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Journal of Physics C Solid State Physics

A self dual relation for a three dimensional assembly

David W. Wood

Published Jul 24, 1972• 16 citations
Publisher

Abstract

An Ising model, which is defined on a face centred cubic lattice in which nearest neighbour spins are coupled via a special four spin coupling term involving the 4-point simplex subgraphs of the lattice, is shown to have a self dual property of the type originally put forward by Onsager for the conventional square net Ising model problem. Assuming that this model has one and only one critical point, the location of the singularity can be determined exactly.

DOI
10.1088/0022-3719/5/14/003

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An Ising model, which is defined on a face centred cubic lattice in which nearest neighbour spins are coupled via a special four spin coupling term involving the 4-point simplex subgraphs of the lattice, is shown to have a self dual property of the type originally put forward by Onsager for the conventional square net Ising model problem. Assuming that this model has one and only one critical point, the location of the singularity can be determined exactly.
We consider a special case of the -component cubic model on the square lattice, for which an expansion exists in Ising-type graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determine the critical points for several values of . Furthermore we determine several universal quantities, including three critical exponents. For , these results agree well with the theoretical predictions for the critical branch. This model is also a special case of the model of Domany and Riedel. It appears that the self-dual plane of the latter model contains the exactly known critical points of the and 2 cubic models. For this reason we have checked whether this is also the case for . However, this possibility is excluded by our numerical results.
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.
A simple renormalisation-group approach based on self-dual clusters is proposed for the two-dimensional nearest-neighbour spin- 1 / 2 Ising model on the square lattice; it reproduces the exact critical point. The authors calculate the internal energy and the specific heat for vanishing external magnetic field, spontaneous magnetisation and the thermal (y T ) and magnetic (y H ) critical exponents. The results obtained from the first four smallest cluster sizes strongly suggest the convergence towards the exact values when the cluster size increases. Even for the smallest cluster, where the calculation is very simple, the results are quite accurate, particularly in the neighbourhood of the critical point.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
For planar lattices, there exists a dual relation between extended (blocked) Ising models in which interactions are expressed by face elements (blocks) instead of bonds. The total interaction for a block contains all even spin products among spins on the face (g-polygon); for g = 2 or 3, it contains only two-spin product, for g = 4 it contains crossed bond interactions and four-spin product, and so on. Formally, there exists a self-dual lattice for every g. The triangular lattice is self-dual for g = 3 and the self-dual lattice for g ≥4 has non integer coordinate number (a rational number) which suggests a universality for the coordination number.