String tension in gonihedric three-dimensional Ising models

(Four) Dual Plaquette 3D Ising Models

A characteristic feature of the 3 d plaquette Ising model is its planar subsystem symmetry. The quantum version of this model has been shown to be related via a duality to the X-Cube model, which has been paradigmatic in the new and rapidly developing field of fractons. The relation between the 3 d plaquette Ising and the X-Cube model is similar to that between the 2 d quantum transverse spin Ising model and the Toric Code. Gauging the global symmetry in the case of the 2 d Ising model and considering the gauge invariant sector of the high temperature phase leads to the Toric Code, whereas gauging the subsystem symmetry of the 3 d quantum transverse spin plaquette Ising model leads to the X-Cube model. A non-standard dual formulation of the 3 d plaquette Ising model which utilises three flavours of spins has recently been discussed in the context of dualising the fracton-free sector of the X-Cube model. In this paper we investigate the classical spin version of this non-standard dual Hamiltonian and discuss its properties in relation to the more familiar Ashkin-Teller-like dual and further related dual formulations involving both link and vertex spins and non-Ising spins.

Dynamics of the two-dimensional gonihedric spin model

In this paper, we study dynamical aspects of the two-dimensional (2D) gonihedric spin model using both numerical and analytical methods. This spin model has vanishing microscopic surface tension and it actually describes an ensemble of loops living on a 2D surface. The self-avoidance of loops is parametrized by a parameter kappa . The kappa=0 model can be mapped to one of the six-vertex models discussed by Baxter, and it does not have critical behavior. We have found that allowing for kappa not equal 0 does not lead to critical behavior either. Finite-size effects are rather severe, and in order to understand these effects, a finite-volume calculation for non-self-avoiding loops is presented. This model, like his 3D counterpart, exhibits very slow dynamics, but a careful analysis of dynamical observables reveals nonglassy evolution (unlike its 3D counterpart). We find, also in this kappa=0 case, the law that governs the long-time, low-temperature evolution of the system, through a dual description in terms of defects. A power, rather than logarithmic, law for the approach to equilibrium has been found.

Multicriticality of the three-dimensional Ising model with plaquette interactions: an extension of Novotny's transfer-matrix formalism

A three-dimensional Ising model with the plaquette-type (next-nearest-neighbor and four-spin) interactions is investigated numerically. This extended Ising model, the so-called gonihedric model, was introduced by Savvidy and Wegner as a discretized version of the interacting (closed) surfaces without surface tension. The gonihedric model is notorious for its slow relaxation to the thermal equilibrium (glassy behavior), which deteriorates the efficiency of the Monte Carlo sampling. We employ the transfer-matrix (TM) method, implementing Novotny's idea, which enables us to treat an arbitrary number of spins N for one TM slice even in three dimensions. This arbitrariness admits systematic finite-size-scaling analyses. Accepting the extended parameter space by Cirillo et al., we analyzed the (multi-) criticality of the gonihedric model for N</=13. Thereby, we found that, as first noted by Cirillo et al. analytically (cluster-variation method), the data are well described by the multicritical (crossover) scaling theory. That is, the previously reported nonstandard criticality for the gonihedric model is reconciled with a crossover exponent and the ordinary three-dimensional-Ising universality class. We estimate the crossover exponent and the correlation-length critical exponent at the multicritical point as phi=0.6(2) and nu; =0.45(15), respectively.

Slow dynamics in the three-dimensional gonihedric model

We study dynamical aspects of three-dimensional gonihedric spins by using Monte-Carlo methods. These models have a purely geometrical motivation, deriving from string and random surface theory. Here, however, we shall analyze this family of models just from a statistical point of view. In particular, we shall be concerned with their ability to exhibit remarkably slow dynamics and seemingly glassy behavior below a certain temperature T(g), without the need of introducing disorder of any kind. We consider first a Hamiltonian that takes into account only a four-spin term (kappa=0), where a first-order phase transition is well established. By studying the relaxation properties at low temperatures, we confirm that the model exhibits two distinct regimes. For T(g)<T<T(c), with long lived metastability and a supercooled phase, the approach to equilibrium is well described by a stretched exponential. For T<T(g), the dynamics appears to be logarithmic. We provide an accurate determination of T(g). We also determine the evolution of particularly long lived configurations. Next, we consider the case kappa=1, where the plaquette term is absent and the gonihedric action consists in a ferromagnetic Ising with fine-tuned next-to-nearest neighbor interactions. This model exhibits a second order phase transition. The consideration of the relaxation time for configurations in the cold phase reveals the presence of slow dynamics and glassy behavior for any T<T(c). Type-II aging features are exhibited by this model.