Absence of anomalous dimension in vertex models: semidilute solution of directed polymers

Abstract

We develop a continuum path-integral approach for the ferroelectric five-vertex model in arbitrary d dimensions by mapping it to a directed polymer problem. A renormalization-group approach with an ε=3-d expansion, 3 being the upper critical dimension, is used to study the polymer solution. The free-energy change due to the interaction of the chains has been computed to O(ε), and the exact expression for the second virial coefficient has been obtained. The fixed point of the problem is found to be exactly 2πε. By use of finite-size-scaling theory and thermodynamics, the exponents for the vertex model are obtained from those of the polymeric system as the specific-heat exponent α=(3-d)/2, and the incommensuration exponent β^-=(d-1)/2. The model is anisotropic with two length-scale exponents v?=1 in one direction and v_⊥ =1/2 in the remaining d-1 directions. It is shown that there are no anomalous dimensions so that the exponents we obtain are exact.