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arXiv:hep-lat/9607078 · Physics Letters A

The phase diagram of the gonihedric 3d Ising model via CVM

E.N.M Cirillo, G Gonnella, D.A Johnston, A Pelizzola

Published Feb 1, 1997• 21 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(96)00918-8

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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],

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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 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.
The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.