We study a model of fermions on the square lattice at half-filling coupled to an Ising gauge theory that was recently shown in Monte Carlo simulations to exhibit Z2 topological order and massless Dirac fermion excitations. On tuning parameters, a confining phase with broken symmetry (an antiferromagnet in one choice of Hamiltonian) was also established, and the transition between these phases was found to be continuous, with coincident onset of symmetry breaking and confinement. While the confinement transition in pure gauge theories is well-understood in terms of condensing magnetic flux excitations, the same transition in the presence of gapless fermions is a challenging problem owing to the statistical interactions between fermions and the condensing flux excitations. The conventional scenario then proceeds via a two-step transition, involving a symmetry-breaking transition leading to gapped fermions followed by confinement. In contrast, here, using quantum Monte Carlo simulations, we provide further evidence for a direct, continuous transition and also find numerical evidence for an enlarged SO(5) symmetry rotating between antiferromagnetism and valence bond solid orders proximate to criticality. Guided by our numerical finding, we develop a field theory description of the direct transition involving an emergent nonabelian [SU(2)] gauge theory and a matrix Higgs field. We contrast our results with the conventional Gross–Neveu–Yukawa transition.
Topological degeneracy (Majorana zero-mode) and 1 + 1D fermionic topological order in a magnetic chain on superconductor via spontaneous $\text{Z}_{\mathbf{2}}^{\text{f}}$ symmetry breaking