Ground State of Hard-Core Lattice Bosons from Algebraic Graph Theory

The hard-core boson problem consists of finding the ground state (and preferably the excitations) of bosons hopping on a lattice, subject to the constraint that no two particles occupy the same site. The Tonks-Girardeau mapping to non-interacting fermions provides an exact solution in one dimensional lattices. For higher dimensions, the fermionic mapping breaks down and no exact solutions are currently known. In algebraic graph theory, the hard-core boson problem is equivalent to finding the maximal eigenvector of the adjacency matrix associated to the symmetric power of a graph. We analyze the Tonks-Girardeau solution in this context without invoking fermions, and suggest a promising route toward generalizing the results to higher-dimensional lattices.