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Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via coupled Heisenberg equations

by Oleksandr Gamayun, Oleg Lychkovskiy

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Submission summary

Authors (as registered SciPost users): Oleksandr Gamayun · Oleg Lychkovskiy
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-05-04
Date submitted: 2022-04-05 16:39
Submitted by: Lychkovskiy, Oleg
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


The Kitaev model on the honeycomb lattice, while being integrable via the spin-fermion mapping, has generally resisted an analytical treatment of the far-from-equilibrium dynamics due to the extensive number of relevant configurations of conserved charges. Here we study a close proxy of this model, the isotropic Kitaev spin-$1/2$ model on the Bethe lattice. Instead of relying on the spin-fermion mapping, we take a straightforward approach of solving Heisenberg equations for a tailored subset of spin operators. The simplest operator in this subset corresponds to the energy contribution of a single bond direction. As an example, we calculate the time-dependent expectation value of this observable for a factorized translation-invariant (or staggered-translation-invariant) initial state with arbitrary initial (staggered) polarization.

Author comments upon resubmission

We thank the Referees for the positive evaluation of our work and apologize for the delayed response. Below we address remarks by the second Referee.

Remark: Include a broader introduction of the model, write the Hamiltonian in Pauli matrices and clarify why the Bethe lattice is easier to access with this method compared to hexagonal and other lattices.

Response: We have extended the Introduction according to this advice. The Hamiltonian in terms of Pauli matrices is now given in eq. (1), see also eq. (2). We have also stated in the Introduction that the important feature of the Bethe lattice is the absence of loops, in contrast e.g. to the honeycomb lattice. We have also added a paragraph in the end of Section 3.2 where we elaborate on why loops complicate the analysis.

Remark: Compare the final result with exact results available for quenches with initial states where single fermionic sector can be used

Response: Unfortunately, we are not aware of a prior analytical work that can be compared to our results. In particular, the settings of refs. [29-33] (that are mentioned in the Introduction) are different in the following ways:
- the observables studied (defect density [29-31], Loschmidt echo [32], work statistics [33]) do not belong to the set of observables our method is applicable to;
- the Hamiltonian is changed slowly in time [29,30,33], while our results are for time-independent Hamiltonian;
- the Kitaev Hamiltonian is anisotropic before and after the quench (with an eye on the phase boundary crossing) [29-31,33], while our results are for the isotropic post-quench Hamiltonian.

Remark: Clarify the meaning of operator of eq 16 and write an explicit derivation of eq 14-19, possible in appendices.

Response: Eq. (18) [eq. (16) in the previous version of the manuscript; observe that all equation numbers have been shifted by 2] simply defines a shorthand notation for strings that are longer by one link than the original string. We have clarified this in the text around eq. (18).

Eqs. (16)-(21), while looking bulky, are just the results of a straightforward commutation between the Hamiltonian and a string. One point possibly calling for clarification is how the ratios of type

sign V^/sign V

appear in these equations. In fact, these ratios equal to \pm 1 and are used to account for a sign emerging from the commutations between Pauli matrices, see the text below eq. (9). This explanation has been added below eq. (21).

Remark: The large time limit of eq 35 can be reduced to a single integral, reminding of diagonal ensemble. Is there a GGE associated to the large time value of correlator?

Response: In fact, in the large-time limit the integral in eq. (37) vanishes altogether, and the remaining equilibrium value is given by the constant term in the second line of this equation. This is noted in the paragraph below eq. (38).

It could be indeed interesting to compare the equilibrium values of observables that can be obtained by our method to the predictions of GGE. Unfortunately, we are not aware of any such predictions in the literature. Furthermore, calculating the partition function of the generalized Gibbs state (and thus applying the GGE in practice) is believed to be a complex task, despite the integrability of the model, due to the emergent disorder, see e.g. a discussion in Phys. Rev. B 79, 214440 (2009) (ref. [45]). We have added this remark to the paragraph below eq. (38).

List of changes

List of changes.

1) Introduction extended, eqs. (1) and (2) added.

2) An explanation of the meaning of the symbol "Ex" is given below eq. (18).

3) Two paragraphs below eq. (21) revised and extended: the appearance of terms of the from (9) explained; complications emerging in lattices with loops highlighted.

4)A remark on the Generalized Gibbs Ensemble added below eq. (38).

5) A number of minor corrections introduced.

Published as SciPost Phys. 12, 175 (2022)

Reports on this Submission

Report 2 by Benjamin Doyon on 2022-4-19 (Invited Report)


The authors have replied to the comments made in a satisfactory fashion I believe; in particular the new introduction is indeed helpful. I am happy with accepting this paper at this point.

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Anonymous Report 1 on 2022-4-12 (Invited Report)


The paper is now ready for publication.

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