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Journal of Physics C Solid State Physics

Phase transition and critical correlation functions in the spin-1/2Ising model on a diamond lattice

Ole G. Mouritsen

Published Jul 20, 1980• 11 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/13/20/013

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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 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.
Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
The singularity of the heat capacity for the spin- 1 / 2 Ising ferromagnet arrayed on simple cubic and diamond lattices is investigated through the analysis of high-precision Monte Carlo data for the internal energy. The analysis incorporates confluent singularities. Two least-squares fitting procedures are considered. In the first one, the leading confluent singularity amplitude is zero and the next confluent singularity is included. For both lattices this procedure gives accordance with: (i) the universal value for the ratio of the leading singularity amplitudes predicted from the epsilon expansion and (ii) the theoretical expectation of equality between the non-singular parts of the heat capacity above and below the critical temperature. In the second least-squares fitting procedure, the leading confluent singularity is finite. This leads to a fit which disagrees with both (i) and (ii). The Monte Carlo results suggest that the failure of the series analysis to accord with (i) and (ii) may be due to neglect of a rather large confluent singularity in the ferromagnetic phase.
Much attention has been paid to the d=3, s= 1 / 2 Ising model on the BCC lattice in recent years because the best high-temperature series expansions were available for this lattice. However, in order to compare series estimates of critical exponents and temperatures with independent evaluations via the Monte Carlo method, lattices of lower coordination number must be considered, since Monte Carlo studies are usually made on these. A study of extant susceptibility series on the FCC, SC, diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent Monte Carlo analyses of the d=3, s= 1 / 2 Ising model on the SC and diamond lattices.