Potts models

The Pink model is found to exhibit a gel-fluid transition for lipids with sufficiently long chains, which is weakly first order. The transition disappears in bilayers of shorter lipids, but it leaves a signature in that one observes strong lateral density fluctuations in a narrow temperature region [200,201]. In later studies, the model has been extended in many ways in order to explore various aspects of gel-fluid transitions [202]. For example, Mouritsen et al. [203] have investigated the interplay between chain melting and chain crystallization by coupling a two-state Doniach model or a ten-state Pink model to a Potts model. (The use of Potts models as models for... 

Models lattice gas models, Potts models, clock models, etc. 94... 

Critical slowing down in Ising and Potts models 140... 

In Eq. (17), each lattice site i can be one of g different states, described by the spin variable Sj. An energy J is won only if two neighboring sites are in the same state. The symbol means a summation over all nearest neighbor pairs. In contrast, the vector Potts model (or clock model) has the Hamiltonian ... 

These ideas have been tested by calculations on the ferromagnetic Ising model and the 10-state Potts model , where exact results on T, +, are available , and excellent agreement with the theoretical predictions Eqs. (45)-(50) was found. As an example. Fig. 10 shows the specific heat for finite 10-state Potts lattices and the scaling representation of these data resulting from Eq. (47). Also experimental data on rounded specific heats of O2 adsorbed on grafoil can be accounted by Eq. (47) quantitatively . 

Immediately when the dynamic interpretation of Monte Carlo sampling in terms of the master equation, Eq. (31), was realized an application to study the critical divergence of the relaxation time in the two-dimensional Ising nearest-neighbor ferromagnet was attempted . For kinetic Ising and Potts models without any conservation laws, the consideration of dynamic universality classespredicts where z is the dynamic exponent , but the... 

Phase diagram based on a four-state Ising-Potts model 626... 

A four-state Ising-Potts model is applied to the [Mn(taa)] system to elucidate thermodynamic relations [21]. An [Mn(taa)] molecule is assumed to take four different microscopic states state 0 is the LS state and states 1-3 are the HS states with the elongation axis parallel to x, y, and z, respectively. Interactions are assumed only between nearest neighbor molecules. Under a mean-field approximation [22], the internal energy of the [Mn(taa)] system is expressed in terms of populations Pi (i = 0,1,2,3) of four microscopic states,... 

Fig. 7. Mean-field phase diagram of the four-state Ising-Potts model (A/kB = 90 K, J lkB = 125 K). The high-spin (HS) phase, the low-spin (LS) phase, and the ferroelectric-ordered (FO) phase are shown. The arrow line corresponds to Jo/kB = — 36 K, appropriate to the [Mn(taa)] system.

Fig. 8. Thermodynamic quantities of the four-state Ising-Potts model along the heating path (arrow line) in Fig. 7.

Figure 15.5 Domain distribution with polarizations PX and Py as indicated by different gray tones in a 4-state random field Potts model at temperatures ksT/ J = 10 (a), 1 (b), 0.8 (c),...

The three-state extended pseudo spin model is equivalent to the Potts model well-known in solid state physics. Its application to JT structural phase transitions was developed by Hock et al. [63]. Also, as a useful tool, it was mentioned by Kugel and Khomskii [9]. It was applied to hexagonal perovskites by Crama and Maaskant [64]. It has multiple applications to layered manganites (e.g., see [65]). 

Salje EKH, Devarajan V (1981) Potts model and phase transitions in lead phosphate Pbs(P04)2. J Phys C 14 L1029-L1035... 

Salje EKH (1995) Chemical mixing and stmctural phase transitions The plateau effect and oscillatoiy zoning near surfaces and interfaces. Eur J Mineral 7 791-806 Salje EKH (1999) Ferroelastic phase transitions and mesoscopic stractures. Ferroelectrics 221 1-7 Salje E, Bismayer U, Wrack B, Hensler J (1991) Influence of lattice imperfections on the transition temperatures of structural phase transitions The plateau effect. Phase Trans 35 61-74 Salje E, Devarajan V (1981) Potts model and phase transition in lead phosphate Pb3(P04)2. J Phys C 14 L1029-L1035... 

Berker AN, Wortis M. Blume-Emery-Griffiths-Potts model in two dimensions phase diagram and critical properties from a position-space renormalization group. Physical Review B. 1976 14 4946-4963. DOI 10.1103/PhysRevB.14.4946. 

Strictly, the Fortuin-Kasteleyn mapping applies to the g-state Potts model, but the Ising model is equivalent to a Potts model with q = 2. 

While cluster updates can solve critical slowing down at second order phase transitions they are usually inefficient at first order phase transitions and in frustrated systems. Let us consider a first order phase transition, such as in a two-dimensional q -state Potts model with Hamilton function... 

The systems exhibiting the >/3 x /3 ordered state are expected to belong to the universality class of the three-state Potts model, while those forming the 2x2 ordered state to the four-state Potts model [138 - 140]. These predictions have been confirmed by theoretical calculations as well as by experiment [103,136,137]. 

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