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Journal of Physics A Mathematical and General

A new method in the many-body problem: II

Igor Krasovsky, V. I. Peresada

Published Jan 7, 1996• 0 citations
Publisher

Abstract

The earlier proposed method for exact determination of the energy spectra of Hamiltonians H and the quantities , where W is a finite-dimensional perturbation, is extended to treat a still larger class of Hamiltonians. In particular, some two- and three-dimensional problems became amenable to this approach. In addition, the expressions for the density and local densities of states are deduced. As an example, an impurity problem is considered in a simple cubic lattice with short-range and in a linear chain with long-range interactions.

DOI
10.1088/0305-4470/29/1/015

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The earlier proposed method for exact determination of the energy spectra of Hamiltonians H and the quantities , where W is a finite-dimensional perturbation, is extended to treat a still larger class of Hamiltonians. In particular, some two- and three-dimensional problems became amenable to this approach. In addition, the expressions for the density and local densities of states are deduced. As an example, an impurity problem is considered in a simple cubic lattice with short-range and in a linear chain with long-range interactions.
An exact real-space rescaling transformation is used to calculate the spectral properties of tight-binding Hamiltonians with an arbitrary range of interaction of two- and three-dimensional lattices. The intrinsic problem of the proliferation of initial short-range interactions under rescaling is shown to be tractable numerically. This method provides a direct way of obtaining the local Green functions and local density of states. Singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation. Band edge fixed points and exponents have been found for the square, triangular and cubic lattices. In two and three dimensions there is rapid convergence to accurate numerical results for the Green function outside the band, while good results have been obtained for the in-band (e.g. density of states) spectra in two dimensions by suitable choices of cluster size and attenuation factor. Insight into the relationship between these factors follows from an investigation of finite-cluster approximations to the exact transformation. It is found that the zeros of the Green functions in the complex plane are Julia sets which have a fractal structure. For a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent.
In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered.The aim is to bring a fermionic Hamiltonian for arbitrary numbers N of particles by analytical means into a shape such that modern numerical methods can successfully be applied.For this purpose the Cook-Schroeck Formalism is taken as starting point.This includes the use of the occupation number representation.It is shown that the N -particle Hamiltonian is determined in a canonical way by a fictional 2particle Hamiltonian.A special approximation of this 2-particle operator delivers an approximation of the N -particle Hamiltonian, which is the orthogonal sum of finite dimensional operators.A complete classification of the matrices of these operators is given.Finally the method presented here is formulated as a work program for practical applications.The connection with other methods for solving the same problem is discussed.
The method of point transforms is extended to the consideration of the truly many-body terms of many-body systems. The point transformation is taken to be regional (one body isolated, two bodies interacting, three bodies interacting, …, N bodies interacting) and continuous in the 3N-dimensional transformed space. The transformed Hamiltonian and wavefunction of an N-body system can then be easily approximated by one-body, two-body, three-body, …, or N-body additivity. As an example of the use of this method, a set of regional point transforms is developed for the short range, strongly repulsive, or hard-core many-body problem. Applying these transforms to the many-body Hamiltonian, we obtain a set of equivalent Hermitian Hamiltonians which are Fourier analyzable and amenable to ordinary perturbation and variational techniques. Assuming two-body additivity, this set is shown to be the set of pairwise point transformations developed by Eger and Gross.
A general nonperturbation problem, described by the Hamiltonian H\mathcal{H}H, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's T4{T}^{4}T4 correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for 〈Sz〉〈{S}^{z}〉〈Sz〉, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.