Dynamical symmetry breaking as the origin of the zero-$dc$-resistance state in an $ac$-driven system

Under a strong $ac$ drive the zero-frequency linear response dissipative resistivity $\rho_{d}(j=0)$ of a homogeneous state is allowed to become negative. We show that such a state is absolutely unstable. The only time-independent state of a system with a $\rho_{d}(j=0)<0$ is characterized by a current which almost everywhere has a magnitude $j_{0}$ fixed by the condition that the nonlinear dissipative resistivity $\rho_{d}(j_{0}^{2})=0$. As a result, the dissipative component of the $dc$ electric field vanishes. The total current may be varied by rearranging the current pattern appropriately with the dissipative component of the $dc$-electric field remaining zero. This result, together with the calculation of Durst \emph{et. al.}, indicating the existence of regimes of applied $ac$ microwave field and $dc$ magnetic field where $\rho_{d}(j=0)<0$, explains the zero-resistance state observed by Mani \emph{et. al.} and Zudov \emph{et. al.}.