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Annual Conference on Magnetism and Magnetic Materials (MMM)

Exact Treatment of Non-Periodic Boundary Conditions for Heisenberg Chains

K. H. Lee, Y. Y. Wang, H. W. White

Published Jan 1, 1974• 0 citations
Publisher

Abstract

The effect of boundary scattering of magnons in Heisenberg chains is studied by use of the thermal Green's function technique. In contrast with the usual approach, the periodic boundary condition is not imposed and the space Fourier transform is not used. The decoupled equation of motion for the two‐spin Green's functions is solved exactly as a system of simultaneous difference equations subject to very general boundary conditions. The solutions contain terms representing all the higher order reflections to infinite order of magnons from the two ends. The effect of chain length on the magnon frequency spectrum and its influence on the specific heat are calculated and discussed.

DOI
10.1063/1.2947301

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The effect of boundary scattering of magnons in Heisenberg chains is studied by use of the thermal Green's function technique. In contrast with the usual approach, the periodic boundary condition is not imposed and the space Fourier transform is not used. The decoupled equation of motion for the two‐spin Green's functions is solved exactly as a system of simultaneous difference equations subject to very general boundary conditions. The solutions contain terms representing all the higher order reflections to infinite order of magnons from the two ends. The effect of chain length on the magnon frequency spectrum and its influence on the specific heat are calculated and discussed.
Abstract The relevance of zero‐energy functions, coming from zero‐energy modes and present in the structure of bosonic Green's functions, is often underestimated. Usually, their values are fixed by assuming the ergodicity of the dynamics, but it can be shown that this is not always correct. As the zero‐energy functions are connected to fundamental response properties of the system under analysis (specific heat, compressibility, susceptibility, etc.), their correct determination is not an irrelevant issue. In this paper we present some results regarding the zero‐energy functions for the Heisenberg chain of spin‐1/2 with periodic boundary conditions as functions of the number of sites, temperature, and magnetic field. Calculations are pursued for finite chains, using equations of motion, exact diagonalization and Lanczos technique, and the extrapolation to the thermodynamic limit is studied.
The Callen and Tyablikov decoupling schemes used in the Green's function method are tested numerically for short Heisenberg linear chains which are first subjected to a constraint. The constraint is imposed to insure that internal excitations alone contribute to the calculated thermal averages so that in this sense these averages correspond to thermodynamic quantities. It is found that improved agreement with the exact results occurs if the Tyablikov decoupling is corrected by a term linear in the temperature.
Abstract New analytical results are obtained for the Green's function of a finite one‐dimensional lattice with nearest neighbor interaction using discrete Fourier transform. The method offers an alternative to the Bethe Ansatz, but does not require any a priori assumption on the form of the wavefunction. This makes it suitable for extensions to nano‐ferromagnets of higher dimensions. Solutions of the Heisenberg spin chain with periodic and open boundary conditions are considered as examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Chains of quantum spins with open ends and isotropic Heisenberg exchange are studied. By diagonalizing the Hamiltonian for chains of finite length N and obtaining all the energy eigenvalues, the magnetic susceptibility X, the specific heat C v , and the partition function Z can be calculated exactly for these chains. The high-temperature series expansions of these are then evaluated. For X and C v it is found that the terms in the series consist of three parts. One is the normal high-T series already known in great detail for the N → ∞ ring (chain with periodic boundary conditions). The other two consist of a 'surface' term and a correction term of order (/T) N . The surface term is found as a series up to and including (1/T) 8 for spin S = 1/2 and 1. Simple Pade approximant formulae are given to extend the range of validity below T = 1.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.
Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.