The dual gonihedric 3D Ising model

Johnston, D.A.; Ranasinghe, R. P. K. C. M.

doi:10.1088/1751-8113/44/29/295004

Cited by 11 publications

(31 citation statements)

References 45 publications

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“…Such effects are very difficult to tackle using canonical Monte Carlo data, as already remarked on by the authors of Ref. [10].…”

Section: Dual Model With Periodic Boundary Conditionsmentioning

confidence: 96%

“…The earlier canonical Monte Carlo simulations of the original plaquette model yielded values of β ∞ = 0.50(1) [8] and more recently canonical simulations of the dual model (3) gave β ∞ = 0.510(2) [10]. Another previous estimate for the infinite-lattice inverse transition temperature, reported by Baig et al [9] from canonical simulations using fixed boundary conditions, β ∞ = 0.54757(63), is much closer to the results here.…”

Section: Original Plaquette Model With Periodic Boundary Conditionsmentioning

confidence: 99%

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Multicanonical analysis of the plaquette-only gonihedric Ising model and its dual

2014

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The three-dimensional purely plaquette gonihedric Ising model and its dual are investigated to resolve inconsistencies in the literature for the values of the inverse transition temperature of the very strong temperature-driven first-order phase transition that is apparent in the system. Multicanonical simulations of this model allow us to measure system configurations that are suppressed by more than 60 orders of magnitude compared to probable states. With the resulting high-precision data, we find excellent agreement with our recently proposed nonstandard finite-size scaling laws for models with a macroscopic degeneracy of the low-temperature phase by challenging the prefactors numerically. We find an overall consistent inverse transition temperature of β ∞ = 0.551334(8) from the simulations of the original model both with periodic and fixed boundary conditions, and the dual model with periodic boundary conditions. For the original model with periodic boundary conditions, we obtain the first reliable estimate of the interface tension σ = 0.12037(18), using the statistics of suppressed configurations.

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“…Such effects are very difficult to tackle using canonical Monte Carlo data, as already remarked on by the authors of Ref. [10].…”

Section: Dual Model With Periodic Boundary Conditionsmentioning

confidence: 96%

Section: Original Plaquette Model With Periodic Boundary Conditionsmentioning

confidence: 99%

Multicanonical analysis of the plaquette-only gonihedric Ising model and its dual

2014

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“…Computer simulation studies of this model were plagued, however, by enduring inconsistencies in the estimates of the transition temperature. The dual to this plaquette gonihedric Hamiltonian can be written as an anisotropic Ashkin-Teller model [14] in which two spins σ, τ live on each vertex, with nearest-neighbour interactions along the x, y, and z-axes,…”

mentioning

confidence: 99%

“…Unusually, the estimates for β C max V and β eqw fall together because of the aforementioned equality of the O(L −4 ) corrections in the scaling ansatz for these quantities in Eqs. (10) and (14). The fits have been carried out according to the non-standard scaling laws with 1/L 2 corrections.…”

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confidence: 99%

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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“…with two flavours of Ising spins σ, τ [54] and possesses the same ground-state degeneracy as the original plaquette Hamiltonian, since it is still possible to flip planes of spins at zero energy cost, see Figure 3. Although it has not been confirmed with a low-temperature expansion in the manner of the plaquette Hamiltonian, the finite-size scaling properties at the first-order transition in the dual model show that the macroscopic degeneracy persists into the low-temperature phase there too, as we discuss below.…”

Section: A Curious Symmetry -Classical Aspectsmentioning

confidence: 99%

Plaquette Ising models, degeneracy and scaling

Johnston¹,

Mueller

Janke

2017

Eur. Phys. J. Spec. Top.

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Abstract. We review some recent investigations of the 3d plaquette Ising model. This displays a strong first-order phase transition with unusual scaling properties due to the size-dependent degeneracy of the low-temperature phase. In particular, the leading scaling correction is modified from the usual inverse volume behaviour ∝ 1/L 3 to 1/L 2 . The degeneracy also has implications for the magnetic order in the model which has an intermediate nature between local and global order and gives rise to novel fracton topological defects in a related quantum Hamiltonian.

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The dual gonihedric 3D Ising model

Cited by 11 publications

References 45 publications

Multicanonical analysis of the plaquette-only gonihedric Ising model and its dual

Multicanonical analysis of the plaquette-only gonihedric Ising model and its dual

Nonstandard Finite-Size Scaling at First-Order Phase Transitions

Plaquette Ising models, degeneracy and scaling

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