Monopole excitations in the $U(1)$ Dirac spin liquid on the triangular lattice

TL;DR

This paper investigates monopole excitations in the $U(1)$ Dirac spin liquid on the triangular lattice, providing numerical evidence for gapless monopoles and supporting the stability of this exotic phase in frustrated quantum magnets.

Contribution

It constructs and analyzes charge-$Q$ monopole excitations in the $J_1$-$J_2$ Heisenberg model, confirming theoretical predictions and demonstrating the stability of the $U(1)$ Dirac spin liquid.

Findings

Singlet and triplet monopoles with $Q=1$ become gapless in the thermodynamic limit.

Energies for generic $Q$ match field-theoretical predictions.

Localized gauge excitations with $ ext{pi}$-fluxes are not relevant at low energies.

Abstract

The $U(1)$ Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in $2+1$ dimensions, where gapless modes exists but spinons and gauge fields are strongly coupled. Its existence has been proposed in frustrated Heisenberg models in presence of frustrating super-exchange interactions, by the (Abrikosov) fermionic representation of the spin operators [X.-G. Wen, \href{https://doi.org/10.1103/PhysRevB.65.165113}{Phys. Rev. B {\bf 65}, 165113 (2002)}], supplemented by the Gutzwiller projection. Here, we construct charge-$Q$ monopole excitations in the Heisenberg model on the triangular lattice with nearest- ($J_1$) and next-neighbor ($J_2$) couplings. In the highly frustrated regime, singlet and triplet monopoles with $Q=1$ become gapless in the thermodynamic limit; in addition, the energies for generic $Q$ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\pi$-fluxes are concentrated in the triangular plaquettes (in analogy with $\mathbb{Z}_2$ visons), showing that these kind of states do not play a relevant role at low energies. All our findings lend support to a stable $U(1)$ Dirac spin liquid in the $J_1-J_2$ Heisenberg model on the triangular lattice.