On chiral magnetic effect in Weyl superfluid ^{3}He-A

G. E. Volovik

*Low Temperature Laboratory, Department of Applied Physics, Aalto University, PO Box 15100, FI-00076 AALTO, Finland*

Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia

**Abstract**

In the theory of the chiral anomaly in relativistic quantum field theories
(RQFT) some results depend on regularization scheme at ultraviolet. In the
chiral superfluid ^{3}He-A, which contains two Weyl points and also
experiences the effects of chiral anomaly, the "trans-Planckian" physics is
known and the results can be obtained without regularization. We discuss
this on example of the chiral magnetic effect (CME), which has been observed
in ^{3}He-A in 90's [Krusius1998]. There are two forms of the
contribution of the CME to the Chern-Simons term in free energy, perturbative
and non-perturbative. The perturbative term comes from the fermions living in
the vicinity of the Weyl point, where the fermions are "relativistic" and
obey the Weyl equation. The non-perturbative term originates from the deep
vacuum, being determined by the separation of the two Weyl points in momentum
space. Both terms are obtained using the Adler-Bell-Jackiw equation for
chiral anomaly, and both agree with the results of the microscopic
calculations in the "trans-Planckian" region.
Existence of the two nonequivalent forms of the Chern-Simons term
demonstrates that the results obtained within the RQFT depend on the specific
properties of the underlying quantum vacuum and may reflect different
physical phenomena in the same vacuum.