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Critical behaviour of the 1D q-state Potts model with long-range interactions

Zvonko Glumac, Katarina Uzelac

Published Jan 1, 1993• 21 citations
OpenAlex

Abstract

The critical behaviour of the one-dimensional q-state Potts model with long-range interactions decaying with distance r as r −(1+σ) has been studied in the wide range of parameters 0 < σ ≤ 1 and 1 16 ≤ q ≤ 64. A transfer matrix has been constructed for a truncated range of interactions for integer and continuous q, and finite range scaling has been applied. Results for the phase diagram and the correlation length critical exponent are presented. Short title: Critical behaviour of the 1D LR q-state Potts model Recently, finite range scaling (FRS) appeared as an efficient method to deal with various discrete one-dimensional models with long range (LR) interactions (Glumac and Uzelac, 1989, 1991). The aim of the present study is to apply it to the onedimensional q-state Potts model with LR interactions, defined by the hamiltonian

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The critical behaviour of the one-dimensional q-state Potts model with long-range interactions decaying with distance r as r −(1+σ) has been studied in the wide range of parameters 0 < σ ≤ 1 and 1 16 ≤ q ≤ 64. A transfer matrix has been constructed for a truncated range of interactions for integer and continuous q, and finite range scaling has been applied. Results for the phase diagram and the correlation length critical exponent are presented. Short title: Critical behaviour of the 1D LR q-state Potts model Recently, finite range scaling (FRS) appeared as an efficient method to deal with various discrete one-dimensional models with long range (LR) interactions (Glumac and Uzelac, 1989, 1991). The aim of the present study is to apply it to the onedimensional q-state Potts model with LR interactions, defined by the hamiltonian
The critical behaviour of the one-dimensional Ising model with long-range ferromagnetic interactions decaying with distance r as 1/r 1+ sigma has been studied by scaling the range of interactions. Exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0< sigma <or=1. Applying the range scaling, the critical temperature, critical exponent nu and the anomalous dimension of the order parameter have been calculated. Additional analysis of the convergence of the method has been performed by applying the Vanden Broeck and Schwartz extrapolation procedure in addition to the simple least-squares approximation and by evaluating the convergence exponents.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.
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.
We investigate the short-time dynamics of the one-dimensional q = 3 Potts model with a power-law decaying interactions of the form r−(1+σ)r− (1+\sigma)r−(1+σ). This model is known to exhibit both first- and second-order transition regimes, which depend on the parameter of range σ\sigmaσ and are separated by a q-dependent tricritical point sigma_c(q) [1]. The location of this point, where the first-order transition becomes extremely weak presents difficulties both for RG approaches and for equilibrium numerical simulations [2-3]. We examine the first-order transition regime and its onset at sigma_c by dynamic Monte Carlo simulations, which recently appeared to be efficient in detecting weak first-order phase transitions [4]. We consider the short-time relaxation processes for magnetization and its higher momenta starting from high and low temperatures. We analyse the appearance of two pseudo critical temperatures characteristic for the first-order transition regime, which should merge at the tricritical point. [1] Z. Glumac and K. Uzelac, Phys. Rev. E 58, 4372 (1998) [2] E. Bayong, H. T. Diep, V. Dotsenko, Phys. Rev. Lett. 83, 14 (1999) ; K. Uzelac and Z. Glumac, Phys. Rev. Lett. 85, 5255 (2000) [3] Sylvain Reynal, Hung-The Diep, Phys. Rev. E 69, 026109 (2004) [4] L. Schuelke nad B. Zhang, Phys. Rev. E 62, 7482 (2000)