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

Thinning degrees of freedom in lattice field theories

John B. Bronzan, Robert L. Sugar

Published Mar 15, 1980• 7 citations
Publisher

Abstract

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-$n$ approximation, where $n$ is the number of field components.

DOI
10.1103/physrevd.21.1564

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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.
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 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.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
Abstract This chapter is devoted to a brief review of general properties of phase transitions in macroscopic physics and, in particular in lattice models. Some of these lattice models actually appear as lattice regularizations of Euclidean (imaginary time) quantum physics theory (QFT). Most of the transitions considered in this work have the following character: spins on the lattice, or macroscopic particles in the continuum, interact through short-range forces, assumed, for simplicity, to decay exponentially. For simple systems, it is possible to find a local observable, called order parameter, whose expectation values depend on the phase in the several phase region, for example, the spin in ferromagnetic systems. In the disordered phase, the connected two-point function decreases exponentially at large distance, at a rate characterized by the correlation length (the inverse of the smallest physical mass in particle physics). In continuous transitions, the correlation length diverges at the critical temperature. Within the mean-field approximation (consistent with Landau's theory of critical phenomena), it can be shown that the singular behaviour of thermodynamic quantities at the critical temperature is universal. These properties can also be reproduced by calculating correlation functions with a perturbed Gaussian measure. It is then shown that the leading corrections to the mean-field approximation, in Ising-like systems, diverge at the critical temperature for dimensions smaller than or equal to 444.
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.