Multicanonical analysis of the plaquette-only gonihedric Ising model and its dual

Abstract

The three-dimensional purely plaquette gonihedric Ising model and its dual are investigated to resolve inconsistencies in the literature for the values of the inverse transition temperature of the very strong temperature-driven first-order phase transition that is apparent in the system. Multicanonical simulations of this model allow us to measure system configurations that are suppressed by more than 60 orders of magnitude compared to probable states. With the resulting high-precision data, we find excellent agreement with our recently proposed nonstandard finite-size scaling laws for models with a macroscopic degeneracy of the low-temperature phase by challenging the prefactors numerically. We find an overall consistent inverse transition temperature of β ∞ = 0.551334(8) from the simulations of the original model both with periodic and fixed boundary conditions, and the dual model with periodic boundary conditions. For the original model with periodic boundary conditions, we obtain the first reliable estimate of the interface tension σ = 0.12037(18), using the statistics of suppressed configurations.

Plaquette Ising models, degeneracy and scaling

“…If we collect the various estimates for physical quantities using the correct, modified leading 1/L 2 scaling corrections for periodic boundaries and 1/L corrections for fixed boundaries we get consistent values across the original plaquette Hamiltonian with both fixed and periodic boundary conditions and the dual Hamiltonian with periodic boundaries as shown in Table 1 [36,37]. We find an overall consistent value for the inverse transition temperature of…”

“…It is also macroscopic, since it is increasing exponentially with the system size, although sub-extensive since it is increases as exp(L) rather than exp(L 3 ). Taking q = 2 3L we can rework the calculation of the leading scaling terms in Equations (10), (12) to take account of the size-dependence [36,37], giving…”

“…Since in references [36][37][38][39]41,42] we employed multicanonical simulations giving a flat energy distribution, reweighting techniques [64][65][66][67] can be used to get an estimator of the energy probability densities at different temperatures. β eqw is chosen systematically to minimise…”

“…For κ = 0 the 3d Hamiltonians H κ display a continuous transition, possibly with 3d Ising exponents which are masked by strong crossover effects from the nearby tricritical point [33,34]. The κ = 0 plaquette Hamiltonian, on the other hand, has a strong first-order phase transition [35] and recent high-precision multicanonical simulations [36][37][38][39] have revealed that it displays non-standard finite-size scaling properties at this transition. In the sequel we will describe how this non-standard scaling is a consequence of a macroscopic low-temperature phase degeneracy in the 3d plaquette Ising model and also discuss the implications of this degeneracy for the nature of the low-temperature phase and the definition of an order parameter [40][41][42].…”

The gonihedric paradigm extension of the Ising model

“…The other interesting property of the system is that the relaxation to the equilibrium state is very slow, like in spin glasses [20,39,41,42,43,44,45,46,51,54,52,55,56,57,58]. The reason is rooted in high symmetry of the system (3.9), its energy states are exponentially degenerated [17,18,19].…”