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Journal of Physics C Solid State Physics

Two-point correlations in random uniaxial systems at marginal dimensionality

Bosiljka Tadić, R. Pirc

Published Feb 29, 1984• 3 citations
Publisher

Abstract

The two-point correlation function for a random Ising model at d=4 is obtained using the solutions of the renormalisation-group recursion relations in the range of temperatures where the system exhibits non-asymptotic critical behaviour, the origin of which lies in a particular degeneracy of the RG recursion relations. The effects of lattice compressibility are also studied with two types of boundary conditions, corresponding to constant volume and pressure, respectively. A comparison with pure Ising systems is made.

DOI
10.1088/0022-3719/17/6/011

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The two-point correlation function for a random Ising model at d=4 is obtained using the solutions of the renormalisation-group recursion relations in the range of temperatures where the system exhibits non-asymptotic critical behaviour, the origin of which lies in a particular degeneracy of the RG recursion relations. The effects of lattice compressibility are also studied with two types of boundary conditions, corresponding to constant volume and pressure, respectively. A comparison with pure Ising systems is made.
Renormalization group methods are applied to study the critical behaviour of a compressible n-component Ising model with short range interactions at d = 4 and a one component Ising model with dipolar interactions at d = 3. The recursion equations are exactly solved in the case of an elastic system of spherical symmetry (d = 4) or cylindrical symmetry (d = 3); new types of logarithmic corrections, corresponding to a Fisher renormalization at marginal dimensions, are found. It is shown that the system exhibits a first-order transition for constant pressure external conditions or when anisotropy is taken into account. The relevance of the calculations to the critical behaviour of uniaxial ferroelectrics is discussed.
Abstract The effect of quenched random fields on the zero‐temperature critical behaviour is studied by the means of the Wilson‐Fisher renormalisation‐group recursion relations to first order in the ϵ‐expansion. A number of quantum models is considered. The simultaneous influence of both, the random‐field and the classical‐to‐quantum dimensional crossover phenomena on the quantum critical behaviour is investigated. A special attention is paid to the properties of some reasonable particular forms of Gaussian random correlations which might be of relevance to the description of natural systems.
The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.