Symmetry-resolved monopoles and stability criteria for an $N_f=12$ Dirac spin liquid on the maple-leaf lattice

Abstract

The $\mathrm{U}(1)$ Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED$_3$ with $N_f=12$ two-component Dirac fermions. The charge-one monopoles span a $924$-dimensional complex representation. In the spin-singlet, zero-momentum, zero-$C_6$-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, $C_6$, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-$N_f$ and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED$_3$ in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the $N_f=12$ Dirac spin liquid.