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Journal of Physics A Mathematical and General

Approximation schemes for finite lattices

A.C. Irving, Chris J. Hamer

Published Mar 11, 1983• 8 citations
Publisher

Abstract

Two algorithms are formulated for systematically truncating the set of basis states required in lattice Hamiltonian calculations, thus allowing larger lattice abort sizes to be treated. They are tested for the case of the (1+1)-dimensional Ising model. It is shown that smoothly convergent finite lattice sequences are obtained, which can successfully the extrapolated to the bulk limit. The authors point out the relevance of the technique for lattice calculations in higher dimensions.

DOI
10.1088/0305-4470/16/4/019

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Two algorithms are formulated for systematically truncating the set of basis states required in lattice Hamiltonian calculations, thus allowing larger lattice abort sizes to be treated. They are tested for the case of the (1+1)-dimensional Ising model. It is shown that smoothly convergent finite lattice sequences are obtained, which can successfully the extrapolated to the bulk limit. The authors point out the relevance of the technique for lattice calculations in higher dimensions.
A method is presented to accelerate the convergence of finite-lattice sequences to their bulk limit. The calculation of highly accurate estimates of the critical parameters of the bulk system is then possible. Applied to the Hamiltonian version of the Z3 model (three-state Potts model) in (1+1) dimensions, these techniques yield estimates for the exponents gamma =1.444+or-0.0001, nu =0.8333+or-0.0003 and alpha =0.33+or-0.01. For the Z5 model, the presence of a Kosterlitz-Thouless transition is confirmed.
We present an extension of past approximations such as the Bethe approximation where we look at generalized Husimi trees. In doing so we obtain a sequence of ever more accurate approximations. The sequence is very systematic, and sequences of as few as three points can be extrapolated to give critical temperature estimates to within 0.1% or better of the exact value for our test cases. In the case of the square lattice, Ising ferromagnet, where five levels of approximation are obtained, estimates within 0.003% are obtained. Critical exponents can also be approximated but with much less accuracy. Much of our attention is devoted to the critical line of the phase diagram of the antiferromagnetic Ising model on the square lattice, where our results compare very favorably with other approximations. The method is general enough to be applicable to any spin lattice system where the spin takes on a discrete set of values.
For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.
This is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices.The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models.After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given.Finite lattice calculations on three quantum spin models are then presented.The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites.Good convergence is found for KB T ~ 0•4 and these results are compared with recent c -axis susceptibility measurements on the compound Cs 2 CoC1 4 .Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model, A perturbation analysis using non degenerate Rayleigh-Schrodingerperturbation theory is followed by a mass gap scaling analysis.These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model.In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian.This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase.Evid~nce is given that the transition from the paramagnetic to modulated phase is exponential in nature.The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,<ij> (Jl s .. s. +J2(~ . .~.)2). 1 J 1 J Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices.Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering.The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance., ACKNOWLEVGEMENTS I would like to express
The method of 'symmetrized' stochastic truncation is applied to the (2+1)-dimensional Ising model, to calculate the ground-state energy, mass gap, magnetization and susceptibility on lattices of sizes up to 6*6 sites. The method is shown to be a powerful and accurate Monte Carlo technique, provided the lattice size is not too large, and the model is suitably 'convergent'. Finite-size scaling estimates of the critical indices are nu =0.627(2), alpha =0.12(2), beta =0.324(3), and delta =1.23(1). An estimate of the combination gamma / nu +2 beta / nu =3.000(1) is obtained, in outstanding agreement with the expected hyperscaling relation.