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The interplay of symmetry and topology has been at the forefront of recentprogress in quantum matter. Here we uncover an unexpected connection betweenband topology and the description of competing orders in a quantum magnet.Specifically we show that aspects of band topology protected by crystallinesymmetries determine key properties of the Dirac spin liquid (DSL) which can bedefined on the honeycomb, square, triangular and kagom\'e lattices. At lowenergies, the DSL on all these lattices is described by an emergent QuantumElectrodynamics (QED$_3$) with $N_f=4$ flavors of Dirac fermions coupled to a$U(1)$ gauge field. However the symmetry properties of the magnetic monopoles,an important class of critical degrees of freedom, behave very differently ondifferent lattices. In particular, we show that the lattice momentum andangular momentum of monopoles can be determined from the charge (or Wannier)centers of the corresponding spinon insulator. We also show that for DSLs onbipartite lattices, there always exists a monopole that transforms triviallyunder all microscopic symmetries owing to the existence of a parent SU(2) gaugetheory. We connect our results to generalized Lieb-Schultz-Mattis theorems andalso derive the time-reversal and reflection properties of monopoles. Ourresults indicate that recent insights into free fermion band topology can alsoguide the description of strongly correlated quantum matter.

From spinon band topology to the symmetry quantum numbers of monopoles in Dirac spin liquids