Maximally random discrete-spin systems with symmetric and asymmetric interactions and maximally degenerate ordering
Discrete-spin systems with maximally random nearest-neighbor interactions that can be symmetric or asymmetric ferromagnetic or antiferromagnetic including off-diagonal disorder are studied for the number of states q = 34 in d dimensions. We use renormalization-group theory that is exact for hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic lattices. For all d > 1 and all nonmfimte temperatures the system eventually renormalizes to a random single state thus signaling q x q degenerate ordering. Note that this is the maximally degenerate ordering. For high-temperature initial conditions the system crosses over to this highly degenerate ordering only after spending many renormalization-group iterations near the disordered (infinite-temperature) fixed point. Thus a temperature range of short-range disorder in the presence of long-range order is identified as previously seen in underfrustrated Ising spin-glass systems. The entropy is calculated for all temperatures behaves similarly for ferromagnetic and antiferromagnetic interactions and shows a derivative maximum at the short-range disordering temperature. With a sharp immediate contrast of infinitesimally higher dimension 1 + epsilon the system is as expected disordered at all temperatures for d = 1.