## SciPost Submission Page

# Topological thermal Hall effect for topological excitations in spin liquid: Emergent Lorentz force on the spinons

### by Yong Hao Gao, Gang Chen

#### This is not the current version.

### Submission summary

As Contributors: | Gang Chen |

Arxiv Link: | https://arxiv.org/abs/1901.01522v1 |

Date submitted: | 2019-07-23 |

Submitted by: | Chen, Gang |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

We study the origin of Lorentz force on the spinons in a U(1) spin liquid. We are inspired by the previous observation of gauge field correlation in the pairwise spin correlation using the neutron scattering measurement when the Dzyaloshinskii-Moriya interaction intertwines with the lattice geometry. We extend this observation to the Lorentz force that exerts on the (neutral) spinons. The external magnetic field, that polarizes the spins, effectively generates an internal U(1) gauge flux for the spinons and twists the spinon motion through the Dzyaloshinskii-Moriya interaction. Such a mechanism for the emergent Lorentz force differs fundamentally from the induction of the internal U(1) gauge flux in the weak Mott insulating regime from the charge fluctuations. We apply this understanding to the specific case of spinon metals on the kagome lattice. Our suggestion of emergent Lorentz force generation and the resulting topological thermal Hall effect may apply broadly to other non-centrosymmetric spin liquids with Dzyaloshinskii-Moriya interaction. We discuss the relevance with the thermal Hall transport in kagome materials volborthite and kapellasite.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 2 on 2019-11-10 Invited Report

### Strengths

Paper is well written

Introduces a novel mechanism to get linear in T thermal Hall effect

### Weaknesses

The mean field spinon tight binding model is derived somewhat hand-wavily.

### Report

This paper introduces an interesting idea for how to get a thermal Hall effect that is linear in T at small T in the spinon Fermi surface spin liquids with U(1) gauge fields. The idea follows naturally from a few previous works, that are acknowledged in the paper, but I still find it important to highlight their mechanism as an interesting way to get substantial thermal Hall effect that scales linearly with T even in the strongly insulating regime.

My main concern is not the quality and novelty of their essential ideas, but rather with the specifics of the example that they chose to illustrate such general ideas. They consider a Hamiltonian given by Eq(1). I suspect that they are correct to assume that such Hamiltonian has a stable Spinon Fermi surface state at least at the level of Slave Fermion Mean Field neglecting the gauge field fluctuations. The pattern of fluxes that they conjecture in the presence of the Zeeman field does not increase the unit cell, so it should be easy to derive this pattern of fluxes as the one that minimises the mean field energy because their Hamiltonian is a spin bilinear. The work would be more complete if they did so and demonstrate explicitly that the pattern of fluxes is the one that minizes the mean field energy. But more than completeness, there are certain issues of consistency with their stated mean field Hamiltonian. For example, their tight binding model, Eq.(6), has conservation of total Sz. This is strange because their original Hamiltonian has DM interactions which break conservation of Sz in general. Are they assuming a restricted form of DM to get such conservation?

Their description of the Berry curvature is also rather limited. It seems they are dealing with a band structure with 3 different bands, and the bands have no touchings at finite B. What are their separate Chern numbers of each band? This is particularly important because the fully occupied bands could be contributing to the Hall effect.

When plotting Figure 3 they don't specify the value of B/t (Zeeman over kinetic energy), but only the value of the flux. These are independent parameters. What is this value?

Also I find very peculiar that their Thermal Hall conductivity reaches a value of order 1 at a temperature scale of the order of the band-width, even for very weak values of flux in their Fig. 3. For temperatures above the Zeeman scale their effect should be washed out by temperature and the system should resemble the case of no Zeeman, which should have zero Hall conductivity, because the Zeeman scale is what self-consistently determines the flux. The ban-width could be huge in comparison to Zeeman since it ie determined by J. But they never specify the value of the Zeeman scale for this plot so it is hard to tell. This is could also be an artefact of their not solving the mean field self-consistently since the value of the effective magnetic field itself should decrease with temperature because the system should have an Sz spin susceptibility that itself decreases with temperature.

### Requested changes

I hope that the authors address my concerns in the Report before recommending publication.

### Anonymous Report 1 on 2019-8-1 Invited Report

### Strengths

1. Very readable paper, nicely written on the whole.

2. Makes a contribution to the literature.

3. No technical faults

### Weaknesses

1. Major weakness is whether the work is of high enough significance to warrant publication in scipost

2. Use of the word "Topological" for this thermal hall effect (despite other papers doing similarthings) seems deceptive and should be avoided.

