SciPost Submission Page

Quenching the Kitaev honeycomb model

by Louk Rademaker

This is not the current version.

Submission summary

As Contributors: Louk Rademaker
Arxiv Link: https://arxiv.org/abs/1710.09761v2
Date submitted: 2018-05-16
Submitted by: Rademaker, Louk
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Condensed Matter Physics - Theory

Abstract

I studied the non-equilibrium response of an initial N\'{e}el state under time evolution with the Kitaev honeycomb model. This time evolution can be computed using a random sampling over all relevant flux configurations. With isotropic interactions the system quickly equilibrates into a steady state valence bond solid. Anisotropy induces an exponentially long prethermal regime whose dynamics are governed by an effective toric code. Signatures of topology are absent, however, due to the high energy density nature of the initial state.

Current status:
Has been resubmitted


Reports on this Submission

Anonymous Report 2 on 2018-6-28 Invited Report

Strengths

1- The setup of beginning with a product state, quenching and time evolving under a Hamiltonian that hosts topological phases, and looking for signatures of topological order in the final state is an excellent one.

2- 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.

3- The arguments given on the behavior of the nonequilibrium energy content as a function of time as well as those of the short versus long range correlations make for nice physical scenarios.

4 – The treatment of the gauge/plaquette degrees of freedom is well executed and exploits an interesting, key feature of the Kitaev honeycomb.

Weaknesses

1- Overall, the paper can be much more strengthened in terms of the actual physics focus and a clear message being given.

2- With regards to the above, the paper does not fully accomplish what it sets about to do. My biggest issue is with the claim of explicitly addressing topological aspects of quenches in topological systems. Entanglement seems to be one prominent measure evoked here. It is neither the case that entanglement need be the prime measure of topological order (say compared to Chern numbers in non-interacting topological insulators/superconductors) nor need highly entangled states be topological. The Kitaev model does have other measures which are strong indicators.
On that note, there is mention of anyons in the conclusion, but just to say that the highly excited state cannot distinguish signatures. If this were a focus with regards to topology (it is a good one), relevant degrees of freedom (e-m excitations, for instance) should have been defined and tracked more explicitly.

3 – It is difficult to distinguish how much is specific to the Kitaev honeycomb and its topological aspects ,as opposed to something as simple as the transverse Ising chain. See further comments in the Report.

4- See Report for other points on the amibiguity with phase transition; language; referencing; and other issues.

Report

Report
1- Overall, the idea of taking a product state, performing a quench using a Kitaev honeycomb Hamiltonian, and investigating features of the resultant state is very interesting. The question of whether topological aspects evolve is an excellent one ot focus on in this instance.

2- Please see Strength and Weakness sections for further comments on the above and the overall manuscript.

To comment on the manuscript in order of appearance:
3- The Abstract is confusing and similarly, throughout, while there is no problem with grammar, the chain of ideas gets a bit confusing here and there. In the abstract, the computational method is not so important and is confusing. I presume anisoptropy versus not refers to the coupling in the Kitaev honeycomb model, but this is not stated explicitly. The last sentence is not very comprehensible; what signatures of topology are being looked for? What is high energy about the initial state and what implication does this have on topology? There are a nice set of concrete results that could have made its way into the abstract.

4- What does ‘non-topological Hamiltonian’ mean? A Hamiltonian that does not realize a topological phase (ground state property?) for any value of coupling parameters? Perhaps better to refer to a state being topological?

5- It would be insightful to see what aspects of this problem are unique to the Kitaev honeycomb, both in terms of topological aspects as well as its unique gauge and plaquette features. In particular, since entanglement is repeatedly mentioned, how much of what is studied would not be seen in quenches, for instance, in the well explored case of the transverse XY system? Here too, a product state could be used (ground state for just one point in the phase diagram) and then acted upon for its time-evolution by the XY spin Hamiltonian having some specific couplings. Undoubtedly one would obtain a highly entangled, excited state. But likely, many features of spin correlations would be different.
Also, entanglement features are basis dependent. For instance, in the XY case, one could go into a fermionic basis (representing the 1D analog of the 2D p-wave topological superconductor represented by the Kitaev honeycomb model) – can the author comment on this? Particularly given that he uses the lovely transformation initially used by Kitaev.

6- End of Sec. 2 – This discussion on the Neel state, and the gauge-matter representation is very nice and it shows how the power of the Kitaev honeycomb can be used for quenches.

7- Sec. 3 – I am confused about the phase transition aspect. Should one be thinking of crossing a phase boundary in the static Kitaev honeycomb phase diagram? Or more in terms of a Floquet-type dynamic evolution of a phase? A more precise description and measure would be welcome.

8- The thermalization paragraphs on p.5 and the physical arguments are very nice. These could be expanded upon, including starting with standard quench measures (again, say comparing with Ising/XY chains) and how this is unique.

9- Fig. 5 Caption – what is meant by the indication of a particular (valence solid) phase in the long time? Generally one would look for ground state properties. But in this case, due to the quench, the system would be surfeit with excitations.

10 - Could the author make the connection to Drude peaks more precise? In particular, I would imagine one would need to define the conductivity?

11- Please see comments in the Weakness section on anyons.
By looking at the right measures, given the exactly solvable nature of the Kitaev honeycomb, there could perhaps be a very nice non-equilibrium calculation to be done here targetting anyon behavior and strengthening the paper’s premise on quench and topology.

12 - Appendix: Very nice to see a solid, explicit derivation!
After Eq. 12 – How accurate a method is it to average over random gauge fields?

13 - References: There are other highly related works. For instance, even a quick search on the arxiv (say abstract- kitaev honeycomb; abstract – quench).

Requested changes

1- The Weakness and Report sections bring up my main issue with topology, entanglement, etc. This is a major concept that I believe should be addressed throughout the paper.

