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Journal of Physics A Mathematical and General

Finite size scaling and universality in the (2+1)D Ising model

Malte Henkel

Published Aug 21, 1987• 29 citations
Publisher

Abstract

The energy and magnetisation density of the (2+1)D Ising model are investigated. For the thermal exponent the author finds x epsilon =1.42+or-0.02, in agreement with hyperscaling. The Privman-Fisher universality hypothesis is confirmed by two methods. First, the author presents, at the critical point, numerical data for the finite-size scaling amplitudes of the (singular) free energy, magnetisation, susceptibility and surface tension and checks them for universality. Second, using the explicitly computed eigenvectors of the Hamiltonian, it is shown that the ratios of matrix elements of the energy and magnetisation density are universal.

DOI
10.1088/0305-4470/20/12/041

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The energy and magnetisation density of the (2+1)D Ising model are investigated. For the thermal exponent the author finds x epsilon =1.42+or-0.02, in agreement with hyperscaling. The Privman-Fisher universality hypothesis is confirmed by two methods. First, the author presents, at the critical point, numerical data for the finite-size scaling amplitudes of the (singular) free energy, magnetisation, susceptibility and surface tension and checks them for universality. Second, using the explicitly computed eigenvectors of the Hamiltonian, it is shown that the ratios of matrix elements of the energy and magnetisation density are universal.
A method of obtaining the spontaneous magnetisation from finite-lattice matrix elements of the magnetic field operator, due to Yang (1952) and Uzelac (1980), is discussed. The method is demonstrated for the case of the Ising model in (1+1) dimensions, and is shown to provide smooth and rapidly convergent finite-lattice sequences. Applied to the case of the three-state Potts (Z 3 ) model in (1+1) dimensions, the method yields an estimate beta =0.111 09+or-0.000 05 for the critical exponent. This confirms Alexander's conjecture of universality with the hard hexagonal model.
The structure of finite-size scaling within the framework of the field-theoretic renormalization-group approach is examined. New size-dependent ultraviolet divergences are introduced into the theory. It is shown, nonetheless, that ``two-scale-factor'' universality reflects itself in the finite-size free energy and produces a scaling form such as that discussed by Privman and Fisher. A consequence is that the singular part of the finite-system free energy at the bulk critical point is universal. Extensions to surface and interfacial free energies, first proposed from the results of Ising simulations, are supported qualitatively.
The five-dimensional Ising model with free boundary conditions has recently received a renewed interest in a debate concerning the finite-size scaling of the susceptibility near the critical temperature. We provide evidence in favour of the conventional scaling picture, where the susceptibility scales as L2 inside a critical scaling window of width 1/L2. Our results are based on Monte Carlo data gathered on system sizes up to L=79 (ca. three billion spins) for a wide range of temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility and also the scaling and distribution of the size of the Fortuin–Kasteleyn cluster containing the origin. The probability of this cluster reaching the boundary determines the correlation length, and its behaviour agrees with the mean field critical exponent δ=3, that the scaling window has width 1/L2.