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

Abstract

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 T _c. It is shown that for p = 3 the moments q ^(k) of the spin-glass order parameter satisfy a simple scaling behavior, $q^{(k)} \alpha N^{--k/3} \tilde f_k \{ N^{1/3} (1--T/T_c )\} ,{\text{ }}k = 1,2,3,...,\tilde f_k$ being the appropriate scaling function and T the temperature. Also the specific heat maxima have a similar behavior, $c_V^{\max } \alpha {\text{ }}const--N^{--1/3}$, while moments of the magnetization scale as $m^{(k)} \alpha N^{--k/2}$. The approach of the positions T _max of these specific heat maxima to T _c as N → ∞ is nonmonotonic. For p = 6 the results are compatible with a first-order transition, q ^(k) → (q _jump)^k as N → ∞ but since the order parameter q _jump at T _c 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 c ^max_V 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 $R_x \equiv [(\langle X\rangle _T - [\langle X\rangle _T ]_{{\text{av}}} )^2 ]_{{\text{av}}} /[\langle X\rangle _T ]_{{\text{av}}}^2$, for various quantities X, to test the possible lack of self-averaging at T _c.