A well-known property of Fermi liquids is that have they have a Fermi surface in momentum space whose volume enclosed is determined by the microscopic electron density according to Luttinger's theorem. The situation for metallic states not described by Fermi liquid theory -- so-called "non-Fermi liquids" -- has so far been far less clear. Do these systems necessarily have Fermi surfaces? What exactly do we even mean by a Fermi surface in systems without quasiparticles? If there is a Fermi surface, does it obey Luttinger's theorem?
In this talk, I will describe a recent work in which ideas previously appearing in the theory of symmetry-protected topological (SPT) phases are applied to these and other related questions. Our results can be viewed as a vast generalization of both Luttinger's theorem and the Lieb-Schultz-Mattis theorem.
Finally, I will discuss implications of these results for the "strange metal" observed in cuprates. Combining with the experimental observations regarding the resistivity of the strange metal, we obtain powerful model-independent conclusions about the nature of the physics.
