Improved phenomenological renormalization schemes

Abstract

An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent.

The anisotropic 3D Ising model

“…The upper bound was found by an analysis of the asymptotic properties of self-avoiding random walks on the lattice under consideration. More recently, Yurishchev [7] derived upper and lower bounds on the critical temperature by using the transfer-matrix technique and an extended phenomenological renormalization group theory approach [8,9]. Yurishchev showed that for a relatively small anisotropy ( 510 À3 ) the accuracy of equation (1) is rather low.…”

Universality in the partially anisotropic three-dimensional Ising lattice

“…Thus, the transfer matrices for which the eigenproblems was solved were dense matrices of sizes up to 65 536 × 65 536. To solve the eigenproblem we took into account the internal and lattice symmetries of subsystems and used the block-diagonalization method (see, e. g., [9,7]). Calculations were performed on an 800 MHz Pentium III PC running the FreeBSD operating system.…”

“…locate T c more accurately in comparison with the ordinary RG equation ( can be evaluated by standard formulas via the eigenvalues and eigenvectors of transfer matrices (see, e. g., [7,8,9]).…”

Double potts chain and exact results for some two-dimensional spin models

“…In other words, the amplitudes of all finite-size corrections to the critical internal energy of a system u ∞ (K c ) are equal to zero. For the square isotropic Ising lattice, the derivative of the inverse correlation length κ L (K) with respect to a temperature-like variable K has a similar property [20,21]:…”