Non-Fermi liquids as ersatz Fermi liquids: general constraints on compressible metals

Understand

A system with charge conservation and lattice translation symmetry has a well-defined filling $\nu$, which is a real number representing the average charge per unit cell.

• We show that if $\nu$ is fractional (i.e.
• not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems.
• The most powerful constraint comes about if $\nu$ is continuously tunable (i.e.