Theory of Competing Orders in Two Dimensional Quantum Magnets

Quantum magnets provide the simplest example of strongly interacting quantum matter, yet they continue to resist a comprehensive understanding above one spatial dimension. We explore a promising framework in 2D lattice , the Dirac spin liquid (DSL), a version of Quantum Electrodynamics (QED3) with four flavors of Dirac fermions coupled to photons. Importantly, its excitations include magnetic monopoles that drive confinement, and the symmetry actions on monopoles on square, honeycomb, triangular and kagome lattices contain crucial information about the DSL states. The underlying band topology of spinon insulators, e.g., wannier insulator protected by rotation, determines the elusive Berry phase of monopole under rotations. The stability of the DSL is enhanced on triangular and kagome lattices compared to bipartite (square and honeycomb) lattices. We obtain the universal signatures of the DSL on triangular and kagome lattices, including those of monopole excitations, as a guide to numerics and experiments on existing materials. Even when unstable, the DSL helps unify and organize the plethora of competing orders in correlated two-dimensional materials.