Topological Properties of the Configuration Space of N Bosons On One-Dimensional Optical Lattices

We analyze the topology of the discrete configuration space of N bosons on finite, one-dimensional lattices with open and periodic boundary conditions. The configuration space is the set of available number states, and a graph topology is provided by the allowable transitions in the Hamiltonian. As a prototypical example, we consider modifications of the Bose-Hubbard model where nearest-neighbor hopping terms can have additional phases, like the recently experimentally realized Anyon-Hubbard model. The first Betti number describes the number of independent loops in the configuration space, and thereby classifies the number of gauge-independent phase fluxes possible for this class of models.This approach represents oriented incidence structure of the ring configuration space for computation of the combinatorial Laplacian. The technique also clarifies the topology of the configuration space for N bosons on a ring, which described the quotient of a discrete torus by the symmetric group. For N = 2, 3, and 4 this configuration space corresponds to the Möbius strip, Penrose triangle, and the disphenoid twisted prism, respectively.