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Physical Review A

Renormalization-group study of the crossover to classical behavior for an Ising model with vacancies

J.J.M. Rijpkema

Published Feb 1, 1984• 3 citations
Publisher

Abstract

By using Kadanoff's variational renormalization-group method an Ising model with vacancies is investigated for dimensions up to $d=4$ to study the crossover to classical behavior. Up to a crossover dimension $d\ensuremath{\simeq}3.4$, both an optimal critical and tricritical fixed point are located with critical indices that agree well with other data. At $d\ensuremath{\simeq}3.4$, both fixed points merge, representing in this way the crossover to classical behavior. At $d>3.4$, indications are found that the two fixed points do not annihilate but form a "marginal" pair that describes both critical and tricritical behavior.

DOI
10.1103/physreva.29.859

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The critical behavior of the Blume-Capel model for d=2,3d=2,3d=2,3 is analyzed using a renormalization transformation based on Kadanoff's lower-bound transformation. There are fixed points associated with first-order, second-order, and tricritical phase transitions. The discontinuity fixed point onto which the line of first-order phase transition is mapped has two relevant eigenoperators with eigenvalue 2d{2}^{d}2d, corresponding to the discontinuities in $〈\ensuremath{\sigma}〉$ and $〈{\ensuremath{\sigma}}^{2}〉$. Quite different values of the variational parameters in the renormalization transformation optimize the free energy at each fixed point, making it difficult to calculate crossover behavior with the present approach.
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Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.