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 antisymmetry and analytic properties. The asymptotic form of the vertex function ${\mathrm{\Gamma }}_{\sigma {\sigma }^{\prime };{\sigma }^{\prime }\sigma }^{}(i\omega ,i{\omega }^{\prime };i{\omega }^{\prime },i\omega )$ is determined up to terms of linear order with respect to the two frequencies $\omega$ and ${\omega }^{\prime }$, as well as the ${\omega }^2$ contribution for antiparallel spins ${\sigma }^{\prime }\ne \sigma$ at ${\omega }^{\prime }=0$. From these results, we also obtain a series of the Fermi-liquid relations beyond those of Yamada-Yosida [Prog. Theor. Phys. 54, 316 (1975)]. The ${\omega }^2$ real part of the self-energy ${\mathrm{\Sigma }}_{\sigma }^{}\left(i\omega \right)$ is shown to be expressed in terms of the double derivative ${\partial }^2{\mathrm{\Sigma }}_{\sigma }^{}\left(0\right)/\partial {\epsilon }_{d\sigma }^2$ with respect to the impurity energy level ${\epsilon }_{d\sigma }^{}$, and agrees with the formula obtained recently by Filippone, Moca, von Delft, and Mora (FMvDM) in the Nozières phenomenological Fermi-liquid theory [Phys. Rev. B 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 $\partial {\chi }_{\uparrow \downarrow }/\partial {\epsilon }_{d,-\sigma }^{}$, where ${\chi }_{\uparrow \downarrow }$ is the static susceptibility between antiparallel spins. We also provide an alternative derivation of the asymptotic form of the vertex function. Specifically, we calculate the skeleton diagrams for the vertex function ${\mathrm{\Gamma }}_{\sigma \sigma ;\sigma \sigma }^{}(i\omega ,0;0,i\omega )$ for parallel spins up to order $U^4$ in the Coulomb repulsion $U$. It directly clarifies the fact that the analytic components of order $\omega$ vanish as a result of the cancellation of four related Feynman diagrams, which are related to each other through the antisymmetry operation.

• Received 9 October 2017
• Revised 20 December 2017

DOI:https://doi.org/10.1103/PhysRevB.97.045406