## SciPost Submission Page

# Quenching the Kitaev honeycomb model

### by Louk Rademaker

#### - Published as SciPost Phys. 7, 071 (2019)

### Submission summary

As Contributors: | Louk Rademaker |

Arxiv Link: | https://arxiv.org/abs/1710.09761v4 |

Date accepted: | 2019-11-20 |

Date submitted: | 2019-09-24 |

Submitted by: | Rademaker, Louk |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

I studied the non-equilibrium response of an initial N\'{e}el state under time evolution with the Kitaev honeycomb model. With isotropic interactions ($J_x = J_y = J_z$) the system quickly loses its antiferromagnetic order and crosses over into a steady state valence bond solid, which can be inferred from the long-range dimer correlations. There is no signature of a dynamical phase transition. Upon including anisotropy ($J_x = J_y \neq J_z$), an exponentially long prethermal regime appears with persistent magnetization oscillations whose period derives from an effective toric code.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 7, 071 (2019)

### Author comments upon resubmission

### List of changes

>> Ref 1, main point; and Ref 2-1: Discussion on prethermalization

Response: I think I finally find the origin of our misunderstanding, leading hopefully to convergence on our understanding of my results.

The ‘prethermal’ regime has two phenomenon that need to be explained:

1) Why does it take exponentially long in Jxy/Jz for the system to relax?

2) Why are there persistent magnetization oscillations in this regime?

In the renewed manuscript, I made this distinction clearer. The answer to the first question lays within the realm of prethermalization theory, in particular Ref. [23].

In the new manuscript, I elaborate more on the answer to the second question, which is largely inspired by the referee’s comments. Indeed, starting with an initial Neel state in the toric code, one would expect T=8Pi Jz^3/Jxy^4 periodic oscillations of the staggered magnetization. The essence of the prethermal regime is that the dynamics are * as if * the system is undergoing evolution with a toric code. Indeed, in the new Fig. 6, you can see that this is the case. When J_xy < 0.2 Jz, the oscillations follow almost perfectly the expectations from a toric code.

I hope that with this extension, the section 3.2 has become suitable for publication.

>> Ref 1, second point: "While I understand that potentially one can use all u_ij to encode the vortex sector, many of these encodings lead to the same physical scenario. It seems to me that the N_c \approx 2^3N number here seems to be a fairly significant ..."

Response: Actually, it is true that the gauge freedom allows you to construct the Néel state with maybe fewer gauge field configurations than I used. However, the construction I chose has the advantage of being very transparent in that it is a equal-weight superposition of all the possible configurations that satisfy the given criteria. I will add a comment along these lines to the new manuscript.

>> Ref 2-2: "There has been considerable discussion in the reports and response with regards to free fermions, etc. I believe that if the initial state chosen were the zero-flux sector, one could get away with an effectively free fermion treatment (since the Kitaev Hamiltonian conserves flux)?"

This is correct.

"But now, the novelty here is the initial Neel state, which has superpositions of different flux sectors, making the gauge coupling important."

Again, correct.

"The paper would benefit greatly from having a discussion on this point in connection to quenches, given the subtle discussions."

I will add a discussion on this at the end of Sec. 2.

>> Ref 2-3. " I think the specific choice of operators, i.e. S^z correlators, and perhaps the magnetization too, overcomes this problem in getting rid of the string in the Majorana basis? Could the author comment on this in the manuscript?"

If I understand the referee correctly, the point is that within my choice of Majorana-Gauge field basis, it is much simpler to compute Sz and correlators of Sz than Sx and Sy-expectation values. I will add a comment on this at the end of Sec. 2.

### Submission & Refereeing History

- Report 2 submitted on 2019-11-15 04:51 by
*Anonymous* - Report 1 submitted on 2019-10-12 15:54 by
*Anonymous*

- Report 2 submitted on 2019-08-08 07:35 by
*Anonymous* - Report 1 submitted on 2019-07-27 15:36 by
*Anonymous*

## Reports on this Submission

### Anonymous Report 2 on 2019-11-15 Invited Report

- Cite as: Anonymous, Report on arXiv:1710.09761v4, delivered 2019-11-15, doi: 10.21468/SciPost.Report.1319

### Strengths

1. As reported earlier, employing the Kitaev honeycomb model and its exactly solvable nature makes for a good approach. The methods used are very nice; expressing the product state in terms of variables used to diagonalize the Kitaev model to solve the quench problem is an innovative addition to the existing literature.

2. This version of the paper addresses all the issues brought up in previous versions. There is very good data presented on post-quench dynamics. There are solid discussions on a phase transition versus crossover, prethermalization, and the persistence of non-trivial dimer correlations.

### Weaknesses

1. Perhaps there could have been an involved discussion on entanglement and anyon dynamics as highlights of the Kitaev honeycomb. But there is enough rich physics already that this is by no means needed. Also, the author has now added a lovely, thought-provoking discussion on these two topics in the concluding sections.

### Report

After the revisions, the author has done a very good job overall of addressing the issues raised by the reviewers. The paper is now a well-presented solid piece of work contributing to the less-studied, important topic of quenches in topological systems. I recommend publication with no reservations.

### Anonymous Report 1 on 2019-10-12 Invited Report

- Cite as: Anonymous, Report on arXiv:1710.09761v4, delivered 2019-10-12, doi: 10.21468/SciPost.Report.1221

### Strengths

Interesting numerical method to model dynamics of the Kitaev honeycomb model.

Results in the anisotropic regime match what one would expect from the Toric code analysis.

### Weaknesses

I'm still a little worried that the so called exponential time scale is shorter that the timescale set by the TC analysis (which is non-exponential).

### Report

One remaining concern is that the pre-thermal exponential time scale seems shorter than precise non-exponential time-scale worked out via degenerate perturbation theory. However, the author seems satisfied that there is no contradiction here so I am happy to recommend for publication.