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

by Yuan Miao, Enej Ilievski, Oleksandr Gamayun

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Authors (as registered SciPost users): Oleksandr Gamayun · Yuan Miao
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-03-26
Date submitted: 2021-03-22 15:38
Submitted by: Miao, Yuan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


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 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 semi-classical quantisation using the Riemann-Hilbert problem approach. We present an expression for the overlap of two semi-classical eigenstates and discuss how correlation functions at the semi-classical level arise from classical phase-space averaging.

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.

  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?

Period $\ell$ prescribes the circumference of the classical phase space. It enters simply as the overall scale in the quasi-momentum. Wavenumbers of nonlinear finite-gap modes are given by $k_i = (2\pi/\ell)n_i$ for integer mode numbers $n_i$. In the one-cut case, we can fix the value of \ell. By choosing a mode number n we can determine the single wavenumber k. In the two-cut case, general quasi-periodic 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 quasi-momentum is also defined from the same algebraic data. Its expansion at infinity generates values for the conserved quantities of the given finite-gap 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 B-cycles may be viewed as the classical version of the Bethe equations. In practice we employ ``reverse procedure'' to find the branch points for 2-cut solutions, as described in Appendix E.2.

  1. 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".

  1. 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.

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 semi-classical 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 form-factors, we are eventually looking at a pronounced finite-size 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) finite-size 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.

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

We thank the referee for spotting this. We have made appropriate adjustments.

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

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).

Published as SciPost Phys. 10, 086 (2021)

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Anonymous Report 1 on 2021-3-22 (Invited Report)


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