Finite-temperature spectral density for the Anderson model

Spectral density of local single-particle excitations for the nondegenerate single-impurity Anderson model is calculated for various values of U/πΔ, $n_f^0$≡$n_f$(T=0), and T, using the finite-T perturbation theory with U/πΔ as the expansion parameter and the Hartree-Fock (HF) approximation for the unperturbed Hamiltonian. For strong Coulomb correlation and at T=0, ${\rho }_f$(ω) is triple peaked in the Kondo region ($n_f^0$≃1), double peaked in the intermediate-valence region (0.8≳$n_f^0$≳0.5) and single peaked in the region of nearly empty (or full) orbital. As T increases, ${\rho }_f$(ω) goes through various regimes, to end up with two peaks of half-width ∼Δ at the single-particle excitation energies ${\varepsilon }_f$ and ${\varepsilon }_f$+U. Thus the only effect of hybridization ($V_{\mathrm{k}f}$) at high T and U≫Δ is the lifetime broadening of two sharp resonances obtained for $V_{\mathrm{k}f}$≡0, all other many-body effects having been washed out. Comparison with finite-T HF results shows that the HF approximation is reasonable (i.e., correlation is weak) for U≲πΔ ${\sin }^2$(1/2π${\mathrm{n}}_{\mathrm{f}}^0$) at low T, but only for U≲Δ at high T.