As Contributors: | Alessio Calzona |

Arxiv Link: | http://arxiv.org/abs/1711.02967v1 |

Date submitted: | 2017-11-09 |

Submitted by: | Calzona, Alessio |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Condensed Matter Physics - Theory |

We investigate the influence of thermal effects on the relaxation dynamics of a one-dimensional quantum system of interacting fermions subject to a sudden quench of the interaction strength. It has been shown that quantum quench in a one-dimensional interacting system induces entanglement between counter-propagating excitations, whose signature is reflected in finite two-point bosonic cross-correlators. At zero temperature, their relaxation dynamics is governed by a universal power law $\propto t^{-2}$, whose behavior can be detected in transport properties. Here, we consider the system initially prepared in a thermal state and we demonstrate that these quench-induced features are stable and robust against thermal effects. Remarkably, we argue that the long-time dynamics of the current injected from a biased probe still exhibits a universal power law relaxation $\propto t^{-2}$, even at finite temperature. This result is in sharp contrast with the non-quenched case, for which the current features a fast exponential relaxation towards its steady value, and thus represents a fingerprint of quench-induced dynamics.

Has been resubmitted

Resubmission 1711.02967v2 (9 February 2018)

Submission 1711.02967v1 (9 November 2017)

Timely subject.

Just a small progress with respect to what have been already presented in Ref.[41] by the authors.

In this paper the authors analyse the relaxation dynamics of a Tomonaga-Luttinger model

after preparing the system in a thermal initial state and suddenly quenching the interaction strength.

In the filed theory, after bosonization, and provided a careful regularisation of the ultraviolet divergencies

(which is encoded into the short-length cutoff “a”), all the calculations can be easily push forward

thanks to the quadratic nature of the bosonic Hamiltonians (cf. eq. (6) and (7)). Indeed, the quench

simply reduces to a sudden change of the Luttinger Liquid parameter “K”.

The authors focus on the diagonal part of the dynamical green function

(the two-point Fermionic correlation function at different times and equal positions).

The main result in this regard is the large-time asymptotic decay (t^-2) which confirms

what has been already found in Ref. [41] for a zero temperature initial state, providing evidence that

temperature does not modify the leading time-dependent contribution into the approaching the stationary state.

The authors stress the fact that this contribution arises from the bosonic cross correlation term which

indeed is related to the quench protocol. Honestly, I’m not much surprised by the fact that the

cross correlation survives even for a thermal initial state, since it is merely a consequence of the

quench protocol.

Regarding this section (namely Sec. 3), I think it’s clearly written, nevertheless,

I was struggling by figuring out how the (\tau/2t)^2 behaviour in Eq.(24)

for \tau << t << T^-1 comes from Eq. (22). Moreover, I would like to draw the attention of

the authors to a strongly related result about the low-energy description of an interaction quench

in the XXZ spin chain (Phys. Rev. B 92, 125131 (2015)).

Let me now comment about Sec. 4. Here the authors propose to investigate the

relaxation dynamics by looking at the transport properties arising by injecting fermions into the system.

They thus engineer a noninteracting probe locally coupled (for t>0 at x=x0) to the original Hamiltonian.

They therefore analyse the total particle current. This calculation

having been done at the first order in perturbation theory with respect to

the probe-system coupling strength. The authors state that the

fermionic field of the probe “chi(x)” is kept at fixed temperature “T”. Now I’m a bit confused:

(1) is T the same temperature at which the original system has been prepared?

What about different temperatures?

(2) When the authors claim that the probe is at thermal equilibrium, what do they exactly mean?

In other words, I suppose the probe field is a new dynamical variable of the new setup,

which evolves according to the new post-quench Hamiltonian. Is this the case?

Otherwise, if the probe is really kept at fixed temperature, then in the new setup,

the system is no longer a closed system. Therefore, although there could be an intermediate regime

for which the system relaxes toward a generalised thermal ensemble, at very large time, due to

the influence of the external bath, I expect the system thermalising. Maybe thermalisation occurring

starting from x_0, with a sort of light-cone effect. Can the authors be more clear about this.

In particular, I’m really curious about the effect of the new setup regarding the “local” quench with

the probe. Indeed, as far as the global quench is joined with a local quench, I expect that, on top of the

homogeneous dynamics induced by the global quench, there should be a sort of spreading of particles density injected in x_0.

This leading to two different stationary descriptions, inside and outside the light cone.

To conclude, even though I appreciated this work, I can support publication only after

Sec. 4 has been largely rewritten in order to address all the points I mentioned before

so as to clarify the setup.

see report.

see report

see report

The questions of if and how observables and correlation functions

of closed one-dimensional quantum many-body systems relax towards

a steady state value after an abrupt change of the amplitude of the

two-particle interaction are currently heavily addressed in model

studies. It is furthermore of interest to clarify if the long-time

asymptotic expectation values, if reached, can be computed using

a properly chose ensemble.

The authors of the present manuscript consider the Tomonaga-Luttinger

(TL) model. The field-theoretical variant is studied, which means that

the Hamiltonian Eqs. (1) and (2) is ultraviolet divergent as written.

To obtain finite results for correlation functions an ad hoc

regularization of integrals must be introduced "by hand". This

procedure should be considered as part of the model, is, however,

left implicit. This is often done when using phenomenological

bosonization (as opposed to constructive bosonization).

