  The algebraic spin liquid is a long-sought-after phase of matter
characterized by the absence of quasiparticle excitations, a low-energy
description in terms of emergent Dirac fermions and gauge fields interacting
according to (2+1)D quantum electrodynamics (QED$_3$), and power-law
correlations with universal exponents. The prototypical algebraic spin liquid
is the Affleck-Marston $\pi$-flux phase, originally proposed as a possible
ground state of the spin-1/2 Heisenberg model on the 2D square lattice. While
the latter model is now known to order antiferromagnetically at zero
temperature, recent sign-problem-free quantum Monte Carlo simulations of
spin-1/2 fermions coupled to a compact U(1) gauge field on the square lattice
have shown that quantum fluctuations can destroy N\'eel order and drive a
direct quantum phase transition to the $\pi$-flux phase. We show this
transition is in the universality class of the chiral Heisenberg
QED$_3$-Gross-Neveu-Yukawa model with a single SU(2) doublet of four-component
Dirac fermions (i.e., $N_f=1$), pointing out important differences with the
corresponding putative transition on the kagome lattice. Using an $\epsilon$
expansion below four spacetime dimensions to four-loop order, and a large-$N_f$
expansion up to second order, we show the transition is continuous and compute
various thermodynamic and susceptibility critical exponents at this transition,
setting the stage for future numerical determinations of these quantities. As a
byproduct of our analysis, we also obtain charge-density-wave and
valence-bond-solid susceptibility exponents at the semimetal-N\'eel transition
for interacting fermions on the honeycomb lattice.
