What is the central charge of the disordered $q$-state Potts model, for large $q$?

Question

The central charge of a model, is, heuristically related to the number of microscopic degrees of freedom. Is there a simple argument for the asymptotic behavior of the central charge for the disordered $q$-state Potts model for large $q$?

Further, is there any other set of critical lattice models, that show different critical behaviors corresponding to some parameter and all of which appear in the unitary series of 2d CFTs?

This post imported from StackExchange Physics at 2014-12-30 13:59 (UTC), posted by SE-user Srivatsan Balakrishnan

Comments

The phase transition of the Potts model is first-order for $q\geq 5$ (in two dimensions), there is no critical point.

This post imported from StackExchange Physics at 2014-12-30 13:59 (UTC), posted by SE-user Yvan Velenik

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Ah, yes. I am supposed to know that! I would still ask the same question, with disorder.

This post imported from StackExchange Physics at 2014-12-30 13:59 (UTC), posted by SE-user Srivatsan Balakrishnan

Answer 1

The phase transition of the pure Potts model is indeed first-order for $q\ge 5$ but a continuous transition is induced for any number of states $q$ when the exchange couplings are made random. In the large $q$-limit, the central charge has been conjectured to follow the law

    $$c(q)={1\over 2}\ln_2 q$$ 

See the reference J.L. Jacobsen, and M. Picco (2000), Phys. Rev. E 61, R13. Note that there are some subtleties in the definition of $c$. The precise CFT of the random Potts model is not known but it cannot be unitary.