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

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's 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 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 nonlocal, their key experimental signature.