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Journal of Physics A Mathematical and General

A new method in the many-body problem

Igor Krasovsky, V. I. Peresada

Published Mar 23, 1995• 4 citations
Publisher

Abstract

A new method is proposed for exact calculation of the energy levels of Hamiltonians H and the quantities Sp(f(H+W)-f(H)), where W is a finite-dimensional perturbation, in the case when H and H+W may be represented by a certain wide class of (2n+1)-diagonal matrices. As an example, two problems for a linear chain of spins with nearest- and next-nearest neighbour interactions described by the Heisenberg Hamiltonian are considered.

DOI
10.1088/0305-4470/28/6/007

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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.

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A new method is proposed for exact calculation of the energy levels of Hamiltonians H and the quantities Sp(f(H+W)-f(H)), where W is a finite-dimensional perturbation, in the case when H and H+W may be represented by a certain wide class of (2n+1)-diagonal matrices. As an example, two problems for a linear chain of spins with nearest- and next-nearest neighbour interactions described by the Heisenberg Hamiltonian are considered.
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.
A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.
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.
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.