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 states Potts model in a d dimensional periodic box of volume L × . . . × L × t (assuming that d ≥ 2 and that q is sufficiently large). I find two simple eigenvalues λ± corresponding to the trivial representation of the global symmetry and an q − 1 fold degenerate eigenvalue λ⊥ corresponding to the remaining irreducible representations of the global symmetry group. The finite-size scaling of the gap ξ−1(L, β) = log(λ+/λ⊥) and of the gap ξ−1 sym(L, β) = log(λ+/λ−) in the symmetric subspace, and their relation to the surface tension, as well as the finite-size scaling of the internal energy Ecyl(L, β) = −L−(d−1)d log λ+/dβ are discussed. As a final application, I discuss the finite-size scaling of the derivative of ξ(L, β). I prove that 1/ν(L) := log[−Ldξ(L, β)/dβ]β=βt(L)/ log L converges to the renormalization group eigenvalue yT = d, if βt(L) is chosen as the point where ξ−1 sym(L, β) is minimal. I also propose other definitions of a finite volume exponent ν(L) which should be more suitable for numerical considerations.