Transition to zero resistance in a two-dimensional electron gas driven with microwaves

High-mobility two-dimensional electron systems in a perpendicular magnetic field exhibit zero-resistance states (ZRSs) when driven with microwave radiation. We study the nonequilibrium phase transition into the ZRS using phenomenological equations of motion to describe the electron current and density fluctuations in the presence of a magnetic field. We focus on two models to describe the transition into a time-independent steady state. In model I the equations of motion are invariant under a global uniform change in the density. This model is argued to describe physics on small length scales where the density does not vary appreciably from its mean. The ordered state that arises in this case spontaneously breaks rotational invariance in the plane and consists of a uniform current and a transverse Hall field. We discuss some properties of this state, such as stability to fluctuations and the appearance of a Goldstone mode associated with the continuous symmetry breaking. Using dynamical renormalization group techniques, we find that with short-range interactions this model can admit a continuous transition described by mean-field theory, whereas with long-range interactions the transition is driven first order. In model II, we relax the invariance under global density shifts as appropriate for describing the system on longer length scales, and in this case we predict a first-order transition with either short- or long-range interactions. We discuss implications for experiments, including a possible way to detect the Goldstone mode in the ZRS, scaling relations expected to hold in the case of an apparent continuous transition into the ZRS, and a possible signature of a first-order transition in larger samples. Our framework for describing the phase transition into the ZRS also highlights the connection of this problem to the well-studied phenomenon of ``bird flocking.''