FINITE-SIZE SCALING FOR POTTS MODELS IN LONG CYLINDERS

Using a recently developed method to rigorously control the finite-size behaviour in long cylinders near first-order phase transitions, I calculate the finite-size scaling of the first q + 1 eigenvalues of the transfer matrix of the q-state Potts model in a d-dimensional periodic box of volume L x ... x L x t (assuming that d greater-than-or-equal-to 2 and that q is sufficiently large). I find two simple eigenvalues lambda+/- corresponding to the trivial representation of the global symmetry and an (q - 1)-fold degenerate eigenvalue lambda(perpendicular-to) corresponding to the remaining irreducible representations of the global symmetry group. The finite-size scaling of the gap xi-1(L, beta) = log(lambda+/lambda(perpendicular-to) and of the gap xi(sym)-1(L, beta) = log(lambda+/lambda-) in the symmetric subspace, and their relation to the surface tension, as well as the finite-size scaling of the internal energy E(cyl)(L, beta) = -L-(d-1)d loglambda+/dbeta are discussed. As a final application, I discuss the finite-size scaling of the derivative of xi(L, beta). I prove that 1/nu(L) := log[-Ldxi-1(L, beta)/dbeta]beta=beta(t)(L)/log L converges to the renormalization group eigenvalue y(T) = d, if beta(t)(L) is chosen as the point where xi(sym)-1(L, beta) is minimal. I also propose other definitions of a finite-volume exponent nu(L) which should be more suitable for numerical considerations.