Potts model

The Potts model, proposed by Renfrey B. Potts in 1952 , is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs . 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

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

• Ashkin-Teller model
• Kac model

References

Related reading

• Nathan Duff and Baron Peters "Nucleation in a Potts lattice gas model of crystallization from solution", Journal of Chemical Physics 131 184101 (2009)