The Luttinger Count is the Homotopy not the Physical Charge: Generalized Anomalies Characterize Non-Fermi Liquids

TL;DR

This paper reveals that the Luttinger count in non-Fermi liquids is governed by a generalized symmetry anomaly, linking deviations from traditional charge counting to topological and homotopy properties rather than physical charge.

Contribution

It introduces a formulation of the Luttinger-Ward functional as a generalized symmetry operator, explaining non-Fermi liquid behavior through anomaly structures.

Findings

The Luttinger-Ward functional acts as a generalized symmetry with a quantized charge.

Divergences in the functional produce anomalies that explain deviations from the Luttinger count.

Non-Fermi liquids are classified by anomaly structures similar to those in particle physics.

Abstract

We show that the Luttinger-Ward functional can be formulated as an operator insertion in the path integral and hence can be thought of as a generalized symmetry. The key result is that the associated charge, always quantized, defines the homotopy, not the physical charge. The disconnect between the two arises from divergences in the functional or equivalently zeros of the single-particle Green function. Such divergences produce an anomaly of the triangle-diagram type. As a result of this anomaly, we are able to account for the various deviations\cite{rosch,dave,altshuler,osborne,l3,l5} of the Luttinger count from the particle density. As a consequence, non-Fermi liquids can be classified generally by the well known anomaly structures in particle physics. Charges descending from generalized symmetries, as in the divergence of the Luttinger-Ward functional, are inherently non-local, their key experimental signature.