[orcid=0000-0001-6891-7008]

[orcid=0000-0002-7313-6094]

[1]

[orcid=0000-0002-3851-9305]

[1] [1]Corresponding author

# Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors

Yusuke Masaki  Department of Physics, Tohoku University, Sendai, Miyagi
980-8578, Japan  Research and Education Center for Natural Sciences, Keio
University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan  Takeshi
Mizushima  Department of Materials Engineering Science, Osaka University,
Toyonaka, Osaka 560-8531, Japan<EMAIL_ADDRESS>Muneto Nitta
Department of Physics, Keio University, Hiyoshi 4-1-1, Japan  International
Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2),
Hiroshima University, Kagamiyama 1-3-2, Higashi-Hiroshima, Hiroshima 739-8511,
Japan<EMAIL_ADDRESS>
###### Abstract

Anyons are particles obeying statistics of neither bosons nor fermions. Non-
Abelian anyons, whose exchanges are described by a non-Abelian group acting on
a set of wave functions, are attracting a great attention because of possible
applications to topological quantum computations. Braiding of non-Abelian
anyons corresponds to quantum computations. The simplest non-Abelian anyons
are Ising anyons which can be realized by Majorana fermions hosted by vortices
or edges of topological superconductors, $\nu=5/2$ quantum Hall states, spin
liquids, and dense quark matter. While Ising anyons are insufficient for
universal quantum computations, Fibonacci anyons present in $\nu=12/5$ quantum
Hall states can be used for universal quantum computations. Yang-Lee anyons
are non-unitary counterparts of Fibonacci anyons. Another possibility of non-
Abelian anyons (of bosonic origin) is given by vortex anyons, which are
constructed from non-Abelian vortices supported by a non-Abelian first
homotopy group, relevant for certain nematic liquid crystals, superfluid 3He,
spinor Bose-Einstein condensates, and high density quark matter. Finally,
there is a unique system admitting two types of non-Abelian anyons, Majorana
fermions (Ising anyons) and non-Abelian vortex anyons. That is ${}^{3}P_{2}$
superfluids (spin-triplet, $p$-wave paring of neutrons), expected to exist in
neutron star interiors as the largest topological quantum matter in our
universe.

###### keywords:

Non-Abelian vortices, non-Abelian anyons, non-Abelian statistics, topological
quantum computation, topological materials, topological superconductors,
topological superfluids, superfluid 3He, spinor Bose-Einstein condensates,
nematic liquid crystals, chiral liquid crystals, quark matter, nuclear matter,
nuclear superfluids

## Key points/objectives

  * •

Unlike fermions and bosons, the exchange of non-Abelian anyons is described by
unitary transformations that operate on topologically protected degenerate
ground states. The adiabatic exchanges of two or more non-Abelian anyons
manipulate quantum information stored in a set of degenerate ground states,
leading to topological quantum computations.

  * •

The simplest non-Abelian anyons are Ising anyons which are predicted to exist
as Majorana zero modes in topological superconductors, fractional quantum Hall
states, Kitaev materials, and dense quark matter.

  * •

Another non-Abelian anyons are vortex anyons, which are constructed from non-
Abelian vortices supported by a non-Abelian first homotopy group. Such vortex
anyons exist in liquid crystals, superfluid 3He, spinor Bose-Einstein
condensates, and high density quark matter.

  * •

${}^{3}P_{2}$ superfluids, which are expected to be realized in neutron stars,
provide unique systems simultaneously admitting two kinds of non-Abelian
anyons, i.e., Ising anyons and vortex anyons.

##  1 Introduction

In three spatial dimensions, all particles are either bosons or fermions in
quantum physics, that is, a wave function of multi-particle states is
symmetric (antisymmetric) under the exchanges of two bosons (fermions). On
contrary, in two spatial dimensions, there exist exotic particles classified
to neither bosons nor fermions, anyons. A wave function of two anyons receives
a nontrivial phase factor under their exchanges Leinaas & Myrheim (1977);
Wilczek (1982). Such exotic particles play essential roles in fractional
quantum Hall states Halperin (1984); Arovas et al. (1984), and have been
experimentally observed for $\nu=1/3$ fractional quantum Hall states
Bartolomei et al. (2020); Nakamura et al. (2020).

Recently, yet exotic particles attracted great attention, that is, non-Abelian
anyons. Non-Abelian anyons are described by a set of multiple wave functions,
and the exchanges of two non-Abelian anyons lead to unitary matrix operations
on a set of wave functions. They have been theoretically predicted to exist in
$\nu=5/2$ fractional quantum Hall states Moore & Read (1991); Nayak & Wilczek
(1996), topological superconductors (SCs) and superfluids (SFs) Read & Green
(2000); Ivanov (2001); Kitaev (2001), and spin liquids Kitaev (2006); Motome &
Nasu (2020), and experimental observation is pursued. Non-Abelian anyons are
attracting significant interests owing to the possibility to offer a platform
of topologically protected quantum computations realized by braiding of non-
Abelian anyons Kitaev (2003); Kitaev & Laumann (2008); Nayak et al. (2008);
Pachos (2012); Sarma et al. (2015); Field & Simula (2018). Since the Hilbert
space and braiding operations are topologically protected, they are robust
against noises in contrast to the conventional quantum computation methods.
Recently, it has been reported that non-Abelian braiding and fusions has been
experimentally realised in a superconducting quantum processor, where the
fusion and braiding protocols are implemented using a quantum circuit on a
superconducting quantum processor Andersen et al. (2022), thereby opening a
significant step to realize topological quantum computations.

One of the main routes to realize non-Abelian anyons is based on Majorana
fermions in topological SCs Ivanov (2001); Kitaev (2001); Alicea (2012);
Leijnse & Flensberg (2012); Beenakker (2013); Silaev & Volovik (2014); Elliott
& Franz (2015); Sato & Fujimoto (2016); Mizushima et al. (2016); Sato & Ando
(2017); Beenakker (2020); Marra (2022). Majorana fermions were originally
proposed in high energy physics to explain neutrinos; they are particles that
coincide with their own anti-particles Majorana (1937). In condensed matter
physics, Majorana fermions are localized at vortices or edge of materials for
which several protocols of non-Abelian braiding were proposed. Non-Abelian
anyons constructed from Majorana fermions are so-called Ising anyons. They are
not enough for universal quantum computations, and thus some non-topological
process should be included Nayak et al. (2008). In contrast, another type of
anyons called Fibonacci anyons Trebst et al. (2008) can offer a promising
platform for universal topological quantum computation that all quantum gates
are implemented by braiding manipulation in a topologically protected way
Preskill (2004); Bonesteel et al. (2005); Hormozi et al. (2007). Such
Fibonacci anyons are proposed to exist in $\nu=12/5$ quantum Hall states, a
junction made of a conventional SC and a $\nu=2/3$ fractional quantum Hall
state Mong et al. (2014), interacting Majorana fermions realized in a
septuple-layer structure of topological SCs Hu & Kane (2018), and Rydberg
atoms in a lattice Lesanovsky & Katsura (2012). Yang-Lee anyons are also
proposed as non-unitary counterparts of Fibonacci anyons, obeying nonunitary
non-Abelian statistics Ardonne et al. (2011); Freedman et al. (2012); Sanno et
al. (2022).

The aforementioned anyons are all quasiparticle excitations composed of
fermions. On the other hand, a different type of non-Abelian anyons can be
composed of bosons in certain ordered states accompanied with symmetry
breakings $G\to H$. They are called non-Abelian vortex anyons, whose exchange
statistics are non-Abelian due to non-Abelian vortices, that is quantum
vortices supported by a non-Abelian first homotopy (fundamental) group of
order parameter manifolds, $\pi_{1}(G/H)$ Bais (1980); Wilczek & Wu (1990);
Bucher (1991); Brekke et al. (1993); Lo & Preskill (1993); Lee (1994); Brekke
et al. (1997).111 The term “non-Abelian” on vortices depends on the context.
In the other contexts (in particular in high energy physics), vortices in a
symmetry breaking $G\to H$ with non-Abelian magnetic fluxes are often called
non-Abelian even though $\pi_{1}(G/H)$ the first homotopy group is Abelian
Hanany & Tong (2003); Auzzi et al. (2003); Eto et al. (2006); Shifman & Yung
(2007); Eto et al. (2014). In condensed matter physics, vortices with Majorana
fermions in their cores are also sometimes called non-Abelian vortices
(because they are non-Abelian anyons). In this article, the term “non-Abelian”
on vortices is used only for vortices with non-Abelian first homotopy group
$\pi_{1}$.  Such non-Abelian vortices exist in liquid crystals Poenaru &
Toulouse (1977); Volovik & Mineev (1977); Mermin (1979); Lavrentovich & Kleman
(2001), 3He SFs Balachandran et al. (1984); Salomaa & Volovik (1987); Volovik
(2003), spinor Bose-Einstein condensates (BECs) Semenoff & Zhou (2007);
Kobayashi et al. (2009, 2012); Borgh & Ruostekoski (2016), and high density
quark (QCD) matter Fujimoto & Nitta (2021a, b, c); Eto & Nitta (2021). Non-
Abelian braiding of vortex anyons in spinor BECs and its application to
quantum computations were proposed Mawson et al. (2019).

In addition to these systems admitting one type of non-Abelian anyons, there
is the unique system simultaneously admitting two kinds of non-Abelian anyons,
Ising anyons based on Majorana fermions and non-Abelian vortex anyons. It is a
${}^{3}P_{2}$ SF, spin-triplet and $p$-wave paring with the total angular
momentum two Hoffberg et al. (1970); Tamagaki (1970); Takatsuka & Tamagaki
(1971); Takatsuka (1972); Richardson (1972). Such ${}^{3}P_{2}$ SFs are
expected to be realized by neutrons, relevant for neutron star interiors
Sedrakian & Clark (2018). ${}^{3}P_{2}$ SFs are the largest topological SFs in
our universe Mizushima et al. (2017) and admit non-Abelian vortices Masuda &
Nitta (2020). Non-Abelian vortices host Majorana fermions in their cores
Masaki et al. (2022), thus behaving as non-Abelian anyons.

The purpose of this article is to summarise these non-Abelian anyons of
various types. After introducing basics of non-Abelian anyons in Sec. 2, we
describe non-Abelian anyons in fermionic and bosonic systems, based on
Majorana fermions and non-Abelian first homotopy group in Secs.3 and 4,
respectively. In Sec. 5, we introduce ${}^{3}P_{2}$ SFs as the unique system
simultaneously admitting two kinds of non-Abelian anyons. We summarize this
article in Sec. 6

##  2 Non-Abelian anyons

###  2.1 Braid group and quantum statistics

Here we consider pointlike topological defects (e.g., vortices in two-
dimensional spinless SCs) which behave as identical particles in a two-
dimensional plane. The exchange of $n$ particles in three or higher dimension
is described by the symmetric group $S_{n}$ Leinaas & Myrheim (1977). There
are two one-dimensional representations of $S_{n}$, $\pm 1$, due to even/odd
permutation and $+1$ ($-1$) corresponds the Bose (Fermi) statistics. Two
dimension is special and the exchange of particles is given by the braid group
$B_{n}$ Wu (1984). The braid of particles is expressed as a set of operators
$T_{k}$ $(1\leq k\leq n-1)$ that exchange the neighboring $k$th and $(k+1)$th
particles in an anticlockwise direction. The operators obey the relations (see
Fig. 1)

$\displaystyle T_{i}T_{j}T_{i}=T_{j}T_{i}T_{j},\quad\mbox{for $|i-j|=1$},$ (1)
$\displaystyle T_{i}T_{j}=T_{j}T_{i},\quad\mbox{for $|i-j|\geq 2$}.$ (2)

Figure 1: Schematic of the braid relations in Eqs. (1) and (2):
$T_{i}T_{j}=T_{j}T_{i}$ for $|i-j|\geq 2$ (left) and
$T_{i}T_{j}T_{i}=T_{j}T_{i}T_{j}$ for $|i-j|=1$ (right).

The exotic statistics of particles represented by the braid group stems from
the relation $T^{-1}_{i}\neq T_{i}$. In the one dimensional representation,
the generator of $T_{i}$ is given by a phase factor that a wave function under
the exchange of particles acquires,
$\tau_{j}\equiv\tau(T_{j})=e^{i\theta_{j}}$ ($0\leq\theta_{j}<2\pi$). The
relation in Eq. (1) implies that the exchange operation of any two particles
induces the same phase factor
$\tau_{1}=\tau_{2}=\cdots=\tau_{n-1}=e^{i\theta}$. The phase factor
characterizes the quantum statistics of particles Wu (1984), and the absence
of the relation $T^{2}_{i}=1$ allows for the fractional (anyon) statistics
with neither $\theta=0$ (bosons) nor $\theta=\pi$ (fermions). In addition to
the one-dimensional representation, the braid group has non-Abelian
representations. In Sec. 3, we will show that the generators of the braid of
Majorana zero modes are noncommutative as $[\tau_{i},\tau_{j}]\neq 0$ for
$|i-j|=1$ and the pointlike defects hosting Majorana modes behave as non-
Abelian anyons.

