Finite-size scaling and phenomenological renormalization (invited)

Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.