Journal of Statistical Physics, Vol. 61, Nos. 1/2, 1990 Conformal Invariance in Incommensurate Phases

We study finite-size corrections to the free energy of free-fermion models on a torus with periodic, twisted, and fixed boundary conditions. Inside the critical (striped-incommensurate) phase, the free energy density f(N, M) on an Nx M square lattice with periodic (or twisted) boundary conditions scales as f(N, M) = f~-A(s) / (NM)+.... We derive exactly the finite-size-scaling (FSS) amplitudes A(s) as a function of the aspect ratio s = M/N. These amplitudes are universal because they do not depend on details of the free-fermion Hamiltonian. We establish an equivalence between the FSS amplitudes of the free-fermion model and the Coulomb gas system with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while electric defect lines are generated by competition between domain wall separation and system size; The FSS behavior of the free-fermion model is consistent with predictions of the theory of conformal invariance with the conformal charge c = 1. For instance, the FSS amplitude on an infinite cylinder with fixed boundary conditions is found to be one-quarter of that with periodic boundary conditions. Finally, w