Although the braid group is trivial in three dimensions, a three-dimensional
model with pointlike topological defects which host Majorana modes and obey
the non-Abelian statistics has been proposed Teo & Kane (2010a). The non-
Abelian statistics of the defects, which behave as hedgehogs, can be
interpreted as the projective ribbon permutation statistics Freedman,
Hastings, Nayak, Qi, Walker & Wang (2011); Freedman, Hastings, Nayak & Qi
(2011). Such non-Abelian statistics enables to construct three-dimensional
networks of topological superconducting wires supporting Majorana modes.

###  2.2 Non-Abelian anyons

In three spatial dimensions, the statistics of particles is determined by
their intrinsic spins. According to the spin statistics theorem, all particles
with integer (half-integer) spin are bosons (fermions). A pair formed by two
fermions with the spin $1/2$ behaves as a boson, and the spin of the composite
particle obeys the “fusion rule” $\frac{1}{2}\otimes\frac{1}{2}=0\oplus 1$,
where ${0}$ and $1$ denote the spin singlet and triplet states, respectively.
When particles are trapped in a two dimensional plane, however, there is
another possibility that is neither fermions nor bosons, i.e., anyons. In
general, anyons can be characterized by the topological charge, the fusion
rule ($N^{c}_{ab}$), the associative law (the $F$-matrix), and the braiding
operation (the $R$-matrix) Preskill (2004); Nayak et al. (2008); Pachos
(2012).

Let ${\bf 1}$ and $\\{a,b,\cdots\\}$ be the vacuum and the different species
of particles, respectively. Consider the anyon model spanned by
$M=\\{{\bm{1}},a,b,\cdots\\}$ and bring two anyons $a$ and $b$ together. The
fused particle also belongs to $M$. The fusion rule is represented by

$\displaystyle a\otimes b=\sum_{c\in M}N_{ab}^{c}c,$ (3)

where fusion coefficients $N_{ab}^{c}$ are non-negative integers. Figure 2(a)
shows the diagrammatic expression of the fusion of two anyons with topological
charges $a$ and $b$ to an anyon with charge $c$. When $N_{ab}^{c}$ is not zero
for only one value of $c$, the fusion of paired $a$ and $b$ anyons is uniquely
determined and the anyon is called the Abelian anyon. The non-Abelian anyons
are characterized by two or more coefficients that satisfy $N_{ab}^{c}\neq 0$.
In the context of quantum computation, the fusion rule determines the Hilbert
space that encodes quantum information, and quantum computation is implemented
by the braiding manipulation of non-Abelian anyons in a topologically
protected way. Figure 2(b) shows the diagrammatic expression of the
$R$-matrix. When one anyon moves around the other, the pairwise anyons acquire
a phase. As mentioned below, the $R$-matrix describes a phase resulting from
the exchange of anyons $a$ and $b$ which fuse to a anyon $c$. Let us also
consider the fusion of three anyons $a$, $b$, and $c$ into an anyon $d$. The
outcome of the fusion process is independent of order in which the anyons are
to be fused. This implies that the fusion process obeys the associative law,
$(a\otimes b)\otimes c=a\otimes(b\otimes c)$, which is characterized by the
$F$-matrix. The $F$ matrix represents the transformation between different
fusion bases or the choice of order of fusion, which is expressed as shown in
Fig. 2(c). The $R$-matrix and the $F$-matrix are the building-blocks for
constructing the braid group in multiple anyon systems Preskill (2004).

Figure 2: (a) Diagram of the fusion of two anyons $a$ and $b$ to an anyon $c$.
Diagrammatic expressions of the $R$-matrix (b) and the $F$-matrix (c).

Ising anyons. An example of the non-Abelian anyons is the Ising anyon Kitaev
(2006); Nayak et al. (2008); Pachos (2012). The Ising anyon model consists of
the vacuum ${\bf 1}$, Ising anyons $\sigma$, and Dirac (complex) fermions
$\psi$, which obey the fusion rules

$\displaystyle\sigma\otimes\sigma={\bf
1}\oplus\psi,\;\sigma\otimes\psi=\sigma,\;\psi\otimes\psi={\bf 1},\;{\bf
1}\otimes x=x,$ (4)

where $x\in\\{{\bf 1},\sigma,\psi\\}$. Consider three Ising anyons, where the
two left-most anyons fusing into either ${\bf 1}$ or $\psi$ (see Fig. 2(c)
with $(a,b,c)\rightarrow\sigma$ and $i,j=\\{{\bf 1},\psi\\}$). The $R$-matrix
and the $F$-matrix are given, respectively, by

$\displaystyle R=\begin{pmatrix}R^{\sigma\sigma}_{\bf 1}&0\\\
0&R^{\sigma\sigma}_{\psi}\end{pmatrix}=e^{-i\pi/8}\begin{pmatrix}1&0\\\
0&i\end{pmatrix},$ (5) $\displaystyle
F^{\sigma}_{\sigma\sigma\sigma}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}.$ (6)

The diagonal components of the $R$-matrix are the phases resulting from the
counterclockwise exchange of two left-most Ising anyons ($\sigma$) fusing to
${\bf 1}$ or $\psi$, while the twice exchange operation of two right-most
Ising anyons is represented by the unitary matrix,
$F^{-1}R^{2}F=e^{-i\pi/4}\sigma_{x}$. Hence, braiding two anyons corresponds
to the implementation of the quantum gates acting on the quantum states
spanned by ${\bf 1}$ and $\psi$.

In general, the anyons are described by conformal field theory, corresponding
to gapless edge states residing in the boundary of two-dimensional gapped
topological phases. For Ising anyons, the theory is the conformal field theory
with the central charge $c=1/2$, which describes critical Ising models such as
the two-dimensional Ising model at the point of second-order phase transition
Di Francesco et al. (1997).

The Moore-Read state in the fractional quantum Hall state at the filling
factor $\nu=5/2$ supports this type of non-Abelian anyons Nayak et al. (2008).
The Ising anyons can also be realized in the Kitaev’s honeycomb model, which
is an exactly solvable model of quantum spin liquid states Kitaev (2006). In
this model, the spins are fractionalized to Majorana fermions coupled to
$\mathbb{Z}_{2}$ gauge fields. The Ising anyons appear as Majorana zero modes
bound to the $\mathbb{Z}_{2}$ flux. Materials including $4d$ or $5d$ atoms
with a strong spin-orbit coupling have been proposed as candidates of Kitaev
magnets, and the half-integer thermal Hall effect was reported in
$\alpha$-RuCl3 Kasahara et al. (2018); Yamashita et al. (2020); Yokoi et al.
(2021); Bruin et al. (2022), which is a signature of Majorana fermions in the
chiral quantum spin liquid phase Vinkler-Aviv & Rosch (2018); Ye et al.
(2018). Apart from materials, Google team reported the demonstration of fusion
and braiding rules of non-Abelian Ising anyons on a superconducting quantum
processor, where the fusion and braiding protocols are implemented using a
quantum circuit on a superconducting quantum processor Andersen et al. (2022).
Another platform to realize Ising anyons is a topological SC. In the context
of topological SCs, the Ising anyons ($\sigma$) appear as Majorana zero modes
bound at their boundaries or topological defects, such as the surface,
interface, and vortices Read & Green (2000); Kitaev (2001); Nayak et al.
(2008); Alicea (2012); Sato & Fujimoto (2016); Mizushima et al. (2016).
Pairwise Majorana zero modes form a complex fermion that can define either the
unoccupied state (${\bf 1}$) or the occupied state (${\psi}$) of the zero
energy eigenstate, implying $\sigma\otimes\sigma={\bf 1}\oplus\psi$. The
vacuum ${\bf 1}$ corresponds to a condensate of Cooper pairs, while $\psi$
represents a Bogoliubov quasiparticle which can pair into a condensate, i.e.,
$\psi\otimes\psi={\bf 1}$. The detailed properties and realization of Majorana
zero modes in topological SCs are described in Secs. 3.1 and 3.2.

Fibonacci anyons. Another example is the Fibonacci anyons Trebst et al.
(2008). The Fibonacci anyon model consists of the vacuum ${\bf 1}$ and the
non-trivial anyon $\tau$, which obey the fusion rules

$\displaystyle\tau\otimes\tau={\bf 1}\oplus\tau,\quad{\bf 1}\otimes x=x,$ (7)

where $x={\bf 1},\tau$. The first rule implies that the fusion of two anyons
may result in either annihilation or creation of a new anyon, and thus the
Fibonacci anyon may be its own anti-particle. Repeated fusions of the $n+1$
$\tau$-anyons result in either the vacuum or the $\tau$ anyon as
$\tau\otimes\tau\otimes\cdots\otimes\tau=a_{n}\cdot{\bf 1}\oplus b_{n}\tau$,
where $a_{n}=1$ for $n=2$ and $a_{n}=n-2$ for $n\geq 3$. The coefficient
$b_{n}$ grows as the Fibonacci series, and the first few values in the
sequence are $b_{2}=1$, $b_{3}=2$, $b_{4}=3$, $b_{5}=5,\cdots$. The $R$-matrix
and the $F$-matrix are given, respectively, by

$\displaystyle R=\begin{pmatrix}R^{\tau\tau}_{\bf 1}&0\\\
0&R^{\tau\tau}_{\tau}\end{pmatrix}=\begin{pmatrix}e^{i4\pi/5}&0\\\
0&-e^{i2\pi/5}\end{pmatrix},$ (8) $\displaystyle
F^{\tau}_{\tau\tau\tau}=\begin{pmatrix}\phi^{-1}&\phi^{-1/2}\\\
\phi^{-1/2}&-\phi^{-1}\end{pmatrix},$ (9)

where $\phi=(1+\sqrt{5})/2$ is the golden ratio. The Fibonacci anyons are
described by the level-1 $G_{2}$ Wess-Zumino-Witten theory with the central
charge $c=14/5$ Mong et al. (2014), where $G_{2}$ is the simplest exceptional
Lie group. While Ising anyons are not sufficient for universal quantum
computation, the Fibonacci anyon systems can offer a promissing platform for
universal topological quantum computation that all quantum gates are
implemented by braiding manipulation in a topologically protected way Preskill
(2004); Bonesteel et al. (2005); Hormozi et al. (2007).

The existence of the Fibonacci anyons is predicted in the $\nu=12/5$
fractional quantum Hall state that is described by the Read-Rezayi state Read
& Rezayi (1999). It is also proposed that a junction made of a conventional SC
and the $\nu=2/3$ fractional quantum Hall state supports the Fibonacci anyons
Mong et al. (2014). Fibonacci anyons can also be made from interacting
Majorana fermions realized in a septuple-layer structure of topological SCs Hu
& Kane (2018) and from Rydberg atoms in a lattice Lesanovsky & Katsura (2012).

Yang-Lee anyons. There are nonunitary counterparts of Fibonacci anyons, which
are referred to as Yang-Lee anyons. The conformal field theory corresponding
to Yang-Lee anyons is nonunitary and Galois conjugate to the Fibonacci
conformal field theory Ardonne et al. (2011); Freedman et al. (2012). Because
of nonunitarity, the central charge and the scaling dimension for the one
nontrivial primary field are negative, $c=-22/5$ and $\Delta=-2/5$,
respectively. As shown in Eq. (9), the $F$-matrix for Fibonacci anyons is
given by the golden ratio $\phi$, i.e., one solution of the equation
$x^{2}=1+x$, which is an algebraic analogue of the fusion rule. The $F$-matrix
for Yang-Lee anyons is obtained from Eq. (9) by replacing
$\phi\rightarrow-1/\phi$ as

$\displaystyle F^{\tau}_{\tau\tau\tau}=\begin{pmatrix}-\phi&i\sqrt{\phi}\\\
i\sqrt{\phi}&\phi\end{pmatrix},$ (10)

as $-1/\phi$ is the other solution of the equation $x^{2}=1+x$. In Eq. (10),
the bases of the $F$-matrix are spanned by the vacuum state ${\bf 1}$ and a
Yang-Lee anyon $\tau$. The $R$-matrix are given by

$\displaystyle R=\begin{pmatrix}R^{\tau\tau}_{\bf 1}&0\\\
0&R^{\tau\tau}_{\tau}\end{pmatrix}=\begin{pmatrix}e^{i2\pi/5}&0\\\
0&e^{i\pi/5}\end{pmatrix}.$ (11)

Unlike the Fibonicci anyons, braiding two Yang-Lee anyons is represented by a
combination of the $R$-matrix and the nonunitary matrix, $F^{-1}RF$. While
Yang-Lee anyons obey the same fusion rule as that of Fibonacci anyons given by
Eq. (7), the $F$-matrix in Eq. (10) is the nonunitary and the Yang-Lee anyons
obey the nonunitary non-Abelian statistics.

The nonunitary conformal field theory with $c=-22/5$ describes the nonunitary
critical phenomenon known as the Yang-Lee edge singularity Cardy (1985). Let
us consider the Ising model with an imaginary magnetic field $ih$
($h\in\mathbb{R}$). For temperatures above the critical temperature, the zeros
of the partition function in the thermodynamic limit, which are referred to as
the Lee-Yang zeros, accumulate on the line $h>h_{\rm c}$ and the edge of the
Lee-Yang zeros corresponds to the critical point $h_{c}$ Lee & Yang (1952). As
$h$ ($>h_{\rm c}$) approaches the edge, the density of zeros has a power-law
behavior as $|h-h_{\rm c}|^{\sigma}$, which characterizes the critical
phenomenon Kortman & Griffiths (1971); Fisher (1978). For instance, the
magnetization exhibits singular behavior with the same critical exponent
$\sigma$. The Yang-Lee edge singularity is also realized by the quantum Ising
model with a real transverse field and a pure-imaginary longitudinal field von
Gehlen (1991).

