The quantum phases of matter

I present a selective survey of the phases of quantum matter with varieties of many-particle quantum entanglement. I classify the phases as gapped, conformal, or compressible quantum matter. Gapped quantum matter is illustrated by a simple discussion of the Z_2 spin liquid, and connections are made to topological field theories. I discuss how conformal matter is realized at quantum critical points of realistic lattice models, and make connections to a number of experimental systems. Recent progress in our understanding of compressible quantum phases which are not Fermi liquids is summarized. Finally, I discuss how the strongly-coupled phases of quantum matter may be described by gauge-gravity duality. The structure of the large N limit of SU(N) gauge theory, coupled to adjoint fermion matter at non-zero density, suggests aspects of gravitational duals of compressible quantum matter.