From Spinon Band Topology to the Symmetry Quantum Numbers of Monopoles in Dirac Spin Liquids

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 kagome lattices. At low energies, the DSL on all of these lattices is described by an emergent quantum electrodynamics (${\mathrm{QED}}_3$) 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.

Popular Summary

In a quantum spin liquid, the spins of the atoms remain randomly oriented even at absolute zero. Originally conceived to be a precursor state to high-temperature superconductivity, spin liquids have inspired great experimental efforts and theoretical breakthroughs in the area of quantum magnets. One particularly important type of quantum spin liquid is the Dirac spin liquid, where relativistic fermions emerge from the spins and interact strongly with photonlike excitations. Here, we study Dirac spin liquids and find that certain subtle but crucial properties can be fruitfully analyzed using tools in topological band theory developed in the last decade.

In Dirac spin liquids, the symmetry properties of a class of important excitations called monopoles remain elusive. Certain monopoles can destroy the photons, which makes the fermions too energetically costly to produce. To determine whether these particular monopoles should be present, and what types of ordered states (like antiferromagnets) would result if they do destroy the state, requires knowledge of symmetry properties of monopoles, such as the angular momenta.

To this end, we first find ideal limits of the fermions, where they stay localized by virtue of topological band theory. Monopoles create a magnetic flux for those fermions, and conversely fermions are sources of angular momenta for them. Different ideal limits relate to different topologies of the fermion band and determine symmetry properties of monopoles.

Our result resolves a long-standing problem concerning the stability and proximate phases in the study of Dirac spin liquids and highlights a connection between band topology for electric insulators and quantum magnetism, two of the most central fields in condensed-matter physics.

Figure 1

An example showing how charge centers of a spinon band determine lattice-symmetry quantum numbers (like the angular momentum) of a monopole (a flux quantum).

Figure 2

For the Dirac spin liquid on bipartite lattices (honeycomb and square), we can view the continuum ${\mathrm{QCD}}_3+$ Higgs field theory (instead of the original lattice theory) as the UV completion of the ${\mathrm{QED}}_3$ theory. The implementation of microscopic symmetries (at the lattice scale) in the QCD theory can be fixed by PSG analysis, while the implementation of QCD symmetries in the QED theory is also uniquely fixed by field theory analysis. This uniquely fixes the implementation of microscopic symmetries in the QED [$U(1)$ Dirac spin-liquid] theory.

Figure 3

Reflection axis to be considered in the main text.

Figure 4

Reflection axis to be considered in the main text.

Figure 5

The simplest spinon (boson) Mott insulator, with every site in sublattice $A$ empty and every site in sublattice $B$ completely filled (recall that the bosons $b_{\alpha }$ are hard core). This case respects $C_6$ rotation since it acts as $b_{\alpha }\to i{\sigma }_{\alpha \beta }^y{b}_{\beta }^{†}$. As a monopole (flux) moves in this background, it sees a fixed gauge charge pattern with $q_A=-1$ and $q_B=+1$. The amount of gauge charge sitting on each rotation center dictates the nontrivial Berry phase accumulated by the monopole as it moves around the center according to $\theta (C_n^r)=e^{i{q}_{r}2\pi /n}$. The translation quantum numbers of the monopole can be easily obtained once the rotation quantum number is known.

Figure 6

The Wannier limit of a quantum spin Hall insulator on four types of lattices when all the symmetries are broken except for charge $U(1)$ conservation and lattice translation or rotation. The solid dot indicates a Wannier center, and empty dot indicates a “minus” charge pertinent to the fragile nature of the band topology. The pattern of $U(1)$ gauge charge is shown in the figure denoted as $q$, with self-explanatory subscripts associated with locations of charges (e.g., $e$, $c$, $v$ mean charges located at center of edges, plaquettes, on vertices, respectively), with the background $-1$ charge on-site subtracted.

Figure 7

Left diagram: The four-site unit cell used to calculate Wannier centers on the triangular lattice. Right diagram: Brillouin zone of the four-site unit-cell parton Hamiltonian. The $-1$ gauge transformation for $C_3$’s is labeled by a solid blue circle on site (those without the circle have trivial gauge transformation). This gauge transform has momentum ($0,\pi$). The right panel is the rotation-invariant Brillouin zone for the four-site unit cell. Marked are three rotation-invariant momenta.

Figure 8

The momenta of monopoles and fermion bilinears in the triangular lattice DSL—signatures of a DSL system in experiments such as neutron scattering. Solid or empty circles represent momenta of spin-triplet or spin-singlet monopoles, distinct from all fermion bilinear operators, and black triangles represent those of fermion bilinears.

Figure 9

Left panel: The gauge choice (blue or black bonds denote negative or positive hopping strength with equal amplitudes) and the Brillouin zone with the $M$ point the gapless point and $k$, $k^{\prime }$ staying invariant under PSG of threefold rotations. Right panel: The energy spectrum of the staggered flux spinon model on the triangular lattice in the reduced Brillouin zone with a four-site unit cell. The spectrum is gapless at momentum ($\pi ,\pi$).