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Physical review. B, Solid state

Application of the Faddeev Method to the Three-Spin Deviation Problem for the Heisenberg Model

Chanchal K. Majumdar

Published Jan 1, 1970• 25 citations
Publisher

Abstract

The three-spin deviation problem for the Heisenberg Hamiltonian is attacked by the Faddeev technique. The fact that the two-particle $t$ matrix is separable facilitates the solution. In one dimension, the resulting equation can be numerically solved to obtain the eigenvalue. In two and three dimensions, integral equations in one vector variable appear, but no progress has so far been made towards their solution.

DOI
10.1103/physrevb.1.287

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The three-spin deviation problem for the Heisenberg Hamiltonian is attacked by the Faddeev technique. The fact that the two-particle ttt matrix is separable facilitates the solution. In one dimension, the resulting equation can be numerically solved to obtain the eigenvalue. In two and three dimensions, integral equations in one vector variable appear, but no progress has so far been made towards their solution.
The problem of three spin deviations from a fully aligned state is studied for the Heisenberg model with next-nearest-neighbour interactions for the case of spin 1. The method used is a straightforward generalization of the equation-of-motion method of Fukuda and Wortis, taking care of the unphysical states. The resulting integral equation is solved in one dimension and the dependence of the bound states on the next-nearest-neighbour interaction discussed. Numerical calculations have also been done for closed chains containing up to 40 spins. By using C N invariance of the Hamiltonian, the dimensionality of the space is substantially reduced. The results of the finite-chain calculation agree well with solutions obtained from the equation-of-motion method.
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.
Three-body problem involving two valence holes and a single conduction electron, which appears in the study of the final state of the resonant photoemission in insulators and semiconductors, is formulated to be solved exactly in a two-band Hubbard model by use of Faddeev's scattering theory. The wave functions and energy eigenvalues are calculated numerically in a one-dimensional model. By use of this exact solution of the three-body problem, resonance photoemission spectra are studied from a weak correlation resion to a strong one. It is shown that the approximation used in the past analysis, in which two valence holes are assumed to be localized, deviates from the present results even in the strong correlation region U v v ∼2 W v with U v v and W v denoting the Coulomb integral and the band width of valence holes, respectively.