Chiral waves on the Fermi-Dirac sea: Quantum superfluidity and the axial anomaly

Abstract

We show that as a result of the axial anomaly, massless fermions at zero temperature define a relativistic quantum superfluid. The anomaly pole implies the existence of a gapless Chiral Density Wave (CDW), i.e. an axion-like acoustic mode of an irrotational and dissipationless Hamiltonian perfect fluid, that is a correlated fermion/anti-fermion pair excitation of the Fermi-Dirac sea. In D=2 dimensions the chiral superfluid effective action coincides with that of the Schwinger model as e→0, and the CDW acoustic mode is precisely the Schwinger boson. Since this identity holds also at zero chiral chemical potential, the Dirac vacuum itself may be viewed as a quantum superfluid state. The CDW collective boson is a U(1) chiral phase field, which is gapless as a result of a novel, non-linear realization of Goldstone's theorem, extended to this case of symmetry breaking by an anomaly. A new local form of the axial anomaly bosonic effective action in any D even spacetime is given, consistent with superfluidity, and its quantization is shown to be required by the anomalous Schwinger terms in fermion current commutators. In QED4 this collective Goldstone mode appears as a massless pole in the axial anomaly triangle diagram, and is responsible for the macroscopic non-dissipative currents of the Chiral Magnetic and Chiral Separation Effects, as well as the Anomalous Hall Effect. In a constant uniform magnetic field an exact dimensional reduction from D=4 to D=2 occurs and the collective e+e− CDW chiral pair excitation propagating along the magnetic field direction is a Chiral Magnetic Wave, which acquires a mass gap M2=e3B/2π2. Possible realizations and tests of the theory of collective bosonic excitations due to the anomaly in Dirac/Weyl materials are briefly discussed.