Stability ofU(1)spin liquids in two dimensions

We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless $U\left(1\right)$ gauge fluctuations can be stable in two dimensions, at least when the physical $SU\left(2\right)$ spin symmetry is generalized to $SU\left(N\right)$. 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 $U\left(1\right)$ symmetry corresponding to the conservation of gauge flux. Beginning with an $SU\left(N\right)$ generalization of the $S=1∕2$ square-lattice Heisenberg antiferromagnet, we consider the $\pi$-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 ${\mathrm{QED}}_3$.