Maximally random discrete-spin systems with symmetric and asymmetric interactions and maximally degenerate ordering

Abstract

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 = 3,4 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 noninfinite temperatures, the system eventually renormalizes to a random single state, thus signaling q × 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 + , the system is as expected disordered at all temperatures for d = 1.

Lower lower-critical spin-glass dimension from quenched mixed-spatial-dimensional spin glasses

“…see Refs. [24][25][26][27][28][29][30][31][32]. Thus, previous works have generally used a hierarchical lattice generated by a single graph and spatial dimensionality that is microscopically uniform throughout the system.…”

“…The local renormalization-group transformation proceeds by b d−1 bond-movings followed b = 3 (to preserve the ferromagnetic-antiferromagnetic symmetry) decimations, generating a distribution of 500 new interactions, which is of course no longer double valued. [32] (In fact, for numerical efficiency, these operations are broken down to binary steps, each involving two distributions of 500 interactions.) In the disordered phase, the interactions converge to zero.…”

“…The bond J ij is ferromagnetic +J > 0 or antiferromagnetic −J with respective probabilities 1 − p and p. This Hamiltonian is lodged on the hierarchical lattice constructed with the two graphs shown For recent exact calculations on hierarchical lattices, see Refs. [24][25][26][27][28][29][30][31][32]. Thus, previous works have generally used a hierarchical lattice generated by a single graph and spatial dimensionality that is microscopically uniform throughout the system.…”

Metastable reverse-phase droplets within ordered phases: Renormalization-group calculation of field and temperature dependence of limiting size

“…After outstanding success in the calculation of critical exponents and in understanding the mechanism underlying the universality of critical exponents, renormalization-group theory has been equally successfully applied to global non-universal properties at and away from critical points, such as entire thermodynamic functions, discontinuities at first-order phase transitions, and entire multicritical phase diagrams, e.g., leading all the way to global spin-glass phase diagrams in the variables of temperature, bond concentration, spatial dimensionality d, and the continuous variation of chaos and its Lyapunov exponent inside spin-glass phases [1,2]. Such wide application has not yet been reached in non-equilibrium systems.…”