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An exact solution is obtained for a model of itinerant electrons coupled toice-rule variables on the tetrahedron Husimi cactus, an analogue of the Bethelattice of corner-sharing tetrahedra. It reveals a quantum critical point withthe emergence of non-Fermi-liquid behavior in melting of the "charge ice"insulator. The electronic structure is compared with the numerical results forthe pyrochlore-lattice model to elucidate the physics of electron systemsinteracting with the tetrahedron ice rule.

Quantum melting of charge ice and non-Fermi-liquid behavior: An exact solution for the extended Falicov-Kimball model in the ice-rule limit