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

Orthogonal transforms and lattice field theories

R. Blankenbecler, John L. Richardson

Published May 15, 1980• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.21.2867

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•Mar 1, 1982•0 citations
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 \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 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.
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.
Real-space renormalization methods which simply shift and rescale the interaction parameters in the nearest-neighbor Heisenberg model are investigated for two-dimensional lattices of equivalent sites. Results are presented for the hexagonal, square-planar, triangular, and Kagome lattices via three different renormalization techniques. The first, which has been studied for some time, uses perturbation theory to evaluate the renormalized interactions. The second uses the variational method to improve on the perturbative results. The third method is based on a cluster expansion and is found to give much improved agreement with numerical results form Monte Carlo calculations, but it does not provide a variational bound to the exact solution. The dependence of all three methods on the size and shape of the renormalized block of sites is also investigated.
In evaluating the critical properties of lattice spin systems in the real-space renormalization-group theory we use the cluster variation method. A configuration in the transformed system is constrained and the probability of occurrence of this configuration is calculated both in the transformed system and in the original system. By equating the two probabilities and forming ratios of two such equalities (for two or more constrained configurations) the fixed point of the renormalization transformation is evaluated. The method can avoid the trouble due to different singularities in the original and transformed systems, and hence can obviate the possible development of spurious singularities in the transformation at low temperatures. The two-dimensional triangular Ising model is treated with numerical results comparable with those obtained by the cluster treatment of Niemeijer and van Leeuwen who used more and larger cluster types than those we introduce.
A real-space renormalization-group transformation is constructed between blocks of linear dimensions bL\mathrm{bL}bL and LLL to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.
The real-space dynamic renormalization group method developed in previous papers is applied to the kinetic Ising model defined on a square lattice. In particular we extend the formalism to the calculation of space- and time-dependent equilibrium averaged correlation functions. We find that conventional methods for implementing the real-space renormalization group via cumulant expansions in terms of the intercell coupling lead to correlation functions which decay algebraically in space at large distances in the disordered phase in qualitative disagreement with the known exponential decay. We indicate how one can develop new perturbation theory expansion methods which lead to the proper exponential decay at large distances and also lead to good quantitative results for other observable quantities like the magnetization, susceptibility, and single spin time autocorrelation function. As the result of a first-order calculation we obtain excellent results for the static critical exponents and a value of z=1.79z=1.79z=1.79 for the dynamic critical exponent. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations.
This dissertation involves the renormalization group theory and its application to two systems exhibiting renormalization symmetry. Both the real space and the field theoretic formulations of the renormalization group theory are discussed. A continuum of real space renormalization transformations is used to study the critical behavior of the two-dimensional Ising model. We find that there exists an optimal transformation that produces sharply more accurate critical properties that the others. We establish a set of conditions that characterize the optimal transformation. These conditions suggest criteria that could enable one to control a given real space renormalization approximation. Field theory renormalization is used to exploit the renormalization symmetry in the excluded volume problem. We develop a self-contained formalism that incorporates the concepts and methods of field theory renormalization directly into the language of the excluded volume problem. This formalism avoids the Lagrangian model isomorphism, thereby providing a simpler and more natural framework for underst and ing the excluded volume problem from first principles. This formalism is used to investigate the spatial correlations in a large self-repelling ring polymer. In particular, we find that the ratio of the radius of gyration of a ring polymer to that of a linear polymer having the same number of monomers is a universal quantity whose value we calculate to first order in (epsilon) (TBOND) 4 - (dimension of space). In addition, we study the mutual excluded volume characterizing a system of two chains with different lengths. The renormalization of the polymer perturbation theory through two-loop order enables us to calculate the universal value of the interpenetration function to second order in (epsilon). Our renormalization group results are in very good agreement with the results obtained from numerical and experimental studies of rings and chains in three dimensional space. We also discuss the connection between real space and field theory renormalization. We outline the basic conceptual ideas that underlie the general theory of renormalization and illustrate how each of these ideas is realized in the real space and the field theoretic formulations.