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Journal of Physics Condensed Matter

An analysis of the Monte Carlo diamond lattice calculation using an exact reduction relation

B. Frank, C Y Cheung

Published May 10, 1993• 1 citation
Publisher

Abstract

A reduction relation is noted which allows one to relate exactly, at any temperature T, two quantities for the Ising model on the diamond lattice: (i) the ratio, essentially, of the average neighbour-neighbour correlation function and the energy per spin, P=T 2 /U 2 and (ii) a ratio of a certain pair of four-spin correlation functions Q=T4/U4. This relation is applied to Mouritsen's Monte Carlo determination of these quantities P c and Q c , at the critical point T c . It is shown that the Monte Carlo values are severely inconsistent with the exact relation. A more reliable value of the neighbour-neighbour correlation function, ( sigma 1 sigma 2 ) approximately=0.283, is calculated from the Monte Carlo data.

DOI
10.1088/0953-8984/5/19/004

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A reduction relation is noted which allows one to relate exactly, at any temperature T, two quantities for the Ising model on the diamond lattice: (i) the ratio, essentially, of the average neighbour-neighbour correlation function and the energy per spin, P=T 2 /U 2 and (ii) a ratio of a certain pair of four-spin correlation functions Q=T4/U4. This relation is applied to Mouritsen's Monte Carlo determination of these quantities P c and Q c , at the critical point T c . It is shown that the Monte Carlo values are severely inconsistent with the exact relation. A more reliable value of the neighbour-neighbour correlation function, ( sigma 1 sigma 2 ) approximately=0.283, is calculated from the Monte Carlo data.
The critical properties of the spin- 1 / 2 Ising ferromagnet on a diamond lattice are calculated using the Monte Carlo technique. The heat capacity, the susceptibility, the magnetisation, and various pair and multispin correlation functions are calculated as functions of the temperature. The critical temperature, and the critical parameters pertaining to the magnetisation, are determined. The results are consistent with series analyses. Using the critical values of the correlation functions, it is shown that the basic assumption in the theory of Frank and Mitran (1978) is not exact for the diamond lattice.
A method of critical index calculation is proposed. The method is based on matching the temperature values of two different size lattices by finding the optimized solution of the equation t'=bTY{\mathit{b}}_{\mathit{T}}^{\mathit{Y}}bTY​t, where t and t' are the reduced temperatures of the lattices, b is the ratio of lattice sizes, and YT{\mathit{Y}}_{\mathit{T}}YT​ is the thermal critical index. The values of t and t' are obtained from the fourth cumulant of the order parameter, which satisfies the relation Qr{\mathit{Q}}_{\mathit{r}}Qr​(t,L)=Qr{\mathit{Q}}_{\mathit{r}}Qr​(t'L') around the critical point. By using this method, the critical indices of the two-dimensional Ising model and the (2+1)-dimensional lattice gauge theory are calculated. The advantage of the method is that one can obtain the critical point and the critical indices with high accuracy by performing a relatively small number of Monte Carlo iterations.
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).
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo 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.
We present a new analysis on Monte Carlo Renormalization Group (MCRG) results obtained earlier by means of the Delft Ising System Processor (DISP). The MCRG data involve a total of 57 coupling constants, 36 even and 21 odd. Simulations were carried out for simple cubic lattices with 643, 323 and 163 spins. The RG transformation is assumed to be analytic. A number of relations exist between correlation functions at different renormalization levels. Some of these involve the derivatives of the stability matrix. These correlation functions enable an analysis of the so-called regular part of the RG transformation. If the Hamiltonian of the original lattice only contains nearest-neighbour couplings then the regular contributions to the specific heat and the magnetic susceptibility can be easily determined. These contributions must depend only weakly on the initial lattice size, at least if the RG transformation is analytic. We investigated whether this is indeed true when the majority-rule is applied. New simulations involving higher-order correlations will enable us to study the analytic contributions in more detail.