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 T_c. It is shown that for p = 3 the moments q^(k) of the spin-glass order parameter satisfy a simple scaling behavior, being the appropriate scaling function and T the temperature. Also the specific heat maxima have a similar behavior, , while moments of the magnetization scale as . 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_V^max 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 , for various quantities X, to test the possible lack of self-averaging at T_c.