The quantum Ising model with an imaginary longitudinal field, which supports
the Yang-Lee anyons, can be constructed from Majorana zero modes in a network
of topological superconducting wires coupled with dissipative electron baths
Sanno et al. (2022). The Majorana modes bound at the end points of one-
dimensional topological SCs constitute spin $1/2$ operators. A coupling of
Majorana zero modes with electrons in a metallic substrate plays an role of
the pure-imaginarly longitudinal field, while the tunneling of Majorana zero
modes between neighboring superconducting wires induces a transverse magnetic
field. Schemes for the fusion, measurement, and braiding of Yang-Lee anyons
are also proposed in Ref. Sanno et al. (2022). As mentioned above, the Yang-
Lee anyons obey the nonunitary non-Abelian statistics. The nonunitary
evolution of quantum states has been discussed in connection with measurement-
based quantum computation Terashima & Ueda (2005); Usher et al. (2017); Piroli
& Cirac (2020); Zheng (2021). Although the Yang-Lee anyons with the nonunitary
$F$-matrix are not suitable for application to unitary quantum computation,
they can be the building-blocks for the construction of measurement-based
quantum computation. In addition, as the nonunitary quantum gates can be
implemented by braiding manipulations, the Yang-Lee anyon systems may offer a
quantum simulator for nonunitary time evolution of open quantum systems in a
controllable way.

Vortex anyons. In this article, we also discuss non-Abelian anyons made of
bosonic (topological) excitations in ordered states. The nontrivial structure
of the order parameter manifold appears in the liquid crystals, spin-2 BECs,
the A phase of SF 3He, dense QCD matter, and ${}^{3}P_{2}$ SFs. The line
defects in such ordered systems, such as vortices, are represented by non-
Abelian first homotopy group and their topological charges are noncommutative.
Such topological defects with noncommutative topological charges behave as
non-Abelian anyons, called the non-Abelian vortex anyons. In Sec. 4, we
demonstrate that order parameter manifolds in nematic liquid crystals and
spin-2 BECs admit the existence of non-Abelian vortices and show the fusion
rules of such non-Abelian vortex anyons.

##  3 Non-Abelian anyons in topological SCs

###  3.1 Majorana zero modes as Ising anyons

Majorana zero modes. An elementary excitation from superconducting ground
states is a Bogoliubov quasiparticle that is a superposition of the electron
and hole. The quasiparticle excitations are described by the Bogoliubov-de
Gennes (BdG) Hamiltonian

$\displaystyle{H}=\sum_{ij}\begin{pmatrix}{\bm{\psi}}^{{\dagger}}_{i},~{}{\bm{\psi}}_{i}\end{pmatrix}\mathcal{H}_{ij}\begin{pmatrix}{\bm{\psi}}_{j}\\\
{\bm{\psi}}^{{\dagger}}_{j}\end{pmatrix},$ (12)
$\displaystyle\mathcal{H}_{ij}=\begin{pmatrix}h_{ij}&\Delta_{ij}\\\
\Delta^{{\dagger}}_{ij}&-h^{\ast}_{ij}\end{pmatrix}.$ (13)

Here, ${\bm{\psi}}_{j}$ is the $N$-component vector of the electron field
operator and $\mathcal{H}$ is the $2N\times 2N$ hermitian matrix, where $N$ is
the sum of the spin degrees of freedom and the number of the lattice sites and
so on. The $N\times N$ hermitian matrix $h$ describes the normal state
Hamiltonian and the superconducting pair potential $\Delta$ obeys $\Delta^{\rm
t}=-\Delta$ because of the Fermi statistics, where $a^{\rm t}$ denotes the
transpose of a matrix $a$. The BdG Hamiltonian naturally holds the particle-
hole symmetry

$\displaystyle\mathcal{C}\mathcal{H}\mathcal{C}^{-1}=-\mathcal{H},$ (14)

where the particle-hole operator $\mathcal{C}=\Theta K$ is an antiunitary
operator composed of the unitary operator $\Theta$ and the complex conjugation
operator $K$. The self-conjugate Dirac fermions are called Majorana fermions,
where the quantized field
${\bm{\Psi}}\equiv({\bm{\psi}},{\bm{\psi}}^{{\dagger}})^{\rm t}$ obeys

$\displaystyle{\bm{\Psi}}=\mathcal{C}{\bm{\Psi}},\quad\mathcal{C}^{2}=+1.$
(15)

We expand the quantized field ${\bm{\Psi}}$ in terms of the energy
eigenstates. The energy eigenstates are obtained from the BdG equation,

$\displaystyle\sum_{j}\mathcal{H}_{ij}({\bm{\varphi}}_{E})_{j}=E({\bm{\varphi}}_{E})_{i},$
(16)

which describes the quasiparticle with the energy $E$ and the wave function
${\bm{\varphi}}_{E}$. Equation (14) guarantees that the quasiparticle state
with $E>0$ and ${\bm{\varphi}}_{E}$ is accompanied by the negative energy
state with $-E$ and ${\bm{\varphi}}_{-E}=\mathcal{C}{\bm{\varphi}}_{E}$. Thus,
the negative energy states are redundant as long as the particle-hole symmetry
is maintained. Let $\eta_{E}$ be the quasiparticle operator which satisfies
the anticommutation relations,
$\\{\eta_{E},\eta^{{\dagger}}_{E^{\prime}}\\}=\delta_{E,E^{\prime}}$ and
$\\{\eta_{E},\eta_{E^{\prime}}\\}=\\{\eta^{{\dagger}}_{E},\eta^{{\dagger}}_{E^{\prime}}\\}=0$
($E,E^{\prime}>0$). The self-charge conjugation relation (15) then implies
that the quasiparticle annihilation operator with a positive energy is
equivalent to the creation with a negative energy as
$\eta_{E}=\eta^{{\dagger}}_{-E}$, and ${\bm{\Psi}}$ is expanded only in terms
of positive energy states as
${\bm{\Psi}}({\bm{r}})=\sum_{E>0}[{\bm{\varphi}}_{E}\eta_{E}+\mathcal{C}{\bm{\varphi}}_{E}\eta^{{\dagger}}_{E}]$.
The condition (15) can be fulfilled by odd-parity SCs. In the absence of spin-
orbit coupling, the spin-singlet pair potential is always invariant under the
spin rotation, and the particle-hole exchange operator is given by
$\mathcal{C}^{2}=-1$ in each spin sector. Hence, spin-singlet SCs cannot
satisfy Eq. (15). Spin-orbit coupling, however, enables even spin singlet SCs
to host Majorana fermions Sato & Ando (2017); Alicea (2012); Sato & Fujimoto
(2016).

Now, let us suppose that a single zero-energy state exists, and
${\bm{\varphi}}_{0}$ is its wave function. Then, we can rewrite the quantized
field to

$\displaystyle{\bm{\Psi}}={\bm{\varphi}}_{0}\gamma+\sum_{E>0}\left[{\bm{\varphi}}_{E}\eta_{E}+\mathcal{C}{\bm{\varphi}}_{E}\eta^{{\dagger}}_{E}\right].$
(17)

We have introduced $\gamma$, instead of $\eta_{E=0}$, to distinguish the zero
mode from other energy eigenstates. Owing to Eq. (14), the zero-energy
quasiparticle is composed of equal contributions from the particle-like and
hole-like components of quasiparticles, i.e.,
$\mathcal{C}{\bm{\varphi}}_{0}={\bm{\varphi}}_{0}$. The self-conjugate
constraint in Eq. (15) imposes the following relations:

$\displaystyle\gamma^{{\dagger}}=\gamma,$ (18)

and $({\color[rgb]{0,0,0}\gamma})^{2}=1$ and
$\\{{\color[rgb]{0,0,0}\gamma},\eta_{E>0}\\}=\\{{\color[rgb]{0,0,0}\gamma},\eta^{{\dagger}}_{E>0}\\}=0$.
The quasiparticle obeying this relation is called the Majorana zero mode. The
zero energy states appear in a topological defect of topological SCs, such as
chiral SCs. In Fig. 3(a), we show the spectrum of Andreev bound states bound
at a vortex in a chiral SC, where the level spacing between the zero mode and
the lowest excitation state is $\Delta E\sim\Delta^{2}_{0}/\varepsilon_{\rm
F}$ Kopnin & Salomaa (1991); Volovik (1999); Read & Green (2000); Matsumoto &
Heeb (2001). In many vortices, the hybridization between neighboring Majorana
modes gives rise to the formation of the band structure with the width $\Delta
E_{\rm M}\sim e^{-D/\xi}$ Cheng et al. (2009); Mizushima & Machida (2010),
where $D$ and $\xi$ is the mean distance of neighboring vortices and
superconducting coherence length, respectively (see Fig. 3(b)).

Figure 3: Schematic of the quasiparticle states bound at a single vortex (a)
and energy spectrum of many vortices (b) in a spinless SC, where the level
spacing of vortex bound states is $\Delta E\sim\Delta^{2}_{0}/\varepsilon_{\rm
F}$ and $\Delta E_{\rm M}$ denotes the band width of Majorana states bound at
each core. (c) Operations of the braiding matrices $U_{12}$ and $U_{23}$ and
the $F$-matrix in four Majorana modes.

The Majorana zero modes exhibit the non-Abelian anyonic behaviors Ivanov
(2001). To clarify this, we start with two Majorana zero modes residing in a
SC. Using two Majorana operators, $\gamma^{1}$ and $\gamma^{2}$, we define the
new fermion operators $c$ and $c^{{\dagger}}$ as

$\displaystyle c=\frac{1}{2}(\gamma^{1}+i\gamma^{2}),\hskip
8.53581ptc^{{\dagger}}=\frac{1}{2}(\gamma^{1}-i\gamma^{2}),$ (19)

which obey the anticommutation relations, $\\{c,c^{{\dagger}}\\}=1$ and
$\\{c,c\\}=\\{c^{{\dagger}},c^{{\dagger}}\\}=0$. The two degenerate ground
states are defined as the vacuum $\ket{0}$ and the occupied state of the zero
energy state $\ket{1}=c^{{\dagger}}\ket{0}$, respectively, where the former
(latter) state is the even (odd) fermion parity. We note that as the BdG
Hamiltonian for superconducting states is generally commutable with the parity
operator, the fermion parity remains as a good quantum number. For the even
(odd) parity sector, the Hilbert space is spanned by using $\ket{0}$
($\ket{1}$) and excited states that are constructed as
$\eta^{{\dagger}}_{E}\eta^{{\dagger}}_{E^{\prime}}\eta^{{\dagger}}_{E^{\prime\prime}}\cdots\ket{0}$
($\eta^{{\dagger}}_{E}\eta^{{\dagger}}_{E^{\prime}}\eta^{{\dagger}}_{E^{\prime\prime}}\cdots\ket{1}$).
The Majorana operators, $\gamma^{1}$, $\gamma^{2}$, and
$i\gamma^{1}\gamma^{2}$, act on the Hilbert space as the Pauli matrices
$\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$, respectively. The eigenstates of
the Majorana operators $\gamma^{1}$ and $\gamma^{2}$ are given by the
superposition of the degenerate states with different fermion parity,
$\ket{0}$ and $\ket{1}$. Hence, the eigenstate of a single Majorana zero mode
cannot be a physical state.

Consider $2N$ Majorana zero modes denoted by $\gamma^{j}$ ($j=1,\cdots,2N$),
where $N$ complex fermions are constructed by the fusion of $i$th and $j$th
Majorana zero modes as $c_{ij}=(\gamma^{i}+i\gamma^{j})/2$. We define the
occupation number operator of the complex fermion,

$\displaystyle n_{ij}\equiv
c^{{\dagger}}_{ij}c_{ij}=\frac{1}{2}(1+i\gamma^{i}\gamma^{j}).$ (20)

In a basis that diagonalizes paired Majorana modes $i\gamma^{i}\gamma^{j}$,
two eigenvalues of the complex fermion, $n_{ij}=0$ and $1$, correspond to the
fusion channels ${\bf 1}$ and $\psi$, respectively. Hence, the Majorana zero
mode is referred to as the Ising anyon. $2^{N}$ degenerate ground states are
expressed in terms of the occupation numbers as $\ket{n_{12},n_{34},\cdots}$,
which are separated to the sectors of the even/odd fermion parity. Here we
assume that the temperature of the system is much lower than the level spacing
($\Delta E$) between the Majorana zero mode and the lowest excitation (non-
Majorana) state. Then, the $2^{N-1}$ degenerate ground states in each fermion
parity sector can be utilized as topological qubits, where quantum information
is stored in a topologically protected way.

Braiding Majorana zero modes. Here we discuss the braiding statistics of
Majorana zero modes $\gamma^{i}$ and $\gamma^{j}$ and show the non-Abelian
statistics of Majorana zero modes Ivanov (2001); Alicea et al. (2011); Clarke
et al. (2011). While we consider the exchange of Majorana zero modes residing
in vortices, the theory is also applicable to Majorana zero modes bound at the
end points of one-dimensional topological SCs.

