Lower-Critical Dimension of the Random-Field XY Model and the Zero-Temperature Critical Line

The random-field XY model is studied in spatial dimensions d=3 and 4, and in-between, as the limit q --> \infty of the q-state clock models, by the exact renormalization-group solution of the hierarchical lattice or, equivalently, the Migdal-Kadanoff approximation to the hypercubic lattices. The lower-critical dimension is determined between 3.81 < d_c <4. When the random-field is scaled with q, a line segment of zero-temperature criticality is found in d=3. When the random-field is scaled with q^2, a universal phase diagram is found at intermediate temperatures in d=3.