# Marginal dimensions of the Potts model with invisible states

M. Krasnytska, Petro Sarkanych, B. Berche, Y. Holovatch, Ralph Kenna

Research output: Contribution to journalArticle

4 Citations (Scopus)

### 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 ${r}_{{\rm{c}}}=3.65(5);$ there is a second-order phase transition there when $r\lt {r}_{{\rm{c}}}$ and a first-order one at $r\gt {r}_{{\rm{c}}}$. We also analyse the region $1\leqslant q\lt 2$ 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. Above the lower value r c1, the order parameters exhibit discontinuities at temperature $\tilde{t}$ below a critical value 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 ${r}_{{\rm{c}}1}\lt r\lt {r}_{{\rm{c}}2}$, the transition at t c remains second order while at $\tilde{t}$ the system undergoes a first order phase transition. As r increases further, $\tilde{t}$ increases, bringing the discontinuity closer to t c. Finally, when r exceeds r c2 $\tilde{t}$ 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.

Original language English 255001 Journal of Physics A: Mathematical and Theoretical 49 25 https://doi.org/10.1088/1751-8113/49/25/255001 Published - 12 May 2016

### Fingerprint

Potts model
Potts Model
First-order Phase Transition
Phase Transition
Phase transitions
Discontinuity
First-order
discontinuity
Mean-field Model
Order Parameter
Discrepancy
Critical value
Exceed
Symmetry
Prediction
Model

### Bibliographical note

Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.

### Cite this

Marginal dimensions of the Potts model with invisible states. / Krasnytska, M.; Sarkanych, Petro; Berche, B.; Holovatch, Y.; Kenna, Ralph.

In: Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 25, 255001, 12.05.2016.

Research output: Contribution to journalArticle

Krasnytska, M. ; Sarkanych, Petro ; Berche, B. ; Holovatch, Y. ; Kenna, Ralph. / Marginal dimensions of the Potts model with invisible states. In: Journal of Physics A: Mathematical and Theoretical. 2016 ; Vol. 49, No. 25.
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AU - Berche, B.

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AU - Kenna, Ralph

N1 - Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.

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N2 - 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 ${r}_{{\rm{c}}}=3.65(5);$ there is a second-order phase transition there when $r\lt {r}_{{\rm{c}}}$ and a first-order one at $r\gt {r}_{{\rm{c}}}$. We also analyse the region $1\leqslant q\lt 2$ 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. Above the lower value r c1, the order parameters exhibit discontinuities at temperature $\tilde{t}$ below a critical value 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 ${r}_{{\rm{c}}1}\lt r\lt {r}_{{\rm{c}}2}$, the transition at t c remains second order while at $\tilde{t}$ the system undergoes a first order phase transition. As r increases further, $\tilde{t}$ increases, bringing the discontinuity closer to t c. Finally, when r exceeds r c2 $\tilde{t}$ 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.

AB - 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 ${r}_{{\rm{c}}}=3.65(5);$ there is a second-order phase transition there when $r\lt {r}_{{\rm{c}}}$ and a first-order one at $r\gt {r}_{{\rm{c}}}$. We also analyse the region $1\leqslant q\lt 2$ 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. Above the lower value r c1, the order parameters exhibit discontinuities at temperature $\tilde{t}$ below a critical value 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 ${r}_{{\rm{c}}1}\lt r\lt {r}_{{\rm{c}}2}$, the transition at t c remains second order while at $\tilde{t}$ the system undergoes a first order phase transition. As r increases further, $\tilde{t}$ increases, bringing the discontinuity closer to t c. Finally, when r exceeds r c2 $\tilde{t}$ 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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