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arXiv:cond-mat/9807416 · Fizika B

First-order phase transition in $1d$ Potts model with long-range interactions

Katarina Uzelac, Zvonko Glumac

Published Jul 31, 1998• 3 citations
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Abstract

The first-order phase transition in the one-dimensional $q$-state Potts model with long-range interactions decaying with distance as $1/r^{1+σ}$ has been studied by Monte Carlo numerical simulations for $0 < σ\le 1$ and integer values of $q > 2$. On the basis of finite-size scaling analysis of interface free energy $ΔF_L$, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for $σ$ below a threshold value $σ_c(q)$.

DOI
10.48550/arxiv.cond-mat/9807416

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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).
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We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0{\mathit{U}}_{0}U0​(tLd{\mathit{tL}}^{\mathit{d}}tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1{\mathit{U}}_{1}U1​(tLd{\mathit{tL}}^{\mathit{d}}tLd) with t=1-Tc{\mathit{T}}_{\mathit{c}}Tc​/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1{\mathit{e}}_{1}e1​/e2{\mathit{e}}_{2}e2​-e2{\mathit{e}}_{2}e2​/e1{\mathit{e}}_{1}e1​)2{)}^{2})2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2){\mathit{T}}_{\mathit{c}}^{(2)}Tc(2)​(L)-Tc{\mathit{T}}_{\mathit{c}}Tc​=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc{\mathit{T}}_{\mathit{c}}Tc​ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.
We investigate the critical behavior of the one-dimensional q-state Potts model with long-range (LR) interactions 1/r(d+sigma), using a multicanonical algorithm. The recursion scheme initially proposed by Berg is improved so as to make it suitable for a large class of LR models with unequally spaced energy levels. The choice of an efficient predictor and a reliable convergence criterion is discussed. We obtain transition temperatures in the first-order regime which are in far better agreement with mean-field predictions than in previous Monte Carlo studies. By relying on the location of spinodal points and resorting to scaling arguments, we determine the threshold value sigma(c)(q) separating the first- and second-order regimes to two-digit precision within the range 3< or =q < or =9. We offer convincing numerical evidence supporting sigma(c)(q)<1.0 for all q, by virtue of an unusual finite-size effect, namely, finite-size scaling predicts a continuous transition in the thermodynamic limit, despite the first-order nature of the transition at finite size. A qualitative account in terms of correlation lengths is provided. Finally, we find the crossover between the LR and short-range regimes to occur inside a narrow window 1.0<sigma<1.2, thus lending strong support to Sak's scenario.