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→∞ 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.