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The stability of the Dirac spin-liquid on two-dimensional lattices has longbeen debated. It was recently demonstrated [Nature Commun. 10, 4254 (2019) andPhys. Rev. B 93, 144411 (2016)] that the staggered $\pi$-flux Dirac spin-liquidphase on the non-bipartite triangular lattice may be stable in the clean limit.However, quenched disorder plays a crucial role in determining whether such aphase is experimentally viable. For SU(2) spin systems, the effectivezero-temperature, low-energy description of Dirac spin-liquids in $(2+1)$dimensions is given by the compact quantum electrodynamics ($\rm cQED_{2+1}$)which admits monopoles. It is already known that generic quenched randomperturbations to the non-compact version of $\rm QED_{2+1}$ (where monopolesare absent) lead to strong-coupling instabilities. In this paper we study $\rmcQED_{2+1}$ in the presence of a class of time-reversal invariant quencheddisorder perturbations. We show that in this model, random non-Abelian vectorpotentials make the symmetry-allowed monopole operators more relevant. Thedisorder-induced underscreening of monopoles, thus, generically makes thegapless spin-liquid phase fragile.

Destabilization of U(1) Dirac spin-liquids on two dimensional non-bipartite lattices by quenched disorder