3. Citation should be added into abstract.

### Report

Overall I liked this paper -- it is clear and well written. My concern, however, is that it is not a sufficient advance to warrant publication in SciPost (which requires a somewhat higher level of work than, for example, PRB, in my understanding). The problem is that the results seems hardly surprising. (a) it is hardly surprising (based on symmetry) that once you add a zeeman field you will generate a chiral order parameter (b) once you have a nonzero chiral order parameter it is hardly surprising that you will have a thermal hall effect roughly proportional linear in the order parameter at least for small values. If either (a) or (b) is surprising, it is not clear to me why, and it is not explained in the manuscript. If the authors can explain this convincingly I would be happy to recommend it for publication. Otherwise, I'm not sure it is interesting enough. So although I will label the paper as "minor revision", unless the authors convince me it is interesting enough, I'm hesitant to recommend publication.

I have two further comments. First, I don't think using the word "topological" is really acceptable here. What is topological in this calculation? You use berry curvature, but this is not topological -- it is a geometric curvature, and unless you integrate over the whole zone to get a chern band, which you don't, it is not topological. There are no topological invariants, and no topological objects. The word topological is simply out of place and it seems like people just insert the word "topological" randomly in order to attract attention. I know the word has been used similarly in other publications, but this does not make it correct.

My final point is very minor. The abstract states that this work is based on prior work (second and third sentences of the abstract). These works should be cited within the abstract to clarify. With abstract citation, the entire publication information should be within the abstract itself. The way it stands it is quite hard for the reader to figure out what reference the authors mean until half-way through the paper.

### Requested changes

1. Clarify why this is important/interesting enough to be in SciPost

2. Remove use of word "topological" where it is not appropriate (including the title)

3. Include appropriate citation in abstract.

(in reply to Report 1 on 2019-08-01)

Referee: Overall I liked this paper -- it is clear and well written. My concern, however, is that it is not a sufficient advance to warrant publication in SciPost (which requires a somewhat higher level of work than, for example, PRB, in my understanding). The problem is that the results seems hardly surprising. (a) it is hardly surprising (based on symmetry) that once you add a zeeman field you will generate a chiral order parameter (b) once you have a nonzero chiral order parameter it is hardly surprising that you will have a thermal hall effect roughly proportional linear in the order parameter at least for small values. If either (a) or (b) is surprising, it is not clear to me why, and it is not explained in the manuscript. If the authors can explain this convincingly I would be happy to recommend it for publication. Otherwise, I'm not sure it is interesting enough. So although I will label the paper as "minor revision", unless the authors convince me it is interesting enough, I'm hesitant to recommend publication.

Response: Thanks for the comment.

Let me address the referee's comment in a bit inspiring way to motivate our work. In magnetic Mott insulator, when inversion symmetry is absent, we will have Dzyaloshinskii-Moriya interaction because this is allowed by symmetry. But the physical mechanism due to the spin-orbit coupling and high order perturbation calculation from Hubbard model by Moriya in 1960 are important and decisive for our understanding of the Dzyaloshinskii-Moriya interaction in transition metal oxides.

More recently, there is a fashion of honeycomb Kitaev material. Based on the referee's logic, it is completely obvious that, any spin-orbit-coupled Mott insulator with a honeycomb geometry will have a Kitaev interaction. This is because of the symmetry. However, the contribution of Jackeli and Khaliullin's microscopic theory is important in deriving the Kitaev interaction.

The above examples are illustrative works that explain why we need a microscopic origin and microscopic understanding in our paper.

Moreover, we disagree with the referee about the statement of our work quality, we are sure that our work is above PRB.

Referee: I have two further comments. First, I don't think using the word "topological" is really acceptable here. What is topological in this calculation? You use berry curvature, but this is not topological -- it is a geometric curvature, and unless you integrate over the whole zone to get a chern band, which you don't, it is not topological. There are no topological invariants, and no topological objects. The word topological is simply out of place and it seems like people just insert the word "topological" randomly in order to attract attention. I know the word has been used similarly in other publications, but this does not make it correct.

Answer: Thanks for the comments. We used the word "topological" is based on the early works of "topological Hall effect in metals skyrmion textures". Topological Hall effect was observed and understood from the scalar spin chirality in metals with local moments. This Hall current effect is termed as "topological Hall effect".

In our context, the origin of thermal Hall effect is also from the spin texture and scalar spin chirality. The difference is that, our systems are Mott insulators, and there is no electric current but thermal current of spinons. Since our origin is from the spin chirality, so we feel it is natural to generalize this concept to thermal Hall effect. This reason that we adopt this notion was explained in page 3.

Referee: My final point is very minor. The abstract states that this work is based on prior work (second and third sentences of the abstract). These works should be cited within the abstract to clarify. With abstract citation, the entire publication information should be within the abstract itself. The way it stands it is quite hard for the reader to figure out what reference the authors mean until half-way through the paper.

Answer: Thanks very much. We are happy to do that. All works are based on the prior work.