Non-Hermitian wave turbulence

Wave turbulence describes the long-time statistical behavior of out-of-equilibrium systems composed of weakly interacting waves. Non-Hermitian media ranging from open quantum systems to active materials can sustain wave propagation in so-called $PT$-symmetric states where gain and loss are effectively balanced. Here, we derive the kinetic equations governing wave turbulence in a prototypical non-Hermitian medium: a three-dimensional fluid with odd viscosity. We calculate its exact anisotropic solution, the so-called Kolmogorov-Zakharov spectrum, and validate the existence of this regime using direct numerical simulations. This non-Hermitian wave turbulence generates a direct cascade that is sustained down to the smallest scales, suppressing the transition to strong turbulence typically observed in rotating fluids and electron magnetohydrodynamics. Beyond odd viscous fluids, this qualitative mechanism applies to any non-linear system of waves where non-Hermitian effects are enhanced at small scales through gradient terms in the dynamical equations, e.g. via odd elastic moduli or other non-reciprocal responses.