Momentum occupation number bounds for interacting fermions

We derive rigorous bounds on the average momentum occupation numbers $\langle n_{\mathbf{k}\sigma}\rangle$ in the Hubbard and Kondo models in the ground state and at non-zero temperature ($T>0$) in the grand canonical ensemble. For the Hubbard model with $T>0$ our bound proves that, when interaction strength $\ll k_B T\ll$ Fermi energy, $\langle n_{\mathbf{k}\sigma}\rangle$ is guaranteed to be close to its value in a low temperature free fermion system. For the Kondo model with any $T>0$ our bound proves that $\langle n_{\mathbf{k}\sigma}\rangle$ tends to its non-interacting value in the infinite volume limit. In the ground state case our bounds instead show that $\langle n_{\mathbf{k}\sigma}\rangle$ approaches its non-interacting value as $\mathbf{k}$ moves away from a certain surface in momentum space. For the Hubbard model at half-filling on a bipartite lattice, this surface coincides with the non-interacting Fermi surface. In the Supplemental Material we extend our results to some generalized versions of the Hubbard and Kondo models. Our proofs use the Fermi statistics of the particles in a fundamental way.