Beyond that, the Report delineates the various parts where changes would be welcome. To summarize:

2- Abstract

3- Clarify role of entanglement; topological Hamiltonian

4 – Compare with non-topological case, say spin chain; Basis dependence?

5- Clarify notion of phase transition

6- Expand on thermalization discussion

7. Caption of Fig. 5

8.- Discussion of Drude peak

9. Include key discussion on anyons or any other topological aspect.

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

Anonymous Report 1 on 2018-6-6 Invited Report

Strengths

1 . A potentially interesting way of simulating prethermal physics.

Weaknesses

1. How the Neel state is supported on the system eigenstructure has not been clearly worked out.
2. The main claim of a prethermal region in the Toric code limit appears to be a trivial consequence of the Neel state projecting onto the highest energy flat band.
3. It has not been properly argued that thermalization can occur in this model. I am therefore concerned that the notion of prethermaliztion is inaccurate in this instance.

Report

In the paper “Quenching the Kitaev honeycomb model” the author examines how the Neel state time-evolves under various parameters of the Kitaev Honeycomb model. The main claims are that in the gapped Toric-Code phase the Neel state displays pre-thermal behavior, whereas in the gapless B-phase the state quickly relaxes to a valence bond solid.

I have a number of serious concerns/doubts regarding the results and their interpretation.

(1) It has not been proven sufficiently that the Neel state is a superposition of states from all vortex sectors. (see comment A below).
(2) It appears that what the author observes as prethermalization (in the Toric-Code region) is simply the result of the Neel state almost projecting onto a nearly flat energy band. (see comment B below)
(3) Irrespective of the two previous critiques, I am also somewhat concerned about calling the main effect observed as prethermalization.

As a result of these concerns I don’t believe the paper can be published at this time. However the paper is not without merit, and I agree it would be interesting to see if one can use the gauge structure of the model to ‘simulate’ thermalization. However it looks to be like the effect observed here can be explained simply by the flat nature of the spectrum in the Toric code (anisotropic) limit and the fact that, in that limit, the Neel state has a large overlap with the eigenstates in the highest energy band. The observed slow dynamics (in that limit) is a direct consequence of the initial state having a large overlap with eigenstates that have very similar energies.

----Additional comments--------
(A) The author makes the argument that the Neel state overlaps with all vortex sectors. The argument stems from a calculation of the expectation value of the Neel state with an arbitrary plaquette operator. However, although the 0-valued expectation value is a necessary condition, it is not sufficient.
An example of another state that gives a 0-valued plaquette expectation value everywhere is an equally weighted superposition of a vortex-free and a vortex-lattice state. Although I suspect the author is probably correct, is there a way to see that the Neel state is not in a more trivial superposition?

(B) This concern relates to how the Neel state is distributed energetically within each gauge sector.

In the Toric-code limit the ground state manifold is a ferromagnetic arrangement of spins on the z-bonds. Fermionic/matter excitations can be understood as approximately anti-ferromagnetic arrangements of these z-bonds, with an energy cost of around $2 Jz$. Indeed, in the limit that Jx=Jy=0, the Neel state projects exactly onto the fully-filled fermionic state (e.g. the highest energy) in each sector.

As the subsequent dynamical evolution depends on the distribution of supporting eigen-energies , it is therefore natural to expect slow dynamics in the TC limit, especially since, in this limit, the initial state more accurately targets the highest energy flat band. All of the results observed seem to fit with this picture and so I am a bit confused about the main point the author is trying to make.

I should note also that in terms of complexity and eigenstate support, the Neel state is not any more complicated than the its lower-energy ferromagnetic counter-part. Indeed I would expect very similar (perhaps even identical) behavior if this state was chosen instead.

(C) Within each sector, the system is fully described by free-fermions hopping on a lattice. Without a term that breaks the extensive plaquette symmetries, all dynamics will therefore take place within a constrained space within each sector. Doesn’t the free-fermion nature of subsequent dynamics preclude any thermalization?

Additional comments:
(D) Figure 1. The reddish lines seem to disappear behind the legend. They don’t reach the x- or y-axis. Also, it would be interesting to see the results as J_xy approach zero.

(E) The sentence after equation 1 is a subjective one and should be rephrased. Indeed, with respect to Ref. 7, there is much more to that paper than the fermionization method.

(F) On the topic of fermionization methods there are quite a few alternative procedures that make the link between the fermions and spins much more transparent. Kitaev’s methodolgy is excellent if all one requires is the single particle excitation spectrum. However it is cumbersome if one needs to connect back to the actual spin-structure of the eigenstates.

(G) Regarding the concluding comment about topological entanglement entropy arising dynamically. The non-trivial topological entanglement entropy arises exclusively from the gauge degrees of freedom (see Ref 27 and Bray-Ali et al Phys. Rev. B 80, 180504(R) (2009)). Therefore, since all the dynamcis takes place here within each matter sector, there is no way in this set-up for non-trivial entanglement entropy to arise dynamically.

Requested changes

1. The author should include more complete proof for why the Neel state is distributed between all gauge sectors.

2. The author needs to justify why the observed slow dynamics is not simply a consequence of initial state projecting onto a set of eigenstates with similar energy.

3. I would like to see a deeper discussion regarding the notion of how prethermalization can even be valid in this model. Perhaps in the end this concern may just be about terminology. However I have not seen the term “prethermal” used in the context of exactly solvable models before. Can the author provide some references to where this type of scenario has been examined previously? If instead this is a new variation of the idea I think author needs to argue it more forecefully.

4. Figure 1 needs to be fixed (see report)

  • validity: low
  • significance: low
  • originality: ok
  • clarity: good
  • formatting: good
  • grammar: good

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