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Higher-order Fermi-liquid corrections for an Anderson impurity away from half-filling II: equilibrium properties

We study the low-energy behavior of the vertex function of a single Anderson impurity away from half-filling for finite magnetic fields, using the Ward identities with careful consideration of the anti-symmetry and analytic properties. The asymptotic form of the vertex function $Γ_{σσ';σ'σ}^{}(iω,iω';iω',iω)$ is determined up to terms of linear order with respect to the two frequencies $ω$ and $ω'$, as well as the $ω^2$ contribution for anti-parallel spins $σ'\neq σ$ at $ω'=0$. From these results, we also obtain a series of the Fermi-liquid relations beyond those of Yamada-Yosida. The $ω^2$ real part of the self-energy $Σ_σ^{}(iω)$ is shown to be expressed in terms of the double derivative $\partial^2Σ_σ^{}(0)/\partial ε_{dσ}^{2}$ with respect to the impurity energy level $ε_{dσ}^{}$, and agrees with the formula obtained recently by Filippone, Moca, von Delft, and Mora in the Nozières phenomenological Fermi-liquid theory [Phys.\ Rev.\ B {\bf 95}, 165404 (2017)]. We also calculate the $T^2$ correction of the self-energy, and find that the real part can be expressed in terms of the three-body correlation function $χ_{\uparrow\downarrow,-σ}^{[3]} = \partial χ_{\uparrow\downarrow}/\partial ε_{d,-σ}^{}$. We also provide an alternative derivation of the asymptotic form of the vertex function. Specifically, we calculate the skeleton diagrams for the vertex function $Γ_{σσ;σσ}^{}(iω,0;0,iω)$ for parallel spins up to order $U^4$ in the Coulomb repulsion $U$. It directly clarifies the fact that the analytic components of order $ω$ vanish as a result of the cancellation of four related Feynman diagrams which are related to each other through the anti-symmetry operation.