Fractonic phases of matter are novel quantum ground states supporting sub-dimensional emergent excitations with mobility restrictions and/or immobile fractons. The ground state degeneracy of such phases is sub-extensive and depends on the geometry of the underlying lattice. Due to these unusual properties, fractonic phases are considered as models for quantum memory or as examples of quantum glassy behaviors. While there exist a number of exactly solvable models with interactions between multiple particles/spins (twelve or more), the realization of such models in real materials is extremely challenging. In this work, we provide a realistic quantum model of quadratic spin interactions on the breathing pyrochlore lattice, inspired by a classical spin model studied earlier. We show that the emergent excitations in this model are immobile when they are present alone. They can only move as a cluster or when they reside at the corners of a membrane excitation. Using the membrane operators acting on the ground state manifold, we construct degenerate ground states with periodic boundary conditions. It is shown that the ground state degeneracy explicitly depends on the lattice geometry. We discuss the implications of these results in light of the rank-2 tensor gauge theory.
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