We present a theory of nonequilibrium quantum criticality in a coupled
bilayer system of itinerant electron magnets. The model studied consists of the
first layer subjected to an inplane current and open to an external substrate.
The second layer is closed and subject to no direct external drive, but couples
to the first layer via short-ranged spin exchange interaction. No particle
exchange is assumed between the layers. Starting from a microscopic fermionic
model, we derive an effective action in terms of two coupled bosonic fields
which are related to the magnetization fluctuations of the two layers. When
there is no interlayer coupling, the two bosonic modes possess different
dynamical critical exponents z with z=2 (z=3) for the first (second) layer.
This results in multi-scale quantum criticality in the coupled system. It is
shown that the linear coupling between the two fields leads to a low energy
fixed point characterized by the larger dynamical critical exponent z=3. The
perturbative renormalization group is used to compute the correlation length in
the quantum disordered and quantum critical regimes. We also derive the
stochastic dynamics obeyed by the critical fluctuations in the quantum critical
regime. Comparing the nonequilibrium situation to the thermal equilibrium
scenario, where the whole system is at a temperature T, we find that the
nonequilibrium drive does not always play the role of temperature.Comment: 20+ pages, 3 figures; Revised version as accepted by PRB, added
figure of mean field phase diagra