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 $L^3$.

• Received 14 October 2009

DOI:https://doi.org/10.1103/PhysRevLett.104.181601