Ground-state phase-space structures of two-dimensional ±J spin glasses: A network approach

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (+/- J) spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.