Marginal dimensions of the Potts model with invisible states

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 $1\le q<2$ and show that the change from second to first order there is manifest through a new mechanism involving {\emph{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 temperature $\tilde{t}$ 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 at $\tilde{t}$. 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.

15 pages, 7 figures, 2 tables

References in corpus (12)

• Quadrupolar correlations and spin freezing in S = 1 triangular lattice antiferromagnets
• Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
• Explosive Percolation: Unusual Transitions of a Simple Model
• First-Order Transition to Incommensurate Phase with Broken Lattice Rotation Symmetry in Frustrated Heisenberg Model
• Phase Transition in Potts Model with Invisible States
• Phase Transition of Generalized Ferromagnetic Potts Model - Effect of Invisible States -
• Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems
• Potts Models with Invisible States on General Bethe Lattices
• Potts Models with (17) Invisible States on Thin Graphs
• Potts model with invisible colours: Random-cluster representation and Pirogov-Sinai analysis
• Marginal dimensions for multicritical phase transitions
• A Method to Control Order of Phase Transition: Invisible States in Discrete Spin Models

Cited by in corpus (5)

• Exact solution of a classical short-range spin model with a phase transition in one dimension: the Potts model with invisible states
• Potts Model with Invisible States: A Review
• Classical phase transitions in a one-dimensional short-range spin model induced by entropy depletion or complex fields
• Ising model with invisible states on scale-free networks
• Changeover phenomenon in randomly colored Potts models