As Contributors: | Márton Kormos |

Arxiv Link: | http://arxiv.org/abs/1704.03744v2 |

Date submitted: | 2017-08-02 |

Submitted by: | Kormos, Márton |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Quantum Physics |

We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan-Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we compute analytically the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.

Has been resubmitted

Dear Editor,

I submit the revised version of the manuscript, following the suggestions of the referees who found my work "interesting" and of high significance. All the referees made valid and very useful comments and asked relevant questions. I believe I managed to answer all the questions and I have implemented the requested changes. In particular, I extended the discussions of the results at several places. Thanks to the remarks of the referees the paper has definitely improved.

As all three referees recommended the publication of the manuscript after minor revision, I hope that the new version meets all the criteria for publication in SciPost.

Yours sincerely,

Marton Kormos

List of changes (typos or minor changes are not included):

Changed the abstract, introduction, and conclusion. References to main results (equations and figures) included in the Conclusions to improve readability.

Added refs. [42,47,48,68,69,70,72,75].

Connection made with Ref. [42] in he Introduction and Sec. 4 (in the first paragraph, below Eq. (37) and below Eq. (39)).

Remark added at the end of Sec. 5.1. about the leading corrections at finite $x/t$.

Remark added at the end of Sec. 5.1. and in the Conclusions about the possible implications of the lattice effect showing up in the leading corrections to the NESS.

Paragraph added below Eq. (55) about the universal nature of the Airy kernel.

Paragraph added below Eq. (56) discussing the correlations that are not described by the Airy kernel near the front (e.g. the fermion density in the ferromagnetic phase).

Changed the derivation of the front structure in Sec 5.2 to correctly take into account the finite asymptotic value of correlations outside the light cone.

Added subsection 5.2.2. on the edge behavior in the critical case.

Added footnote 7 making it clear that the boundary edge mode was included in the numerical calculation for $h<1$.

Changed Fig. 1a and 2b to show the critical case.

The author addressed the majority of the points I raised and substantially improved the paper. Before publication, however, I think that the author should mention Ref.[75] when discussing the critical case in the newly introduced Section 5.2.2 and in all the passages of Sec. 6 and 8 where he mentions the critical case. Indeed, it is true that the first version of the current paper appeared at the same time as Ref. [75] but in that version there was no mention to the critical case, which was instead thoroughly studied in Ref. [75].

I also suggest to address the following additional points.

1) The condition for the appearance of bound states mentioned after eq 11 is still wrong. As I mentioned in my previous report, the correct one is $h<L/(L+1)$. Regarding the effects of boundary bound states on physical observables, my opinion is that the contribution is negligile because of the particular form (eq. 25) of the initial states considered in this work.

2) In the revised Section 5.2 (and in his response) the author says that the correlator $<A_nB_n>$ is related to the density of Majorana fermions. What is the density of Majornana fermions? I think that the author meant the density of Jordan-Wigner fermions (those described by the operators $c_n$ (cf. eq 2)).

3) At the end of the newly introduced Section 5.2.2 the author writes "... the scaling is $\sim t^{-2/3}$ instead of $\sim t^{-1/3}$". I think that the word scaling here is misleading, the width of the region is still scaling as $t^{1/3}$. I suggest to write something on the lines of "decays with a different power of t".

4) At the beginning of the Conclusion when describing the initial states considered I suggest to mention the condition in eq. 23, i.e. the zero anomalous correlation request. This would also make clearer the final part of the Conclusion when the author discusses the possible generalizations.

- validity: high
- significance: high
- originality: good
- clarity: high
- formatting: excellent
- grammar: good

The author has carefully considered all the remarks raised by the referees and revised the manuscript accordingly.

- validity: -
- significance: -
- originality: -
- clarity: -
- formatting: -
- grammar: -

## Author Márton Kormos on 2017-08-23

(in reply to Report 2 on 2017-08-19)I thank the Referee for his/her second report. I am happy that in his/her opinion I have "substantially improved the paper".

I considered all the comments and requests of the Referee and modified the manuscript accordingly. In particular, I cite Ref. [75] in Sec. 5.2.2, at the end of Sec. 6.2 and in Sec. 8.

1) The condition for the appearance of bound states mentioned after eq 11 is still wrong. As I mentioned in my previous report, the correct one is $h<L/(L+1)$. Regarding the effects of boundary bound states on physical observables, my opinion is that the contribution is negligile because of the particular form (eq. 25) of the initial states considered in this work.

I agree with the Referee. I have corrected the condition and added a footnote on page 6: "The edge modes can be more important for other initial states."

2) In the revised Section 5.2 (and in his response) the author says that the correlator

$\langle A_nB_n\rangle$ is related to the density of Majorana fermions. What is the density of Majornana fermions? I think that the author meant the density of Jordan-Wigner fermions (those described by the operators $c_n$ (cf. eq 2)).

Indeed, this is what I meant. I corrected the sentence in Sec. 5.2.

3) At the end of the newly introduced Section 5.2.2 the author writes "... the scaling is

$\sim t^{−2/3}$ instead of $\sim t^{−1/3}$". I think that the word scaling here is misleading, the width of the region is still scaling as $\sim t^{1/3}$. I suggest to write something on the lines of "decays with a different power of t".

I have changed the quoted sentence.

4) At the beginning of the Conclusion when describing the initial states considered I suggest to mention the condition in eq. 23, i.e. the zero anomalous correlation request. This would also make clearer the final part of the Conclusion when the author discusses the possible generalizations.

I thank the Referee for this suggestion, I have added a sentence about it at the beginning of the Conclusion.