Potts models with invisible states on general Bethe lattices

Ananikian, N. S.; Izmailyan, N.Sh.; Johnston, D.A.; Kenna, Ralph; Ranasinghe, R. P. K. C. M.

doi:10.1088/1751-8113/46/38/385002

Cited by 14 publications

(26 citation statements)

References 31 publications

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“…Here, we determine a more precise estimate for the value for q = 2, namely r c ≃ 3.65(5). This estimate is in excellent agreement with the z → ∞ limit of the result obtained for the Bethe lattice with z nearest neighbours [25]. We then considered the region 1 ≤ q < 2.…”

Section: Discussionsupporting

confidence: 85%

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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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Section: Discussionsupporting

confidence: 85%

“…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: 87%

Marginal dimensions of the Potts model with invisible states

Krasnytska

Sarkanych

Berche

et al. 2016

J. Phys. A: Math. Theor.

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“…The multispin extension of this model possesses a re-entrant phase transition and is in good agreement with experimental observations for polymer transitions [31,32]. The (2, r)-state Potts model without external fields is equivalent to the Blume-Emery-Grifiths (BEG) model [19,33,34] with a temperature dependent external field. Furthermore, the general q and r case can be interpreted as a diluted Potts model [19,29] …”

Section: Introductionsupporting

confidence: 82%

Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states

2017

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We present the exact solution of the 1D classical short-range Potts model with invisible states. Besides the q states of the ordinary Potts model, this possesses r additional states which contribute to the entropy, but not to the interaction energy. We determine the partition function, using the transfer-matrix method, in the general case of two ordering fields: h 1 acting on a visible state and h 2 on an invisible state. We analyse its zeros in the complex-temperature plane in the case that h 1 = 0. When Im h 2 = 0 and r ≥ 0, these zeros accumulate along a line that intersects the real temperature axis at the origin. This corresponds to the usual "phase transition" in a 1D system. However, for Im h 2 = 0 or r < 0, the line of zeros intersects the positive part of the real temperature axis, which signals the existence of a phase transition at non-zero temperature.

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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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Potts models with invisible states on general Bethe lattices

Cited by 14 publications

References 31 publications

Marginal dimensions of the Potts model with invisible states

Marginal dimensions of the Potts model with invisible states

Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states

Ising model with invisible states on scale-free networks

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