Monopoles in Dirac spin liquids I: from spinon band topology to monopole quantum numbers

We explore a low-energy theory for 2D quantum magnets, the Dirac spin liquid (DSL), a version of Quantum Electrodynamics ( QED$_3$) with four flavors of Dirac fermions coupled to photons. We study the spatial/time-reversal symmetry properties of the magnetic monopoles, an important class of excitations that drive confinement. We show that the underlying band topology of spinon insulators, e.g., quantum spin hall phase/wannier insulator protected by rotation provides crucial information on the tricky Berry phase of monopole (under time-reversal / rotations, respectively). We also prove the existence of a trivial monopole on bipartite lattices by invoking its ascension to a QCD with SU(2) gauge structure.