We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in Z 3. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the selfavoidance coupling κ is varied. The value of the magnetization critical exponent β = 0.062 ±0.003, calculated with the cluster variation–Padè approximant method, is also close to the simulation results. 1 The Model In this paper we discuss the use of the cluster variation method (CVM) in mapping out the phase diagram of the gonihedric 3d Ising model. The gonihedric 3d Ising model is a generalization of the usual 3d Ising model where planar Peierls boundaries between + and − spins can be created at zero energy cost. It has been introduced in [1] in relations with string theory. The CVM method in other contexts has shown itself to be an accurate and economical way of describing the phase diagram both for first order and continuous transitions. As both the model and the method may be unfamiliar, we outline both in turn before going on to describe our results. The genesis of the model is in a novel discretized random surface theory, the so-called gonihedric string [1],
We use the cluster variation method (CVM) to investigate the phase structure of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The geometrical spin cluster boundaries in these systems serve as models for the string worldsheets of the gonihedric string embedded in Z 3. The models are interesting from the statistical mechanical point of view because they have a vanishing bare surface tension. As a result the action depends only on the angles of the discrete surface and not on the area, which is the antithesis of the standard 3d Ising model. The results obtained with the CVM are in good agreement with Monte Carlo simulations for the critical temperatures and the order of the transition as the selfavoidance coupling κ is varied. The value of the magnetization critical exponent β = 0.062 ±0.003, calculated with the cluster variation–Padè approximant method, is also close to the simulation results. 1 The Model In this paper we discuss the use of the cluster variation method (CVM) in mapping out the phase diagram of the gonihedric 3d Ising model. The gonihedric 3d Ising model is a generalization of the usual 3d Ising model where planar Peierls boundaries between + and − spins can be created at zero energy cost. It has been introduced in [1] in relations with string theory. The CVM method in other contexts has shown itself to be an accurate and economical way of describing the phase diagram both for first order and continuous transitions. As both the model and the method may be unfamiliar, we outline both in turn before going on to describe our results. The genesis of the model is in a novel discretized random surface theory, the so-called gonihedric string [1],
For the three-dimensional gonihedric Ising models defined by Savvidy and Wegner the bare string tension is zero and the energy of a spin interface depends only on the number of bends and self-intersections, in antithesis to the standard nearest-neighbour three-dimensional Ising action. When the parameter , weighting the self-intersections, is small the model has a first-order transition and when it is larger the transition is continuous. In this paper we investigate the scaling of the renormalized string tension, which is entirely generated by fluctuations, using Monte Carlo simulations for , 0.1, 0.5 and 1.0. The scaling of the string tension allows us to obtain an estimate for the critical exponents and using both finite-size scaling and data collapse for the scaling function. The behaviour of the string tension when the self-avoidance parameter is small also clearly demonstrates the first-order nature of the transition in this case, in contrast to larger values. Direct estimates of are in good agreement with those obtained from the scaling of the string tension. We have also measured .
The gonihedric Ising Hamiltonians defined in three and higher dimensions by Savvidy and Wegner provide an extensive, and little explored, catalogue of spin models on (hyper)cubic lattices with many interesting features. In three dimensions the kappa=0 gonihedric Ising model on a cubic lattice has been shown to possess a degenerate low-temperature phase and a first order phase transition, as well as interesting dynamical properties. The dual Hamiltonian to this may be written as an anisotropic Ashkin-Teller model and also has a degenerate low-temperature phase as a result of similar symmetries to the original plaquette action. It is possible to write an alternative dual formulation which utilizes three flavours of spins, rather than the two of the Ashkin-Teller model. This still possesses anisotropic couplings, but all the interaction terms are now four spin couplings and it acquires an additional, local gauge symmetry. We investigate this alternative dual Hamiltonian using zero temperature and mean-field methods together with Monte-Carlo simulations and discuss its properties and the relation to the Ashkin-Teller variant.
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.
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.
A 3D Ising model with a purely plaquette, 4-spin interaction displays a planar flip symmetry intermediate between a global and a gauge symmetry and as a consequence has a highly degenerate low temperature phase and no standard magnetic order parameter. This plaquette Hamiltonian is a particular case of a family of 3D gonihedric Ising models defined by Savvidy and Wegner. An anisotropic variant of the purely plaquette gonihedric model, originally discussed as the ‘Fuki-Nuke’ model by Suzuki, is non-trivially equivalent to a stack of 2D Ising models, each of which can magnetize independently at the phase transition point. Consideration of this anisotropic model suggests that a suitable order parameter in the isotropic case may also be constructed using a form of planar magnetization, in which nearest neighbour correlators 〈σ i σ j 〉 summed over planes replace a sum over spin values 〈σ i 〉. We conduct Monte-Carlo simulations to investigate this and related candidate order parameters in a toy model of gonihedric ground states, the Fuki-Nuke model and the isotropic plaquette gonihedric model itself.
We report results of the application of a recently proposed technique, which combines the cluster variation method (CVM) and Pad\'e approximants, to the analysis of the critical behavior of the face-centered-cubic Ising model. The highest-order CVM approximation ever considered for this lattice is used to estimate magnetization and susceptibility, which are then analyzed by means of Dlog Pad\'e approximants in order to extract information about the critical behavior, taking also into account the effects of confluent singularities. Our results are in good agreement with the most recent simulation and series expansion estimates and require a remarkably small numerical effort. \textcopyright{} 1996 The American Physical Society.