Universal scaling of quench-induced correlations in a one-dimensional channel at finite temperature

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.

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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.

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