Sum rule violation in self-consistent hybridization expansions

We show that for multi-orbital quantum impurity models the non-crossing approximation and one-crossing approximation versions of the self-consistent hybridization expansions violate the sum rules relating the coefficients of the high-frequency expansion of the self energy and the product of the self energy and Green function to thermodynamic expectation values. Comparison of non-crossing/one-crossing results to numerically exact quantum Monte-Carlo calculations shows that the consistency with sum rules provides a useful estimate of the reliability of the approximations. The sum rule violations are more pronounced, and therefore the quality of the non-crossing/one-crossing approximation is poorer, in situations with multiple orbitals and away from particle-hole symmetry but becomes less severe as the correlation strength increases. The one crossing approximation is markedly superior to the non-crossing approximation.