# Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain

### Submission summary

 As Contributors: Oleksandr Gamayun · Yuan Miao Arxiv Link: https://arxiv.org/abs/2010.07232v4 (pdf) Date submitted: 2020-11-10 10:40 Submitted by: Miao, Yuan Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

Using the algebro-geometric approach, we study the structure of semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic Landau-Lifshitz equation arise as coherent macroscopic low-energy fluctuations of the ferromagnetic grounds state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. To resolve their internal magnon structure, we carry out the semi-classical quantisation of classical spin waves using the Riemann-Hilbert problem approach. We describe overlaps of semi-classical eigenstates employing functional methods and correlation functions of these states with aid of classical phase-space averaging.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2010.07232v4 on 10 November 2020

## Reports on this Submission

### Strengths

1. Thorough analysis supported by numerics
2. Complete classification of semi-classical states of the XXZ spin chain
3. Many examples considered in detail

1. none

### Report

The authors study semi-classical states of the Heisenberg model using Bethe ansatz and the finite-gap integration method. Their analysis is very thorough and comprehensive. Along with the general formalism a few examples are studied at length, such as macroscopic magnetic waves and the bion solution.

The paper can be published as is, but I believe it will benefit from a few minor improvements.

### Requested changes

1. What determines the spacial period of the solution? From the discussion on p.20 and eqs. 4.43, 4.44 it seems to be the asymptotics of the quasi-momentum at infinity. This is indeed the case for the 1-cut solution, but for the 2-cut case the period depends on moduli. Why?

2. The authors call $n_j$ winding numbers and mode intermittently. These are physically distinct notions. The term better reflecting the physics should be used uniformly.

3. Deviations of the $\sigma^z \sigma^z$ correlator in fig. 11b from 8.5 are way too large to be accounted for by finite-size effects, if estimated from $\sigma^x \sigma^x$ where the agreement is almost perfect. This difference needs to be explained, or else the authors should provide a reliable estimate of finite-size corrections. Otherwise the discrepancy casts doubts on the conjectural relation between the phase average and quantum correlators.

4. Limits of integration in 2.38 and 2.41 are inverted.

5. I'd suggest to proofread the text English-wise.

• validity: high
• significance: top
• originality: good
• clarity: high
• formatting: good
• grammar: below threshold

### Strengths

Highly detailed
Pedagogically written
Carefully written
Important results

Originality

### Report

The paper gives a semiclassical treatment in the anisotropic Heisenberg quantum chain. The paper incorporates the most important methods to give semiclassical results in exactly solvable system, by using namely the algebro-geometrical approach, semi-classical quantization, and an astute use of determinantal formulas. All these methods are carefully woven together to give a rather comprehensive treatment of this model in the semiclassical limit. The paper also contains a good introduction based on the harmonic oscillator, which is also very useful. As a whole I think the paper will be an invaluble reference in its field. I recommend the paper to be published in SciPost.

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