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Journal of Physics A Mathematical and General

Magnetisations from finite-size scaling

Chris J. Hamer

Published Dec 1, 1982• 25 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/15/12/005

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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 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.
The critical exponents of the non-interacting hard-square lattice gas model are determined by means of series analysis and finite-size scaling. The series analysis leads to results for the exponents that are consistent with the expected Ising universal values. A scaling analysis is performed on different types of numerical finite-size data obtained by means of the transfer-matrix method. The results support Ising universality. A procedure based on conformal invariance reproduces the known Ising temperature and magnetic exponents within 10 -6 . Furthermore we estimate the critical density of the hard-square model as rho c =0.367 743 000 (5).
The finite lattice method of series expansion is generalised to the qqq-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2q=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26}u26 from u20u^{20}u20. The high-temperature series for the zero-field partition function is extended from v18v^{18}v18 to v22v^{22}v22. Subsequent analysis gives critical exponents in agreement with those from field theory.
A method for obtaining critical properties of physical systems is presented. Based on a recursive relation involving a physical parameter of the system, it drives the system spontaneously to the critical point, providing an efficient way to estimate critical properties. The method is illustrated for several ferromagnetic Ising systems on well-known Bravais lattices. A finite-size scaling approach is performed, by applying the method on lattices of different sizes. The efficiency of the method is confirmed by evaluating critical temperatures, as well as critical exponents, that turn up to be in good agreement with those available in the literature, with a relatively small computational effort.