Grand canonical finite size numerical approaches in one and two dimensions: Real space energy renormalization and edge state generation

The grand canonical numerical analysis recently developed for quantum many-body systems on a finite cluster is the technique to efficiently obtain the physical quantities in an applied field. There, the observables are the continuous and real functions of fields, mimicking their thermodynamic limit, even when a small cluster is adopted. We develop a theory to explain the mechanism of this analysis based on the deformation of the Hamiltonian. The deformation spatially scales down the energy unit from the system center toward zero at the open edge sites, which introduces the renormalization of the energy levels in a way reminiscent of Wilson's numerical renormalization group. However, compared to Wilson's case, our deformation generates a number of far well-localized edge states near the chemical potential level, which are connected via a very small quantum fluctuation in k space with the "bulk" states which spread at the center of the system. As a response to the applied field, the particles on the cluster are self-organized to tune the particle number of the bulk states to their thermodynamic limit by using the "edges" as a buffer. We demonstrate the present analysis in two-dimensional quantum spin systems on square and triangular lattices, and determine the smooth magnetization curve with a clear 13 plateau structure in the latter.