Free-Energy Barriers in the Sherrington-Kirkpatrick Model

– The Sherrington-Kirkpatrick spin-glass model is investigated by means of Monte Carlo simulations employing a combination of the multi-overlap algorithm with parallel tempering methods. We investigate the finite-size scaling behaviour of the free-energy barriers which are visible in the probability density of the Parisi overlap parameter. Assuming that the mean barrier height diverges with the number of spins N as N, our data show good agreement with the theoretical value α = 1/3. Despite three decades of research the nature of the “glassy” low-temperature phase of finite-dimensional spin-glass systems remains a major open problem in statistical physics. It is still unresolved whether the replica symmetry-breaking theory or the phenomenological droplet picture yields the correct description (for reviews, see refs. [1–4]). Even at the meanfield level, only very recently a mathematical proof [5] of Parisi’s replica solution [6] for the Sherrington-Kirkpatrick (SK) model [7] was given. In the thermodynamic limit the frozen phase of the mean field spin glass shows many stable and metastable states. Such a feature is the consequence of the disorder and the frustration characterizing spin glasses in general and leading to a rugged free-energy landscape with probable regions (low free energy) separated by rare-event states (high free energy). But also for finite systems the free-energy landscape shows an intricate, corrugated structure. Therefore, it is hard to measure the free-energy barriers by means of conventional Monte Carlo simulations directly. The aim of this letter is to study the free-energy barriers of the SK mean field spin-glass model using the multi-overlap Monte Carlo algorithm [8] and to compare our results with previous findings for finite-range spin glasses [9]. For analyzing the barriers, we used precisely the same method as introduced in ref. [9] for the Edwards-Anderson (EA) nearest-neighbor model [10], where the scaling of the mean barrier height with the number of spins was found to clearly deviate from the theoretical mean-field prediction in both three and four dimensions. To exclude the possibility that this deviation could, in principle, be caused by employing different definitions in the theoretical and computational work, it is important to apply precisely the same numerical procedure to the SK model. The Hamiltonian of the SK model reads H = − ∑