The authors first compute the time evolution of the single-particle

lesser Green function with respect to the TL Hamiltonian with a

final interaction strength. The starting state is the thermal

equilibrium one (initial temperature T>0) with respect to an

initial interaction. Only the case with equal positions of the two

fermionic fields entering the Green function is considered. As

their main result the authors find that the correlation between

the left and right moving chiral bosonic fields decays as t^{-2}

with the absolute time t after the interaction quench. This cross

correlator enters the expression for the Green function which

thus features a term showing the same type of long-time decay.

Similar calculations were performed in many earlier publications,

e.g. in Ref. [41] by (a subclass of) the present authors themselves.

It is e.g. well established that in the long-time limit the

expectation values of local observables and correlation functions

of the TL model can be understood in terms of a generalized

Gibbs ensemble (GGE). I am in this respect puzzled by the authors

statement that the GGE of the TL model is "...characterized by an

infinite number of local conserved quantities..." (see the second

sentence of the third paragraph on page 2). The conserved mode

occupancies naturally appearing in the GGE are spatially non-local.

Although the GGE description might not be unique I am not aware that

for the TL model a GGE build out of spatially local conserved

quantities was constructed.

In Ref. [41] the same model and quench protocol was discussed but

for initial temperature T=0 leading to the same result for the cross

correlator decay and thus the Green function as found in the present

manuscript for T>0. As discussed in Ref. [41] in the non-equilibrium

spectral function, a reasonable measurable quantity derived from the

Green function by Fourier transform with respect to the relative

time of the two fields, this "universal" decay is masked by other

terms. These show typical Luttinger liquid power-law decay in t

with interaction dependent exponents which turn out to be generically

larger than -2. Surprisingly, this problem is not even mentioned in

the present manuscript. However, up to this point, that is Section 3,

the setup is transparent, the calculations are simple and

conceptually straightforward.

To construct an observable which reveals the above "universal" t^{-2}

decay the authors suggest a modified setup. In this the one-dimensional

(1D) interacting system is at the quench time t=0 locally tunnel coupled

to a (chiral 1D) probe (reservoir) system. The total current injected

into the system is computed using lowest-order perturbation theory in

the tunnel coupling; the expression Eq. (30) for the current then

involves the lesser Green function of the isolated 1D system. Besides

this Eq. (30) contains the greater Green function of the isolated probe.

In Eq. (31) the authors give an analytic expression for this. I am puzzled

that via \omega_f and K_f this (non-interacting) Green function contains

information about the interaction strength in the 1D system? In fact,

in Eq. (A5) of Ref. [41], a publication which already contains the idea

of the modified setup, the authors present an expression for the greater

Green function of the isolated probe (in this case for T=0) which is

independent of the interaction in the system. To me this appears to be

more reasonable. Can the authors comment on this?

After modifying the setup by including the probe a conceptual

difficulty arises. With the coupling of the 1D interacting system

to the infinite probe reservoir held in thermal equilibrium the

authors no longer consider an isolated quantum system but rather

an open one. One would thus expect that the system asymptotically

reaches the equilibrium thermal state imprinted by the reservoir

and not the "closed system" state for which local observables can be

computed employing the non-equilibrium GGE. One would furthermore

expect that the approach towards this equilibrium state is dominated

by an exponential time dependence with rates set by the reservoir-system

coupling (and modified by temperature). In that sense the system

dynamics is heavily affected by the reservoir. However, this change of

the entire dynamics is neither mentioned in the present manuscript

(or Ref. [41] for that matter) nor does it seem to be producable by

the authors perturbative approach to the tunnel coupling. I thus

suspect that what the authors hope for is the following: The time

scales are sufficiently separated such that first the GGE "closed system"

state develops (with the t^{-2} behavior of the cross correlator as

computed for the closed system) while the coupling to the reservoir

affects the dynamics only at a well separated later stage. If my

suspicion is correct the authors must provide arguments that hoping

for this type of time scale separation is reasonable in realistic

setups. On what time scales can one expect to detect the behavior

discussed in Ref. [41] and the present manuscript? Is this a reasonable

scale in the light of cold Fermi gas experiments? Can one perform

improved calculations which show that such simple estimates are

reasonable? If I am mistaken the authors must provide an alternative

way how to circumvent this conceptual problem of using closed system

results in an open system setup. In any case I am very much surprised

that the authors do not explicitly mention this type of conceptual

difficulty.

The coupling to the probe in addition induces a local inhomogeneity

to the Luttinger liquid which might affect the dynamics. It is well

established that local inhomogeneities strongly change the equilibrium

low-energy physics of Luttinger liquids. Can the authors exclude that

this is an issue in the non-equilibrium dynamics of the suggested

setup as well? Again the computational tool, namely perturbation theory

in the system-reservoir coupling might be insufficient to capture

and/or detect the proper impurity physics. The quench studied is not

only one of the global interaction but at the same time a local

single-particle parameter is changed (tunneling). Quenches of local

parameters in the TL model (and related lattice models) were studied

earlier. These studies might provide guidance for what to expect in

the present case.

Even if one ignores the above issues for the moment one might be

tempted to conclude that the progress presented in the present

manuscript (t^{-2} decay of the cross correlator for T>0) is rather

small as compared to what (a subclass of) the authors already

reported on in Ref. [41] (t^{-2} decay of the cross correlator

for T=0). Can the authors make a stronger point why the extension

of the T=0 result justifies another publication?

After the authors have properly responded to all the issues raised

above I am very much willing to reconsider my current decision to

not recommend publication.

see report