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Semiclassical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain
by Yuan Miao, Enej Ilievski, Oleksandr Gamayun
 Published as SciPost Phys. 10, 086 (2021)
Submission summary
As Contributors:  Oleksandr Gamayun · Yuan Miao 
Arxiv Link:  https://arxiv.org/abs/2010.07232v5 (pdf) 
Date accepted:  20210326 
Date submitted:  20210322 15:38 
Submitted by:  Miao, Yuan 
Submitted to:  SciPost Physics 
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 ground state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. The internal magnon structure of classical spin waves is resolved by performing the semiclassical quantisation using the RiemannHilbert problem approach. We present an expression for the overlap of two semiclassical eigenstates and discuss how correlation functions at the semiclassical level arise from classical phasespace averaging.
Published as SciPost Phys. 10, 086 (2021)
Author comments upon resubmission
We are grateful to the Referees for their assessment of our manuscript and for their valuable comments and suggestions. Our replies to the queries are given below.
 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?
Period $\ell$ prescribes the circumference of the classical phase space. It enters simply as the overall scale in the quasimomentum. Wavenumbers of nonlinear finitegap modes are given by $k_i = (2\pi/\ell)n_i$ for integer mode numbers $n_i$. In the onecut case, we can fix the value of \ell. By choosing a mode number n we can determine the single wavenumber k. In the twocut case, general quasiperiodic solutions depend on two complex branch points (four real parameters). By setting the period $\ell$ fixed, this does not yield periodic solutions in general. Those are only a subset of solutions for which wavenumbers are integer multiples of $2\pi / \ell$. In our case, the algebraic data can be read off from the coefficients of the polynomial $\mathcal{R}(\lambda)$ (see Eq. (4.15)). The spatial periods are defined in Eqs. (4.29) and (4.33) via Abelian integrals. The quasimomentum is also defined from the same algebraic data. Its expansion at infinity generates values for the conserved quantities of the given finitegap solution. Given the mode numbers (together with their partial filling fractions), there is no algebraic procedure to compute the associated branch points for cases with more than 2 cuts. Indeed, ensuring integrality of the Bcycles may be viewed as the classical version of the Bethe equations. In practice we employ ``reverse procedure'' to find the branch points for 2cut solutions, as described in Appendix E.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.
We followed the advice and uniformly replaced "winding numbers" by "mode numbers".
 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.
Unfortunately, we presently only have access to the correlation functions of eigenstates with at most M=6 Bethe roots (for the 1/3 filling fraction). This is arguably "far away" from an approximate semiclassical regime. While the mismatch in the longitudinal correlators is admittedly quite large, the observed matching in the transversal sector suggests that our conjectural prescription is accurate. In our opinion, guided also by Smirnov's results on the formfactors, we are eventually looking at a pronounced finitesize effect. We do not know of any better computational scheme to further improve our numerical results and study the convergence. We cannot think of any feasible (let alone reliable) finitesize analysis for such small system sizes in the absence of any natural expansion parameter. Our hope is that this section of the paper, despite being conjectural, can at least stimulate further research on this aspect. In the revised version we have entirely rewritten Section 8 to emphasise clearly what has been computed numerically.
 Limits of integration in 2.38 and 2.41 are inverted.
We thank the referee for spotting this. We have made appropriate adjustments.
 I'd suggest to proofread the text Englishwise.
In the revised version we have substantially improved the text and resolved grammatical issues.
List of changes
List of changes:
1. We have corrected several typographical and grammatical mistakes/errors of the article. We have redrafted Section 8 as suggested by the referee.
2. We have changed the use of winding number to mode numbers.
3. For Eq. (2.38) and (2.41), we change two limits in the equations to a single limit.
4. We correct a sign typo in Eq. (3.15), (3.16) and (3.21).
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