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

### Submission summary

 As Contributors: Alessio Calzona Arxiv Link: http://arxiv.org/abs/1711.02967v3 Date accepted: 2018-04-19 Date submitted: 2018-04-10 Submitted by: Calzona, Alessio Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Condensed Matter Physics - Theory

### Abstract

It has been shown that a quantum quench of interactions in a one-dimensional fermion system at zero temperature induces a universal power law $\propto t^{-2}$ in its long-time dynamics. In this paper we demonstrate that this behaviour is robust even in the presence of thermal effects. The system is initially prepared in a thermal state, then at a given time the bath is disconnected and the interaction strength is suddenly quenched. The corresponding effects on the long times dynamics of the non-equilibrium fermionic spectral function are considered. We show that the non-universal power laws, present at zero temperature, acquire an exponential decay due to thermal effects and are washed out at long times, while the universal behaviour $\propto t^{-2}$ is always present. To verify our findings, we argue that these features are also visible in transport properties at finite temperature. The long-time dynamics of the current injected from a biased probe exhibits the same universal power law relaxation, in sharp contrast with the non-quenched case which features a fast exponential decay of the current towards its steady value, and thus represents a fingerprint of quench-induced dynamics. Finally, we show that a proper tuning of the probe temperature, compared to that of the one-dimensional channel, can enhance the visibility of the universal power-law behaviour.

###### Current status:

We would like to thank both Referees for their overall positive assessment of our work.

In this new version, following the referee suggestions, we have made small amendments and corrected some typos, as detailed below. In the following, we reply in details to the criticisms raised in "Anonymous Report 2 on 2018-3-28".
With best regards,

Alessio Calzona on behalf of all the Authors.

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Anonymous Report 2 on 2018-3-28

R: "To enhance the relevance of the work which was questioned by both referees they added a chapter on the absolute time-dependence of the local "spectral function" (Fourier transform of the local Green function with respect to the relative time). This indeed provides further insights. However, one might argue that the chapter on the transport setup (see 2.) is now superfluous. With respect to this new chapter I am puzzled by the first line of Eq. (35). While on the left hand side the limit t -> oo was taken, t appears explicitly on the right hand side!?"

A: We thank the Referee for pointing out this typo. Indeed, Eq. (35) describes the long time limit behavior of the non-equilibrium spectral function. In the revised manuscript we have added the limit t->\infty also in the right hand side of Eq. (35). Concerning the chapter on the transport setup, we strongly believe that it contains important information and results on its own. Indeed, although the universal power-law decay emerges already in the spectral properties of the system, transport properties represent a convenient way to actually probe the universal behavior.

R: "2. The authors made an effort to now explain more clearly the transport setup. Both referees were confused by the original description. However, I am still not fully convinced that the field theory the authors study can be realized in any microscopic model (not to speak of experiments). In PRL 105, 266404 mentioned in the authors reply it is emphasized that "...we have shown that the picture of tunneling into a LL is qualitatively modified when the tunneling amplitude is not treated as infinitesimally small. The conventional FP [fixed point] has a finite basin of attraction only in the model of the point tunnel contact, but taking a finite size of the contact (or any perturbation induced by the contact in the wire) into account makes it unstable." If the "conventional FP" is generically unstable (as stated in this paper) the probe will generically be invasive and the physics on asymptotic time scales will not be the one of the unperturbed system (the probe does not simply probe the system). In any case the authors revisions are in this respect inappropriate:
"It is worth to underline that the probe is a tool to inspect the intrinsic properties of the fermionic channel out-of-equilibrium and, thus, it is supposed to be as non-invasive as possible. A perturbative approach in the weak tunnel coupling is therefore fully justified in evaluating the related transport properties."
This paragraph does not contain any arguments that a proper "non-invasive" limit exists at all. The latter is the question of utter relevance (see my quote from the paper brought up by the authors). At the minimum the authors must properly revise the wording and backup their view by the proper references. "

A: We do certainly agree with the Referee when he quotes the paper by Aristov et al. [PRL 105, 266404 (2010)]. However, we would like to emphasize that the fact that the conventional fixes point (FP) is generically unstable is not in contradiction with the existence of a "non-invasive probe limit". Indeed, in real experiments the presence of both a finite temperature and a finite bias fix an energy scale at which the renormalization group (RG) flow is cut off. Therefore, the (stable) FP associated with the breakup of the wire into two independent semi-infinite wires is, in general, never reached by the RG flow. That is what actually makes it possible to perform experiments such as the one in Ref. 62 [Nano Letters 25, 3684 (2015)], where an STM tip is used to probe a quantum wire without breaking it into two uncoupled pieces. Moreover, as discussed in PRB 91, 235126 (2015), we know that the quench introduces a further energy scale which stops the flow to the breakup FP. All this means that a set of parameters (temperature, bias, tunnel-coupling, quench) in which the presence of the probe is non-invasive must exist. To clarify this point we have added a note in our manuscript describing in more details what we mean by "non-invasive" probe.

### List of changes

- we have added $lim_{t\to\infty}$ in the right hand side of Eq. (35)
- we have added a note at the beginning of Section 5 which better clarifies the concept of "non-invasive" probe.
- we have added Ref. 42, 60 and 62.

### Submission & Refereeing History

Resubmission 1711.02967v3 (10 April 2018)
Resubmission 1711.02967v2 (9 February 2018)
Submission 1711.02967v1 (9 November 2017)