This was the second of two papers studying quantum critical points between gapless \(\mathbb{Z}_2\) spin liquids and conventionally ordered phases on the square lattice. Here, we study a Higgs transition between a \(U(1)\) and \(\mathbb{Z}_2\) gauge theory with \(N_f = 4\) massless Dirac fermions (spinons) coupled to a charge-2 Higgs boson by a particular choice of Yukawa coupling. See my summary of our first work for an explanation of why we argue for this critical theory as a description of the phase transitions in square lattice antiferromagnetism.

In contrast to the first critical theory we studied, which featured emergent subsystem symmetries and UV/IR mixing, this one was a breath of fresh air due to it "only" involving Lorentz symmetry breaking. We study this theory in a \(N_f^{-1} \) expansion. The key result we find is that a velocity anisotropy term for the spinons, which is allowed by the microscopic square lattice symmetries but is normally irrelevant in the absence of a critical Higgs field, acquires a non-zero fixed point value. This is reflected in physical correlation functions as a lack of emergent continuous rotational symmetry. For example, the two-point correlation function of the Neel and VBS order parameters have power law decay with respect to the radial separation, but their angular dependence is non-trivial and is plotted below - "perturbative" and "non-perturbative" are with respect to the velocity anisotropy, as the VBS angular dependence can only analytically be calculated to leading order. The key point is that the Neel correlations are generally enhanced along the diagonals, whereas VBS correlations are enhanced along the cardinal directions. We also see that VBS order actually vanishes and then switches signs along the diagonals - we think this is likely just an artifact of treating the velocity anistropy perturbatively, as similar behavior happens in the Neel correlations if we only go to first order.

We also find differing critical exponents for the Neel and VBS order parameters, in contrast to the pure \(U(1)\) spin liquid which hosts an emergent symmetry that rotates between the two. Unfortunately, the leading-order correction in a \(N_f^{-1}\) expansion happens to be very small - on the order of 0.05 for our \(N_f\) of interest. This arises due to a competition between scaling corrections from the critical Higgs field and the \(U(1)\) gauge field, which differ in sign and just happen to be nearly equal in magnitude at our fixed point.