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arXiv:hep-lat/9703008 · Journal of Physics A Mathematical and General

String tension in gonihedric three-dimensional Ising models

M. Baig, Domènec Espriu, Desmond A. Johnston, Ranasinghe P.K.C. Malmini

Published Nov 21, 1997• 17 citations
arXivHTMLPDF

Abstract

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 .

DOI
10.1088/0305-4470/30/22/009

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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 .
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],
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 investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
We consider the three-dimensional Ising model slightly below its critical temperature, with boundary conditions leading to the presence of an interface. We show how the interfacial properties can be deduced starting from the particle modes of the underlying field theory. The product of the surface tension and the correlation length yields the particle density along the string whose propagation spans the interface. We also determine the order parameter and energy density profiles across the interface, and show that they are in complete agreement with Monte Carlo simulations that we perform.
The single and joint distribution functions of the nearest-neighbor cell magnetization of the three-dimensional Ising model have been studied by Monte Carlo methods. Near the critical point, both distribution functions tend, for large LLL, towards scaled universal forms. Estimates of the order parameter and susceptibility are obtained from the distribution functions. The critical temperature is estimated by a method independent of the estimate of the critical exponents. The critical exponent $\ensuremath{\nu}$ is obtained with the use of finite-size scaling arguments. The joint distribution function shows a double-well structure for T<TcT<{T}_{c}T<Tc​ and can be represented by a coarse-grained effective Hamiltonian of two cell variables. We have determined the dependence of this functional on the coarse-graining cell size for several temperatures. In particular, we have calculated the dependence of the "spinodal curve" on the (scaled) coarse-graining size. The relevance of this work to the kinetics of first-order phase transitions is also discussed.
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.