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

Abstract

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 ${\Gamma }_{\sigma {\sigma }^{\prime };{\sigma }^{\prime }\sigma }^{}(i\omega ,i{\omega }^{\prime };i{\omega }^{\prime },i\omega )$ 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 anti-parallel 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. The ${\omega }^2$ real part of the self-energy ${\Sigma }_{\sigma }^{}(i\omega )$ is shown to be expressed in terms of the double derivative ${\partial }^2{\Sigma }_{\sigma }^{}(0)/\partial {\epsilon }_{d\sigma }^2$ with respect to the impurity energy level ${\epsilon }_{d\sigma }^{}$, and agrees with the formula obtained recently by Filippone, Moca, von Delft, and Mora 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 ${\chi }_{\uparrow \downarrow ,-\sigma }^{[3]}=\partial {\chi }_{\uparrow \downarrow }/\partial {\epsilon }_{d,-\sigma }^{}$. We also provide an alternative derivation of the asymptotic form of the vertex function. Specifically, we calculate the skeleton diagrams for the vertex function ${\Gamma }_{\sigma \sigma ;\sigma \sigma }^{}(i\omega ,0;0,i\omega )$ 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 anti-symmetry operation.

15 figures, 19pages, and supplemental material (15 pages), typo found in (6.15) has been corrected

References in corpus (1)

• Higher-order Fermi-liquid corrections for an Anderson impurity away from half-filling III: non-equilibrium transport

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