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Physical review. B, Condensed matter

Accurate first-order transition points from finite-size data without power-law corrections

Wolfhard Janke E

Published Jun 1, 1993• 78 citations
Publisher

Abstract

Traditional observables used to locate first-order phase-transition points from Monte Carlo simulations on finite lattices usually exhibit powerlike finite-size scaling behavior. For systems in a box of volume V with periodic boundary conditions, several definitions of finite volume (pseudo-) transition temperatures ${\mathit{T}}_{0}$(V) have recently been proposed which involve only exponential corrections with respect to ${\mathit{T}}_{0}$\ensuremath{\equiv}${\mathit{T}}_{0}$(\ensuremath{\infty}). One of these approaches suggests also a different and efficient way to compute the latent heat. These propositions are tested for the temperature-driven first-order phase transitions in the two-dimensional q-state Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update algorithm and histogram reweighting techniques.

DOI
10.1103/physrevb.47.14757

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Physical review. B, Condensed matter•Mar 1, 1991•54 citations
Traditional observables used to locate first-order phase-transition points from Monte Carlo simulations on finite lattices usually exhibit powerlike finite-size scaling behavior. For systems in a box of volume V with periodic boundary conditions, several definitions of finite volume (pseudo-) transition temperatures T0{\mathit{T}}_{0}T0​(V) have recently been proposed which involve only exponential corrections with respect to T0{\mathit{T}}_{0}T0​\ensuremath{\equiv}T0{\mathit{T}}_{0}T0​(\ensuremath{\infty}). One of these approaches suggests also a different and efficient way to compute the latent heat. These propositions are tested for the temperature-driven first-order phase transitions in the two-dimensional q-state Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update algorithm and histogram reweighting techniques.
We consider a temperature-driven first-order phase transition describing the coexistence of q ordered low-temperature phases and one disordered high-temperature phase at the infinite-volume transition temperature T0{\mathit{T}}_{0}T0​. Analyzing the exponential corrections to the periodic partition function in a box of volume V, Zper{\mathit{Z}}_{\mathrm{per}}Zper​=tsumm{\mathit{tsum}}_{\mathit{m}}tsumm​ exp(-\ensuremath{\beta}fm{\mathit{f}}_{\mathit{m}}fm​V), where fm{\mathit{f}}_{\mathit{m}}fm​ is the (metastable) free energy of the phase m, we propose several definitions of a finite-volume transition temperature T0{\mathit{T}}_{0}T0​(V) which involve only exponential corrections with respect to T0{\mathit{T}}_{0}T0​. We test our propositions in the d=2 Potts model for q=5, 8, and 10 by means of Monte Carlo simulations, using the single-cluster update procedure.
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 study the effects of frozen boundaries in a Monte Carlo simulation near a first-order phase transition. Recent theoretical analysis of the dynamics of first-order phase transitions has enabled us to state the scaling laws governing the critical regime of the transition. We check these new scaling laws performing a Monte Carlo simulation of the two-dimensional, eight-state spin Potts model. In particular, our results support a pseudocritical $\ensuremath{\beta}(L)$ finite-size scaling of the form $\ensuremath{\beta}(\ensuremath{\infty}{)+a}_{1}{/L+a}_{2}{/L}^{2},$ instead of $\ensuremath{\beta}(\ensuremath{\infty})+{\ensuremath{\theta}}_{1}{/L}^{d}+{\ensuremath{\theta}}_{2}{/L}^{2d}.$ Moreover, we obtain a latent heat ${\ensuremath{\Lambda}}_{\mathrm{FBC}}=0.294(11),$ which does not coincide with the latent heat analytically derived for the same model if periodic boundary conditions are assumed, ${\ensuremath{\Lambda}}_{\mathrm{PBC}}=0.486358\dots{}.$
We reexamine the behavior near a first-order phase-transition point of the fourth-order energy cumulant VL{\mathit{V}}_{\mathit{L}}VL​ introduced by Binder, Challa, and Landau. Our prediction is tested on the ten-states Potts model in two dimensions by Monte Carlo simulation using the Ferrenberg-Swendsen technique. We extract a latent heat in agreement with the exact result of Baxter.
A new kind of finite-size phenomenon in the five-state Potts model in two dimensions, a ``false'' ground state, was recently reported by Katznelson and Lauwers based on Monte Carlo studies of large lattices. We reexamine the behavior of this weakly first-order system by performing extensive Monte Carlo simulations, and demonstrate that the ``novel'' states previously observed in internal-energy-distribution histograms tend to disappear when runs of sufficient length are used. We present evidence that very strong critical slowing down and a pseudodivergent correlation length are responsible for the observed effects. Our additional analysis of finite-size effects in the ten-state Potts model on small lattices (L\ensuremath{\le}10) shows a crossover of scaling from ${L}^{\mathrm{\ensuremath{-}}d}$ behavior (observed earlier on larger lattices) to a nonanalytic form. This result is used to discuss the observed nonanalytic scaling of thermodynamic quantities in the five-state Potts model.
Monte Carlo simulations are performed for first-order phase-transition models. The three-dimensional three-state Potts model has a weak first-order transition. For this model we calculate the density of states on L3{\mathit{L}}^{3}L3 block lattices (L up to size 36) and obtain high-precision estimates for the leading partition-function zeros. The finite-size-scaling analysis of the first zero exhibits the expected convergence of the critical exponent \ensuremath{\nu} toward 1/D for large L; in particular, we find \ensuremath{\nu}=2.955(26) from our two largest lattices. Analysis of our specific-heat Cv{\mathit{C}}_{\mathit{v}}Cv​ data yields l=0.080 31(26) for the latent heat. Along another line of approach, we calculate the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} is the correlation length) for cylindrical L2{\mathit{L}}^{2}L2Lz{\mathit{L}}_{\mathit{z}}Lz​ lattices (L up to 24 and Lz{\mathit{L}}_{\mathit{z}}Lz​=256). The finite size-scaling analysis of these results is also consistent with the convergence of \ensuremath{\nu} toward 1/D, but that the limiting value is 1/D is not yet conclusively established. Some theoretical arguments favor \ensuremath{\nu}\ensuremath{\rightarrow}0 in case of a first-order transition in a cylindrical ${\mathit{L}}^{\mathit{D}\mathrm{\ensuremath{-}}1}$\ensuremath{\infty} geometry. Therefore, we also applied our approach to the 2D ten-state Potts model, which is known to have a strong first-order transition. In this case we find unambiguous evidence in favor of 1/D as the limiting value.