Sketch of momentum dependence of real and imaginary parts of poles of the Green's function, in the presence of pumping and decay. At zero momentum, there is a free phase, i.e.a mode with vanishing real and imaginary part. For small momentum both modes are diffusive, with no real part, but a non-zero imaginary part. Only above some critical momentum does a real part develop. Such results are seen both in the calculations in Refs.
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