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Physical review. B, Solid state

Finite-lattice approximations to renormalization groups

Thomas L. Bell, Kenneth Geddes Wilson

Published May 1, 1975• 114 citations
Publisher

Abstract

Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter $b$. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of $b$ around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of $b$ used in the calculation. We study the problem of identifying the correct value of $b$.

DOI
10.1103/physrevb.11.3431

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Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter bbb. The exact transformation has fixed points only for b=2\ensuremath−52b={2}^{\ensuremath{-}\frac{5}{2}}b=2\ensuremath−25​. The approximate transformation has fixed points for a range of values of bbb around 2\ensuremath−52{2}^{\ensuremath{-}\frac{5}{2}}2\ensuremath−25​. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.
Most renormalization groups studied heretofore are linear, in that the block-spin variables are linearly related to the old spin variables. These renormalization groups all require an adjustable parameter that must be properly fixed in order that the renormalization-group transformation have a useful fixed point. Niemeijer and van Leeuwen, however, have investigated the critical behavior of the two-dimensional Ising model with a nonlinear renormalization group, and find a fixed point without introducing such a parameter. We introduce here another nonlinear renormalization group and study it in detail for the Gaussian model. We find that (i) within certain limits, no such parameter need be adjusted in order to reach a fixed point; (ii) an eigenvalue of the linearized transformation with no physical significance depends on the nonlinearity of the transformation; and (iii) a physically significant eigenvalue is unchanged (to the order examined).
We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.
Matching procedures, related to the group structure of block-spin renormalization, are proposed for an optimal choice of parameter-dependent transformations. The method also allows to determine the parameter's dependence on the reduced coupling near the fixed point. Applications to linear transformations for two- and three-dimensional Ising models and two-dimensional three-, four-, and five-state Potts systems are reported. With simple approximations good estimates of critical couplings and exponents are obtained. Results for the two-dimensional spin-\textonehalf{} XY\mathrm{XY}XY system and their possible implications, as far as an eventual transition is concerned, are discussed. Some applications to nonlinear transformations are also considered.
The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
Abstract In Chapter 8 we gave an introduction to the formalism of the renormalization group. A renormalization transformation changes the scale of a system by a factor b. Systems at criticality flow to fixed points under the transformation and their critical exponents and scaling behaviour are determined by linearizing the recursion equations about the fixed point. The aim of this chapter is two-fold: firstly to illustrate these ideas by giving several examples of renormalization group transformations and secondly to demonstrate the ways in which they can be used to calculate critical properties. Our examples are divided into three classes. In Section 9.1, we shall consider a one-dimensional model where the recursion relations can be written down exactly. The drawback of this as a canonical example is that any critical behaviour is at zero temperature. The decimation transformation presented in one dimension is an example of a real-space transformation and the second task, in Section 9.2, is to show how these may be used approximately to treat models in higher dimensions. Thirdly, in Section 9.5, we shall mention the E-expansion where the renormalization takes place in momentum space. This enables an understanding of the crossover to mean-field behaviour. However, the formalism is somewhat involved and the details are beyond the scope of this book.