I want to know how to obtain the universality class of the phase transition from the central charge "c" in one dimensional model. If c is less than 1, there is a one-to-one correspondence. But if c is large or equal to 1, then how can I determine the universality class of that phase transition? In particular, when c=1, how many candidate universality class one can obtain?
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$c=1$ rational CFTs have been completely classified, and the full list was first given in http://www.sciencedirect.com/science/article/pii/0550321388902490. There are infinite number of them. This should not be too surprising, since free compact bosons with different compactification radius already give you infinite examples. The rest are mostly orbifolds of free bosons. So just knowing the central charge does not uniquely determine the universality class. Much less is known for $c>1$ CFTs.
answered Jul 30, 2015 at 3:33