First-Order Transition in Potts Models with "Invisible" States: Rigorous Proofs

Enter, Aernout van; Iacobelli, Giulio; Taati, Siamak

doi:10.1143/ptp.126.983

Cited by 12 publications

(18 citation statements)

References 8 publications

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“…[9] for q > 1 and sufficiently large r. The transmutation of some second-order transitions into first-order transition by invisible states is therefore well established. In Ref.…”

Section: Introductionmentioning

confidence: 93%

Potts models with invisible states on general Bethe lattices

Ananikian¹,

Izmailyan²,

Johnston³

et al. 2013

J. Phys. A: Math. Theor.

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Abstract.The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q = 2 case, a Bragg-Williams, mean-field approach necessitates four such invisible states while a 3-regular, random-graph formalism requires seventeen. In both of these cases, the changeover from second-to first-order behaviour induced by the invisible states is identified through the tricritical point of an equivalent BlumeEmery-Griffiths model.Here we investigate the generalised Potts model on a Bethe lattice with z neighbours. We show that, in the q = 2 case, r c (z) = 4z 3(z − 1)states are required to manifest the equivalent Blume-Emery-Griffiths tricriticality. When z = 3, the 3-regular, random-graph result is recovered, while z → ∞ delivers the Bragg-Williams, mean-field result.arXiv:1307.2803v1 [cond-mat.stat-mech]

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“…[9] for q > 1 and sufficiently large r. The transmutation of some second-order transitions into first-order transition by invisible states is therefore well established. In Ref.…”

Section: Introductionmentioning

confidence: 93%

Potts models with invisible states on general Bethe lattices

Ananikian¹,

Izmailyan²,

Johnston³

et al. 2013

J. Phys. A: Math. Theor.

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“…In this limit the system becomes one of non-interacting particles. Note that for the 2D Potts model with invisible states on a square lattice T c vanishes for large (q + r) as T c ≈ 2/ ln(q + r) [23].…”

Section: The Case Q =mentioning

confidence: 99%

“…Numerical simulations of the Potts model with invisible states in d = 2 dimensions gave solid evidence that the model exhibits a first-order phase transition at q = 2, 3, 4 for high values of r but it still remains a challenge for numerics to get a more precise estimate of r c [16]. Rigorous results prove the existence of a first-order regime for any q > 0, provided that r is large enough [23,24]. Exact results for the value of r c for the model are known for a Bethe lattice [25] too.…”

Section: Introductionmentioning

confidence: 95%

Marginal dimensions of the Potts model with invisible states

Krasnytska

Sarkanych

Berche

et al. 2016

J. Phys. A: Math. Theor.

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Abstract.We reconsider the mean-field Potts model with q interacting and r non-interacting (invisible) states. The model was recently introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where the Z q -symmetry is spontaneously broken. We analyse the marginal dimensions of the model, i.e., the value of r at which the order of the phase transition changes. In the q = 2 case, we determine that value to be r c = 3.65(5); there is a second-order phase transition there when r < r c and a first-order one at r > r c . We also analyse the region 1 ≤ q < 2 and show that the change from second to first order there is manifest through a new mechanism involving two marginal values of r. The q = 1 limit gives bond percolation and some intermediary values also have known physical realisations. Above the lower value r c1 , the order parameters exhibit discontinuities at temperaturet below a critical value t c . But, provided r > r c1 is small enough, this discontinuity does not appear at the phase transition, which is continuous and takes place at t c . The larger value r c2 marks the point at which the phase transition at t c changes from second to first order. Thus, for r c1 < r < r c2 , the transition at t c remains second order while the order parameter has a discontinuity att. As r increases further,t increases, bringing the discontinuity closer to t c . Finally, when r exceeds r c2 t coincides with t c and the phase transition becomes first order. This new mechanism indicates how the discontinuity characteristic of first order phase transitions emerges.

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“…This model was originally suggested to explain why the phase transition with the q−fold symmetry breaking undergoes a different order than predicted theoretically [7,8,9]. Analysis of this model on different lattices has been a subject of intensive analytic [10,11,12,13,14,15,16] and numerical [7,8,9] studies. It has been shown that the number of invisible states (r) plays the role of a parameter, whose increase makes the phase transition sharper.…”

Section: Introductionmentioning

confidence: 99%

Ising model with invisible states on scale-free networks

Sarkanych

Krasnytska

2019

Physics Letters A

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We consider the Ising model with invisible states on scale-free networks. Our goal is to investigate the interplay between the entropic and topological influence on a phase transition. The former is manifest through the number of invisible states r, while the latter is controlled by the network node-degree distribution decay exponent λ. We show that the phase diagram in this case is characterised by two marginal values r c1 (λ) and r c2 (λ), which separate regions with different critical behaviours. Below the r c1 (λ) line the system undergoes only second order phase transition; above the r c2 (λ) -only a first order phase transition occurs; and in-between the lines both of these phase transitions occur at different temperatures. This behaviour differs from the one, observed on the lattice, where the Ising model with invisible states is only characterised with one marginal value r c 3.62 separating the first and second order regimes.

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First-Order Transition in Potts Models with "Invisible" States: Rigorous Proofs

Cited by 12 publications

References 8 publications

Potts models with invisible states on general Bethe lattices

Potts models with invisible states on general Bethe lattices

Marginal dimensions of the Potts model with invisible states

Ising model with invisible states on scale-free networks

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