Stability of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>U</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>spin liquids in two dimensions

We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless $U(1)$ gauge fluctuations can be stable in two dimensions, at least when the physical $SU(2)$ spin symmetry is generalized to $SU(N)$. Equivalently, we show that compact ${\mathrm{QED}}_{3}$ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological $U(1)$ symmetry corresponding to the conservation of gauge flux. Beginning with an $SU(N)$ generalization of the $S=1∕2$ square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-$N$ spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact ${\mathrm{QED}}_{3}$.