Pis’ma v ZhETF, vol. 109, iss. 8, pp. 509 - 510
© 2019
April 25
Two roads to antispacetime in distorted B-phase of3He
G. E. Volovik1)
Low Temperature Laboratory, Aalto University, School of Science and Technology, FI-00076 AALTO, Finland
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
Submitted 19 March 2019
Resubmitted 19 March 2019
Accepted 20
March 2019
DOI: 10.1134/S0370274X19080022
The topological materials with emergent analogs of
γ5 = -iγ0γ1γ2γ3 = τ3.
(3)
gravity demonstrate the possibility of realization of dif-
In some phases of superfluid3He, the Green’s func-
ferent exotic spacetimes, including the transition to an-
tion for fermionic Bogoliubov quasiparticles is similar to
tispacetime, see, e.g., [1] and references therein. There
that in Eq. (1). Now instead of the mass function M(p2),
are several routes to the effective gravity. One of them
the energy of quasiparticles in the normal Fermi liquid
is the tetrad gravity emerging in the vicinity the Weyl
enters, ǫ(p) = vF(|p| - pF). The spin matrices σa act
or Dirac points [2-5] - the exceptional crossing points
in the spin space of3He atoms; the matrices τb act in
in the fermionic spectrum [6, 7]. Also the degenerate
the isotopic Bogoliubov-Nambu space. The function Z
2 + 1 gravity emerges near the Dirac nodal line in the
can be ignored. The tetrads come from the spin-triplet
spectrum [1]. Another important source of gravity is the
p-wave order parameter in3He superfluids - the 3 × 3
formation of the tetrads as bilinear combinations of the
matrix Aia with spin index a = (1, 2, 3) and orbital in-
∑
(
)
fermionic fields [8-10].
dex i = (1, 2, 3):k ki 〈akαa-kβ 〉 ∼ Aia
σaσ2
. For
αβ
Emergent gravity provides different types of the an-
the time reversal symmetric phases [20]:
tispacetime obtained by the space reversal P and time
reversal T operations, including those where the deter-
Aia = pFeiΦeia , a, i = (1, 2, 3).
(4)
minant of the tetrads e changes sign [9-12]. In cosmol-
The tetrads eia emerge due to the spontaneously broken
ogy, the antispacetime Universe was in particular sug-
symmetries SO(3)S ×SO(3)L under spin and orbital ro-
gested as analytic continuation of our Universe across
tations. This is analogous to the formation of the tetrads
the Big Bang singularity [13]. There were speculations,
in relativistic theories as bilinear combinations of the
that antispacetime may support nonequilibrium states
fermionic fields [9, 10]. In addition to tetrads, the order
with negative temperature as a result of analytic contin-
parameter (4) contains the phase Φ coming from spon-
uation across the singularity [14, 15]. Here we consider
taneous breaking of U(1)-symmetry, and the Green’s
the antispacetime realized in experiments [16] on the
function depends both on eµa and on Φ:
analog of cosmological walls bounded by strings [17] -
Kibble walls (KWs).
S(eµa, Φ) = e-γ0Φ/2S(eµa)eγ0Φ/2.
(5)
In notations [18] used in [19], the Green’s function
For Φ = π the symmetry transformation e-γ0Φ/2 is
of the relativistic massive Dirac particle has the form:
equivalent to the conventional space reversal transfor-
Z(p2)
mation - the parity P = e-γ0π/2 = γ0, with P2 = -1.
S =
(1)
-iγaeapµ + M(p2)
This suggests that in relativistic theories the discrete
symmetry, such as the space inversion P, could be the
Here eµa are tetrads with µ, a = 0, 1, 2, 3; the residue
residual Z2 symmetry after breaking of the more funda-
Z(p2) and the mass M(p2) are the functions of p2 =
mental symmetry group.
= gµνpµpν, where gµν = eµaeνbηab. It is convenient to
In the time reversal symmetric states realized in ex-
express γ-matrices in terms of two sets of Pauli matri-
periments [16, 21] the tetrads are:
ces: σ1, σ2 and σ3 for conventional spin, and τ1, τ2, τ3
for the isospin in the left-right space:
eia = c1faxi + c2ĝaŷi + c3 dazi, (a, i) = (1, 2, 3),
(6)
where
d,
f and ĝ are orthogonal unit vectors in spin
γ0 = -iτ1, γa = τ2σa, a = (1, 2, 3).
(2)
space; x,
ŷ and z are orthogonal unit vectors in or-
1)e-mail: volovik@boojum.hut.fi
bital space; c1, c2 and c3 are “speeds of light”. In the
Письма в ЖЭТФ том 109 вып. 7 - 8
2019
509
510
G. E. Volovik
pure B-phase |c1| = |c2| = |c3|; in the polar phase
formation of the discrete symmetry - the parity P in
[21] c1 = c2 = 0; in the polar disorted B phase (PdB)
particle physics - from the continuous symmetry exist-
|c2| = |c1| < |c3|. The particular states of these phases:
ing on the more fundamental level.
This work has been supported by the European
eµa = diag(-1, c1, c2, c3).
(7)
Research Council (ERC) under the European Union’s
Horizon
2020
research and innovation programme
In the PdB phase, the states with c2 = +c1 and
(Grant Agreement # 694248).
c2 = -c1 in Eq.(7) can be separated by the nontopo-
logical domain wall - the analog of the KW bounded by
Full text of the paper is published in JETP Letters
strings [17]. The KW typically appears in the two phase
journal. DOI: 10.1134/S0021364019080034
transitions: at first transition the linear defect becomes
topologically stable; at the second transition the linear
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Fig. 1. Roads to antispacetime: the safe route around the
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Alice string (along C1) or dangerous route along C2 across
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(8)
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lar distorted B-phase also suggests the scenario of the
Письма в ЖЭТФ том 109 вып. 7 - 8
2019