Let $T_{ij}$ be the braid operators that satisfy Eqs. (1) and (2) and
transform $\gamma^{i}$ and $\gamma^{j}$ to $e^{\theta_{i}}\gamma^{j}$ and
$e^{i\theta_{j}}\gamma^{i}$, respectively. The unitary time-evolution of
Majorana zero modes is governed by the Heisenberg equation,
$i\frac{d}{dt}\gamma^{j}(t)=[\gamma^{j}(t),\mathcal{H}(t)]$. The positions of
two Majorana modes are adiabatically exchanged in the time interval $[0,T]$.
The adiabatic condition defines the lower bound for the time scale of the
braiding operation, $T$, so that $T$ is much longer than the inverse of the
level spacing between the Majorana zero mode and the lowest excitation (non-
Majorana) state, $\Delta E$. In addition, the upper bound is associated with
the band width of Majorana modes $\Delta E_{\rm M}\sim e^{-D/\xi}$ (see Fig.
3(b)), i.e., $\Delta E^{-1}\ll T\ll\Delta E^{-1}_{\rm M}$. Within the
adiabatic condition, the braiding dynamics of Majorana zero modes can be
regarded as the unitary time evolution,
$\gamma^{j}(t)=U^{{\dagger}}_{ij}(t)\gamma^{j}(0)U_{ij}(t)$. After the
braiding operation, $\gamma^{i}$ ($\gamma^{j}$) changes to $\gamma^{j}$
($\gamma^{i}$) with an additional phase shift. Then, the braiding operation is
represented by

$\displaystyle
U^{{\dagger}}_{ij}\gamma^{i}U_{ij}=e^{i\theta_{i}}\gamma^{j},\quad
U^{{\dagger}}_{ij}\gamma^{j}U_{ij}=e^{i\theta_{j}}\gamma^{i},$ (21)

where $U_{ij}\equiv U_{ij}(T)$ is the unitary operator which describes the
exchange operation of two Majorana zero modes $\gamma^{i}$ and $\gamma^{j}$,
i.e., the representation of $T_{ij}$. According to the conditions,
$(e^{i\theta_{i}}\gamma^{j})^{2}=(U^{{\dagger}}_{ij}\gamma^{i}U_{ij})^{2}=1$
and $(\gamma^{j})^{2}=1$, the phase shifts obey $\theta_{i}=n\pi$ and
$\theta_{j}=m\pi$, where $n,m\in\mathbb{Z}$. The braiding operations must not
change the parity of the occupation number defined in Eq. (20) and thus
satisfy $U^{{\dagger}}_{ij}n_{ij}U_{ij}=n_{ij}$, which imposes the condition,
$\theta_{1}+\theta_{2}=(2n+1)\pi$, on the phase shift. As a result, the
exchange operation of two Majorana zero modes obtains the following braiding
rules:

$\displaystyle U^{{\dagger}}_{ij}\gamma^{i}U_{ij}=\gamma^{j},\quad
U^{{\dagger}}_{ij}\gamma^{j}U_{ij}=-\gamma^{i}.$ (22)

Consider four Majorana zero modes denoted by $\gamma^{1}$, $\gamma^{2}$,
$\gamma^{3}$, and $\gamma^{4}$, which form two complex fermions as
$c_{12}\equiv(\gamma^{1}+i\gamma^{2})/\sqrt{2}$ and
$c_{34}\equiv(\gamma^{3}+i\gamma^{4})/\sqrt{2}$ (Fig. 3(c)). When the Majorana
mode “3” adiabatically encircles the Majorana mode “2”, both Majorana modes
operators acquire the $\pi$ phase shift, $\gamma^{2}\mapsto-\gamma^{2}$ and
$\gamma^{3}\mapsto-\gamma^{3}$, corresponding to the twice operation of Eq.
(22). Therefore, the braiding operation changes the occupation numbers of the
complex fermion $n_{12}\equiv c^{{\dagger}}_{12}c_{12}$ and $n_{34}\equiv
c^{{\dagger}}_{34}c_{34}$. For example, the above braiding generates a pair of
the complex fermions $\ket{11}$ from their vacuum $\ket{00}$. Here these two
states are orthogonal, $\braket{11}{00}=0$. The braiding rule can be
generalized to $2N$ Majorana modes. The $2N$ Majorana modes are fused to $N$
complex fermions, leading to the $2^{N-1}$-fold degeneracy of ground states
while preserving fermion parity. As discussed above, when the $i$th and $j$th
Majorana modes are exchanged with each other, their operators behave as
$\gamma_{i}\mapsto\gamma_{j}$ and $\gamma_{j}\mapsto-\gamma_{i}$. The
representation of the braid operator $T_{ij}$ that satisfies Eq. (22) is given
in terms of the zero mode operators as Ivanov (2001)

$\displaystyle
U_{ij}=e^{i\theta}\exp\left(\frac{\pi}{4}\gamma_{j}\gamma_{i}\right)=e^{i\theta}\frac{1}{\sqrt{2}}\left(1+\gamma_{j}\gamma_{i}\right).$
(23)

From now on, we omit the overall Abelian phase factor $e^{i\theta}$ as it is
not important for quantum computation. Equation (23) also holds in the case of
the Moore-Read state Nayak & Wilczek (1996). For $N=1$, there is only a single
ground state in each sector with definite fermion parity, and the exchange of
two vortices results in the global phase of the ground state by $e^{i\pi/4}$.
One can easily find that for $N\geq 2$ the exchange operators $U_{ij}$ and
$U_{jk}$ do not commute to each other, $[U_{ij},U_{jk}]\neq 0$, implying the
non-Abelian anyon statistics of the Majorana zero modes.

For four Majorana zero modes ($N=2$), twofold degenerate ground states exist
in each ferimon-parity sector: $\ket{00}\equiv\ket{{\rm vac}}$ and
$\ket{11}=c^{{\dagger}}_{12}c^{{\dagger}}_{34}\ket{\rm vac}$ in the sector of
even fermion parity, and $\ket{10}=c^{{\dagger}}_{12}\ket{\rm vac}$ and
$\ket{01}=c^{{\dagger}}_{34}\ket{\rm vac}$ in the sector of odd fermion
parity. For the even-parity sector, the representation matrix for the exchange
of $1\leftrightarrow 2$ and $3\leftrightarrow 4$ [Fig. 3(c)] is given by

$\displaystyle
U_{12}=U_{34}=e^{-i\frac{\pi}{4}}\ket{00}\bra{00}+e^{i\frac{\pi}{4}}\ket{11}\bra{11}.$
(24)

This merely rotates the phase of the ground state as in the $N=1$ case. In
contrast, the representation matrix for the intervortex exchange
[$2\leftrightarrow 3$ in Fig. 3(c)] has the mixing terms of the two degenerate
ground states $\ket{00}$ and $\ket{11}$,

$\displaystyle U_{23}=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\ket{00}\bra{00}-i\ket{00}\bra{11}\right.$
$\displaystyle\hskip
22.76219pt+\left.\ket{11}\bra{11}-i\ket{11}\bra{00}\right].$ (25)

We note that the choice of the pairing to form the complex fermion is
arbitrary. The change of the fused Majorana modes corresponds to the change of
the basis from one which diagonalizes $i\gamma^{1}\gamma^{2}$ and
$i\gamma^{3}\gamma^{4}$ to another which diagonalizes $i\gamma^{1}\gamma^{3}$
and $i\gamma^{2}\gamma^{4}$. The basis transformation is represented by the
$F$-matrix as

$\displaystyle F=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}.$
(26)

The braiding matrix in Eq. (24) implies that the exchange of two Majorana zero
modes fusing to $\psi$ ($\ket{11}$) acquires an additional $\pi/2$ phase
compared to the fusion channel to ${\bf 1}$ ($\ket{00}$). Hence, Eq. (24)
satisfies the property of the $R$-matrix in Eq. (5), i.e.,
$R^{\sigma\sigma}_{\bf 1}=-iR^{\sigma\sigma}_{\psi}$.

###  3.2 Platforms for Majorana zero modes

The realization of non-Abelian anyons requires to freeze out the internal
degrees of freedom of Majorana modes. The simplest example is spinless
$p$-wave SCs/SFs, which emerge from the low-energy part of spinful chiral
$p$-wave SCs. To clarify this, we start with spin-triplet SCs, whose pair
potential is given by a $2\times 2$ spin matrix Leggett (1975)

$\displaystyle\hat{\Delta}({\bm{k}})=$
$\displaystyle\begin{pmatrix}\Delta_{\uparrow\uparrow}({\bm{k}})&\Delta_{\uparrow\downarrow}({\bm{k}})\\\
\Delta_{\downarrow\uparrow}({\bm{k}})&\Delta_{\downarrow\downarrow}({\bm{k}})\end{pmatrix}$
$\displaystyle=$ $\displaystyle
i\sigma_{\mu}\sigma_{y}d_{\mu}({\bm{k}})=i\sigma_{\mu}\sigma_{y}A_{\mu
i}\hat{k}_{i},$ (27)

where $\hat{k}_{i}\equiv k_{i}/k_{\rm F}$ is scaled with the Fermi momentum
$k_{\rm F}$, and the repeated Greek/Roman indices imply the sum over $x,y,z$.
Here we omit the spin-singlet component. Owing to the Fermi statistics, the
spin-triplet order parameter, ${\bm{d}}({\bm{k}})$, obeys
${\bm{d}}({\bm{k}})=-{\bm{d}}(-{\bm{k}})$. For spin-triplet $p$-wave pairing,
the most general form of the order parameter is given by a $3\times 3$ complex
matrix, $A_{\mu i}\in\mathbb{C}$, where the components are labelled by
$\mu,i\in\\{x,y,z\\}$.

HQVs with Majorana zero modes in SF 3He. We consider the Anderson-Brinkman-
Morel (ABM) state Anderson & Morel (1961); Anderson & Brinkman (1973), as a
prototypical example of chiral $p$-wave states hosting Majorana zero modes.
The ABM state is realized in the A-phase of the SF 3He, which appears in high
pressures and high temperatures Vollhardt & Wölfle (1990); Volovik (2003). At
temperatures above the SF transition temperature, $T>T_{\rm c}\approx
1$–$2~{}{\rm mK}$, the normal Fermi liquid 3He maintains a high degree of
symmetry

$\displaystyle G={SO}(3)_{\bm{L}}\times{SO}(3)_{\bm{S}}\times{U}(1),$ (28)

where ${SO}(3)_{\bm{L}}$, ${SO}(3)_{\bm{S}}$, and ${U}(1)$ are the rotation
symmetry in space, the rotational symmetry of the nuclear spin degrees of
freedom, and the global gauge symmetry, respectively. The tensor $A_{\mu i}$
transforms as a vector with respect to index $\mu$ under spin rotations, and,
separately, as a vector with respect to index $i$ under orbital rotations. The
order parameter of the ABM state is then given by the complex form

$\displaystyle A_{\mu j}=\Delta
e^{i\varphi}\hat{d}_{\mu}(\hat{m}_{j}+i\hat{n}_{j}).$ (29)

The ABM state is the condensation of Cooper pairs with the “ferromagnetic”
orbital, and spontaneously breaks the time-reversal symmetry. The orbital part
of the order parameter is characterized by a set of three unit vectors forming
the triad $(\hat{\bm{m}},\hat{\bm{n}},\hat{\bm{l}})$, where
$\hat{\bm{l}}=\hat{\bm{m}}\times\hat{\bm{n}}$ denotes the orientation of the
orbital angular momentum of Cooper pairs. The remaining symmetry in the ABM
state is $H_{\rm
A}={SO}(2)_{S_{z}}\times{SO}(2)_{L_{z}-\varphi}\times\mathbb{Z}_{2}$, where
${SO}(2)_{S_{z}}$ is the two-dimensional rotation symmetry in the spin space.
The ABM state is also invariant under ${SO}(2)_{L_{z}-\varphi}$ which is the
combined gauge-orbital symmetry, where the ${U}(1)$ phase rotation,
$\varphi\rightarrow\varphi+\delta\varphi$, is compensated by the continuous
rotation of the orbital part about $\hat{\bm{l}}$,
$\hat{m}_{j}+i\hat{n}_{j}\rightarrow
e^{-i\delta\varphi}(\hat{m}^{\prime}_{j}+i\hat{n}^{\prime}_{j})$. In addition,
$\mathbb{Z}_{2}$ is the mod-2 discrete symmetry
$(\hat{\bm{d}},\hat{\bm{m}},\hat{\bm{n}})\rightarrow(-\hat{\bm{d}},-\hat{\bm{m}},-\hat{\bm{n}})$.
The manifold of the order parameter degeneracy is then given by

$\displaystyle R_{\rm A}\simeq G/H_{\rm A}\simeq
S^{2}_{\bm{S}}\times{SO}(3)_{L_{z},\varphi}/\mathbb{Z}_{2}.$ (30)

The two-sphere, $S^{2}_{\bm{S}}$, is associated with the variation of
$\hat{\bm{d}}$. The degeneracy space has an extra $\mathbb{Z}_{2}$ symmetry
that the change from $\hat{\bm{d}}$ to $-\hat{\bm{d}}$ can be compensated by
the phase rotation $\varphi\rightarrow\varphi+\pi$.

