MagnetoHydrodynamics 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 CMHD as a low-energy effective theory based on a generalized derivative expansion. We demonstrate that 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. We further study the real-time properties of chiral fluid by solving linearized CMHD equations. We discover a remarkable "transition" at an intermediate axial chemical potential $\mu_{A}$ between a stable Chiral fluid at low $\mu_{A}$ and an unstable Chiral fluid at large $\mu_{A}$. We summarize this transition in a "phase diagram" in terms of $\mu_{A}$ and the angle of the wavevector relative to the magnetic field. In the unstable regime, there are four collective modes carrying both magnetic and fluid helicity, in contrary to MHD waves which are unpolarized. The half of the helical modes grow exponentially in time, indicating the instability, while the other half become dissipative.