Sum-rules and bath-parametrization for quantum cluster theories

We analyze cellular dynamical mean-field theory (CDMFT) and the dynamical cluster approximation (DCA). We derive exact sum-rules for the hybridization functions and give examples for DMFT, CDMFT, and DCA. For impurity solvers based on a Hamiltonian, these sum-rules can be used to monitor convergence of the bath-parametrization. We further discuss how the symmetry of the cluster naturally leads to a decomposition of the bath Green matrix into irreducible components, which can be parametrized independently, and give an explicit recipe for finding the optimal bath-parametrization. As a benchmark we revisit the one-dimensional Hubbard model. We carefully analyze the evolution of the density as a function of chemical potential and find that, close to the Mott transition, convergence with cluster size is unexpectedly slow. In two dimensions we find, that we need so many bath-sites to obtain a reliable parametrization that Lanczos calculations are hardly feasible with current computers. For such large baths our symmetry-adapted approach should prove crucial for finding a reliable bath-parametrization.