Vortex lines in the three-dimensional XY model with random phase shifts

The stability of the ordered phase of the three-dimensional XY model with random phase shifts is studied by considering the roughening of a single stretched vortex line due to disorder. It is shown that the vortex line may be described by a directed polymer Hamiltonian with an effective random potential that is long-range correlated. A Flory argument estimates the roughness exponent to zeta=3/4 and the energy fluctuation exponent to omega=1/2, thus fulfilling the scaling relation omega=2 zeta-1. The Schwartz-Edwards method as well as a numerical integration of the corresponding Burger's equation confirms this result. Since zeta<1, the ordered phase of the original XY model is stable.