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physica status solidi (b)

Green's function and excitation spectrum of finite lattices

S. Cojocaru, Victor Bârsan, Arnout Jozef Ceulemans

Published Jun 1, 2006• 1 citation
Publisher

Abstract

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)

DOI
10.1002/pssb.200541484

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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)
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.
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.
Abstract In this work, the Ising model on a trilayer Bethe lattice is studied in terms of recursion relations for which the nearest‐neighbor spins of each layer are coupled ferromagnetically and the adjacent spins of the nearest‐neighbor layers are coupled either ferromagnetically or antiferromagnetically depending on the sign of the bilinear interactions for given values of the coordination number q. Besides the ground‐state phase diagram, the phase diagrams of the model are also calculated for given values of the system parameters. It is found that the model presents both first‐ and second‐order phase transitions for all values of q. In order to construct the phase diagrams, we have also studied the thermal and interlayer coupling constant changes of the layer magnetizations, the spin–spin correlation functions between the nearest‐neighbor spins of the adjacent layers and the free energy of the model. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)