Compressible Quantum Matter: General Constraints, Emergent Symmetries, and Anomalies

I will discuss properties of phases of matter in systems with a global U(1) symmetry and a (possibly discrete) translation symmetry. The low energy theory of such systems is strongly constrained by these symmetries. I will give a framework to understand such constraints in great generality, vastly generalizing old theorems. The most powerful constraint comes about if the system is compressible, in which case the low-energy theory must have a very large emergent symmetry group -- larger than any compact Lie group. A familiar example is a Landau Fermi liquid, whose properties I will revisit from a modern point of view of characterizing its emergent symmetry and the associated 't Hooft anomaly. Many, if not all, non-Fermi liquids will have the same emergent symmetry group/anomaly as a Fermi liquid (even though they could have very different dynamics), and this determines some of their universal properties. I will discuss the implications for understanding the famous "strange metal" physics observed in experiments in some condensed matter systems.