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 dynamical mean-field theory, 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. Going from one to two dimensions we find that fitting the bath becomes in general significantly more difficult, requiring a large number of bath sites. For such large baths our symmetry-adapted approach should prove crucial for finding a reliable bath-parametrization.