Magnetohydrodynamics with chiral anomaly: Phases of collective excitations and instabilities

We study 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 the chiral magnetic effect. We further study the real-time properties of a 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 high mu(A). We summarize this transition in a phase diagram in terms of mu(A) and the angle of the wave vector relative to the magnetic field. In the unstable regime, four collective modes carry both magnetic and fluid helicity, in contrary to MHD waves, which are unpolarized. Half of the helical modes grow exponentially in time, indicating the instability, while the other half become dissipative.