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Journal of Physics A Mathematical and General

Systematic lower bounds for lattice Hamiltonians

T Almassy, A. Patkós

Published May 21, 1989• 2 citations
Publisher

Abstract

The variational version of the Weinstein-Aronszajn method (1935), for constructing sequences of intermediate Hamiltonians with spectra converging to the exact eigenvalues from below, is revived for lattice theories. A first test is performed on the critical Ising quantum chain and U(1)-invariant gauge systems. Special emphasis is put on applications to degenerate levels. A related upper bound construction is also discussed.

DOI
10.1088/0305-4470/22/10/013

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The variational version of the Weinstein-Aronszajn method (1935), for constructing sequences of intermediate Hamiltonians with spectra converging to the exact eigenvalues from below, is revived for lattice theories. A first test is performed on the critical Ising quantum chain and U(1)-invariant gauge systems. Special emphasis is put on applications to degenerate levels. A related upper bound construction is also discussed.
In the context of the Hamiltonian variational approach we propose a systematic method for the improvement of any given trial wave function. The approach has many similarities with the Lanczos scheme and it may be used for quantum-mechanical problems as well as field-theory and statistical-mechanics systems. We apply the method to the Mathieu equation and to the Ising one-dimensional model in a finite lattice. The agreement between our results and the exact ones is excellent in the whole range of parameters. We also briefly discuss the application of these ideas to lattice gauge theories and the similarities with other recently proposed methods.
Weinstein's modification of the Ritz principle is used to derive (1) a lower bound for the nnn-th energy-level of a quantum mechanical system if a lower bound for the $(n+1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known; and (2) an upper bound for the nnn-th level if an upper bound for the $(n\ensuremath{-}1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known.
The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving