symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a second phase transition at low temperatures into a nematic spiral phase which only breaks one

{\mathbb{Z}}_{2}

symmetry. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the fourfold pinch-point singularities across the phase diagram.

DOI: 10.1103/physrevb.110.085106

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{\mathbb{Z}}_{2}

{\mathbb{Z}}_{2}

\mathbb{Z}_2

symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a re-entrance phenomenon at low temperatures into a nematic spiral phase. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the four-fold pinch point singularities across the phase diagram

{C}_{4}

lattice and spin rotation symmetry of the compass model but is free of its fine-tuned subsystem symmetries. Using both field theoretic arguments and Monte Carlo simulations, we find that our model possesses a line of critical points with continuously varying exponents of the Ashkin-Teller type terminating at the four-state Potts point. Further, our Monte Carlo study uncovers that beyond the four-state Potts point, the line of phase transition is connected to the lattice-nematic Ising phase transition in the square lattice compass model through a region of first-order transitions.

(2+1)

-dimensional quantum electrodynamics

({\mathrm{QED}}_{3})

, and power-law correlations with universal exponents. The prototypical algebraic spin liquid is the Affleck-Marston $\ensuremath{\pi}$-flux phase, originally proposed as a possible ground state of the spin-

\frac{1}{2}

Heisenberg model on the two-dimensional 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-

\frac{1}{2}

fermions coupled to a compact

\text{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 $\ensuremath{\pi}$-flux phase. We show this transition is in the universality class of the chiral Heisenberg

{\mathrm{QED}}_{3}

-- Gross-Neveu-Yukawa model with a single

\text{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 $\ensuremath{\epsilon}$ expansion below four space-time 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 by-product 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.