Maximally Random Discrete-Spin Systems with Symmetric and Asymmetric Interactions and Maximally Degenerate Ordering

Discrete-spin systems with maximally random nearest-neighbor interactions that are symmetric or asymmetric, ferromagnetic or antiferromagnetic, including off-diagonal disorder, are studied for q=3,4 states in d dimensions, using renormalization-group theory exact for hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic lattices. For all d>1 and all non-infinite temperatures, the system eventually renormalizes to a random single state, signaling qxq degenerate ordering, which is maximally degenerate ordering. For high-temperature initial conditions, the system crosses over to this highly degenerate ordering after many renormalization-group iterations near the disordered infinite-temperature fixed point. Thus, a temperature range of short-range disorder in presence of long-range order occurs, as previously seen in underfrustrated Ising spin-glasses. The calculated entropy behaves similarly for ferromagnetic and antiferromagnetic interactions and shows a derivative maximum at the short-range disordering temperature. The system is disordered at all temperatures for d=1.
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