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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 submitted: 2021-10-27 10:37
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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-12-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2110.13123v2, delivered 2021-12-19, doi: 10.21468/SciPost.Report.4064


- innovative idea to access time evolution of kitaev model and in general of a number of interesting many-body system


- clarifications and some extra checks needed


The idea of using Heisenberg time evolution in many-body systems is usually overlooked, and this paper is a great example of how instead it can be a very useful tool. I recommend publication provided the following points are carefully addressed:
- 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.
- compare the final result with exact results available for quenches with initial states where single fermionic sector can be used
- clarify the meaning of operator of eq 16 and write an explicit derivation of eq 14-19, possible in appendices.
- 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?

Requested changes

see report.

  • validity: high
  • significance: high
  • originality: top
  • clarity: good
  • formatting: excellent
  • grammar: excellent

Report 1 by Benjamin Doyon on 2021-12-17 (Invited Report)

  • Cite as: Benjamin Doyon, Report on arXiv:2110.13123v2, delivered 2021-12-17, doi: 10.21468/SciPost.Report.4062


Clear and to the point
Interesting technical idea, showing that it works (proof of concept)
Potential for many new results of interest in quantum many-body


very special model, still need to see how general the technique can be


In this paper, the authors provide exact solutions to the Heisenberg equations for a particular subset of operators in the Kitaev model on the Bethe lattice. The purpose is to illustrate that it is possible to evaluate exactly time evolution of nontrivial and interesting operators from nontrivial states in spin models, without resorting to exact diagonalisation, and, for instance in the case of the Kitaev model, to mapping to free fermions. This is a short paper, to the point, presenting a clear calculation with explicit results. I find it very well written, and the idea is extremely interesting. The construction is quite stunning, and this, along with previous works by the authors, seems to open up many possibilities, going beyond the standard solution methods. The model here is relatively simple, and the lattice is a regular tree, which drastically simplifies the calculation. Nevertheless, I think the results, and especially the techniques introduces, are extremely interesting.

I do not have any specific comment for improvement, I believe this is publishable as is.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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