Phase-Space Networks of Geometrical Frustrated Systems

Geometric frustration leads to complex phases of matter with exotic properties. Antiferromagnets on triangular lattices and square ice are two simple models of geometrical frustration. We map their highly degenerated ground-state phase spaces as discrete networks such that network analysis tools can be introduced to phase-space studies. The resulting phase spaces establish a novel class of complex networks with Gaussian spectral densities. Although phase-space networks are heterogeneously connected, the systems are still ergodic except under periodic boundary conditions. We elucidate the boundary effects by mapping the two models as stacks of cubes and spheres in higher dimensions. Sphere stacking in various containers, i.e. square ice under various boundary conditions, reveals challenging combinatorial questions. This network approach can be generalized to phase spaces of some other complex systems.