Nonstandard Finite-Size Scaling at First-Order Phase Transitions

Abstract

We note that the standard inverse system volume scaling for finite-size corrections at a firstorder phase transition (i.e., 1/L 3 for an L × L × L lattice in 3D) is transmuted to 1/L 2 scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exemplar of just such a system. We use multicanonical simulations of this model to generate high-precision data which provides strong confirmation of the non-standard finite-size scaling law. The dual to the gonihedric model, which is an anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also displays the non-standard scaling.PACS numbers: 05.50.+q, 05.70.Jk, 75.10.Hk First-order phase transitions are ubiquitous in nature [1]. Pioneering studies of finite-size scaling for first-order transitions were carried out in [2] and subsequently pursued in detail in [3]. Rigorous results for periodic boundary conditions were further derived in [4,5]. It is possible to go quite a long way in discussing the scaling laws for such first-order transitions using a simple heuristic twophase model [6]. We assume that a system spends a fraction W o of the total time in one of the q ordered phases and a fraction W d = 1 − W o in the disordered phase with corresponding energiesê o andê d , respectively. The hat is introduced for quantities evaluated at the inverse transition temperature of the infinite system, β ∞ . Neglecting all fluctuations within the phases and treating the phase transition as a sharp jump between the phases, the energy moments become e n = W oê, where the disordered and ordered peaks of the energy probability density have equal weight. The probability of being in any of the ordered states or the disordered state is related to the free energy densitiesf o ,f d of the states,and by construction the fraction of time spent in the ordered states must be proportional to qp o . Thus for the ratio of fractions we find (up to exponentially small corrections in L [4-7]). Taking the logarithm of this ratio gives ln(WAt the specific-heat maximum W o = W d , so we find by an expansion around βwhich can be solved for the finite-size peak location of the specific heat:Although this is a rather simple toy model, it is known to capture the essential features of first-order phase transitions and to correctly predict the prefactors of the leading finite-size scaling corrections for a class of models with a contour representation, such as the q-state Potts model, where a rigorous theory also exists [5]. Similar calculations give [6,8] for the location β Normally the degeneracy q of the low-temperature phase does not change with system size and the generic finite-size scaling behaviour of a first-order transition thus has a leading contribution proportional to the inverse volume L −d . We can see from Eqs. (4), (5) that if the degeneracy q of the low-temperature phase depends exponentially on the system size, q ∝ e L , this would be modified. One model with preci...

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].…”