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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Tricriticality and the Blume-Capel model: A Monte Carlo study within the microcanonical ensemble

Markus Deserno

Published Nov 1, 1997• 75 citations
Publisher

Abstract

The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.

DOI
10.1103/physreve.56.5204

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The microcanonical partition function \ensuremathΩ\ensuremathα,N(E){\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)\ensuremathΩ\ensuremathα,N​(E) of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios \ensuremathα\ensuremath{\alpha}\ensuremathα close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiL spins with L\ensuremath∈8,10,12,14,18L\ensuremath{\in}{8,10,12,14,18}L\ensuremath∈8,10,12,14,18 was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields \ensuremathα(t)=2.84479\ifmmode±\else\textpm\fi0.00030{\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030\ensuremathα(t)=2.84479\ifmmode±\else\textpm\fi0.00030 for the tricritical coupling ratio, and kBT(t)/J=1.4182\ifmmode±\else\textpm\fi0.0055{k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055kB​T(t)/J=1.4182\ifmmode±\else\textpm\fi0.0055 for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.
We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.
Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.
A microcanonical Monte Carlo algorithm has been developed to calculate the density of states of a two-dimensional Blume-Capel model in zero field. The full density of states is calculated for a 16*16 lattice as a function of the two spin summations which appear in the Hamiltonian. This permits the partition function and related thermodynamic functions to be evaluated for any temperature and any single site anisotropy from a single data set. Results are presented for the pseudo-critical transition temperature and the maximum in the specific heat as a function of the single site anisotropy parameter. The technique will allow the region near the tricritical point to be explored in detail and will permit the future determination of the zeros of the partition function. It should be noted that the microcanonical sampling method of Lee (1990) was found to be unusably slow for the S=1 problem in regions were the density of states had a significant slope and the microcanonical sampling described in this work is a significant improvement on his sampling method.
We use a thermodynamically self-consistent theory to obtain the phase diagram of the ferromagnetic spin-3/2 Blume-Capel model on the simple cubic lattice. The theory is based on an Ornstein-Zernike approximation where the direct correlation function is truncated and the dependence upon the thermodynamic variables is determined by a set of two coupled partial differential equations. Within this framework, we localize the critical line in zero external field with high accuracy and in good agreement with previous Monte Carlo analysis. At low temperature, in contrast with Monte Carlo results, we find a first-order transition line ending at a critical end point whose coordinates are given by (k(B)T(c)/Jc=0.213+/-0.003,Delta(c)/Jc=0.491+/-0.001).
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (Tt{T}_{t}Tt​=0.610\ifmmode\pm\else\textpm\fi{}0.005, Dt{D}_{t}Dt​=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.