Unusual changeover in the transition nature of local-interaction Potts models

Abstract

We present a novel combinatorial approach which allows the determination of the critical temperature and the phase-transition order of Potts models with a round-the-face interaction. Using this approach, it is demonstrated that for some two-dimensional ferromagnetic Potts models with completely local interaction there is a changeover in the transition order at a critical integer qc ≤ 3, where a first order transition is observed for q > qc and a second order transition is assumed at least for q = qc. This stands in contrast to the standard two-spin interaction Potts model where the critical integer value for which the transition is continuous is qc = 4.

Changeover phenomenon in randomly colored Potts models

“…where k #(k, n) ∼ μ n and the growth number μ in general depends on δ [23]. In making such fractals monochromatic, the entropy of the system is changed in the amount of ln #(k, n)q −k 0 q −(n−k) .…”

“…More precisely, two dimensional systems accompanying the spontaneous breaking of the qfold Potts symmetry, are expected to maintain the changeover behavior of the standard model for a wide range of local interaction patterns. There are, however, a few counterexamples [23][24][25].…”

Changeover phenomenon in randomly colored Potts models

“…where k #(k, n) ∼ µ n for some 1 < µ ≤ λ depending on δ [38] and λ is the growth constant of all the fractals with n sites. The expected change in the number of states is given by #(k, n) q −k 0 q −(n−k) so that, assuming #(k, n) is narrowly distributed around its mean and using (3), the free energy per site can be written…”

“…Strictly speaking, if at β solving f sim = 0 we have f frac ≥ 0, then it is entropically disadvantageous for the system to possess large fractals at that temperature. Instead, large simple monochromatic clusters are formed and the system undergoes a first order transition [38] at…”

“…The WL method is specially designed to approximate the entropy S(E) ∝ ln Ω(E) where Ω(E) is the number of states with energy E, at any desired accuracy. This has been exploited to calculate the energy probability-distribution-function (PDF) for samples of various sizes L at pseudo-criticality, e.g., at a temperature where the specific heat is maximal [38,39]. While for a second order transition the pseudo-critical (PC) energy PDF is expected to display a single peak centred around the PC energy, in the first order transition case a double peaked distribution centred around the energies of the coexisting ordered and disordered states is conventional [40].…”