Finite-Size Scaling in the p-State Mean-Field Potts Glass: A Monte Carlo Investigation

The p-state mean-field Potts glass with bimodal bond distribution (±J) is studied by Monte Carlo simulations, both for p = 3 and p = 6 states, for system sizes from N = 5 to N = 120 spins, considering particularly the finite-size scaling behavior at the exactly known glass transition temperature Tc. It is shown that for p = 3 the moments q(k) of the spin-glass order parameter satisfy a simple scaling behavior, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$q^{(k)} \alpha N^{--k/3} \tilde f_k \{ N^{1/3} (1--T/T_c )\} ,{\text{ }}k = 1,2,3,...,\tilde f_k $$ \end{document} being the appropriate scaling function and T the temperature. Also the specific heat maxima have a similar behavior, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$c_V^{\max } \alpha {\text{ }}const--N^{--1/3} $$ \end{document}, while moments of the magnetization scale as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$m^{(k)} \alpha N^{--k/2} $$ \end{document}. The approach of the positions Tmax of these specific heat maxima to Tc as N → ∞ is nonmonotonic. For p = 6 the results are compatible with a first-order transition, q(k) → (qjump)k as N → ∞ but since the order parameter qjump at Tc is rather small, a behavior q(k) ∝ N-k/3 as N → ∞ also is compatible with the data. Thus no firm conclusions on the finite-size behavior of the order parameter can be drawn. The specific heat maxima cVmax behave qualitatively in the same way as for p = 3, consistent with the prediction that there is no latent heat. A speculative phenomenological discussion of finite-size scaling for such transitions is given. For small N (N ≤15 for p = 3, N ≤ 12 for p = 6) the Monte Carlo data are compared to exact partition function calculations, and excellent agreement is found. We also discuss ratios \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$R_x \equiv [(\langle X\rangle _T - [\langle X\rangle _T ]_{{\text{av}}} )^2 ]_{{\text{av}}} /[\langle X\rangle _T ]_{{\text{av}}}^2 $$ \end{document}, for various quantities X, to test the possible lack of self-averaging at Tc.