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Physical review. B, Solid state

Nonlinear renormalization groups

Thomas L. Bell, Kenneth Geddes Wilson

Published Nov 1, 1974• 95 citations

Abstract

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).

DOI: 10.1103/physrevb.10.3935

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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).

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

.

A renormalization-group technique is used to study the critical behavior of spin models in which each interaction has a small independent random width about its average value. The cluster approximation of Niemeyer and Van Leeuwen indicates that the two-dimensional Ising model has the same critical behavior as the homogeneous system. The $\ensuremath{\epsilon}$ expansion for

n

-component continuous spins shows that this behavior holds to first order in $\ensuremath{\epsilon}$ for

n>4

. For

n<4

, there is a new stable fixed point with $2\ensuremath{\nu}=1+[\frac{3n}{16(n\ensuremath{-}1)}]\ensuremath{\epsilon}$.

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{}

\mathrm{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 two-dimensional

\mathrm{XY}

and Heisenberg models are studied with a quantum-mechanical generalization of the Niemeijer and Van Leeuween renormalization-group method. In second-order cumulant expansion a fixed point exists for the

\mathrm{XY}

model, giving a critical temperature at $\frac{J}{k{T}_{c}}\ensuremath{\cong}0.87\ifmmode\pm\else\textpm\fi{}0.05$. For the Heisenberg model we find a fixed point at

{K}^{*}=2.8

. This result may not be reliable, since the truncation of the cumulant expansion is only valid for small

K

.