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Journal of Physics C Solid State Physics

A Monte Carlo study of the critical behaviour of compressible Ising systems

Bikas K. Chakrabarti, N Bhattacharyya, Suman Sinha

Published Aug 20, 1982• 14 citations
Publisher

Abstract

The discontinuous (first-order) transition behaviour of compressible Ising systems described by Hamiltonians of the form H=-J s Sigma (ij) S i S j -J 4 Sigma (ij) Sigma (kl) S i S j S k S l has been studied for the first time using Monte Carlo techniques for lattices of sizes 100 2 and 20 3 . The results have been compared with those obtained using the renormalisation group technique.

DOI
10.1088/0022-3719/15/23/010

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The discontinuous (first-order) transition behaviour of compressible Ising systems described by Hamiltonians of the form H=-J s Sigma (ij) S i S j -J 4 Sigma (ij) Sigma (kl) S i S j S k S l has been studied for the first time using Monte Carlo techniques for lattices of sizes 100 2 and 20 3 . The results have been compared with those obtained using the renormalisation group technique.
A position space renormalisation group procedure is developed for the compressible Ising model. The procedure is essentially a generalisation of the cell variable approach of Niemeijer and Van Leeuwen (1974) in that vibrational cell variables are introduced in addition to spin cell variables. The procedure is applied to the two-dimensional triangular lattice, the shearless square lattice. In both cases the rigid Ising fixed point is stable to spin-lattice coupling as long as the latter is small. Beyond a critical strength of the spin-lattice coupling, the system develops instability leading to, presumably, a first-order transition. For the shearless square lattice the conclusion is at variance with the exact result of Baker and Essam (1971), so the approximation is regarded as unreliable for larger values of the spin-lattice coupling (1971).
Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 and 12\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is Kc{\mathit{K}}_{\mathit{c}}Kc​=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.
The Hamiltonian H=-J 2 Sigma (ij) sigma i sigma j -(J 4 /N) Sigma (ij)(kl) sigma i sigma j sigma k sigma l where the first term is a nearest-neighbour pair interaction and the second term is an infinite-range pair-pair interaction, is proposed as a model of an Ising magnet with lattice coupling. The free energy is reduced to that of a rigid Ising model on the same lattice but with a modified pair interaction. In two and three dimensions, the phase transition is found to be first order for all J 4 . The relationship between this result and recent renormalization group calculations is discussed.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.