We study possible quantum U ( 1 ) spin liquids in three dimensions with time-reversal symmetry. We find a total of seven families of such U ( 1 ) spin liquids, distinguished by the properties of their emergent electric or magnetic charges. We show how these spin liquids are related to each other. Two of these classes admit nontrivial protected surface states which we describe. We show how to access all of the seven spin liquids through slave particle (parton) constructions. We also provide intuitive loop gas descriptions of their ground-state wave functions. One of these phases is the “topological Mott insulator,” conventionally described as a topological insulator of an emergent fermionic “spinon.” We show that this phase admits a remarkable dual description as a topological insulator of emergent fermionic magnetic monopoles. This results in a new (possibly natural) surface phase for the topological Mott insulator and a new slave particle construction. We describe some of the continuous quantum phase transitions between the different U ( 1 ) spin liquids. Each of these seven families of states admits a finer distinction in terms of their surface properties, which we determine by combining these spin liquids with symmetry-protected topological phases. We discuss lessons for materials such as pyrochlore quantum spin ices which may harbor a U ( 1 ) spin liquid. We suggest the topological Mott insulator as a possible ground state in some range of parameters for the quantum spin ice Hamiltonian.
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