Transition between algebraic and $\mathbb{Z}_2$ quantum spin liquids at large $N$

We present a field theory description of a quantum phase transition in two spatial dimensions between a $U(1)$ algebraic spin liquid with $N$ flavors of gapless two-component Dirac fermionic spinons and a gapped $\mathbb{Z}_2$ spin liquid. This transition is driven by spinon pairing and concomitant Higgsing of the emergent $U(1)$ gauge field. For sufficiently large $N$ we find a quantum critical point with non-Gaussian exponents that is stable against instanton proliferation. We compute critical exponents using either $1/N$ or $\epsilon$ expansions, and give estimates of the critical value of $N$ below which the quantum critical point disappears.