Abstract
The interplay of symmetry and topology has been at the forefront of recent progress in quantum matter. Here we uncover an unexpected connection between band topology and the description of competing orders in a quantum magnet. Specifically we show that aspects of band topology protected by crystalline symmetries determine key properties of the Dirac spin liquid (DSL) which can be defined on the honeycomb, square, triangular and kagom\'e lattices. At low energies, the DSL on all these lattices is described by an emergent Quantum Electrodynamics (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 on different lattices. In particular, we show that the lattice momentum and angular momentum of monopoles can be determined from the charge (or Wannier) centers of the corresponding spinon insulator. We also show that for DSLs on bipartite lattices, there always exists a monopole that transforms trivially under all microscopic symmetries owing to the existence of a parent SU(2) gauge theory. We connect our results to generalized Lieb-Schultz-Mattis theorems and also derive the time-reversal and reflection properties of monopoles. Our results indicate that recent insights into free fermion band topology can also guide the description of strongly correlated quantum matter.
Highlights
Quantum spin liquids represent a class of exotic quantum phases of matter beyond the traditional Landau symmetry-breaking paradigm
We show that aspects of band topology protected by crystalline symmetries determine key properties of the Dirac spin liquid (DSL), which can be defined on the honeycomb, square, triangular, and kagome lattices
In this work, we uncover a close and unexpected connection between the symmetry properties of monopoles and fermion band topology. This connection allows us to build on recent progress understanding band topology protected by crystalline symmetries, to develop a systematic analytical approach to calculate the monopole symmetry quantum numbers on essentially any lattice— we focus on the physically relevant ones, including square, honeycomb, triangular, and kagome lattices
Summary
Quantum spin liquids represent a class of exotic quantum phases of matter beyond the traditional Landau symmetry-breaking paradigm. In this work, we uncover a close and unexpected connection between the symmetry properties of monopoles and fermion band topology This connection allows us to build on recent progress understanding band topology protected by crystalline symmetries, to develop a systematic analytical approach to calculate the monopole symmetry quantum numbers on essentially any lattice— we focus on the physically relevant ones, including square, honeycomb, triangular, and kagome lattices. Armed with this deeper understanding and analytical machinery, we are able to obtain a complete understanding of the symmetry action on monopoles. This connection leads to a different, and simpler, way of calculating monopole quantum numbers on bipartite lattices, with results that are consistent with the band topology approach
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