# Luttinger theorem and imbalanced Fermi systems

## TL;DR

This paper extends the proof of the Luttinger theorem to spin-imbalanced Fermi systems within approximate many-body theories, relevant for understanding normal phases of superfluid atomic gases with unequal spin populations.

## Contribution

It provides a detailed extension of the Luttinger theorem proof to spin-imbalanced systems using conserving approximations, including the self-consistent t-matrix method.

## Key findings

- Luttinger theorem holds for each spin population in imbalanced Fermi systems.
- The proof applies to any $\
- $ ext{Φ}$-derivable approximation, ensuring broad applicability.

## Abstract

The proof of the Luttinger theorem, which was originally given for a normal Fermi liquid with equal spin populations formally described by the exact many-body theory at zero temperature, is here extended to an approximate theory given in terms of a "conserving" approximation also with spin imbalanced populations. The need for this extended proof, whose underlying assumptions are here spelled out in detail, stems from the recent interest in superfluid trapped Fermi atoms with attractive inter-particle interaction, for which the difference between two spin populations can be made large enough that superfluidity is destroyed and the system remains normal even at zero temperature. In this context, we will demonstrate the validity of the Luttinger theorem separately for the two spin populations for any "$\Phi$-derivable" approximation, and illustrate it in particular for the self-consistent $t$-matrix approximation.