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Journal of Physics C Solid State Physics

Gauge equivalence between one-dimensional Heisenberg ferromagnets with single-site anisotropy and nonlinear Schrodinger equations

Katsuhiro Nakamura, Tomohei Sasada

Published Sep 20, 1982• 53 citations
Publisher

Abstract

The concept of gauge equivalence is introduced for the classical one-dimensional Heisenberg ferromagnets with single-site anisotropy. The classical continuum version of the one-dimensional anisotropic Heisenberg antiferromagnet (ferromagnet) of spin 1 / 2 is shown to be gauge-equivalent to the nonlinear Schrodinger equation with the repulsive (attractive) interaction, which is consistent with the Yang and Yang statement on thermodynamical equivalence between the corresponding quantum systems.

DOI
10.1088/0022-3719/15/26/006

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The concept of gauge equivalence is introduced for the classical one-dimensional Heisenberg ferromagnets with single-site anisotropy. The classical continuum version of the one-dimensional anisotropic Heisenberg antiferromagnet (ferromagnet) of spin 1 / 2 is shown to be gauge-equivalent to the nonlinear Schrodinger equation with the repulsive (attractive) interaction, which is consistent with the Yang and Yang statement on thermodynamical equivalence between the corresponding quantum systems.
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.
The one-dimensional S=1 quantum spin system has been mapped onto the equivalent two-dimensional Ising model on the basis of the equivalence theorem of Suzuki. The classical representation of the partition function has been expressed in terms of the maximum eigenvalue of the transfer matrix and evaluated by means of the numerically exact transfer-matrix method. The physical meaning of two kinds of correlation functions in RSD and CBD has been made clear in terms of the concept of the maximum eigenstate. Calculations have been made for both ferromagnetic and antiferromagnetic Heisenberg chains with several single-site anisotropy values, both positive and negative. Good agreements are found with some existing numerical results.
Effective nonlinear field theory for a semiclassical long-wavelength description of one-dimensional S=1/2 Heisenberg antiferromagnets with dimerized ground states is derived within an approach using two-site coherent states. The method is applied to bond-alternating anisotropic spin chains in presence of external magnetic field and next-nearest-neighbor interactions. Linear and nonlinear excitations in singlet and magnetic phases are studied. Soliton solutions are found and their properties under semiclassical quantization are discussed. \textcopyright{} 1996 The American Physical Society.