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Geometric effects on critical behaviours of the Ising model

Hiroyuki Shima, Yasunori Sakaniwa

Published May 5, 2006• 29 citations
OpenAlex

Abstract

We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.

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We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.
We investigate the dynamic critical exponent of the two-dimensional Ising model defined on a curved surface with constant negative curvature. By using the short-time relaxation method, we find a quantitative alteration of the dynamic exponent from the known value for the planar Ising model. This phenomenon is attributed to the fact that the Ising lattices embedded on negatively curved surfaces act as ones in infinite dimensions, thus yielding the dynamic exponent deduced from mean field theory. We further demonstrate that the static critical exponent for the correlation length exhibits the mean field exponent, which agrees with the existing results obtained from canonical Monte Carlo simulations.
The critical properties of the xy model with nearest-neighbour interactions on a two-dimensional square lattice are studied by a renormalization group technique. The mean magnetization is zero for all temperatures, and the transition is from a state of finite to one of infinite susceptibility. The correlation length is found to diverge faster than any power of the deviation from the critical temperature. Analogues of the strong scaling laws are derived and the critical exponents, eta , and delta , are the same as for the two-dimensional Ising model.
We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power κ of the distance l from the free surface with an oscillating amplitude Al=(−1)flAA_l = ( - 1)^{f_l } AAl​=(−1)fl​A where f l = 0, 1 follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent κ, the wandering exponent ω which governs the fluctuation of the sequence and the bulk correlation length exponent ν. Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the (κ, ω)-plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.