The topologically stable linear defects in the ABM state are characterized by
the group of the integers modulo 4 Vollhardt & Wölfle (1990); Volovik (1992);
Salomaa & Volovik (1987); Volovik (2003),

$\displaystyle\pi_{1}(R_{\rm
A})\simeq\pi_{1}({SO}(3)/\mathbb{Z}_{2})\simeq\mathbb{Z}_{4}.$ (31)

There exist four different classes of topologically protected linear defects
in the dipole-free case. The four linear defects can be categorized by the
fractional topological charge,

$\displaystyle N_{\rm A}=0,\;\frac{1}{2},\;1,\;\frac{3}{2},$ (32)

where $N_{\rm A}=3/2$ is topologically identical to $N_{\rm A}=-1/2$. The
representatives of $N_{\rm A}=0$ and $N_{\rm A}=1/2$ classes include
continuous vortex such as the Anderson-Toulouse vortex and half quantum vortex
(HQV), respectively. Owing to $N_{\rm A}=2=0$, a pure phase vortex with
winding number $2$ is continuously deformed into a nonsingular vortex without
a core, that is, the Anderson-Toulouse vortex Anderson & Toulouse (1977). The
$N_{\rm A}=1/2$ vortex is a combination of the half-wound
${\bm{d}}$-disgyration with a half-integer value of the $U(1)$ phase winding
(Fig. 4). The extra $\mathbb{Z}_{2}$ symmetry allows us to take the half-
integer value of the topological charge, because the $\pi$-phase jump arising
from the half-winding of the $U(1)$ phase ($\varphi=\theta/2$) can be canceled
out by the change in the orientation of $\hat{\bm{d}}$
($\hat{\bm{d}}\rightarrow-\hat{\bm{d}}$). The $N_{\rm A}=1$ class includes a
pure phase vortex with odd winding number and the radial/tangential
disgyrations without phase winding. The latter was originally introduced by de
Gennes as ${\bm{l}}$-textures with a singularity line De Gennes (1973);
Ambegaokar et al. (1974).

The vortex with the fractional charge $N_{\rm A}=1/2$ is a harbor for spinless
Majorana zero modes. We introduce the center-of-mass coordinate of Cooper
pairs, ${\bm{R}}=(\rho,\theta,z)$, as
$\hat{\Delta}({\bm{k}})\rightarrow\hat{\Delta}({\bm{k}},{\bm{R}})$, where
$\rho=\sqrt{x^{2}+y^{2}}$. The vortex core is located at $\rho=0$. The vortex
state is subject to the boundary condition at $\rho\rightarrow\infty$ where
the orbital angular momentum of the Cooper pair is aligned to the $z$-axis
($\hat{\bm{l}}\parallel\hat{\bm{z}}$) and the $U(1)$ phase $\varphi$
continuously changes from 0 to $2\pi\kappa$ along the azimuthal ($\theta$)
direction,

$\displaystyle A_{\mu i}(\rho=\infty,\theta)=\Delta
e^{i\kappa\theta}\hat{d}_{\mu}(\theta)(\hat{x}_{j}+i\hat{y}_{j}).$ (33)

Owing to the spontaneous breaking of the gauge-orbital symmetry, there are two
classes for the vorticity $\kappa$: Integer quantum vortices with
$\kappa\in\mathbb{Z}$ and HQVs with $\kappa\in\mathbb{Z}/2$. In the HQVs, both
the $U(1)$ phase and $\hat{\bm{d}}$ rotate by $\pi$ about the vortex center
(see Fig. 4). In genral, the $\hat{\bm{d}}$-texture for HQVs is obtained as

$\displaystyle\hat{\bm{d}}(\theta)=\cos(\kappa^{\rm
sp}\theta)\hat{\bm{x}}+\sin(\kappa^{\rm sp}\theta)\hat{\bm{y}},$ (34)

where $\kappa^{\rm sp}$ denotes the winding of $\hat{\bm{d}}$. The HQV is
characterized by $(\kappa,\kappa^{\rm sp})=(1/2,\pm 1/2)$, while the integer
quantum vortex has $(\kappa,\kappa^{\rm sp})=(1,0)$. It is remarkable to
notice that since the ABM state is the equal spin pairing state, the order
parameter for the HQV is recast into the representation in the spin basis as

$\displaystyle\hat{\Delta}=\Delta\left[e^{i(\kappa-\kappa^{\rm
sp})\theta}\ket{\uparrow\uparrow}+e^{i(\kappa+\kappa^{\rm
sp})\theta}\ket{\downarrow\downarrow}\right].$ (35)

For the HQV with $(\kappa,\kappa^{\rm sp})=(1/2,1/2)$ the
$\ket{\uparrow\uparrow}$ Cooper pair possesses the spatially uniform phase,
while the $\ket{\downarrow\downarrow}$ pair has the phase winding of $2\pi$
around the vortex as in a conventional singly quantized vortex. Thus, the
vortex-free state in the $\uparrow$ spin sector exhibits fully gapped
quasiparticle excitations, while the low-lying structures in half quantum
vortex are effectively describable with a singly quantized vortex in the
$\downarrow$ spin sector, i.e., the spin-polarized chiral $p$-wave SC. An odd-
vorticity vortex in the spin-polarized chiral system hosts a single spinless
Majorana zero mode that obeys non-Abelian statistics. The existence of non-
Abelian anyonic zero modes in half quantum vortices was first revealed by
Ivanov, who developed the non-Abelian braiding statistics of vortices with
spinless Majorana zero modes Ivanov (2001).

Figure 4: Schematic of the HQV realized in the ABM state. The color map shows
the ${\rm U}(1)$ phase winding from $\varphi=0$ (red) at $\theta$ to $\pi$
(blue) at $\theta=2\pi$, and the arrows represent the texture of the
$\hat{\bm{d}}$-vectors shown in Eq. (34) with $\kappa^{\rm sp}=1/2$.

In the bulk A-phase of the SF 3He, an obstacle to realizing the HQVs is the
formation of continuous vortices with the $\hat{\bm{l}}$-texture, which are
characterized by the topological charge $N=0$. As the orientation of the
$\hat{\bm{l}}$-vector is associated with the orbital motion of the Cooper
pair, the $\hat{\bm{l}}$-texture can be uniformly aligned in a parallel plate
geometry with thickness $D$. The motion of Cooper pairs is confined in the
two-dimensional plane and $\hat{\bm{l}}$ is locked perpendicular to the
plates. Applying a magnetic field further restricts the orientation of
$\hat{\bm{d}}$ to the two-dimensional plane perpendicular to the applied
field, which is a favorable situation to stabilize the HQVs. In the parallel
plate geometry with $D=12.5~{}{\mu}{\rm m}$, the measurements of NMR frequency
shift observed that the $\hat{\bm{d}}$-vectors are confined to the two-
dimensional plane perpendicular to $\hat{\bm{l}}$ Yamashita et al. (2008). The
experiment was performed in a rotating cryostat at ISSP, University of Tokyo.
The parallel plates are rotated at the angular velocity $\lesssim 12{\rm
rad}/{\rm s}$ but no conclusive evidence of HQVs was observed Yamashita et al.
(2008). Although the SF 3He-A thin film provides an ideal platform for HQVs
with Majorana zero modes, the realization of HQVs remains as a challenging
task. We also note that point-like solitons in a superfluid 3He film, such as
spin disclinations, were shown to behave as anyons with fractional charges
Volovik & Yakovenko (1989).

Figure 5: (a) The experimental setup and the phase diagram in the liquid 3He
with nematic disorders and (b) NMR spectra at pressure $P=7~{}{\rm bar}$ and
$T=0.60T_{\rm c}$ in the presence of a magnetic field perpendicular to the
anisotropy direction of nematic disorders Mäkinen et al. (2019), where $T_{\rm
c}$ is the critical temperature of the bulk SF 3He without disorders. The
disorders consist of nearly parallel Al2O3 strands, where the diameter and
mean distance are $d_{2}\approx 8~{}{\rm nm}$ and $d_{1}\approx 50~{}{\rm
nm}$, respectively. In (b), the main peaks with green and magenta colors
correspond to the signal of the bulk PdA phase, while the satellite peaks
originate from the spin excitation bound to the $\hat{\bm{d}}$-solitons
connecting pairs of HQVs. The satellite peak remains unchanged after the
thermal cycling illustrated by purple arrows in (a). Both figures are taken
from Ref. Mäkinen et al. (2019).

Another promising route to realize HQVs with Majorana zero modes is to
artificially introduce well-controlled disorders with high-porosity aerogel.
In particular, the polar phase was observed in anisotropic aerogels consisting
of uniaxially ordered alumina strands, called the nematic aerogels Dmitriev et
al. (2015); Halperin (2019) [See Fig. 5(a)]. The order parameter of the polar
phase is given by $A_{\mu i}=\Delta e^{i\varphi}\hat{d}_{\mu}\hat{z}_{i}$,
where the orbital state of the Cooper pair ($\hat{z}_{i}$) is confined by the
uniaxially anisotropic disorders. As in the ABM state in the bulk 3He, the
texture of the $\hat{\bm{d}}$-vector concomitant with the half-integer
vorticity can realize HQVs in the polar phase. In the polar phase, HQVs are
energetically preferable to integer quantum vortices at zero magnetic fields
and magnetic fields applied along the uniaxial anisotropy Nagamura & Ikeda
(2018); Mineev (2014); Regan et al. (2021). Indeed, the HQVs were
experimentally observed in nematic aerogels under rotation Autti et al.
(2016). The HQVs were also created by temperature quench via the Kibble-Zurek
mechanism Rysti et al. (2021). As shown in Fig. 5(a), the superfluid phase
diagram in nematic aerogels is drastically changed from that of the bulk 3He
without disorders, where the polar-distorted A and B (PdA and PdB) phases are
stabilized in addition to the polar phase. The NMR measurements performed in
Ref. Mäkinen et al. (2019) observed that the HQVs survive across the phase
transition to the PdA phase. As shown in Fig. 4, the HQV is accompanied by the
$\hat{\bm{d}}$-soliton in which the $\hat{\bm{d}}$ orientation rotates. Figure
5(b) shows the observed NMR spectra which have satellite peaks in addition to
the main peak around the Larmor frequency. The main peak is a signal of the
bulk PdA phase. The satellite peak originates from the spin excitation
localized at the $\hat{\bm{d}}$-solitons connecting pairs of HQVs. The order
parameter of the PdA phase is given by $A_{\mu i}=\Delta
e^{i\varphi}\hat{d}_{\mu}(\hat{\bm{m}}+i\varepsilon\hat{\bm{n}})_{i}$, where
$\hat{\bm{m}}$ is aligned along the axis of nematic aerogels and
$\varepsilon\in(0,1)$ is the temperature- and pressure-dependent parameter on
the distortion of Eq. (29) by nematic aerogels. Although the HQVs in the polar
phase host no Majorana zero modes, the low energy structure of the HQVs in the
PdA phase is essentially same as that of HQVs in the ABM state and each core
may support the existence of spinless Majorana zero modes Mäkinen et al.
(2019, 2022). Hence, the survival of the HQVs in the PdA phase is important as
a potential platform for topological quantum computation. The elucidation of
the nature of the fermionic excitations bound to the HQVs remains as a future
problem.

HQVs in SCs. In contrast to the SF 3He, it is difficult to stabilize HQVs in
SCs. The difficulty is attributed to the fact that the mass (charge) current
is screened exponentially in the length scale of the penetration depth
$\lambda$, while such a screening effect is absent in the spin current Chung
et al. (2007). The HQV is accompanied by the spin current in addition to the
mass current, while in the integer quantum vortex, the integer vorticity
($\kappa\in\mathbb{Z}$) and uniform $\hat{\bm{d}}$-texture in Eq. (33) induce
no spin current. Therefore, the absence of the screening effect on the spin
current is unfavorable for the stability of the HQVs relative to the integer
quantum vortices, where the later may host spin-degenerate Majorana zero
modes. It has also been proposed that configurational entropy at finite
temperatures drives the fractionalization of the integer quantum vortices into
pairwise HQVs Chung & Kivelson (2010). However, no firm experimental evidence
for HQVs has been reported in SCs.

Superconducting nanowires. Here we briefly mention recent progress on the
search of Majorana zero modes bound at the end points of superconducting
nanowires, although we mainly focus on non-Abelian anyons emergent in vortices
in this article.

The first idea was proposed by Kitaev that a spinless one-dimensional SC is a
platform to host a Majorana zero mode Kitaev (2001). Although it was initially
thought to be a toy model, it was recognized that a system with the same
topological properties can be realized by a semiconductor/SC heterostructure
Alicea (2012); Sato & Ando (2017). The key ingredients are a semiconductor
nanowire and a large Zeeman magnetic field in addition to the proximitized
superconducting gap $\Delta$ Sato et al. (2009, 2010). Consider the
Hamiltonian of the semiconductor nanowire under a magnetic field $B$,
$h(p)=p^{2}/2m-\mu+\lambda p\sigma_{y}-B\sigma_{z}$, which appears in the
diagonal part of Eq. (13), where $m$ is the mass of an electron and $\lambda$
is the strength of the spin-orbit coupling. The spin-orbit coupling is not
only a source of nontrivial topology, but is also necessary to realize
spinless electronic systems together with the large Zeeman field ($B>\mu$). In
fact, if the proximitized superconducting gap is taken into account, a
combination of spin-orbit coupling with the superconducting gap induces
topological odd-parity superconductivity. The topological phase with the
spinless Majorana zero mode appears in the higher magnetic field satisfying
$B>B_{\rm c}=\sqrt{\Delta^{2}+\mu^{2}}$.

