Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics

Statistical Mechanics of a Classical Heisenberg Chain with Anisotropies

Kazuo Sasaki

Published Nov 1, 1982• 23 citations
Publisher

Abstract

Statistical mechanics of a classical Heisenberg ferromagnetic chain with weak anisotropies, which exhibits the crossover behavior from the XY- to Ishing-like systems as an anisotropy parameter is varied, are studied by means of the transfer integral method. Analytic expressions for the free energy and correlation functions are obtained at low temperatures. The kink and spin wave contributions to these quantities are identified.

DOI
10.1143/ptp.68.1518

Related papers

Statistical mechanics of classical one-dimensional Heisenberg ferromagnets with single-site anisotropy

Kazuo Torii Nakamura, Tomohei Sasada

The exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with an easy plane. The problem has been managed on the basis of the functional integral method. The calculated results have a few leading terms in series-expansion with respect to the reduced temperature, exhibiting the characteristic points borne out in the recent numerical studies. In particular, the longitudinal correlation function and susceptibility of the system with an easy axis show Ising-like behaviours and the transverse counterparts of the system with an easy plane show isotropic XY- or Heisenberg-like behaviours. Physical implications of the calculated results are also discussed.

Journal of Physics C Solid State Physics•Jan 28, 1978•61 citations

Statistical mechanics of classical spin-field model of the one-dimensional Heisenberg ferromagnet in a magnetic field

Kazuo Torii Nakamura, Tomohei Sasada

The exact analytical forms of the low-temperature free energy, and longitudinal and transverse correlation functions are obtained for the continuum model of the classical one-dimensional Heisenberg ferromagnet in a magnetic field. The calculated results show a few leading terms in series expansion with respect to the reduced temperature. It is also pointed out that the low-temperature free energy can not be identified with its counterpart for the XY model in a magnetic field.

Journal of Physics C Solid State Physics•Mar 14, 1978•4 citations

Critical Behavior of Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low Temperatures

Miki Yamada, Minoru Takahashi

A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in \(\sqrt{T/|J|}\) about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.

Journal of the Physical Society of Japan•Jun 15, 1986•86 citations

The free energy of the classical Heisenberg model with anisotropic interactions

J Rae

Exact expressions are given for the free energy of the classical Heisenberg model in one lattice dimension and two or three spin dimensions with general anisotropy; the two-dimensional result, already known, is in terms of Mathieu functions and the three-dimensional result in terms of Lame wavefunctions. The method presented solves the eigenvalue problem for an appropriate transfer matrix by relating it to solutions of the Helmholtz equation in suitable curvilinear coordinate systems. High and low temperature limits are discussed for the three-dimensional spin case and the connection given with already known results for partial anisotropy.

Journal of Physics A Mathematical Nuclear and General•Jul 21, 1974•7 citations

The low-temperature magnetic properties of the one-dimensional anisotropic Heisenberg ferromagnet in a magnetic field

Liangjie Fu

The spin-1/2 quantum Heisenberg ferromagnet with anisotropic spin exchange interactions in one dimension is investigated by means of the Green's function method and the spin-wave theory. In this paper we use the Holstein-Primakoff transformation to get the approximately Hamiltonian. We show that the magnetic field and anisotropy can have a rather drastic effect on the correlation functions and susceptibilities in the low-temperature region. Our results show that inverse correlation lengths kx and ky are independent of the temperature in the weak magnetic field. kx is less dependent on the anisotropy than ky. These results are consistent with the theoretical and experimental results.

Journal of Guangzhou University•Jan 1, 2009•0 citations

Low-temperature thermodynamics and correlation functions for a classical Heisenberg chain with two anisotropies

Hans C. Fogedby, Per Hedegård, A. Svane

The authors derive the thermodynamics and the static correlation functions for the classical Heisenberg chain with the anisotropies within a low-temperature steepest-descent approach. As regards the thermodynamics, they find agreement with transfer matrix and 'ideal-gas' phenomenological results. For the longitudinal correlation function the authors find an antiferromagnetic-type behaviour with a correlation length inversely proportional to the domain wall density. The transverse in-plane correlation function exhibits a damped oscillatory behaviour due to domain-wall-spin interference. The transverse out-of-plane correlations, on the other hand, are exponentially damped with the correlation length increasing as the easy-axis limit is approached.

Journal of Physics C Solid State Physics•Jul 10, 1984•9 citations
Statistical mechanics of a classical Heisenberg ferromagnetic chain with weak anisotropies, which exhibits the crossover behavior from the XY- to Ishing-like systems as an anisotropy parameter is varied, are studied by means of the transfer integral method. Analytic expressions for the free energy and correlation functions are obtained at low temperatures. The kink and spin wave contributions to these quantities are identified.
The exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with an easy plane. The problem has been managed on the basis of the functional integral method. The calculated results have a few leading terms in series-expansion with respect to the reduced temperature, exhibiting the characteristic points borne out in the recent numerical studies. In particular, the longitudinal correlation function and susceptibility of the system with an easy axis show Ising-like behaviours and the transverse counterparts of the system with an easy plane show isotropic XY- or Heisenberg-like behaviours. Physical implications of the calculated results are also discussed.
A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in T/∣J∣\sqrt{T/|J|}T/∣J∣​ about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.
Exact expressions are given for the free energy of the classical Heisenberg model in one lattice dimension and two or three spin dimensions with general anisotropy; the two-dimensional result, already known, is in terms of Mathieu functions and the three-dimensional result in terms of Lame wavefunctions. The method presented solves the eigenvalue problem for an appropriate transfer matrix by relating it to solutions of the Helmholtz equation in suitable curvilinear coordinate systems. High and low temperature limits are discussed for the three-dimensional spin case and the connection given with already known results for partial anisotropy.
The spin-1/2 quantum Heisenberg ferromagnet with anisotropic spin exchange interactions in one dimension is investigated by means of the Green's function method and the spin-wave theory. In this paper we use the Holstein-Primakoff transformation to get the approximately Hamiltonian. We show that the magnetic field and anisotropy can have a rather drastic effect on the correlation functions and susceptibilities in the low-temperature region. Our results show that inverse correlation lengths kx and ky are independent of the temperature in the weak magnetic field. kx is less dependent on the anisotropy than ky. These results are consistent with the theoretical and experimental results.
The authors derive the thermodynamics and the static correlation functions for the classical Heisenberg chain with the anisotropies within a low-temperature steepest-descent approach. As regards the thermodynamics, they find agreement with transfer matrix and 'ideal-gas' phenomenological results. For the longitudinal correlation function the authors find an antiferromagnetic-type behaviour with a correlation length inversely proportional to the domain wall density. The transverse in-plane correlation function exhibits a damped oscillatory behaviour due to domain-wall-spin interference. The transverse out-of-plane correlations, on the other hand, are exponentially damped with the correlation length increasing as the easy-axis limit is approached.