Executive Summary : | The well-known exponential growth of the many-particle Hilbert space inhibits even modest system size study of even the simplest simplest case of spinless fermions in one dimension. Such simple models are however relevant for studying many particle properties such as many-body localization, stability of few-body bound and forms the basis of understanding core-hole spectroscopies. For access to full spectrum one typically resorts to Exact Diagonalization (ED). The more popular Lanczos based diagonalization provides ground state and few excited states on large systems but is well-known to miss features of the spectrum without adequate care. Density matrix renormalization group (DMRG) is numerically exact; however, it is limited to quasi one-dimensional systems due to area law growth if entanglement entropy. The challenge is to develop a-numerical scheme to start with exact, like ED, but which is both numerically efficient and allows for systematic approximations allowing access to large one and two dimensional lattices as well. This proposal presents such a scheme in terms of many-particle Green's functions. We also discuss how experimentally relevant observable such as single and few body excitations of the many particle ground states can be extracted. |