A direct signature of Majorana zero mode is the quantization of a zero-bias
peak in differential conductance, where the height of the zero-bias peak
stemming from the Majorana zero mode is predicted to be quantized to the
universal conductance value of $2e^{2}/h$ at zero temperature Akhmerov et al.
(2011); Fulga et al. (2011). Consider a charge-transfer process through a
normal metal-SC junction. When an electron with the incident energy $E$ enters
from the normal metal to the SC, it forms the Cooper pair with an electron in
the Fermi sea at the interface and a hole is created in the normal side as a
consequence of momentum conservation. Such a process in which electrons are
retroreflected as holes is called Andreev reflection. Let $\phi_{E}^{\rm
in,out}=[u_{E},v_{E}]^{\rm t}$ be the two-component wavefunctions of the
incident and reflected particles, respectively, where $u_{E}$ ($v_{E}$)
accounts for the electron (hole) component. We now consider the scattering
problem ${\phi}^{\rm out}_{E}({\bm{x}})=S(E){\phi}_{E}^{\rm in}({\bm{x}})$.
The $S$-matrix is defined in the particle-hole space as

$\displaystyle S(E)=\begin{pmatrix}r^{\rm ee}(E)&r^{\rm eh}(E)\\\ r^{\rm
he}(E)&r^{\rm hh}(E)\end{pmatrix},$ (36)

which is a unitary matrix $S\in{U}(2)$. The coefficient $r^{\rm ee}$ ($r^{\rm
eh}(E)$) is the amplitude of the normal (Andreev) reflection, and the others
are the reflection coefficients of incident holes.

The conductance $G(E)$ at the normal/SC interface is obtained with the
conductance quantum $G_{0}=e^{2}/h$ per spin as $G(E)=2G_{0}|r^{\rm
eh}(E)|^{2}$. The particle-hole symmetry imposes on the $S$ matrix the
constraint, $\mathcal{C}S(E)\mathcal{C}^{-1}=S(-E)$, leading to $\det
S(0)=|r^{\rm ee}(0)|^{2}-|r^{\rm eh}(0)|^{2}$ at $E=0$. Let $V$ be the unitary
matrix defined by $V\phi_{0}=({\rm Re}u,{\rm Im}u)$. In this basis, the
scattering matrix $S(0)$ is transformed as $S^{\prime}(0)=VS(0)V^{{\dagger}}$,
where $S^{\prime}(0)\in{O}(2)$ is an orthogonal matrix satisfying
$S^{\prime{\dagger}}=S^{\prime-1}=S^{\prime{\rm tr}}$ and $\det S(0)=\det
S^{\prime}(0)=\pm 1$. Hence, there are two different processes, a perfect
normal reflection process ($\det S(0)=+1$) and a perfect Andreev reflection
process ($\det S(0)=-1$). This discrete value, $\det S(0)=\pm 1$, is a
topological invariant representing the parity of the number of Majorana zero
modes residing at the interface. When $\det S(0)=-1$, there exists at least
one Majorana zero mode, which involves the perfect Andreev reflection process
and the quantized conductance,

$\displaystyle G(0)=\frac{2e^{2}}{h}.$ (37)

The analytical expression of the conductance is obtained by solving the BdG
eqution and quasiclassical Usadel equation at a interface of the normal metal
and unconventional SCs (e.g., $p$\- and $d$-wave SCs) Tanaka & Kashiwaya
(1995); Kashiwaya & Tanaka (2000); Tanaka et al. (2004, 2005), where the the
conductance per spin reaches $2e^{2}/h$. The quantized conductance is also
related to the Atiyah-Singer index when the Hamiltonian holds the chiral
symmetry Ikegaya et al. (2016).

The conductance has recently been measured in a junction system of InSb-Al
hybrid semiconductor-SC nanowire devices. The differential conductance yields
the plateau behavior where the peak height reaches values $2e^{3}/h$ Zhang et
al. (2021). In the device used in the experiment, the tunnel barrier is
controlled by applying a gate voltage to the narrow region between the
electrode and the region where the proximity effect occurs in the
semiconductor wire. In the vicinity of the barrier, the unintentionally formed
quantum dot or nonuniform potential formed in the vicinity of the interface
gives rise to the formation of nearly zero-energy Andreev bound states. Such
Andreev bound states mimic the plateau of $G(0)=2e^{2}/h$ even in the
topologically trivial region of the magnetic field ($B<B_{\rm c}$) Liu et al.
(2017); Prada et al. (2020); Cayao & Burset (2021); Yu et al. (2021);
Valentini et al. (2021). From the plateau of $G(0)$, it is not possible to
identify whether the observed $G(0)\sim 2e^{2}/h$ is due to Majorana zero mode
or the effect of accidentally formed s bound states. Alternative experimental
schemes to distinguish the trivial and topological bound states have been
proposed Liu et al. (2018); Ricco et al. (2019); Yavilberg et al. (2019);
Awoga et al. (2019); Zhang & Spånslätt (2020); Schulenborg & Flensberg (2020);
Pan et al. (2021); Liu et al. (2021); Ricco et al. (2021); Thamm & Rosenow
(2021); Chen et al. (2022); Sugeta et al. (2022).

Vortices in Fe(Se,Te). The iron-based SC Fe(Se,Te) is a candidate of
topologically nontrivial superconducting state supporting Majorana zero modes.
In the parent material FeSe, the $3d$ electrons of Fe near the Fermi level
mainly contribute to superconductivity, and the $p_{z}$ orbitals of Se appear
on the higher energy. Substitution of Se with Te causes a band inversion
between the $p_{z}$ and $3d$ orbitals. As a result of the band inversion, the
normal state of FeTe1-xSex is topologically nontrivial, and accompanied by the
surface Dirac fermions Zhang et al. (2018). Below the superconducting critical
temperature, the superconducting gap is proximitized to the surface Dirac
states. It was theoretically pointed out that the topological phase with a
spinless Majorana zero mode can be realized when the superconducting proximity
effect occurs in the Dirac fermion system Sato (2003); Fu & Kane (2008).
Therefore, the surface state of Fe(Se,Te) is a topological superconucting
state. When a magnetic field is applied to Fe(Se,Te), the vortex lines
penetrate to the surface state and the zero energy state appears. The wave
function of the zero mode is tightly bound at the intersection of the vortex
line and the surface Hosur et al. (2011); Kawakami & Hu (2015).

When a magnetic field is applied to a SC, a quantized vortex penetrates. The
magnetic flux is accompanied by the $2\pi m$ winding of the $U(1)$ phase and
the superconducting gap vanishes. In other words, a quantized vortex can be
regarded as a quantum well with a radius of $\xi$ and a height of $\Delta$.
Hence, the Andreev bound states are formed and the level spacing between them
is on the order of $\Delta^{2}/\varepsilon_{\rm F}$, where the level spacing
is about $100$–$200~{}\mu{\rm eV}$ in Fe(Se,Te). In the topological phase, the
lowest level has the exact zero energy, and the quasiparticle behaves as
Majorana fermion. Ultra-low temperature STM/STS with high energy resolution
observed a pronounced zero-bias conductance peak Machida et al. (2019). If a
Majorana zero mode is bound to the vortex, the conductance should be quantized
to the universal value $2e^{2}/h$ independent of tunnel barriers. In the
tunneling spectroscopy performed while changing the distance between the
sample surface and the STM tip, the height of the zero-bias conductance peak
is approximately $0.6$ times as large as the universal value stemming from the
Majorana zero mode Zhu et al. (2020). Although $2e^{2}/h$ has not been
reached, this plateau structure strongly suggests the existence of Majorana
zero modes because it cannot be explained by non-Majorana vortex bound states.

###  3.3 Topological quantum computation

Quantum computation based on the braiding of Majorana zero modes is basically
along the two directions: implementation of quantum gates and measurement-
based quantum computation. The unitary evolution of the qubits is controlled
by a set of discrete unitary operations, i.e., quantum gates. The simple
example of the quantum gates acting on a single qubit is the set of the Pauli
gates, which are given by the three Pauli matrices $X\equiv\sigma_{x}$,
$Y\equiv\sigma_{y}$, and $Z\equiv\sigma_{z}$. The multi-qubit gate operation
can be implemented by a combination of a set of single-qubit gates and the
controlled-NOT (CNOT) gate. According to the Solovay-Kitaev theorem Nielsen &
Chuang (2010), an arbitrary single-qubit gate can be approximated by a
sequence of the discrete gate operations $(H,S,T)$, where $H=(X+Z)/\sqrt{2}$,
$S=\sqrt{Z}$, and $T=\sqrt{S}$ are the Hadamard gate, the single-qubit $4\pi$
rotation, and the $T$-gate ($\pi/8$-gate), respectively. Universal quantum
computation can be implemented by a sequence of the single-qubit gates
$(H,S,T)$ and the CNOT gate.

In Majorana qubits, the Pauli gates are expressed in terms of the braiding
operations as $X=iU^{2}_{23}$, $Y=iU^{2}_{31}$, and $Z=iU^{2}_{12}$, and the
Hadamard and $S$ gates are implemented by a combination of such operations as
$H=iU_{12}U_{23}U_{12}$ and $S=e^{i\pi/4}U_{12}$. In addition, the CNOT gate
can be implemented by a combination of the measurement and braiding operations
in two Majorana qubits (i.e., 8 Majorana zero modes) with a single ancilla
Majorana qubit Bravyi (2006). All the Clifford gates ($H$, $S$, CNOT) are
realized as a composition of braiding operators in a topologically protected
way. According to the Gottesman-Knill theorem, however, quantum circuits only
using the elements of Clifford group can be efficiently simulated in
polynomial time on a classical computer Nielsen & Chuang (2010). Indeed,
degenerate quantum states composed of multiple Ising anyons offer tolerant
storage of quantum information and their braiding operations provide all
necessary quantum gates in a topologically protected way, except for the non-
Clifford $T$-gate. The $T$-gate can be implemented only in a topologically
unprotected way. For instance, Karzig et al. proposed a scheme to implement
the $T$-gate in a Y-shaped junction accompanied by 4 Majorana zero modes
Karzig et al. (2016).

As mentioned in Sec. 2, Fibonacci anyon systems can offer a platform for
universal topological quantum computation, where all the single-qubit gates
$(H,S,T)$ and the CNOT gate are implemented by the braiding manipulations of
the anyons in a topologically protected way Preskill (2004); Bonesteel et al.
(2005); Hormozi et al. (2007).

Another way to realize universal quantum computation in Majorana qubits is
measurement-based quantum computation. It was realized that all braiding
manipulations can be implemented in a topologically protected manner by the
measurements of topological charges of pairwise anyons Bonderson et al. (2008,
2009). The process of the quantum information encoded to Majorana qubits can
take place by a series of simple projective measurements without braiding
Majorana modes. The building block is a Coulomb blockaded Majorana box which
is made from even numbers of Majorana modes in a floating topological SC
Plugge et al. (2017); Karzig et al. (2017); Oreg & von Oppen (2020).

###  3.4 Symmetry-protected non-Abelian anyons

Symmetry-protected non-Abelian anyons in spinful SCs and SFs were discussed
for class D Ueno et al. (2013); Sato et al. (2014); Fang et al. (2014) and for
class DIII Liu et al. (2014); Gao et al. (2016). In the former (latter) case,
mirror reflection symmetry (time reversal symmetry) plays an essential role on
the topological protection of non-Abelian nature. In addition, unitary
symmetry-protected non-Abelian statistics has been theoretically proposed Hong
et al. (2022). The non-Abelian statistics of vortices supporting multiple
Majorana zero modes has also been discussed in Refs. Yasui et al. (2011) and
Hirono et al. (2012) in the context of high-energy physics.

Mirror and unitary symmetries. As the simple example of symmetry-protected
non-Abelian anyons, we consider integer quantum vortices in chiral $p$-wave
SCs and 3He-A Sato et al. (2014), where each vortex core supports spinful
Majorana zero modes. Contrary to the standard wisdom, a pair of Majorana zero
modes in integer quantum vortices can be protected by the mirror symmetry and
obey the non-Abelian statistics Ueno et al. (2013); Sato et al. (2014).

