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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Anharmonic analysis of lattice field theories

John L. Richardson, R. Blankenbecler

Published Sep 15, 1979• 11 citations
Publisher

Abstract

A new calculational procedure for polynomial lattice field theories is discussed that utilizes an anharmonic basis and a general orthogonal transformation of coordinates. The standard blocking procedure is shown to correspond to a discrete Haar transform of the field coordinates. Some generalizations of the Haar transform are given which allow one to block an arbitrary number of sites. This is applied to both the energy density and to the correlation functions. In this paper only an "unperturbed" problem will be discussed, but the unperturbed Hamiltonian will be chosen using a variational principle---it includes couplings and nonharmonic effects in a very nontrivial way. Numerical results will be given for certain critical indices for a ${\ensuremath{\varphi}}^{4}$ theory in one space dimension.

DOI
10.1103/physrevd.20.1351

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•Jan 1, 1984•0 citations
A new calculational procedure for polynomial lattice field theories is discussed that utilizes an anharmonic basis and a general orthogonal transformation of coordinates. The standard blocking procedure is shown to correspond to a discrete Haar transform of the field coordinates. Some generalizations of the Haar transform are given which allow one to block an arbitrary number of sites. This is applied to both the energy density and to the correlation functions. In this paper only an "unperturbed" problem will be discussed, but the unperturbed Hamiltonian will be chosen using a variational principle---it includes couplings and nonharmonic effects in a very nontrivial way. Numerical results will be given for certain critical indices for a \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 theory in one space dimension.
In a previous paper we showed that a class of real-space renormalization-group calculations could be reinterpreted as an orthogonal transformation of coordinates followed by an approximate (variational) evaluation of the resulting Hamiltonian. Here we shall discuss the orthogonal-transform method and concentrate on analytic rather than numerical results. The transform method may ultimately be more accurate because even though similar approximations are made in coupling the lattice oscillators, at no time is the system split into noninteracting finite-sized blocks. Approximate variational solutions to a ${\ensuremath{\varphi}}^{4}$ lattice field theory in 2, 3, and 4 spacetime dimensions are constructed using an anharmonic-oscillator basis. The solutions exhibit an ordered and a disordered phase. Analytic expressions are obtained for certain critical surfaces of these theories in regions of parameter space where a spin approximation would be invalid. The continuum limit of our solutions is discussed. The resultant perturbation expansion is also discussed and a method for its evaluation is described.
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
We discuss the calculation of critical behavior by thinning degrees of freedom in lattice field theories. We point out the tendency of simple approximations to give incorrect critical indices, even for free-field theory. We introduce a precise definition of the thinning transformation and analyze how it induces interactions between distant lattice points. We introduce three procedures for calculating the parameters of the Hamiltonian in the thinned space, all of which give correct free-field-theory indices. For one of them, we study the interacting theory and obtain the critical indices of the large-nnn approximation, where nnn is the number of field components.
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.
This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can, in principle, be an exact real-space renormalization-group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest are achieved by introducing "look-ahead" algorithms which allow us to maintain a renormalization-group interpretation while introducing long-distance physics into the determination of variational parameters. We find that algorithms incorporating both these features are much more powerful than previous truncation algorithms. For pedagogical reasons we present our algorithms using as examples two well-understood theories, namely, free scalar field theory and the Ising model in a transverse field in 1 + 1 dimensions. Discussion of higher dimensions and less trivial theories is included in the final section. An appendix of generalized mean-field theory is also included.
In this work, we develop a new technique for the numerical study of quantum field theory. The procedure, borrowed from non-relativistic quantum mechanics, is that of finding the eigenvalues of a finite Hamiltonian matrix. The matrix is created by evaluating the matrix elements of the Hamiltonian operator on a finite basis of states. The eigenvalues and eigenvectors of the finite dimensional matrix become an accurate approximation to those of the physical system as the finite basis of states is extended to become more complete. We study a model of scalars coupled to fermions in 0+1 dimensions as a simple field theory to consider in the course of developing the technique. We find in the course of studying this model a change of basis which diagonalizes the Hamiltonian in the large coupling limit. The importance of this transformation is that it can be generalized to higher dimensional field theories involving a trilinear coupling between a Bose and a Fermi field. Having developed the numerical and analytical techniques, we consider a Fermi field coupled to a Bose field in 1+1 dimensions with the Yukawa coupling λΨΨΨ. We extend the large coupling limit basis of the 0+1 dimensional model to this case using a Bogoliubov transformation on the fermions. Although we do not use this basis in the numerical work due to its complexity, it provides a handle on the behavior of the system in the large coupling limit. In this model we consider the effects of renormalization and the generation of bound states.