The Potts model |
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The Q-state Potts model is a lattice model.
Each spin on the lattice can take on an integer value between 1 and Q.
The particular value Q=2 reduces to the Ising model.
The model Hamiltonian is given by:
where the sum is over every pair of nearest-neighbors, and the minus sign
ensures that interactions are indeed ferromagnetic.
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Experiments and applications : (in the following, D denotes the spatial dimension)
- (D=2, Q=3) transitions in monolayers adsorbed on a surface (lattice gas);
- (D=2) the Q=0 and Q=1 limits describe infinite resistor networks ("spanning trees") and percolation processes, respectively.
- (D=3) magnetism in some alloys (e.g., CoCs2Br5, CoCs3Cl5, DyPO4);
- (D=3) structural transitions (e.g., SrTiO3)
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Thermodynamic properties :
the two-dimensional Potts model exhibits a first-order phase transition
whenever q > 4, and a second-order transition if q < 4
In two dimensions, i.e., with a coordinance number z=4,
mean-field theory (Kihara et al., 1954) yields
the following transition temperature:
- Q=2 (Ising) : Tc = 2.0
- Q>2 : Tc = 2(q-2)/[(q-1)log(q-1)]
(to be compared with the exact result 1/Tc = ln(1+sqrt(q))).
| Q | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Tc(MF) | 2.0 | 1.44 | 1.21 | 1.08 | 0.99 | 0.93 | 0.88 | 0.84 |
| Tc(exact) | 1.1346 | 0.99497 | 0.91024 | 0.85153 | 0.80761 | 0.77306 | 0.74490 | 0.72135 |
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MF theory obviously leads to overestimated temperatures wrt actual transition temperatures.
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Bibliography :
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