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Journal of Applied Physics

Potts model with algebraically decaying interactions in two dimensions

E. Bayong, H. T. Diep

Published May 1, 2000• 0 citations
Publisher

Abstract

We study the 2d Potts model with long-range interaction algebraically decayed as 1/rd+σ using extensive Monte Carlo simulations to determine the nature of the phase transition with histogram technique. The model is shown to exhibit either a first-order transition or a second-order phase transition with critical exponents depending on σ and q. For q=3 there is a critical value of σ below which a first-order transition occurs. For q⩾5 the transition is of first order for any σ.

DOI
10.1063/1.372922

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First-order phase transition in 1d1d1d Potts model with long-range interactions
The first-order phase transition in the one-dimensional qqq-state Potts model with long-range interactions decaying with distance as 1/r1+σ1/r^{1+σ}1/r1+σ has been studied by Monte Carlo numerical simulations for 0<σ≤10 < σ\le 10<σ≤1 and integer values of q>2q > 2q>2. On the basis of finite-size scaling analysis of interface free energy ΔFLΔF_LΔFL​, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for σσσ below a threshold value σc(q)σ_c(q)σc​(q).
Many condensed matter systems, ranging from adsorbed surfaces to bulk magnets, are microscopically modelled by interacting q-state Potts spins, arrayed in d dimensions. A changeover from second-order phase transitions at q≤qc(d) to first-order transitions at q≳qc can be understood as a condensation of effect vacancies, which are patches of local disorder favored by entropy. Accordingly, the renormalization-group treatment of Potts models is within context of Potts-lattice-gas models, where critical and tricritical fixed points occur at low q, but merge and annihilate at qc. This picture has led to exact tricritical exponents in two dimensions. It is also consistent with recent experimental results on intercalated systems in three dimensions. Effective vacancies in pure Potts models have also been studied by Monte Carlo simulation. Their effective chemical potential can be controlled by a four-point interaction, which proved useful in Monte Carlo renormalization-group studies.
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Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.