Quantum criticality and fractional charge excitations in itinerant ice-rule systems

``Ice rule'' is a configurational constraint on Ising-type variables defined on tetrahedron-based lattices, such as a pyrochlore lattice, so that two out of the four sites on a tetrahedron are in the opposite state to the other two. This concept plays an important role in many systems, such as water ice I$_h$, magnetite Fe$_3$O$_4$, and spin ice materials Ho(Dy)$_2$Ti$_2$O$_7$. Under the ice-rule constraint, the ground state is disordered and retains macroscopic degeneracy. Nevertheless, the ice-rule configuration is not completely random but has a peculiar spatial structure with quasi-long-range correlation. It is interesting to ask how itinerant electrons change their properties by coupling to this anomalous spatial structure. To answer this problem, we adopt an extended Falicov-Kimball model as a minimal model, in which itinerant electrons interact with localized charge degrees of freedom under the ice rule. We exactly solve this model on a loop-less variant of the tetrahedron-based lattices, a tetrahedron Husimi cactus and clarify the ground-state phase diagram. The exact solution reveals a quantum critical point separating two insulating phases, where a novel non-Fermi-liquid behavior emerges. We also discuss the nature of fractional excitations breaking the ice-rule manifold.