)req , (C1)∂t r = ⎝ 0−θ̇ (t)2E(t)−γ2 (t)PHYSICAL REVIEW B 92, 014306 (2015)

where θ (t) = arctan[ξk (t)/k ] and req = ( tanh[βE(t)],0,0).
The terms proportional to θ̇ describe coherent nonadiabatic
evolution, while the time-dependent relaxation and dephasing
rates areγ1 (t) = cos2 [θ (t)]S̄[2E(t)],(C2)γ2 (t) = 12 γ1 (t) + sin2 [θ (t)]S̄[0],(C3)where S̄[ω] = 2π J [ω] coth βω/2 is the symmetrized spectral
density of the bath noise.
For a given quench protocol, i.e., ξk (t), we solve theadiabatic-Markovian master equation (C1) using a standard
fourth-order Runge-Kutta scheme [35]. Thus we get the
time-dependent statistical operator for each k and, in turn,the probability P (k,v; γ ,T ) for a given effective TLS to end
up in the excited state at the end of the quench. To obtainthe total excitation density produced during the quench, we
integrate over all k using a standard Romberg scheme [35].
APPENDIX D: FINITE BATH SPATIAL CORRELATIONSAs discussed in the main text, our results for a spatially
uniform dissipative bath remain valid in the case in which thebath noise has a finite spatial correlation length ξN , as longas this length is much larger than the Kibble-Zurek length
ξKZ ; one does not need ξN to approach the size of the system.
To make this precise, we generalize the system-bath couplingso that 
there is a distinct bath noise operator on each site,
ĤSB = j σ̂jz X̂j . As usual, we take the bath to be an infinite
collection of harmonic oscillators in thermal equilibrium, andwe take the X̂j to be linear in the bath creation and destruction
operators. We also take the bath to be in a translationallyinvariant state (unlike the model presented in Ref. [13]). We
consider a generic situation in which (like the main text) theon-site noise is still described by an Ohmic spectral density
J [ω], but where the noise correlation decays exponentiallywith distance,{X̂j [ω],X̂k [ω ]}ω
= J [ω] cothe−|j −k|/ξN . (D1)
2π δ(ω + ω )2kB T
The above form implies that we can express the Fouriertransformed bath noise operators as1  −iql†X̂l = qX̂q ≡ √eλν (b̂q,ν + b̂−q,ν ), (D2)N l
νwhere the b̂q,ν describe Einstein phonons with
 energy ων [q] =ων . We use a normalization such as (1/N ) q |q |2 = 1. The
spectral function J [ω] associated with the noise on any givensite is then identical to that used in the main text.
We next specialize to exponentially decaying spatial correlations, with a correlation length much smaller than the systemsize. This implies|q |2 =2ξN /a.
1 + q 2 ξN2(D3)

Unlike the global-bath model in the main text, the finite
correlation length here means that the bath can exchange
nonzero momentum with the system. We follow Ref. [13],and we use the weakness of the system-bath coupling to014306-6

QUANTUM KIBBLE-ZUREK PHYSICS IN THE PRESENCE . . .

PHYSICAL REVIEW B 92, 014306 (2015)

treat the system perturbatively, using a self-consistent Born
approximation for the Keldysh self-energy of the fermion
Green functions. Within this approximation, the Keldysh
self-energy for a fermion with momentum k is given selfconsistently by1 |q |2 Gk+q (t2 ,σ2 ; ,t1 ,σ1 )k (t2 ,σ2 ; ,t1 ,σ1 ) =N q

ξT . In
Consider next the opposite regime, in which ξN
this case, the structure factor q will suppress the contributionof large momentum transfers in the self-energy, as opposedto the bath temperature: the largest contributing |q| will be∼1/ξN . If in addition we have ξN
ξKZ , then we can also
ignore the q dependence of the fermion propagator Gk+q
in Eq. (D4). To understand this point, note that without
dissipation, quasiparticle modes with k  1/ξKZ will evolve
diabatically during the quench (and become excited), whilemodes with k
1/ξKZ will evolve adiabatically (and hence
remain unpopulated). Correspondingly, for small k, the quasiparticle modes k and k + q behave almost identically when|q|  1/ξKZ , implying Gk ∼ Gk+q .
Thus, for ξN
ξKZ ,ξT , the q summation in the self-energyof Eq. (D4) can be estimated as× Dq0 (t2 ,σ2 ; ,t1 ,σ1 ),(D4)

where σj = ± denotes the forward and backward Keldysh
contours, Gk [t,σ ; t  σ  ] is a dressed fermion Keldysh Greenfunction, and Dq0 [t,σ ; t  σ  ] is the unperturbed bosonic (equilibrium) Keldysh Green function for the bath operator X̂q /q .
In our model (where we assume Einstein phonons, corresponding to a frequency-independent ξN ), this Green function is
independent of q.
We are now in a position to make estimates concerning
the role of ξN , based on the behavior of the imaginary partof the self-energies (which control bath-induced transitions).
We will focus on transitions that are thermally enhanced, i.e.,
those that involve the absorption or emission of bath phonons
having ω < kB T . For a bath-induced scattering event taking aquasiparticle from momentum k to k + q, energy conservationand the fermion dispersion relation will determine the energyof the bath phonon involved. Assuming kB T  J as always,
this then naturally leads to the thermal length ξT ≡ aJ /kB T :the only transitions that are thermally enhanced involve
momentum transfers with |q|  1/ξT .
The simplest regime is the one in which ξN  ξT . In
this case, the only thermally enhanced transitions have q 
1/ξT  1/ξN , and the q dependence of the structure factor q
plays no role: we can safely replace q by q=0 . In this case,the q integral in the self-energy of Eq. (D4) can be estimatedas 1/ξT π/aaa|0 |2|q |2 Gk+q
Gk+q (D5)2π −π/a
2π
−1/ξT

a
2π

 π/a

|q | Gk+q
2

−π/a

a
Gk
2π
Gk

a
2π

 π/a

|q |2(D7)|q |2

Gk . (D8)

