DCA$^+$: Dynamical Cluster Approximation with continuous lattice self-energy

Abstract: The dynamical cluster approximation (DCA) is a systematic extension beyond the single site approximation in dynamical mean field theory (DMFT), to include spatially non-local correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in oder to achieve better convergence with cluster size. The new method, which we call DCA$^+$, cures the cluster shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that in the DCA$^+$ algorithm, the self-energy and pseudo-gap temperature $T^*$ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-rage correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA$^+$ algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.