Generalized Modular Transformations in (3+1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding

In topologically ordered quantum states of matter in (2+1)D (spacetime dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property that is directly related to global transformations of the ground-state wave functions on a torus (the modular transformations). On the other hand, there are theoretical descriptions of various topologically ordered states in (3+1)D, which exhibit both pointlike and looplike excitations, but systematic understanding of the fundamental physical distinctions between phases, and how these distinctions are connected to quantum statistics of excitations, is still lacking. One main result of this work is that the three-dimensional generalization of modular transformations, when applied to topologically ordered ground states, is directly related to a certain braiding process of looplike excitations. This specific braiding surprisingly involves three loops simultaneously, and can distinguish different topologically ordered states. Our second main result is the identification of the three-loop braiding as a process in which the worldsheets of the three loops have a nontrivial triple linking number, which is a topological invariant characterizing closed two-dimensional surfaces in four dimensions. In this work, we consider realizations of topological order in (3+1)D using cohomological gauge theory in which the loops have Abelian statistics and explicitly demonstrate our results on examples with Z(2) x Z(2) topological order.