# Fermi Liquids and the Luttinger Integral

## TL;DR

This paper demonstrates that the Luttinger integral, traditionally associated with singular Fermi liquids, also characterizes regular Fermi liquid phases and quantum critical points as topological features.

## Contribution

It reveals that the Luttinger integral can distinguish phases with regular low energy behavior, extending its significance beyond singular Fermi liquids.

## Key findings

- The Luttinger integral takes values ±π/2 in certain regular Fermi liquid phases.
- Quantum critical points can be interpreted as topological transitions.
- The Luttinger integral serves as a phase characterizer in impurity models.

## Abstract

The Luttinger Theorem, which relates the electron density to the volume of the Fermi surface in an itinerant electron system, is taken to be one of the essential features of a Fermi liquid. The microscopic derivation of this result depends on the vanishing of a certain integral, the Luttinger integral $I_{\rm L}$, which is also the basis of the Friedel sum rule for impurity models, relating the impurity occupation number to the scattering phase shift of the conduction electrons. It is known that non-zero values of $I_{\rm L}$ with $I_{\rm L}=\pm\pi/2$, occur in impurity models in phases with non-analytic low energy scattering, classified as singular Fermi liquids. Here we show the same values, $I_{\rm L}=\pm\pi/2$, occur in an impurity model in phases with regular low energy Fermi liquid behavior. Consequently the Luttinger integral can be taken to characterize these phases, and the quantum critical points separating them interpreted as topological.