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Journal of Physics A Mathematical and General

Critical phenomena during a dimensional crossover

Denjoe O’Connor, Christopher Rhodes Stephens

Published Jan 7, 1992• 21 citations
Publisher

Abstract

The critical behaviour of a system undergoing a dimensional crossover is investigated. A renormalization group equation is obtained that interpolates between different dimensions. The susceptibility of an Ising-like system is treated in some detail, in particular and effective critical exponent for the susceptibility is computed, which for the particular crossover examined interpolates between the susceptibility exponents appropriate for a (4- in )-dimensional and a (3- in )-dimensional Ising model.

DOI
10.1088/0305-4470/25/1/014

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The critical behaviour of a system undergoing a dimensional crossover is investigated. A renormalization group equation is obtained that interpolates between different dimensions. The susceptibility of an Ising-like system is treated in some detail, in particular and effective critical exponent for the susceptibility is computed, which for the particular crossover examined interpolates between the susceptibility exponents appropriate for a (4- in )-dimensional and a (3- in )-dimensional Ising model.
The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.
The equality φ = γ between the exponent φ, which describes crossover from zero to finite-random-field critical behaviour, and the order parameter susceptibility exponent, γ, is shown to be replaced by φ > γ when the random exchange dominates. Rough renormalization group estimates yield φ/γ ≃ 1.1 for three-dimensional random Ising models. The implications to experiments on dilute antiferromagnets in uniform fields are discussed. Similarly to spin glasses, the nonlinear susceptibility of the latter is shown to diverge as ( T - T N ) -0.7 .
Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.
We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .
The renormalization group approach to critical phenomena is developed for quantum mechanical problems with non-commuting operators. Applying the theory to the spin 1 / 2 Ising model with a transverse field Gamma it is found that the critical exponents are those of the Ising model with Gamma =0 if the transition occurs at T>0. However, for transitions at T=0 the critical behaviour of the d-dimensional transverse system corresponds to that of the (d+1)-dimensional Ising model with Gamma =0, in agreement with series expansion predictions. At T=0 the dynamic, as well as static, critical behaviour is given by mean field theory for d>3.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.