Maximally random systems, maximally degenerate ordering, and lower lower-critical spin-glass dimension

Official URL: http://risc01.sabanciuniv.edu/record=b1817195 (Table of Contents)

Discrete-spin systems with maximally random nearest-neighbor interactions that is symmetric or asymmetric, ferromagnetic or antiferromagnetic, including off-diagonal disorder, are studied, for q=3,4 states in d dimensions. Using renormalization-group theory, for all d≥1 and all noninfinite temperatures, the system eventually renormalizes to a random single state, thus signaling q×q degenerate ordering, which is 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 behaves similarly for ferromagnetic and antiferromagnetic interactions, and shows a derivative maximum at the short-range disordering temperature. As expected, the system is disordered at all temperatures for d=1 . By quenched-randomly mixing local units of different spatial dimensionalities, we also have studied Ising spin-glass systems on hierarchical lattices continuously in dimensionalities 1≤d≤3. The global phase diagram in temperature, antiferromagnetic bond concentration, and spatial dimensionality is calculated. We found, the spin-glass phase disappears to zero temperature at the lower-critical dimension d c=2.431. This sets an upper limit to the lower-critical dimension in general for the Ising spin-glass phase. As dimension is lowered towards dc, the spin-glass critical temperature continuously goes to zero, but the spin-glass chaos fully sustains to the brink of the disappearance of the spin-glass phase. The Lyapunov exponent, measuring the strength of chaos, is thus largely unaffected by the approach to dc and shows a discontinuity to zero at dc