Emergent anomalies and generalized Luttinger theorems in metals and semimetals
Luttinger's theorem connects a basic microscopic property of a given metallic crystalline material, the number of electrons per unit cell, to the volume, enclosed by its Fermi surface, which defines its low-energy observable properties. Such statements are valuable since, in general, deducing a low-energy description from microscopics, which may perhaps be regarded as the main problem of condensed matter theory, is far from easy. In this paper, we present a unified framework which allows one to discuss Luttinger theorems for ordinary metals as well as closely analogous exact statements for topological (semi)metals, whose low-energy description contains either discrete points or continuous line nodes. This framework is based on the 't Hooft anomaly of the emergent charge conservation symmetry at each point on the Fermi surface, a concept recently proposed by Else et al. Here we find that the Fermi surface codimension p plays a crucial role for the emergent anomaly. For odd p, such as ordinary metals (p = 1) and magnetic Weyl semimetals (p = 3), the emergent symmetry has a generalized chiral anomaly. For even p, such as graphene and nodal line semimetals (both with p = 2), the emergent symmetry has a generalized parity anomaly. When restricted to microscopic symmetries, such as U(1) and lattice symmetries, the emergent anomalies imply (generalized) Luttinger theorems, relating Fermi surface volume to various topological responses. The corresponding topological responses are the charge density for p = 1, Hall conductivity for p = 3, and polarization for p = 2. As a by-product of our results, we clarify exactly what is anomalous about the surface states of nodal line semimetals.