  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. the
system is charge-compressible), in which case we show that the low-energy
theory must have a very large emergent symmetry group -- larger than any
compact Lie group. An example is the Fermi surface of a Fermi liquid, where the
charge at every point on the Fermi surface is conserved. We expect that in
many, if not all, cases, even exotic non-Fermi liquids will have the same
emergent symmetry group as a Fermi liquid, even though they could have very
different dynamics. We call a system with this property an "ersatz Fermi
liquid". We show that ersatz Fermi liquids share a number of properties in
common with Fermi liquids, including Luttinger's theorem (which is thus
extended to a large class of non-Fermi liquids) and periodic "quantum
oscillations" in the response to an applied magnetic field. We also establish
versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall
systems and for spinon Fermi surfaces in Mott insulators. Our work makes
connection between filling constraints and the theory of symmetry-protected
topological (SPT) phases, in particular through the concept of " 't Hooft
anomalies".
