THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

SYSTEMS WITH FERMI POINTS 101
under such a perturbation the general form of eqn (8.18) is preserved. The only
thing that the perturbation can do is to shift locally the position of the Fermi
point p (a)
µ in momentum space and to deform locally the vierbein e µ
b (which in
particular includes slopes of the energy spectrum).
This means that the low-frequency collective modes in such Fermi systems
are the propagating collective oscillations of the positions of the Fermi point and
the propagating collective oscillations of the slopes at the Fermi point (Fig. 8.6).
The former is felt by the right- or the left-handed quasiparticles as a dynamical
gauge (electromagnetic) field, because the main effect of the electromagnetic field
Aµ =(A0, A) is just the dynamical change in the position of zero in the energy
spectrum: in the simplest case, (E − qaA0) 2 = c2 (p − qaA) 2 .
The collective modes related to a local change of the vierbein e µ
b correspond
to the dynamical gravitational field g µν . The quasiparticles feel the inverse tensor
gµν as the metric of the effective space in which they move along the geodesic
curves
ds 2 = gµνdx µ dx ν . (8.20)
Therefore, the collective modes related to the slopes play the part of a gravity
field.
Thus near a Fermi point the quasiparticle is a chiral massless fermion moving
in the effective dynamical electromagnetic and gravitational fields generated by
the low-frequency collective motion of the vacuum.
8.2.8 Fermi points and their physics are natural
From the topological point of view the Standard Model and the Lorentz noninvariant
ground state of 3He-A belong to the same universality class of systems
with topologically non-trivial Fermi points, though the underlying ‘microscopic’
physics can be essentially different. Pushing the analogy further, one may conclude
that classical (and quantum) gravity, as well as electromagnetism and weak
interactions, are not fundamental interactions. If the vacuum belongs to the universality
class with Fermi points, then matter (chiral particles and gauge fields)
and gravity (vierbein or metric field) inevitably appear together in the lowenergy
corner as collective fermionic and bosonic zero modes of the underlying
system.
The emerging physics is natural because vacua with Fermi points are natural:
they are topologically protected. If a pair of Fermi points with opposite topological
charges exist, it is difficult to destroy them because of their topological
stability: the only way is to annihilate the points with opposite charges. Vacua
with Fermi points are not as abundant as vacua with Fermi surfaces, since the
topological protection of the Fermi surface is more powerful. But they appear
in most of the possible phases of spin-triplet superfluidity. They can naturally
appear in semiconductors without any symmetry breaking (see Nielsen and Ninomiya
(1983) where the Fermi point is referred to as a generic degeneracy
point). Possible experimental realization of such Fermi points in semiconductors
was discussed by Abrikosov (1998) and Abrikosov and Beneslavskii (1971). In