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

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 2{π}{ε}. 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 ν?=1 in one direction and ν_⊥ =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.