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arXiv:hep-lat/0702018 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Normalized entropy density of the 3D 3-state Potts model

Alexei Bazavov, Bernd A. Berg

Published May 31, 2007• 14 citations
arXivHTMLPDF

Abstract

Using a multicanonical Metropolis algorithm we have performed Monte Carlo simulations of the 3D 3-state Potts model on ${L}^{3}$ lattices with $L=20$, 30, 40, 50. Covering a range of inverse temperatures from ${\ensuremath{\beta}}_{\mathrm{min}}=0$ to ${\ensuremath{\beta}}_{\mathrm{max}}=0.33$ we calculated the infinite volume limit of the entropy density $s(\ensuremath{\beta})$ with its normalization obtained from $s(0)=\mathrm{ln}3$. At the transition temperature the entropy and energy endpoints in the ordered and disordered phase are estimated employing a novel reweighting procedure. We also evaluate the transition temperature and the order-disorder interface tension. The latter estimate increases when capillary waves are taken into account.

DOI
10.1103/physrevd.75.094506

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Using a multicanonical Metropolis algorithm we have performed Monte Carlo simulations of the 3D 3-state Potts model on L3{L}^{3}L3 lattices with L=20L=20L=20, 30, 40, 50. Covering a range of inverse temperatures from \ensuremathβmin=0{\ensuremath{\beta}}_{\mathrm{min}}=0\ensuremathβmin​=0 to \ensuremathβmax=0.33{\ensuremath{\beta}}_{\mathrm{max}}=0.33\ensuremathβmax​=0.33 we calculated the infinite volume limit of the entropy density s(\ensuremathβ)s(\ensuremath{\beta})s(\ensuremathβ) with its normalization obtained from s(0)=ln3s(0)=\mathrm{ln}3s(0)=ln3. At the transition temperature the entropy and energy endpoints in the ordered and disordered phase are estimated employing a novel reweighting procedure. We also evaluate the transition temperature and the order-disorder interface tension. The latter estimate increases when capillary waves are taken into account.
Traditional definition of phase transition involves an infinitely large system in thermodynamic limit. Finite systems such as biological proteins exhibit cooperative behavior similar to phase transitions. We employ recently discovered analysis of inflection points of microcanonical entropy to estimate the transition temperature of the phase transition in q state Potts model on a finite two dimensional square lattice for q=3 (second order) and q=8 (first order). The difference of energy density of states (DOS) Δ ln g(E) = ln g(E+ ΔE) −ln g(E) exhibits a point of inflexion at a value corresponding to inverse transition temperature. This feature is common to systems exhibiting both first as well as second order transitions. While the difference of DOS registers a monotonic variation around the point of inflexion for systems exhibiting second order transition, it has an S-shape with a minimum and maximum around the point of inflexion for the case of first order transition.
Using multicanonical Monte Carlo simulations, we have calculated the interface free energy between the coexisting ordered and disordered phase at the temperature driven first-order phase transition of the two-dimensional 7-state Potts model. Our result 2f3= 0.0241±0.0010 is about an order of magnitude smaller than other estimates in the recent literature. We discuss the differences in analysis and give a possible explanation for the discrepancy. Finally, we briefly mention similar investigations at the deconfining phase transition of SU(3) lattice gauge theory.
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.
In this investigation we show that the entropy of the two-dimensional infinite-state Potts model is linear in configurational energy in the thermodynamic limit. This is a direct consequence of the local convexity of the microcanonical entropy, associated with a finite system undergoing a first-order transition. For a sufficiently large number of states , this convexity spans the entire energy range of the model. In the thermodynamic limit, the convexity becomes insignificant, and the microcanonical entropy (the logarithm of the density of states) tends to a straight line. In order to demonstrate the behaviour of the convexity, we use the Wang-Landau Monte-Carlo technique to numerically calculate the density of states for a few finite but high values of . Finally, we calculate the free energy and discuss the generality of our results.
We develop an interface approach to bulk phase transitions and ordered states by using Monte Garlo simulations, and apply it to antiferromagnetic q -state Potts models with q =3∼6 on the simple cubic lattice. A stiffness exponent a defined by Δ F ∼ L a , where Δ F is the interface free energy for a system of size L , is introduced as a measure of the stiffness of the ordered phase against an external stress. Applying finite-size scaling to Δ F and to the squared order parameter enable one to determine each of T c , ν, β and γ in order where T c is the critical temperature and the others are critical exponents. This approach also provides a means to study properties of the ordered state through a and interface profiles. In the q =3 and 4 models the estimated exponents ν, β, γ and a indicate new universality classes. The q =5 model shows a phase transition, while the q =6 model does not. The q =3∼5 models have non-integer values of a at low temperatures. The phase transitions in these models are confirmed to be due to entropy gains.