Compact z=2 Electrodynamics in 2+1 Dimensions: Confinement with Gapless Modes

We consider 2 + 1-dimensional compact U(1) gauge theory at the Lifshitz point with a dynamical critical exponent z = 2. As in the usual z = 1 theory, monopoles proliferate the vacuum for any value of the coupling, generating a mass scale. The theory of the dilute monopole gas is written in terms of a nonrelativistic sine-Gordon model with two real fields. While monopoles remove some of the massless poles of the perturbative field strength propagator, a gapless mode representing the incomplete screening of monopoles remains, and is protected by a shift invariance of the original theory. Timelike Wilson loops still obey area laws, implying that minimal charges are confined, but the action of spacelike Wilson loops of linear size L goes instead as L(3).