## SciPost Submission Page

# Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone

### by Nils O. Abeling, Lorenzo Cevolani, Stefan Kehrein

#### - Published as SciPost Phys. 5, 052 (2018)

### Submission summary

As Contributors: | Lorenzo Cevolani |

Arxiv Link: | https://arxiv.org/abs/1707.02328v3 |

Date accepted: | 2018-11-14 |

Date submitted: | 2018-08-21 |

Submitted by: | Cevolani, Lorenzo |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Quantum Physics |

### Abstract

In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the time evolution of the correlation function of two local disjoint observables if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.

###### Current status:

### Ontology / Topics

See full Ontology or Topics database.### Author comments upon resubmission

1) We specified what assumptions are made such that the bound is optimal and elaborate the derivation appendix A.

2) We added a comparison of the bound in chapter 2.

3) Changed the reference to Ref. 56. Removed Ref. 23..

- Fixed the citations and their titles. Fixed the equations.

- Fixed typos

- Removed the references 51-56 from the bibliography

Response to referee 2:

1) Page 1: Removed the "anyway".

2) Page 1: Changed all citations to look properly.

3) Page 1: Changed to "allowed".

4) Page 2: Added a hint.

5) Page 5: We provided more details about the choice of the optima l_A and l_B parameters in the Appendix A. The main point is that in some limits the exponentially decaying function of the Lieb-Robinson bound can be ignored after a constraining the optimization space of the previous variables. In this way it is possible to find an explicit solution of the minimisation problem.

6) Page 5: Corrected.

7) Page 6: Extended the proof and moved it to the appendix.

8) Page 8: Corrected.

10) Page 10: Corrected.

11) Page 10: All wrong references have been fixed.

12) Page 11: All equations that overlapped into the margin have been rearranged.

13) Page 12: Fixed the duplicate symbols. The estimation has been made more precise and moved to the appendix. All necessary assumptions have been added to the text.

14) Page 12: Fixed.

15) Page 13: Changed to "expectation value".

16) Page 13: Removed the misleading equation and added an explanation in

the text.

17) Page 14: The equation describes the function that the approximation in the deep space-like region converges to. In that sense, it can be rather understood as a definition of the space-like behavior In the deep

space-like regime.

18) Page 14: Fixed.

19) Figures 2,3,4: Added y-axis label.

20) Figure 3: Added description of blue dashed line.

21) Page 19: Fixed.

22) Page 20: Added description of the formula.

### List of changes

1) In Appendix A we present in more detail the derivation of our bound for non-equal time correlation functions.

2) In Appendix B we extend the result of Bravy et al (Phys. Rev. Lett. 97, 050401, 2006) to non equal time correlation functions.

3) We fixed typos in the manuscript

4) We fixed the bibliography

### Submission & Refereeing History

- Report 2 submitted on 2018-11-08 06:57 by
*Anonymous* - Report 1 submitted on 2018-11-05 15:12 by
*Anonymous*

## Reports on this Submission

### Anonymous Report 2 on 2018-11-8 Contributed Report

### Strengths

N/A

### Weaknesses

N/A

### Report

This report is entered by the editor for one of the original referees that submitted a report on the resubmission by email, and not via the online interface. They are happy with the changes made, and recommend acceptance.

### Requested changes

N/A

### Anonymous Report 1 on 2018-11-5 Invited Report

- Cite as: Anonymous, Report on arXiv:1707.02328v3, delivered 2018-11-05, doi: 10.21468/SciPost.Report.635

### Strengths

1 - Clear message, clear presentation

2 - technically sound

### Weaknesses

1 - The research presented in this work is not highly original, but rather incremental beyond existing work.

### Report

The paper closes an open gap in the existing Lieb-Robinson (LR) literature, as it proves LR bounds for unequal times also in the case of initial states with power-law correlations. The paper is technically sound, clearly presented and can be published in SciPost Physics. I recommend that the authors have a look at the recent experimental work https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.021070, which illustrates

LR bounds in two-dimensional systems, and which seems to have escaped the authors attention.

### Requested changes

none