Planar spin glasses on d-dimensional hierarchical lattices: a Migdal-Kadanoff renormalization-group analysis

The p-state clock spin glass is studied on d-dimensional hierarchical lattices by means of a Migdal-Kadanoff renormalization-group approach. Considering such procedure as an approximation for Bravais lattices, the spin-glass critical temperature is computed for several values of p as a function of the dimension. Our main results are: (a) the XY limit (p → ∞) presents a small, but finite, critical temperature in three dimensions; (b) the Ising and XY spin glasses exhibit the same lower critical dimensions (dℓ ≈ 2.46, in our case) as predicted by early arguments.