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Physical review. B, Condensed matter

Linked-cluster series analysis of the Blume-Capel model

Yung‐Li Wang, Felix Lee

Published May 1, 1984• 16 citations
Publisher

Abstract

Linked-cluster series have been obtained for the Blume-Capel model in an fcc lattice to the eighth order. We show that the linked-cluster series aided by the standard extrapolation techniques provide an effective method in the study of models which display first-order phase transitions as well as second-order phase transitions. The full phase diagram and the tricritical phenomena have been studied. Results are compared with those of the Monte Carlo simulation and the high-temperature and low-temperature series analysis.

DOI
10.1103/physrevb.29.5156

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Linked-cluster series have been obtained for the Blume-Capel model in an fcc lattice to the eighth order. We show that the linked-cluster series aided by the standard extrapolation techniques provide an effective method in the study of models which display first-order phase transitions as well as second-order phase transitions. The full phase diagram and the tricritical phenomena have been studied. Results are compared with those of the Monte Carlo simulation and the high-temperature and low-temperature series analysis.
The thermodynamic behavior of the fcc Blume-Capel ferromagnet, $\mathcal{H}=\ensuremath{-}J\ensuremath{\Sigma}\stackrel{}{〈12〉}{S}^{z}(1){S}^{z}(2)+\ensuremath{\Delta}\ensuremath{\Sigma}\stackrel{}{1}{[{S}^{z}(1)]}^{2}\ensuremath{-}h\ensuremath{\Sigma}\stackrel{}{1}{S}^{z}(1), {S}^{z}=0, \ifmmode\pm\else\textpm\fi{}1,$ is studied by series-extrapolation techniques. By using both high- and low-temperature series, we are able to trace first- and second-order branches of the phase boundary and examine behavior at temperatures both above and below the phase transition. We find a tricritical point at $\frac{{k}_{B}{T}_{t}}{12J}=0.2615\ifmmode\pm\else\textpm\fi{}0.0070$, $\frac{{\ensuremath{\Delta}}_{t}}{12J}=0.4716\ifmmode\pm\else\textpm\fi{}0.0010$. Tricritical exponents are consistent with ${\ensuremath{\gamma}}_{t}={\ensuremath{\gamma}}_{t}^{\ensuremath{'}}=1$, ${\ensuremath{\beta}}_{t}=\frac{1}{4}$, ${\ensuremath{\phi}}_{t}={\ensuremath{\nu}}_{t}={\ensuremath{\alpha}}_{t}={\ensuremath{\alpha}}_{t}^{\ensuremath{'}}=\frac{1}{2}$, in good agreement with tricritical mean-field theory and the Gaussian tricritical fixed point of Riedel and Wegner.
A linked-cluster expansion technique for spin systems has been used to obtain the free energy, the magnetization, and the susceptibility series for three Ising systems: (1) The Ising model; (2) The Blume-Capel model; and (3) The Blume-Emery-Griffiths (BEG) model. Each Hamiltonian is first divided into a single-ion potential and a term describing the interaction of the spin fluctuations. Only the latter is treated as perturbation in the series expansion. In the parametric phase our series reduces to the exact high-temperature series. However, in the ordered phase the same series can be used to study the low temperature behavior, such as the variation of the magnetization with temperature. The analysis of the eighth-order series for systems (1) and (2) show results in excellent agreement with the previous ones obtained from the high-temperature and low-temperature series. The series analysis for the BEG model shows quite different behavior from that predicted by the mean-field approximation.
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.
The Cluster Variation Method (CVM), in its recent formulation using the Moebius Inversion, is applied to study the critical properties of the face-centered-cubic Blume-Capel model in the triangle and tetrahedron approximations. We develop a procedure which preserves the characteristics of Kikuchi’s Natural Iteration Method, moreover reducing considerably the number of involved variables. The resulting phase diagram is compared with that obtained from lower orders of CVM approximation (single-site and pair) and from other methods (series expansions, Monte Carlo, …). The tetrahedron approximation shows a good agreement with these high-precision methods, in particular for the location of the tricritical point. The behavior of order parameters and multisite (up to four-body) correlation functions is determined for all significant regions of the phase space.
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.