Exact diagonalization using hierarchical wave functions and calculation of topological entanglement entropy

In this work we describe a technique for numerical exact diagonalization. The method is particularly suitable for cold bosonic atoms in optical lattices, in which multiple atoms can occupy a lattice site. We describe the use of the method for Bose-Hubbard model of a two-dimensional square lattice system as an example; however, the method is general and can be applied to other lattice models and can be adapted to three-dimensional systems. The proposed numerical technique focuses in detail on how to construct the basis states as a hierarchy of wave functions. Starting from single-site Fock states, we construct the basis set in terms of row states and multirow states. This simplifies the application of constraints and calculation of the Hamiltonian matrix. The approach simplifies the calculation of the reduced density matrices, and this has applications in characterizing the topological entanglement of the state. Each step of the method can be parallelized to accelerate the computation. As a case study, we discuss the computation of the spatial bipartite entanglement entropy in the correlated $\ensuremath{\nu}=1/2$ fractional quantum Hall state.