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Physical Review B

Classical Heisenberg lattice-gas model: Thermodynamics and phase diagrams

Hassan Chamati, Silvano Romano

Published Aug 12, 2005• 8 citations
Publisher

Abstract

The present paper addresses a simple lattice gas model, consisting of particles that carry a three-component classical spin (i.e., a three-component unit vector), and associated with a simple-cubic lattice; each site can host one particle at most, thus implicitly allowing for hard-core repulsion; the pair interaction, restricted to nearest neighbors, is ferromagnetic and isotropic in spin space. We have studied thermodynamic properties and phase diagrams of the model by means of mean-field and two-site cluster treatments, and identified the tricritical points. Note that the fluctuating occupation numbers give rise to additional fluidlike observables in comparison with the usual saturated-lattice situation. The case of zero chemical potential has been investigated by grand-canonical Monte Carlo simulations, so as to obtain a quantitative test of the named approximations, as well as a more detailed physical characterization. The mean-field treatment was found to yield only qualitative agreement with simulation results, and the two-site cluster treatment recognizably improved upon it. Extensive simulations point to a second-order transition, whose universality class was found to be the same as its saturated-lattice counterpart.

DOI
10.1103/physrevb.72.064424

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The present paper addresses a simple lattice gas model, consisting of particles that carry a three-component classical spin (i.e., a three-component unit vector), and associated with a simple-cubic lattice; each site can host one particle at most, thus implicitly allowing for hard-core repulsion; the pair interaction, restricted to nearest neighbors, is ferromagnetic and isotropic in spin space. We have studied thermodynamic properties and phase diagrams of the model by means of mean-field and two-site cluster treatments, and identified the tricritical points. Note that the fluctuating occupation numbers give rise to additional fluidlike observables in comparison with the usual saturated-lattice situation. The case of zero chemical potential has been investigated by grand-canonical Monte Carlo simulations, so as to obtain a quantitative test of the named approximations, as well as a more detailed physical characterization. The mean-field treatment was found to yield only qualitative agreement with simulation results, and the two-site cluster treatment recognizably improved upon it. Extensive simulations point to a second-order transition, whose universality class was found to be the same as its saturated-lattice counterpart.
We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.
We have considered a classical lattice-gas model, consisting of a two-dimensional lattice Z2, each site of which hosts at most one two-component unit vector; particles occupying pairs of nearest-neighbouring sites interact via the ferromagnetic potential [Formula: see text] where νj=0,1 denotes occupation numbers, uj are the unit vectors (classical spins) and ∊ is a positive constant setting energy and temperature scales; the total Hamiltonian is given by [Formula: see text] where ∑{j<k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model, where all sites are occupied, supports the well-known Berezhinskiĭ–Kosterlitz–Thouless transition; we report here a simulation study, carried out for both μ= 0.1 and μ=-0.2, showing evidence of a transition of this kind, in broad qualitative agreement with previous Renormalization-Group studies.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.
PACS. 02.70.Lq(Monte Carlo and statistical methods) PACS. 05.50.+q(Lattice theory and statistics; Ising problems) PACS. 68.35.Rh(Phase transitions and critical phenomena) Abstract. We perform Monte Carlo simulations of the hard-sphere lattice gas on the body-centred cubic lattice with nearest neighbour exclusion. We get the critical exponents, β/ν = 0.311(8) and γ/ν = 2.38(2), where β, γ, and ν are the critical exponents of the staggered density, the staggered compressibility, and the correlation length, respectively. The values of the hard-sphere lattice gas on the simple cubic lattice agree with them but those of the threedimensional Ising model do not. This supports that the hard-sphere lattice gas does not fall into the Ising universality class in three dimensions. In this letter we study the hard-sphere lattice gas whose atoms interact with infinite repulsion of nearest neighbour pairs. The grand partition function is ΞV (z) = ∑ z N ZV (N), (1) N where z is an activity and ZV (N) is the number of configurations in which there are N atoms in the lattice of V sites. There are many studies on:
The classical nnn-vector model on the face-centered cubic lattice with ferromagnetic and antiferromagnetic interactions is studied by using the framework of an effective field theory approach in cluster with two spins. For n=1n=1n=1 (Ising), n=2n=2n=2 (XYXYXY-planar rotator), n=3n=3n=3 (Heisenberg), and $n\ensuremath{\rightarrow}\ensuremath{\infty}$ (spherical) models the ferromagnetic system undergoes a second-order phase transition while the antiferromagnetic system presents a first-order behavior. The first-order character for n=3n=3n=3, which is inconclusive in the literature, seems actually to be the case for the isotropic Heisenberg antiferromagnet, and the same is expected to hold for the less studied XYXYXY and spherical models. In addition, the transition temperatures are quantitatively comparable to the exact one (when available) and to those from Monte Carlo simulations and series expansion.
The adsorbed atoms exhibit tendency to occupy a triangular lattice formed by periodic potential of the underlying crystal surface. Such a lattice is formed by, e.g., a single layer of graphane or the graphite surfaces as well as (111) surface of face-cubic center crystals. In the present work, an extension of the lattice gas model to S=1/2 fermionic particles on the two-dimensional triangular (hexagonal) lattice is analyzed. In such a model, each lattice site can be occupied not by only one particle, but by two particles, which interact with each other by onsite U and intersite W1 and W2 (nearest and next-nearest-neighbor, respectively) density-density interaction. The investigated hamiltonian has a form of the extended Hubbard model in the atomic limit (i.e., the zero-bandwidth limit). In the analysis of the phase diagrams and thermodynamic properties of this model with repulsive W1>0, the variational approach is used, which treats the onsite interaction term exactly and the intersite interactions within the mean-field approximation. The ground state (T=0) diagram for W2≤0 as well as finite temperature (T>0) phase diagrams for W2=0 are presented. Two different types of charge order within 3×3 unit cell can occur. At T=0, for W2=0 phase separated states are degenerated with homogeneous phases (but T>0 removes this degeneration), whereas attractive W2<0 stabilizes phase separation at incommensurate fillings. For U/W1<0 and U/W1>1/2 only the phase with two different concentrations occurs (together with two different phase separated states occurring), whereas for small repulsive 0<U/W1<1/2 the other ordered phase also appears (with tree different concentrations in sublattices). The qualitative differences with the model considered on hypercubic lattices are also discussed.