Potts model

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The Potts model, proposed by Renfrey B. Potts in 1952 [1][2], is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs [3][4]. In practice one has a lattice system. The sites of the lattice can be occupied by particles of different species,  S=1,2, \cdots, q .

The energy of the system,  E , is defined as:

 E =  - K \sum_{ \langle ij \rangle } \delta (S_i,S_j)

where  K is the coupling constant,  \langle ij \rangle indicates that the sum is performed exclusively over pairs of nearest neighbour sites, and  \delta(S_i,S_j) is the Kronecker delta. Note that the particular case  q=2 is equivalent to the Ising model.

Phase transitions[edit]

Considering a symmetric situation (i.e. equal chemical potential for all the species):

 \mu_1 = \mu_2 = \cdots = \mu_q ;

the Potts model exhibits order-disorder phase transitions. For space dimensionality  d=2 , and low values of  q the transitions are continuous ( E(T) is a continuous function), but the heat capacity,  C(T) = (\partial E/\partial T) , diverges at the transition temperature. The critical behaviour of different values of  q belong to (or define) different universality classes of criticality For space dimensionality  d=3 , the transitions for  q \ge 3 are first order ( E shows a discontinuity at the transition temperature).

See also[edit]


Related reading