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Semiclassical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain
by Yuan Miao, Enej Ilievski, Oleksandr Gamayun
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Submission summary
Authors (as registered SciPost users):  Oleksandr Gamayun · Yuan Miao 
Submission information  

Preprint Link:  https://arxiv.org/abs/2010.07232v4 (pdf) 
Date submitted:  20201110 10:40 
Submitted by:  Miao, Yuan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Using the algebrogeometric approach, we study the structure of semiclassical eigenstates in a weaklyanisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic LandauLifshitz equation arise as coherent macroscopic lowenergy 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 semiclassical quantisation of classical spin waves using the RiemannHilbert problem approach. We describe overlaps of semiclassical eigenstates employing functional methods and correlation functions of these states with aid of classical phasespace averaging.
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Reports on this Submission
Anonymous Report 2 on 20201229 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2010.07232v4, delivered 20201229, doi: 10.21468/SciPost.Report.2343
Strengths
1. Thorough analysis supported by numerics
2. Complete classification of semiclassical states of the XXZ spin chain
3. Many examples considered in detail
Weaknesses
1. none
Report
The authors study semiclassical states of the Heisenberg model using Bethe ansatz and the finitegap 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 quasimomentum at infinity. This is indeed the case for the 1cut solution, but for the 2cut 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 finitesize 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 finitesize 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 Englishwise.
Anonymous Report 1 on 2020124 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2010.07232v4, delivered 20201204, doi: 10.21468/SciPost.Report.2253
Strengths
Highly detailed
Pedagogically written
Carefully written
Important results
Weaknesses
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 algebrogeometrical approach, semiclassical 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.