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arXiv:0704.2898 · International Journal of Modern Physics B

GREEN'S FUNCTION OF A FINITE CHAIN AND THE DISCRETE FOURIER TRANSFORM

S. Cojocaru

Published Feb 20, 2006• 10 citations

Abstract

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.

DOI: 10.1142/s0217979206033401

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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.

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)

The method of discrete Fourier transform (DFT) is applied to obtain the exact one- or bi-magnonic states in several finite one-dimensional spin systems. The advantage of DFT method, compared to Bethe ansatz, is that no assumption is made on the form of the wave function and that its limit of infinite system reduces to the standard approach to excitation spectra in condensed matter. So the method has, in principle, no limitation on lattice dimensionality and its physical interpretation is relatively transparent. It is demonstrated that the two approaches (DFT and BA) give identical results for the solution of the Schrodinger equation on the 1D lattice, although the structure of the methods is rather different. The excitation spectrum of the XXZ chain with arbitrary end fields is analyzed in detail and an analogy with atomic wires is briefly discussed.

The Heisenberg XXZ model is a chain model with nearest-neighbor interactions. Its thermodynamics is exactly obtained via Bethe ansatz. Recently, we developed a method to derive the high-temperature expansion of the grand potential per site of translationally invariant chain models, with periodic boundary conditions. Here we apply this approach to the XXZ model with periodic boundary conditions for the Ising limit case (t = 0) and the free fermion case (delta = 0 and h = 0), obtaining results in agreement with the literature. In this new way of obtaining the coefficients of the high-temperature expansion of the grand potential, the coefficients are derived from an auxiliary function written only in terms of open connected sub-chains.

We consider an arbitrary translationally invariant chain model, with nearest neighbors interaction and satisfying periodic boundary condition. The approach developed here allows a thermodynamic description of the chain model directly in terms of grand potential per site. This thermodynamic function is derived from an auxiliary function constructed only from open connected sub-chains. To exemplify its application we consider the Heisenberg XXZ model in two limit cases: the fermionized version Ising model (t = 0) and the free fermion case ( ∆ = 0). In both cases we recover the known expressions of the corresponding thermodynamical potential.

We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.