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Chinese Physics Letters

Calculation of Free Energy of the Integrable Landau-Lifshitz Model

Cai Cong-Zhong, Wanlu Wang, Chen Yu-Zong

Published Jun 17, 2003• 0 citations
Publisher

Abstract

The generalized Bethe-ansatz method of thermodynamic analysis of integrable systems was employed to compute the free energy of a classical integrable model, i.e., the Landau-Lifshitz model. Using the action-angle variables of the model and by imposing a periodic boundary condition, we derive a phase-shifted density of states for the excitations of the system. The free energy, in the thermodynamic limit, can be expressed analytic in terms of two coupled nonlinear integral equations of the finite temperature excited energy for effective phonons and kinks (antikinks). We solve these equations iteratively for a special case that the model is in the limit of anisotropic strong yz coupling.

DOI
10.1088/0256-307x/20/7/311

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The generalized Bethe-ansatz method of thermodynamic analysis of integrable systems was employed to compute the free energy of a classical integrable model, i.e., the Landau-Lifshitz model. Using the action-angle variables of the model and by imposing a periodic boundary condition, we derive a phase-shifted density of states for the excitations of the system. The free energy, in the thermodynamic limit, can be expressed analytic in terms of two coupled nonlinear integral equations of the finite temperature excited energy for effective phonons and kinks (antikinks). We solve these equations iteratively for a special case that the model is in the limit of anisotropic strong yz coupling.
The generalized Bethe-ansatz method of thermodynamic analysis of integrable systems was employed to compute the free energy of a classical integrable model,i.e.the Landau-Lifshitz model.Using the action-angle variables of the model and by imposing a periodic boundary condition.we derive a phase-shifted density of states for the excitations of the system.The free energy,in the thermodynamic limit,can be expressed analytic in terms of two coupled nonlinear integral equations of the finie temperature excited energy for effective phonons and kinks (antikinks).we solve these equations iteratively for a special case that the model is in the limit of anisotropic strong yz coupling.
We give two fundamental methods for evaluation of classical free energies of all the integrable models admitting soliton solutions; the sine-Gordon equation is one example. Periodic boundary conditions impose integral equations for allowed phonon and soliton momenta. From these, generalized Bethe-Ansatz and functional-integration methods using action-angle variables follow. Results for free energies coincide, and coincide with those that we find by transfer-integral methods. Extension to the quantum case, and quantum Bethe Ansatz, on the lines to be reported elsewhere for the sinh-Gordon equation, is indicated.
We use the quantum Bethe ansatz method to compute the free energy of the sine-Gordon model in the classical limit. In this limit the number of breathers, and hence the number of coupled integral equations to be solved, diverges. By linearizing the breather mass spectrum and phase shifts, extending a method of Maki, we can reduce the breather ladder to anharmonic phonons. The divergent set of integral equations is reduced to only two, for interacting phonons and solitons. We solve these equations iteratively to give the free energy in a double series in the temperature t and the soliton density ${e}^{\mathrm{\ensuremath{-}}1/t}$ which agrees to high order with classical transfer integral results.
We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature ttt and the soliton density ${e}^{\frac{\ensuremath{-}1}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.
The classical Landau-Lifshitz equation-the simplest model of a ferromagnet-provides an archetypal example for studying transport phenomena. In one-spatial dimension, integrability enables the classification of linear and nonlinear mode spectrum. An exact characterization of finite-temperature thermodynamics and transport has nonetheless remained elusive. We present an exact description of thermodynamic equilibrium states in terms of interacting modes. This is achieved by retrieving the classical Landau-Lifschitz model through the semiclassical limit of the integrable quantum spin-S anisotropic Heisenberg chain at the level of the thermodynamic Bethe ansatz description. In the axial regime, the mode spectrum comprises solitons with unconventional statistics, whereas in the planar regime we find two additional types of modes of radiative and solitonic type. Our framework enables analytical study of unconventional transport properties: as an example we study the finite-temperature spin Drude weight, finding excellent agreement with Monte Carlo simulations.
We solve the finite-temperature Bethe-ansatz (BA) integral equations with coupling constant $\ensuremath{\mu}=\ensuremath{\pi}(1\ensuremath{-}{n}^{\ensuremath{-}1})+0$, in the semiclassical limit $n=\mathrm{integer}\ensuremath{\rightarrow}\ensuremath{\infty}$. After eliminating long strings and then an exact bosonization, transforming $n\ensuremath{-}2$ fermion strings into a single boson, the BA integral equations are reduced to two, coupling bosons and two fermions (kinks and antikinks). In the classical limit these iterate exactly to the sine-Gordon free energy found by the transfer integral method and by other methods recently reported. We suggest that our result can be extended to general values of $\ensuremath{\mu}$.