Effective Minkowski to Euclidean signature change of the magnon BEC pseudo-Goldstone mode in polar ^{3}He

J. Nissinen^{+}, G. E. Volovik^{+* 1)}

^{+}Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland

^{*}Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

**Abstract**

We discuss the effective metric experienced by the
Nambu-Goldstone
mode propagating in the broken symmetry spin-superfluid state of coherent
precession of magnetization. This collective mode represents the phonon in
the RF driven or pulsed out-of-equilibrium Bose-Einstein condensate (BEC) of
optical magnons. We derive the effective BEC free energy and consider the
phonon spectrum when the spin superfluid BEC is formed in the anisotropic
polar phase of superfluid ^{3}He, experimentally observed in uniaxial aerogel
^{3}He-samples. The coherent precession of magnetization experiences an
instability at a critical value of the tilting angle of external magnetic
field with respect to the anisotropy axis. From the action of quadratic
deviations around equilibrium, this instability is interpreted as a
Minkowski-to-Euclidean signature change of the effective phonon metric. We
also note the similarity between the magnon BEC in the unstable region and an
effective vacuum scalar "ghost" condensate.