−π/a

 ∞

−∞

This is identical to having taken the global-coupling ξN → ∞limit from the outset, i.e., having used
q =

2π
δ(q).
a

(D9)

In this limit in which q can be treated as a constant, werecover the local-bath model studied in Ref. [13], where there
are no spatial correlations between bath noise operators X̂j .
This mapping to uncorrelated noise is√valid irrespective of thevalue of ξN /ξKZ , where ξKZ = aJ / v is the Kibble-Zurek
length introduced in the main text. Note the explicit factor of
T that emerges from the momentum summation.

Thus, when ξN is the largest length scale in the problem (i.e.,
ξN
ξKZ ,ξT ), the system does not know about the finite bathspatial correlation length ξN , and one gets the same results as
for a model in which ξN → ∞. Note that in this large ξN limit,
the summation over q gives a temperature-independent result,in contrast to the small ξN estimate in Eq. (D6). This difference
is at the heart of why our global-coupling result for the thermal
defect density scales as a lower power of temperature thanthe corresponding result found in Refs. [12,13] for a locallycoupled bath.
Finally, there is the remaining case ξT  ξN  ξKZ . In
this case, the bath correlation length cuts off large momentum
transfers (as opposed to temperature). However, this cutoff is
not large enough to prevent coupling between very different
fermionic modes, i.e., adiabatic and nonadiabatic modes. In
this crossover regime, neither the local model studied inRef. [12] nor the global bath model studied in the main text
are appropriate.

[1] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore,
Rev. Mod. Phys. 83, 863 (2011).
[2] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
[3] A. Polkovnikov, Phys. Rev. B 72, 161201 (2005).
[4] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95,
105701 (2005).
[5] J. Dziarmaga, Adv. Phys. 59, 1063 (2010).
[6] A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T.-F.
Rosenbaum, and D. Sen, arXiv:1012.0653.
[7] T. W. B. Kibble, J. Phys. A 9, 1387 (1976).

[8] T. W. B. Kibble, Phys. Rep. 67, 183 (1980).
[9] W. H. Zurek, Nature (London) 317, 505 (1985).
[10] W. H. Zurek, Phys. Rep. 276, 177 (1996).
[11] Note that v is the rate of change of an energy that is externallyvaried as part of the quench protocol [in our case, the transverse
field h(t) ∝ vt].
[12] D. Patanè, A. Silva, L. Amico, R. Fazio, and G. E. Santoro,
Phys. Rev. Lett. 101, 175701 (2008).
[13] D. Patanè, L. Amico, A. Silva, R. Fazio, and G. E. Santoro,
Phys. Rev. B 80, 024302 (2009).

∝ (ξN /a)T .

(D6)014306-7

P. NALBACH, SMITHA VISHVESHWARA, AND AASHISH A. CLERK[14] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge
University Press, Cambridge, 2011).
[15] A. Chandran, A. Erez, S. S. Gubser, and S. L. Sondhi, Phys.
Rev. B 86, 064304 (2012).
[16] L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).
[17] C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).
[18] E. C. G. Stueckelberg, Helv. Phys. Acta 5, 369 (1932).
[19] E. Majorana, II Nuovo Cimento 9, 43 (1932).
[20] P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).
[21] We focus on the weak dissipation limit and the leading-ordercontribution of the bath to the defect density; any bath-induced
modification of the critical behavior will be a higher-order effect.
[22] P. Ao and J. Rammer, Phys. Rev. Lett. 62, 3004 (1989).
[23] P. Ao and J. Rammer, Phys. Rev. B 43, 5397 (1991).
[24] E. Shimshoni and Y. Gefen, Ann. Phys. (NY) 210, 16 (1991).
[25] Y. Kayanuma and H. Nakayama, Phys. Rev. B 57, 13099 (1998).
[26] M. Wubs, K. Saito, S. Kohler, P. Hänggi, and Y. Kayanuma,
Phys. Rev. Lett. 97, 200404 (2006).

PHYSICAL REVIEW B 92, 014306 (2015)[27] P. Nalbach and M. Thorwart, Phys. Rev. Lett. 103, 220401(2009).
[28] P. Nalbach and M. Thorwart, Chem. Phys. 375, 234(2010).
[29] P. P. Orth, A. Imambekov, and K. Le Hur, Phys. Rev. A 82,
032118 (2010).
[30] P. P. Orth, A. Imambekov, and K. Le Hur, Phys. Rev. B 87,
014305 (2013).
[31] L. Henriet and K. Le Hur, arXiv:1502.06863.
[32] P. Nalbach, J. Knörzer, and S. Ludwig, Phys. Rev. B 87, 165425(2013).
[33] P. Nalbach, Phys. Rev. A 90, 042112 (2014).
[34] A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011).
[35] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B.
P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge,2007).

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