Phase transition in lattice surface systems with gonihedric action

Pietig, R.; Wegner, Franz

doi:10.1016/0550-3213(96)00072-7

Cited by 41 publications

(43 citation statements)

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“…Thus (5) will be arbitrarily small for sufficiently large β. The same type of argument was used in [33] to show the existence of a phase transition in the gonihedric model for k > 0. The argument given there however contains a flaw, since the edge diagrams are not independent from each other, if k > 0 3 .…”

Section: Introductionmentioning

confidence: 99%

Low temperature expansion of the gonihedric Ising model

Pietig

Wegner

1998

Nuclear Physics B

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We investigate a model of closed (d − 1)-dimensional soft-self-avoiding random surfaces on a d-dimensional cubic lattice. The energy of a surface configuration is given by E = J(n 2 + 4k n 4 ), where n 2 is the number of edges, where two plaquettes meet at a right angle and n 4 is the number of edges, where 4 plaquettes meet. This model can be represented as a Z 2 -spin system with ferromagnetic nearest-neighbour-, antiferromagnetic next-nearest-neighbour-and plaquette-interaction. It corresponds to a special case of a general class of spin systems introduced by Wegner and Savvidy. Since there is no term proportional to the surface area, the bare surface tension of the model vanishes, in contrast to the ordinary Ising model. By a suitable adaption of Peierls argument, we prove the existence of infinitely many ordered low temperature phases for the case k = 0. A low temperature expansion of the free energy in 3 dimensions up to order x 38 (x = e −βJ ) shows, that for k > 0 only the ferromagnetic low temperature phases remain stable. An analysis of low temperature expansions up to order x 44 for the magnetization, susceptibility and specific heat in 3 dimensions yields critical exponents, which are in agreement with previous results.

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Section: Introductionmentioning

confidence: 99%

Low temperature expansion of the gonihedric Ising model

Pietig

Wegner

1998

Nuclear Physics B

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“…An alternative approach to discretizing the linear size would be to restrict the allowed surfaces to the plaquettes of a (hyper)cubic lattice, which corresponds to also discretizing the target space. Savvidy and Wegner [3,4,5,6] did this and rewrote the resulting model as an equivalent generalized Ising model using the geometrical spin cluster boundaries to define the surfaces. The energy of a surface on a cubic lattice is then given by E = n 2 + 4κn 4 , where n 2 is the number of links where two plaquettes meet at a right angle, n 4 is the number of links where four plaquettes meet at right angles, and κ is a free parameter which determines the relative weight of a self-intersection of the surface.…”

Section: The Modelmentioning

confidence: 99%

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

et al. 1997

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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.

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“…(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 precisely this feature is a 3D plaquette (4-spin) interaction Ising model on a cubic lattice where q = 2 3L on an L 3 lattice [9]. This is a member of a family of so-called gonihedric Ising models [10] whose Hamiltonians contain, in general, nearest i, j , next-tonearest i, j and plaquette interactions [i, j, k, l].…”

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Nonstandard Finite-Size Scaling at First-Order Phase Transitions

Mueller

Janke

Johnston³

2014

Phys. Rev. Lett.

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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...

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Phase transition in lattice surface systems with gonihedric action

Cited by 41 publications

References 13 publications

Low temperature expansion of the gonihedric Ising model

Low temperature expansion of the gonihedric Ising model

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

Nonstandard Finite-Size Scaling at First-Order Phase Transitions

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