Symmetry enriched U(1) quantum spin liquids

We classify and characterize three dimensional U(1) quantum spin liquids (deconfined U(1) gauge theories) with time reversal and SO(3) spin rotational symmetries. We find there are 15 distinct such quantum spin liquids based on the properties of bulk excitations, and we show how to interpret them as gauged symmetry-protected topological states (SPTs). By examining the properties of the monopoles of an SO(3) gauge field to which the quantum spin liquid can couple, we identify 11 other anomalous states that can be categorized into 3 classes. When the surface properties of these quantum spin liquids are also of interests, the classification is further enriched by weakly coupling these quantum spin liquids to bosonic SPTs with the same symmetry. Taking this into account, we find all distinct such U(1) quantum spin liquids.