Unconstrained compact lattice QED$_{2+1}$ coupled to phonons: Gauss sectors, orthogonal semimetal, and deconfined criticality

Abstract

Gauge theoretic descriptions of 2D quantum magnets are defined on a restricted physical Hilbert space by imposing Gauss's law. In this work, we study an unconstrained compact lattice QED$_{2+1}$ coupled to bond phonons, in which local Gauss operators are conserved, but their eigenvalues are not a priori fixed. Using exact auxiliary-field quantum Monte Carlo simulations at $N_f = 2$, we investigate how the physical Gauss sector becomes energetically favourable, and how this process is connected to confinement and symmetry breaking. We identify an orthogonal semimetallic (OSM) phase, in which fermions striped off their gauge charge and form a Dirac liquid. This phase mixes various Gauss sectors, and becomes unstable when Gauss's law is explicitly enforced. Upon the imposition of Gauss's law, compactness allows for monopole excitations carrying antiferromagnetic (AFM) and valence-bond-solid (VBS) quantum numbers. Gauge field fluctuations and spinon-phonon coupling tune between the competition of such monopoles and generate a phase diagram with OSM, AFM and VBS orders. The transitions out of the OSM phase coincide with the dynamical generation of Gauss's law, while the competition between AFM and VBS charged monopoles produces a transition consistent with deconfined quantum criticality. Thus our results establish that upon the projection to the physical Hilbert space, deconfinement in compact lattice QED$_{2+1}$ is always unstable towards either AFM or VBS order.