Magnetoydroynamics with chiral anomaly: phases of collective excitations and instabilities

We study the relativistic hydrodynamics with chiral anomaly and dynamical electromagnetic fields, namely Chiral MagnetoHydroDynamics (CMHD). We formulate the CMHD as a low-energy effective theory based on a derivative expansion. We demonstrate the modification of ordinary MagnetoHydroDynamics (MHD) due to chiral anomaly can be obtained from the second law of thermodynamics and is tied to chiral magnetic effect with the universal coefficient. When axial charge imbalance becomes larger than a critical value, a new type of collective gapless excitation in the CMHD appears, as a result of the interplay among magnetic field, flow velocity, and chiral anomaly; we call it "Chiral MagnetoHelical Mode" (CMHM). These modes carry definite magnetic and fluid helicities and will either grow exponentially or dissipate in time, depending on the relative sign between their helicity and axial charge density. The presence of exponentially growing CMHM indicates a hydrodynamic instability.