Let us consider two-dimensional superconducting film and assume that the
normal state holds the mirror symmetry with respect to the $xy$-plane,
$M_{xy}h({\bm{k}})M^{{\dagger}}_{xy}=h({\bm{k}})$, where
${\bm{k}}=(k_{x},k_{y})$ is the two-dimensional momentum and the mirror
operator is defined as $M_{xy}=i\sigma_{z}$. The nontrivial topological
properties are characterized by the first Chern number in each mirror
subsector. When $\Delta({\bm{k}})$ obeys $M_{xy}\Delta({\bm{k}})M^{\rm
t}_{xy}=\eta\Delta({\bm{k}})$ ($\eta=\pm$), the BdG Hamiltonian is commutable
with the mirror reflection operator in the particle-hole space,
${\mathcal{M}}^{\eta}$, as

$\displaystyle\left[{\mathcal{M}}^{\eta},\mathcal{H}({\bm{k}})\right]=0,\quad{\mathcal{M}}^{\eta}=\begin{pmatrix}M_{xy}&0\\\
0&\eta M^{\ast}_{xy}\end{pmatrix}.$ (38)

Thus, the eigenstates of $\mathcal{H}({\bm{k}})$ are the simultaneous
eigenstates of ${\mathcal{M}}^{\eta}$. Let
$\ket{u^{({\lambda})}_{n}({\bm{k}})}$ and $E^{(\lambda)}_{n}$ be the
wavefunction and eigenenergy of the Bogoliubov quasiparticles in each mirror
subsector, where the $\lambda=\pm i$ are the eigenvalues of
$\mathcal{M}^{\eta}$. Then, the first Chern number is defined in each mirror
subsector,

$\displaystyle{\rm
Ch}^{(\lambda)}_{1}=\frac{i}{2\pi}\int\mathcal{F}^{(\lambda)}\in\mathbb{Z},$
(39)

where $\mathcal{F}^{(\lambda)}=d\mathcal{A}^{(\lambda)}$ is the Berry
curvature in each mirror subsector and
$\mathcal{A}^{(\lambda)}_{\mu}({\bm{k}})=\sum_{E^{(\lambda)}_{n}<0}\braket{u^{({\lambda})}_{n}({\bm{k}})}{\partial_{k_{\mu}}u^{({\lambda})}_{n}({\bm{k}})}$.
The nonzero value ensures the existence of the zero energy state in the
$\lambda$ subsector. For the 3He-A thin film and chiral $p$-wave SCs, $|{\rm
Ch}_{1}^{(\lambda)}|=1$ in each mirror subsector, implying that integer
quantum vortices support two Majorana zero modes per each core.

Multiple Majorana zero modes behave as non-Abelian anyons only when the mirror
subsector holds the particle-hole symmetry, which are referred to as mirror
Majorana zero modes. The condition for each mirror subsector to maintain the
particle-hole symmetry is given by Ueno et al. (2013); Sato et al. (2014)

$\displaystyle\left\\{\mathcal{C},\mathcal{M}^{\eta}\right\\}=0.$ (40)

For integer quantum vortices, the condition depends on the orientation of the
${\bm{d}}$-vector. For $\hat{\bm{d}}\parallel\hat{\bm{z}}$, the mirror
subsector supports its own particle-hole symmetry, but otherwise the subsector
does not maintain the particle-hole symmetry and belongs to class A.

Figure 6: Complex fermions formed by intervortex pairs (a) and intravoretx
pairs (b), where $\gamma^{i}_{a}$ denotes a Majorana zero mode with internal
degrees of freedom $a$ (e.g., spin) bound at the $i$th vortex. (c) Schematic
of the exchange of two vortices with multiple Majorana zero modes. (d) When
the system holds a unitary symmetry such as the mirror symmetry in Eq. (38),
the operation (c) is equivalent to the exchange of two Majorana zero modes in
each sector.

Now let us clarify the non-Abelian statistics of spin-degenerate Majorana
modes. Consider $2N$ integer quantum vortices. Two Majorana zero modes bound
at the $i$th vortex are denoted by $\gamma^{i}_{\lambda}$ with mirror
eigenvalues $\lambda=\pm i$. The Majorana zero modes satisfy the self-
conjugate condition $\gamma_{\lambda}^{i}=(\gamma_{\lambda}^{i})^{{\dagger}}$
and the anticommutation relation,
$\\{\gamma_{\lambda}^{i},\gamma_{\lambda^{\prime}}^{j}\\}=2\delta_{i,j}\delta_{\lambda,\lambda^{\prime}}$.
In contrast to spinless Majorana zero modes, there are two possibilities to
form a complex fermion: (i) intervortex pairs,
$c^{\lambda}_{ij}\equiv(\gamma_{\lambda}^{i}+i\gamma_{\lambda}^{j})/2$ and
(ii) intravortex pairs,
$\psi_{i}\equiv(\gamma_{\lambda=i}^{i}+i\gamma_{\lambda=-i}^{i})/2$ [See Figs.
6(a,b)]. In the former case (i), the exchange operators of the $i$th and $j$th
vortices are defined in a manner similar to that for spinless Majorana zero
modes as

$\displaystyle
U^{\lambda{\dagger}}_{ij}\gamma_{\lambda}^{i}U^{\lambda}_{ij}=\gamma_{\lambda}^{j},\quad
U^{\lambda{\dagger}}_{ij}\gamma_{\lambda}^{j}U^{\lambda}_{ij}=-\gamma_{\lambda}^{i}.$
(41)

The above transformation is realized by the unitary operator,

$\displaystyle
U^{\lambda}_{ij}=\exp\left(\frac{\pi}{4}\gamma_{\lambda}^{j}\gamma_{\lambda}^{i}\right)=\frac{1}{\sqrt{2}}\left(1+\gamma_{\lambda}^{j}\gamma_{\lambda}^{i}\right),$
(42)

Similarly to $U_{ij}$ and $U_{jk}$ in spinless Majorana zero modes, the
exchange operators $U^{\lambda}_{ij}$ and $U^{\lambda}_{jk}$ defined in each
mirror subsector do not commute to each other, which implies the non-Abelian
anyon statistics of integer quantum vortices hosting mirror Majorana zero
modes.

In the case (ii) [Fig. 6(b)], a complex fermion is formed as a local pair of
two mirror Majorana modes, which also obeys the non-Abelian statistics Yasui
et al. (2012); Sato et al. (2014). The expression of the exchange operator is
obtained as
$U_{ij}=1+\psi_{j}\psi_{i}^{{\dagger}}+\psi_{j}^{{\dagger}}\psi_{i}-\psi^{{\dagger}}_{i}\psi_{i}-\psi^{{\dagger}}_{j}\psi_{j}+2\psi^{{\dagger}}_{j}\psi_{j}\psi^{{\dagger}}_{i}\psi_{i}$.
In general, the exchange operators are not commutable with each other,
$[U_{ij},U_{jk}]\neq 0$. As the operator preserves the fermion number $N_{\rm
f}=\sum_{i}\psi_{i}^{\dagger}\psi_{i}$, the degenerate ground states are
expressed in terms of the occupation number of the complex fermion in each
vortex. For example, let us consider the four-vortex state $\ket{1100}$ where
the first and second vortices are accompanied by the Dirac zero modes, while
the third and fourth are not. Up to a phase factor, this state changes under
$U_{12}$ and $U_{34}$ as $\ket{1100}\rightarrow\ket{1100}$ and
$\ket{1100}\rightarrow\ket{1010}$, respectively. The complete representations
of the exchange operators $U_{ij}$ are presented in Ref. Yasui et al. (2012).
The extension to non-Abelian statistics of multiple complex fermions was
discussed in Ref. Yasui et al. (2013).

When the BdG Hamiltonian maintains the mirror symmetry in Eq. (38), as
mentioned above, the braiding of vortices supporting spinful Majorana modes is
decomposed into two individual exchanges between Majorana zero modes
$\gamma^{i}_{\lambda}$ and $\gamma^{j}_{\lambda}$ in each mirror subsector
[see also Figs. 6(c) and 6(d)] and the exchange matrices are represented by
Eqs. (41) and (42). This argument is applicable for the systems hosting
multiple Majorana modes protected by unitary symmetries Hong et al. (2022),
where the couplings between Majorana zero modes in each core are excluded by
the unitary symmetries. The braiding of two vortices with $n$ unitary-
symmetry-protected multiple Majorana modes generically reduces to the $n$
independent braiding operations in each sector as

$\displaystyle U_{ij}=\prod_{a}U_{ij}^{a},$ (43)

where $a$ denotes the internal degrees of freedom of Majorana zero modes. The
unitary time evolution representing the braiding process does not dynamically
break the unitary symmetry, which prohibits the coupling between Majorana
modes in different subsectors labeled by $a$. The mirror reflection symmetry
in Eq. (38) is an example of such symmetry-protected non-Abelian anyons.
Another example is the non-Abelian vortices in dense QCD matter which trap
multiple Majorana zero modes in each core Yasui et al. (2010); Fujiwara et al.
(2011). As discussed below, such vortices with multiple Majorana zero modes
obey the non-Abelian statistics and the braiding matrices are identical with
the elements in the Coxeter group Yasui et al. (2011); Hirono et al. (2012).

Contrary to unitary symmetries, the braiding process, characterized as a
unitary evolution, may dynamically break antiunitary symmetry such as the
time-reversal symmetry. We note that in contrast to time-reversal -broken
topological SCs such as superconducting nanowires, topological superconductors
with time-reversal symmetry do not require an external magnetic field for
realizing the topological phase and can keep the topological gap close to the
proximity-induced superconducting gap Haim & Oreg (2019); Wong & Law (2012);
Zhang et al. (2013). Hence, the Majorana-Kramers pairs leads to longer
decoherence time of the stored quantum information Liu et al. (2014). As
mentioned above, however, the braiding process of the Majorana-Kramers pairs
may dynamically break the antiunitary symmetry. Such dynamical symmetry
breaking gives rise to the local mixing of the Majorana-Kramers pairs Wölms et
al. (2014, 2016); Knapp et al. (2020). Gao et al. proposed the conditions to
protect the non-Abelian statistics of the Majorana-Kramers pairs Gao et al.
(2016). In addition, Tanaka et al. examined by numerical simulations the
tolerance of non-Abelian braidings of Majorana-Kramers pairs against two types
of perturbations which may cause decoherence of Majorana-Kramers quits Tanaka
et al. (2022): (i) Applied magnetic fields, and (ii) the effect of a gate-
induced inhomogeneous potential at junctions of superconducting nanowires. The
former break time-reversal symmetry and generate the energy gap of Majorana
states, while the latter gives rise to non-Majorana low-energy Andreev bound
states Kells et al. (2012); Liu et al. (2017); Moore, Zeng, Stanescu & Tewari
(2018); Moore, Stanescu & Tewari (2018); Pan et al. (2020); Pan & Das Sarma
(2020). The numerical simulation revealed that the non-Abelian braiding is
successful when the direction of the applied magnetic field preserves the
chiral symmetry in the initial and final states of a braiding process, where
the chiral symmetry is a combination of time-reversal symmetry and mirror
symmetry Tanaka et al. (2022). Remarkably, this tolerance is preserved even
when the intermediate states of the braiding process breaks this symmetry. As
for (ii), non-Majorana bound states emergent in gate-induced inhomogeneous
potentials make Majorana-Kramers qubits vulnerable to quasiparticle poisoning
and disturbe the braiding protocol. However, the influence can be ignored when
the width of the gate-induced potential is sufficiently smaller than the
superconducting coherence length.

Multiple Majorana zero modes and Coxeter group. It has been demonstrated that
the non-Abelian statistics of vortices hosting three Majorana zero modes has a
novel structure, when the Majorana modes have additional ${SO}(3)$ symmetry
Yasui et al. (2011); Hirono et al. (2012). The representation of braiding
operations in four vortices is given by a tensor product of two matrices,
where one is identical to the matrix for the braiding of spinless Majorana
zero modes, and the other is a generator of the Coxeter group.

Let $\gamma_{a}^{i}$ ($a=1,2,3$) be the operators of three Majorana zero modes
in the $i$th vortex, belonging to the triplet of ${SO}(3)$. Consider four
vortices hosting three Majorana zero modes in each core. The braiding of these
vortices is decomposed into the three individual exchanges between Majorana
zero modes $\gamma^{i}_{a}$ and $\gamma^{j}_{a}$ [see Figs. 6(c,d)]. For each
component $a$, Majorana zero modes $\gamma_{a}^{i}$ and $\gamma_{a}^{j}$ under
braiding manipulations transforms in the same way as Eq. (41) and the exchange
matrices, $U^{a}_{ij}$, are obtained by $U_{ij}^{\lambda}\rightarrow
U^{a}_{ij}$ in Eqs. (41) and (42). Then, the representations of $U_{ij}$ for
vortices with triplet Majorana zero modes are obtained by the $64\times 64$
matrix, $U_{ij}=\prod_{a=1,2,3}U^{a}_{ij}$, as in Eq. (43).

