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arXiv:1601.00515 · Physical Review E

Reentrant disorder-disorder transitions in generalized multicomponent Widom-Rowlinson models

Roman Krčmár, Ladislav Šamaj

Published Nov 4, 2015• 4 citations

Abstract

In the lattice version of the multicomponent Widom-Rowlinson (WR) model, each site can be either empty or singly occupied by one of M different particles, all species having the same fugacity z. The only nonzero interaction potential is a nearest-neighbor hard-core exclusion between unlike particles. For M<M(0) with some minimum M(0) dependent on the lattice structure, as z increases from 0 to ∞ there is a direct transition from the disordered (gas) phase to a demixed (liquid) phase with one majority component at z>z(d) (M). If M≥M(0), there is an intermediate ordered "crystal phase" (composed of two nonequivalent even and odd sublattices) for z lying between z(c)(M) and z(d)(M) which is driven by entropy. We generalize the multicomponent WR model by replacing the hard-core exclusion between unlike particles by more realistic large (but finite) repulsion. The model is solved exactly on the Bethe lattice with an arbitrary coordination number. The numerical calculations, based on the corner transfer matrix renormalization group, are performed for the two-dimensional square lattice. The results for M=4 indicate that the second-order phase transitions from the disordered gas to the demixed phase become of first order, for an arbitrarily large finite repulsion. The results for M≥M(0) show that, as the repulsion weakens, the region of the crystal phase diminishes itself. For weak enough repulsions, the direct transition between the crystal and demixed phases changes into a separate pair of crystal-gas and gas-demixed transitions; this is an example of a disorder-disorder reentrant transition via an ordered crystal phase. If the repulsion between unlike species is too weak, the crystal phase disappears from the phase diagram. It is shown that the generalized WR model belongs to the Ising universality class.

DOI: 10.1103/physreve.92.052103

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In the lattice version of the multicomponent Widom-Rowlinson (WR) model, each site can be either empty or singly occupied by one of M different particles, all species having the same fugacity z. The only nonzero interaction potential is a nearest-neighbor hard-core exclusion between unlike particles. For M<M(0) with some minimum M(0) dependent on the lattice structure, as z increases from 0 to ∞ there is a direct transition from the disordered (gas) phase to a demixed (liquid) phase with one majority component at z>z(d) (M). If M≥M(0), there is an intermediate ordered "crystal phase" (composed of two nonequivalent even and odd sublattices) for z lying between z(c)(M) and z(d)(M) which is driven by entropy. We generalize the multicomponent WR model by replacing the hard-core exclusion between unlike particles by more realistic large (but finite) repulsion. The model is solved exactly on the Bethe lattice with an arbitrary coordination number. The numerical calculations, based on the corner transfer matrix renormalization group, are performed for the two-dimensional square lattice. The results for M=4 indicate that the second-order phase transitions from the disordered gas to the demixed phase become of first order, for an arbitrarily large finite repulsion. The results for M≥M(0) show that, as the repulsion weakens, the region of the crystal phase diminishes itself. For weak enough repulsions, the direct transition between the crystal and demixed phases changes into a separate pair of crystal-gas and gas-demixed transitions; this is an example of a disorder-disorder reentrant transition via an ordered crystal phase. If the repulsion between unlike species is too weak, the crystal phase disappears from the phase diagram. It is shown that the generalized WR model belongs to the Ising universality class.

Using transfer matrix and finite-size scaling methods, we study the thermodynamic behavior of a lattice gas with two kinds of particles on the square lattice. Only excluded volume interactions are considered, so that the model is athermal. Large particles exclude the site they occupy and its four first neighbors, while small particles exclude only their site. Two thermodynamic phases are found: a disordered phase where large particles occupy both sublattices with the same probability and an ordered phase where one of the two sublattices is preferentially occupied by them. The transition between these phases is continuous at small concentrations of the small particles and discontinuous at larger concentrations, both transitions are separated by a tricritical point. Estimates of the central charge suggest that the critical line is in the Ising universality class, while the tricritical point has tricritical Ising (Blume-Emery-Griffiths) exponents. The isobaric curves of the total density as functions of the fugacity of small or large particles display a minimum in the disordered phase.

We present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.

We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.

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.

Using phenomenological renormalization (transfer-matrix scaling), we have reexamined the phase transition of a triangular lattice gas with particles having both nearest- and second-nearest-neighbor exclusions. Widely accepted classical studies indicated that disordering of the ordered ["$p(2\ifmmode\times\else\texttimes\fi{}2)$"] state is first order. In contradiction, we show that the transition is second order; its exponents are consistent with the four-state Potts model universality class, in accord with its Landau-Ginzburg-Wilson Hamiltonian classification.

The nature of phase transitions occurring in the two-dimensional spin-1 lattice model with the first- and second-nearest-neighbor interactions is studied using Monte Carlo methods. This system models the adsorption of rigid diatomic molecules $(A\text{\ensuremath{-}}B)$ on (100) surfaces of crystals, assuming that each molecule is oriented perpendicularly to the crystal surface, and the binding energy depends on whether the

A

atom or

B

atom is adsorbed. It is shown that under the condition of a fully filled lattice, the order-disorder transition of the superantiferromagnetic (SAF) phase may occur via a continuous as well as a first-order phase transition, depending on the model parameters. The continuous order-disorder transition of the SAF phase is found to be nonuniversal. The nature of the various phase transitions is analyzed by the finite size scaling method. The application of this method is also demonstrated for the case of a second-order transition from one ordered phase to another ordered phase. We have also demonstrated that the phase diagram topology changes with the model parameters. While at low temperatures a first-order transition between the SAF phase and the $c(2\ifmmode\times\else\texttimes\fi{}2{)}_{\mathrm{AF}}$-ordered phase corresponding to a half-filled lattice occurs, this transition line at higher temperatures may terminate either by a triple point or a tricritical point. For the latter case a phase diagram with a different type of multicritical point is suggested.