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Journal of Physics A Mathematical and General

Linear defects in two dimensional systems: a finite size investigation

M. P. Nightingale, Henk W. J. Blöte

Published Jan 1, 1982• 36 citations
Publisher

Abstract

Finite-size scaling theory is employed to study non-universal and crossover critical behaviour in two-dimensional systems with linear defects. The finite-size method is shown to reproduce very accurately the non-universality as known to the present in the case of the Ising model. The calculations show that the non-universal behaviour of all the models investigated in the Ising universality class can be described approximately by a single universal curve.

DOI
10.1088/0305-4470/15/1/007

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Finite-size scaling theory is employed to study non-universal and crossover critical behaviour in two-dimensional systems with linear defects. The finite-size method is shown to reproduce very accurately the non-universality as known to the present in the case of the Ising model. The calculations show that the non-universal behaviour of all the models investigated in the Ising universality class can be described approximately by a single universal curve.
Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent zzz of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for zzz; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate zzz. Even in this case we find that zzz depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.
Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.
Based on Novotny's transfer-matrix method, we simulated the (stacked) triangular Ising anti-ferromagnet embedded in the space with the dimensions variable in the range 2 < or = d < or = 3. Our aim is to investigate the criticality of the XY universality class for 2 < or = d < or = 3. For that purpose, we employed an extended version of the finite-size-scaling analysis developed by Novotny, who utilized this scheme to survey the Ising criticality (ferromagnet) for 1 < or = d < or = 3. Diagonalizing the transfer matrix for the system sizes N up to N = 17 , we calculated the d -dependent correlation-length critical exponent nu(d). Our simulation result nu(d) appears to interpolate smoothly the known two limiting cases, namely, the Kosterlitz-Thouless (KT) and d = 3 XY universality classes, and the intermediate behavior bears close resemblance to that of the analytical formula via the 1/N-expansion technique. Methodological details including the modifications specific to the present model are reported.
We determine the scaling functions describing the crossover from Ising-like critical behavior to classical critical behavior in two-dimensional systems with a variable interaction range. Since this crossover spans several decades in the reduced temperature as well as in the finite-size crossover variable, it has up to now largely evaded a satisfactory numerical determination. Using a dedicated Monte Carlo method, we could obtain accurate results for sufficiently large interaction ranges. Our data cover the full crossover region both above and below the critical temperature and support the hypothesis that the crossover functions are universal. Also the so-called effective exponents are discussed and we show that these can vary nonmonotonically in the crossover region.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.