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Anisotropic Landau-Lifshitz Model in Discrete Space-Time

by Žiga Krajnik, Enej Ilievski, Tomaž Prosen, Vincent Pasquier

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Authors (as registered SciPost users): Žiga Krajnik
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-05-05 18:12
Submitted by: Krajnik, Žiga
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2021-7-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2104.13863v1, delivered 2021-07-18, doi: 10.21468/SciPost.Report.3255


The authors construct an integrable space-time discretization of the classical anisotropic Landau-Lifshitz model. The paper is timely, interesting and well-written. I have only a couple of scientific questions/comments:

1) On p6-8 the authors work with “Sklyanin variables” rather than canonical spin variables, which they justify on the grounds that it is “to facilitate the computation”. However, the space-time continuum limits of the models in question are usually discussed in terms of standard spin variables. Also, the explicit formulas for the propagator in Appendix B seem much more complicated than the expressions that arise at the isotropic point in terms of standard spin variables Eqs. D3 and D4. Could the authors clarify why Sklyanin variables are more natural here, and what advantage they offer compared to ordinary spins?

2) On p.18-19 the authors conjecture that (quasi)-local charges protect the easy-plane spin current, as for the spin-1/2 quantum XXZ model. They also note that for their model, the standard local charges arising from the transfer matrix are spin-flip even. However, this is also true in the quantum case. Could the authors elaborate on the obstacles to writing down classical Z-charges? These have been conjectured to arise as a semiclassical limit of quantum Z-charges for some time (e.g. Ref 71).

Finally, I noticed the following typos:
p2: Can be both -> can both be,
p2, p13,p29: Landau-Lifhsitz -> Landau-Lifshitz
Fig. 2 caption: comprising of -> comprising
p6: Algebra A_q becomes non-degenerate-> The algebra A_q…
p8: To enforce it, variable x -> …the variable x
p8: formuale -> formulae, p20: formuae -> formulae
p10: simplifies considerably -> simplify considerably
p11: There is a number -> there are a number
p15: converging -> converges
p15,16,17,20: consistently - > consistent
p20: corresponding the easy-axis -> corresponding to the…

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Žiga Krajnik  on 2021-07-27  [id 1618]

(in reply to Report 1 on 2021-07-18)
answer to question

We thank the referee for his careful reading of the manuscript and the positive appraisal of our work.
The answers to the referee's questions follow below:

(1) One can quickly recognize that, upon deforming the model, canonical spins are no longer the most natural variables to work with. We remind the reader that in the quantum setting, the Sklyanin algebra specifies the commutation relations for an elliptic quantum group, reducing to the widely known q-deformed su(2) algebra in the trigonometric limit. In the paper, we deal with Sklynain's Poisson algebra, which is its classical counterpart. The classical Lax matrix (and likewise the Hamiltonian) can be elegantly written in terms of Sklyanin variables (see also the textbook [61]), and this further facilitates its decomposition in terms of Weyl variables. The most economic explicit parametrization of the two-body map we could achieve was in terms of Sklyanin variables. We could not recognize any advantage of working with canonical spins. Those are only used at the very end in applications, given that we wish to interpret our model as discrete dynamics of magnetization.

(2) To avoid running to excessive length we had to come to a stop and not pursue this matter further. We note that the whole discussion regarding the existence and physical significance of "classical Z-charges" has been given before, e.g. in Ref.[71] (as also noted by the referee) which invokes the Mazur-Suzuki bound and invariance under the spin-reversal transformation. Moreover in the present paper we do not imply that there is a fundamental obstacle to constructing such charges. We have made substantial progress on this question after the completion of the study. There are nonetheless certain delicate technical aspects that deserve a careful and focused discussion which we prefer to report elsewhere. We hope to have an update soon.

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