# Non-perturbative conserving approximations and Luttinger's sum rule

## TL;DR

This paper investigates the conditions under which Luttinger's sum rule holds in non-perturbative conserving approximations, using theoretical analysis and numerical examples within the self-energy-functional framework.

## Contribution

It demonstrates that Luttinger's sum rule can be violated in certain non-perturbative conserving approximations and discusses implications for cluster dynamical mean-field theory.

## Key findings

- Sum rule validity depends on approximation details.
- Numerical tests show potential violations of the sum rule.
- Implications for cluster extensions of DMFT are discussed.

## Abstract

Weak-coupling conserving approximations can be constructed by truncations of the Luttinger-Ward functional and are well known as thermodynamically consistent approaches which respect macroscopic conservation laws as well as certain sum rules at zero temperature. These properties can also be shown for variational approximations that are generated within the framework of the self-energy-functional theory without a truncation of the diagram series. Luttinger's sum rule represents an exception. We analyze the conditions under which the sum rule holds within a non-perturbative conserving approximation. Numerical examples are given for a simple but non-trivial dynamical two-site approximation. The validity of the sum rule for finite Hubbard clusters and the consequences for cluster extensions of the dynamical mean-field theory are discussed.