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Physical review. E

Athermal tricriticality

Mauro Sellitto

Published Apr 7, 2025• 0 citations
Publisher

Abstract

I construct a two dimensional monodisperse lattice gas model with selective geometric constraints featuring a fluid-solid continuous phase transition on the square lattice and a dimerized ground state with fourfold degeneracy. Finite-size scaling analysis confirms that the lattice gas belongs to the universality class of two-dimensional four-state Potts model, and since its thermodynamics is governed by entropy alone, this provides possibly the simplest statistical mechanical realization of an athermal tricritical point.

DOI
10.1103/physreve.111.l042101

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I construct a two dimensional monodisperse lattice gas model with selective geometric constraints featuring a fluid-solid continuous phase transition on the square lattice and a dimerized ground state with fourfold degeneracy. Finite-size scaling analysis confirms that the lattice gas belongs to the universality class of two-dimensional four-state Potts model, and since its thermodynamics is governed by entropy alone, this provides possibly the simplest statistical mechanical realization of an athermal tricritical point.
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 finite-size-scaling study of a two-dimensional square-lattice-gas model with nearest-neighbor repulsion and next-nearest-neighbor attraction. This system may be used as a simplified model of adsorbates on transition-metal (100) surfaces. An accurate determination of the tricritical temperature is obtained by the utilization of the degeneracy of the three largest eigenvalues of the transfer matrix at a tricritical point. We obtain the phase diagram and critical and tricritical exponents, all quite consistent with previously known results. We also find that a recent conjecture relating the critical exponent $\ensuremath{\eta}$ to the correlation-length amplitude seems to work well even at the tricritical point.
We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field Δ. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of Δ and a parallelized variant of the multicanonical approach performed at constant temperature T. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.
Phase transitions in centered-rectangular- and square-lattice-gas models have been studied by transfer-matrix methods, with the use of finite-size scaling ideas. The transition lines and critical exponents have been obtained for the $p(2\ifmmode\times\else\texttimes\fi{}1)$ and $p(2\ifmmode\times\else\texttimes\fi{}2)$ structures for the centered-rectangular model with several pairwise interactions and a three-body interaction. This model is of interest in that the $p(2\ifmmode\times\else\texttimes\fi{}1)$ and $p(2\ifmmode\times\else\texttimes\fi{}2)$ transitions belong to the universality class of the XY\mathrm{XY}XY model with cubic anisotropy and hence the critical exponents are nonuniversal. As well, it is a simple model of O on W(110). The transition lines have also been obtained for a square-lattice-gas model with competing interactions, for the (2\ifmmode\times\else\texttimes\fi{}1) and (2\ifmmode\times\else\texttimes\fi{}2) ordered structures. The importance of finite-size effects is demonstrated for both models.
We investigate several tricritical models on the square lattice by means of Monte Carlo simulations. These include the Blume-Capel model, Baxter's hard-square model, and the q=1 , 3, and 4 Potts models with vacancies. We use a combination of the Wolff and geometric cluster methods, which conserves the total number of vacancies or lattice-gas particles and suppresses critical slowing down. Several quantities are sampled, such as the specific heat C and the structure factor C(s) , which accounts for the large-scale spatial inhomogeneity of the energy fluctuations. We find that the constraint strongly modifies some of the critical singularities. For instance, the specific heat C reaches a finite value at tricriticality, while C(s) remains divergent as in the unconstrained system. We are able to explain these observed constrained phenomena on the basis of the Fisher renormalization mechanism generalized to include a subleading relevant thermal scaling field. In this context, we find that, under the constraint, the leading thermal exponent y(t1) is renormalized to 2- y(t1) , while the subleading exponent y(t2) remains unchanged.