Potts model: Difference between revisions

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m (Trivial tidy up.)
m (some details about phase behavior)
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where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
where <math> K </math> is the coupling constant, <math> \langle ij \rangle </math> indicates
that the sum is performed exclusively over pairs of nearest neighbour sites,  and <math> \delta(S_i,S_j) </math> is the [[Kronecker delta|Kronecker delta]].
that the sum is performed exclusively over pairs of nearest neighbour sites,  and <math> \delta(S_i,S_j) </math> is the [[Kronecker delta|Kronecker delta]].
Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]]
Note that the particular case <math> q=2 </math> is equivalent to the [[Ising Models | Ising model]].
 
Considering a symmetric situation (i.e. equal chemical potential for all the species):
 
<math> \mu_1 = \mu_2 = \cdots = \mu_q </math>;
 
the Potts models exhibit order-disorder phase transitions. For space dimensionality <math> d=2 </math>, and low
values of <math> q </math> the transitions are continuous (<math> E(T) </math> is a continuous function), but the heat capacity
<math> C(T) = (\partial E/\partial T) </math> diverges at the transition temperature. The critical behavior of
different values of <math> q </math> corresponds to different critical universality classes.
 
For space dimensionality <math> d=3 </math>, the transitions for <math> q \ge 3 </math> are first order (<math> E </math>
shows a discontinuity at the transition temperature).
 
==See also==
==See also==
*[[Ashkin-Teller model]]
*[[Ashkin-Teller model]]

Revision as of 10:59, 7 July 2008

The Potts model was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs. 2 and 3. In practice one has a lattice system. The sites of the lattice can be occupied by particles of different species, .

The energy of the system, , is defined as:

where is the coupling constant, indicates that the sum is performed exclusively over pairs of nearest neighbour sites, and is the Kronecker delta. Note that the particular case is equivalent to the Ising model.

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

;

the Potts models exhibit order-disorder phase transitions. For space dimensionality , and low values of the transitions are continuous ( is a continuous function), but the heat capacity diverges at the transition temperature. The critical behavior of different values of corresponds to different critical universality classes.

For space dimensionality , the transitions for are first order ( shows a discontinuity at the transition temperature).

See also

References

  1. Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society 48 pp. 106−109 (1952)
  2. F. Y. Wu "The Potts model", Reviews of Modern Physics 54 pp. 235-268 (1982)
  3. F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics 55 p. 315 (1983)