We now introduce the complex fermion operators as the nonlocal pairs of
Majorana zero modes, $c_{ij}^{a}=(\gamma_{a}^{i}+i\gamma^{j}_{a})/2$. For two
vortices (i.e., six Majorana modes), this leads to the $2^{3}$-fold degenerate
ground states and the Hilbert space is spanned by the four basis sets:
singlet-even $\ket{{\bm{1}}_{0}}\equiv\ket{0}$ (vacuum) and triplet-even
$\ket{{\bm{3}}_{2}}\equiv\frac{1}{2!}\epsilon^{abc}c^{b{\dagger}}_{ij}c^{c{\dagger}}_{ij}\ket{0}$
(occupied by two fermions) for even fermion parity and singlet-odd
$\ket{{\bm{1}}_{3}}\equiv\frac{1}{3!}\epsilon^{abc}c^{a{\dagger}}_{ij}c^{b{\dagger}}_{ij}c^{c{\dagger}}_{ij}\ket{0}$
(occupied by three fermions) and triplet-odd $\ket{{\bm{3}}_{1}}\equiv
c^{a{\dagger}}_{ij}\ket{0}$ (occupied by single fermion). In the case of four
vortices, the ground state degeneracy is $2^{6}=64$ and the bases of the
Hilbert space are singlet (${\bm{1}}$), triplet (${\bm{3}}$), and quintet
(${\bm{5}}$) states, which are further divided in terms of the fermion parity.
The exchange operator is expressed as a product of two $SO(3)$ invariant
unitary operators,

$\displaystyle U_{ij}=\prod_{a=1,2,3}U^{a}_{ij}=\sigma_{ij}h_{ij}$ (44)

where both matrices are given in terms of $\gamma^{a}_{i}$ as

$\displaystyle\sigma_{ij}=\frac{1}{2}\left(1-\gamma^{1}_{j}\gamma^{2}_{j}\gamma^{1}_{i}\gamma^{2}_{i}-\gamma^{2}_{j}\gamma^{3}_{j}\gamma^{2}_{i}\gamma^{3}_{i}-\gamma^{3}_{j}\gamma^{1}_{j}\gamma^{3}_{i}\gamma^{1}_{i}\right),$
(45)

and

$\displaystyle
h_{ij}=\frac{1}{\sqrt{2}}\left(1-\gamma^{1}_{j}\gamma^{2}_{j}\gamma^{3}_{j}\gamma^{1}_{i}\gamma^{2}_{i}\gamma^{3}_{i}\right).$
(46)

While the matrices $h_{ij}$ are a natural extension of that originally
introduced in Ref. Ivanov (2001), the matrices $\sigma_{ij}$ are proper to
vortices with three or more Majorana zero modes. It was demonstrated that the
matrices $\sigma_{ij}$ are identified with the elements in the Coxeter group
which satisfies the relations

$\displaystyle(\sigma_{ij})^{2}=1,\;(\sigma_{ij}\sigma_{jk})^{3}=1,\;(\sigma_{ij}\sigma_{kl})^{2}=1.$
(47)

The Coxeter group is a symmetry group of polytopes in high dimensions such as
a triangle and a tetrahedron. The exchange operators acting on the singlet
(${\bf 1}$) and triplet (${\bf 3}$) are identified with the elements in the
Coxeter group, such as a 2-simplex (triangle) under the reflections for the
singlet and a 3-simplex (tetrahedron) under the reflections for the triplet
Yasui et al. (2011). The decomposition of the braiding operators $U_{ij}$ in
the ${SO}(3)$ triplet was generalized to the Majorana zero modes of arbitrary
odd $n_{\rm M}\geq 3$ with ${SO}(n_{\rm M})$ symmetry Hirono et al. (2012).

Such multiple Majorana zero modes were first found in Refs. Yasui et al.
(2010); Fujiwara et al. (2011) inside the cores of color flux tubes (“non-
Abelian” vortices) Balachandran et al. (2006); Nakano et al. (2008); Eto &
Nitta (2009); Eto et al. (2014) in high density quark (QCD) matter Alford et
al. (2008). They were also discussed in edges of a topological superconducting
wire coupled to a normal lead Kashuba & Timm (2015).

##  4 Non-Abelian vortex anyons

Here we introduce non-Abelian anyons made of bosonic excitations, which are
non-Abelian vortices obeying a non-Abelian statistics Bais (1980); Wilczek &
Wu (1990); Bucher (1991); Brekke et al. (1993); Lo & Preskill (1993); Lee
(1994); Brekke et al. (1997); Mawson et al. (2019). These non-Abelian vortices
are sometime called fluxons.222Here, the term “flux” originally comes from the
fact that they were proposed in gauge theory where non-Abelian symmetry is
gauged in the context of high energy physics Bais (1980). In this article, we
consider only global non-Abelian symmetry relevant for condensed matter
physics and they do not carry non-Abelian fluxes.

###  4.1 Non-Abelian vortices

When a symmetry $G$, that is either global or local, is spontaneously broken
into its subgroup $H$, there appears an order parameter space

$\displaystyle M\simeq G/H$ (48)

parameterized by Nambu-Goldstone modes. The first homotopy group of the order
parameter space

$\displaystyle\pi_{1}(G/H)\simeq\pi_{0}(H)$ (49)

is responsible for the existence of quantum vortices, where we have assumed
that $G$ is semisimple. We further have

$\displaystyle\pi_{0}(H)\simeq H/H_{0}(\simeq H\mbox{ for a discrete group
}H)$ (50)

with a normal subgroup $H_{0}$ of $H$. When $\pi_{1}(G/H)$ is Abelian (and
thus $H/H_{0}$ (for semisimple $G$) is Abelian), vortices are Abelian, while
when it is non-Abelian, vortices are non-Abelian Balachandran et al. (1984);
Alford et al. (1990, 1991, 1992).

One of characteristic features of non-Abelian vortices is that $H$ is not
globally defined (or multi-valued): the unbroken symmetry $H_{\theta}$ depends
on the azimuthal angle $\theta$ around a non-Abelian vortex as
$H_{\theta}=g(\theta)H_{\theta=0}g^{-1}(\theta)$ with $g(\theta)\in G$,
$g(0)={\bf 1}$ and $g(2\pi)$ taking a disconnected component of $H/H_{0}$, see
Eq. (50). Then, after an adiabatic transport along a loop encircling the non-
Abelian vortex, the unbroken symmetry at $\theta=2\pi$ does not have to come
back: $H_{\theta=2\pi}\neq H_{\theta=0}$. This is called topological
obstruction, and the symmetry $H$ is topologically broken down to a subgroup
$K\equiv\\{h|[h,g(2\pi)]=0,h\in H\\}$ which is single-valued and globally
defined Balachandran et al. (1984); Alford et al. (1990). This is also called
topological symmetry breaking. When $G$ and $H$ are local gauge symmetries, it
gives a non-Abelian Aharonov-Bohm effect Bais (1980); Wilczek & Wu (1990);
Bucher (1991); Alford et al. (1990, 1991, 1992). Another related feature is
the existence of non-local charges, called Cheshire charges, existing among
separated vortices Preskill & Krauss (1990); Alford et al. (1990); Bucher et
al. (1992).

Figure 7: Exchange of two vortices. (a) Two closed paths $\alpha$ and $\beta$
encircle two vortices. (b) Exchange them counterclockwisely.

Let $\alpha,\beta\in\pi_{1}(G/H)$ being closed passes enclosing two vortices.
Then, if one adiabatically transports the vortex $\alpha$ around the vortex
$\beta$ counterclockwisely, the vortex $\alpha$ transforms as (see Fig. 7(b))

$\displaystyle\alpha\to\beta\alpha\beta^{-1}.$ (51)

Therefore, when one exchanges a pair $(\alpha,\beta)$ counterclockwisely, they
transform as

$\displaystyle(\alpha,\beta)\to(\beta,\beta\alpha\beta^{-1}).$ (52)

The first homotopy group $\pi_{1}$ is a based fundamental group for which
every loop starts from a fixed point $O$. Instead of it, the conjugacy classes

$\displaystyle[\alpha]=\beta\alpha\beta^{-1},\;\forall\beta\in\pi_{1}(G/H)$
(53)

characterize individual vortices as can be seen from Eqs. (51) and (52).  Two
vortices corresponding to two different elements in the same conjugacy class
are indistinguishable even though they are not the same.

Non-Abelian first homotopy groups significantly control the dynamics of
vortices. In two spatial dimensions, scattering of such non-Abelian vortices
was discussed in the case that $G$ and $H$ are local gauge symmetries Wilczek
& Wu (1990); Bucher (1991); Lo & Preskill (1993). In three spatial dimensions,
vortices are lines. When two vortex lines collide in three spatial dimensions,
the fate is determined by corresponding elements of the first homotopy group.
When these elements commute, the two vortices pass through or reconnect each
other if these elements are identical. On the other hand, when these elements
are non-commutative, the third vortex bridging them must be created Poenaru &
Toulouse (1977); Mermin (1979); Kobayashi et al. (2009).

###  4.2 Examples of non-Abelian vortices

Nematic liquid crystals. Let us start from nematic liquid crystals focusing on
uniaxial nematics (UNs), and $D_{2}$\- and $D_{4}$-biaxial nematics (BNs).

UN liquid crystals. The order parameter of UNs is an unoriented rod. The
rotational symmetry $G=SO(3)$ is spontaneously broken down to $H\simeq O(2)$,
which implies the order parameter manifold

$\displaystyle\frac{G}{H}=\frac{SO(3)}{O(2)}\simeq\frac{SU(2)}{Pin(2)}\simeq
S^{2}/{\mathbb{Z}}_{2}\simeq{\mathbb{R}}P^{2},$ (54)

which is a real projective space of two dimensions, where $SU(2)$ and $Pin(2)$
are double covering groups of $SO(3)$ and $O(2)$, respectively.  Since the
fundamental group $\pi_{1}({\mathbb{R}}P^{2})={\mathbb{Z}}_{2}$ is Abelian,
this does not admits non-Abelian vortices. These vortices are called Alice
strings (when $G$ and $H$ are local gauge symmetries) Schwarz (1982).  Around
the Alice string, $SO(2)\subset H$ is not globally defined and $H$ is
topologically broken to $K\simeq{\mathbb{Z}}_{2}$.

$D_{2}$-BN liquid crystals Poenaru & Toulouse (1977); Volovik & Mineev (1977);
Mermin (1979); Balachandran et al. (1984).

$g(2\pi)\in\pi_{1}(G/H)$ | ${\bf 1}_{2}$ | $-{\bf 1}_{2}$ | $\pm i\sigma_{x}$ | $\pm i\sigma_{y}$ | $\pm i\sigma_{z}$
---|---|---|---|---|---
$g(\theta)$ | ${\bf 1}_{2}$ |  $\exp\left({\color[rgb]{0,0,0}\pm}i{\color[rgb]{0,0,0}\frac{\theta}{2}}{\bf n}\cdot{\mathbb{\sigma}}\right)$ | $\exp\left(\pm i{\color[rgb]{0,0,0}\frac{\theta}{4}}\sigma_{x}\right)$ | $\exp\left(\pm i{\color[rgb]{0,0,0}\frac{\theta}{4}}\sigma_{y}\right)$ | $\exp\left(\pm i{\color[rgb]{0,0,0}\frac{\theta}{4}}\sigma_{z}\right)$
$O(2\pi)$ | ${\bf 1}_{3}$ | ${\bf 1}_{3}$ | $I_{x}$ | $I_{y}$ | $I_{z}$
$O(\theta)$ | ${\bf 1}_{3}$ |  $\exp(i\theta{\bf n}\cdot{\bf L})$ | $\exp\left(\pm i\frac{\theta}{2}L_{x}\right)$ | $\exp\left(\pm i\frac{\theta}{2}L_{y}\right)$ | $\exp\left(\pm i\frac{\theta}{2}L_{z}\right)$
Table 1: Asymptotic configurations of spin vortices in $D_{2}$-BNs:  the
fundamental group elements $g(2\pi)\in\pi_{1}(G/H)$, the $SU(2)$ elements
$g(\theta)\in SU(2)$, $O(2\pi)$, and the $SO(3)$ group elements $O(\theta)\in
SO(3)$, acting on the the order parameter $A$ as Eq. (60). The group actions
$O(\theta)$ and $g(\theta)$ for a spin vortex of $2\pi$ rotation ($-{\bf
1}_{2}$) can be a rotation around any axis along the unit vector
${\mathbb{n}}$. $L_{i}$ are $3\times 3$ spin-1 matrices
$[L_{i}]_{jk}=-i\epsilon_{ijk}$.

Cholesteric liquid crystals (chiral nematic liquid crystals) can also have the
same order parameter space with $D_{2}$-BN liquid crystals Lavrentovich &
Kleman (2001) and all the following discussions hold.  The order parameter of
BNs is a set of two unoriented rods of different lengths orthogonal to each
other. The (unbroken) symmetry is the dihedral group $D_{2}$ keeping a
rectangular invariant. Thus, the rotational symmetry $G=SO(3)$ is
spontaneously broken down to

$\displaystyle H\simeq D_{2}$ $\displaystyle=\left\\{{\bf
1}_{3},I_{x}=\begin{pmatrix}{\color[rgb]{0,0,0}+}1&0&0\\\
0&{\color[rgb]{0,0,0}-}1&0\\\ 0&0&{\color[rgb]{0,0,0}-}1\end{pmatrix},\right.$
$\displaystyle\left.\quad I_{y}=\begin{pmatrix}{\color[rgb]{0,0,0}-}1&0&0\\\
0&{\color[rgb]{0,0,0}+}1&0\\\
0&0&{\color[rgb]{0,0,0}-}1\end{pmatrix},I_{z}=\begin{pmatrix}{\color[rgb]{0,0,0}-}1&0&0\\\
0&{\color[rgb]{0,0,0}-}1&0\\\
0&0&{\color[rgb]{0,0,0}+}1\end{pmatrix}\right\\}.$ (55)

Note that $D_{2}\simeq{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ is Abelian.  The
order parameter manifold is

$\displaystyle G/H=SO(3)/D_{2}\simeq SU(2)/{\mathbb{Q}},$ (56)

where the quaternion group ${\mathbb{Q}}$ are universal (double) covering
group of $D_{2}$:

$\displaystyle{\mathbb{Q}}=\left\\{\pm{\bf 1}_{2},\pm i\sigma_{x},\pm
i\sigma_{y},\pm i\sigma_{z}\right\\}.$ (57)

Note that $H^{*}\simeq{\mathbb{Q}}$ is a non-Abelian group while $H\simeq
D_{2}$ is an Abelian group. This breaking can occur for instance by five real-
scalar order-parameters belonging to spin-2 representation of $SO(3)$, which
is a traceless symmetric $3\times 3$ tensor $A$ with real components, taking a
form of

$\displaystyle A={\rm diag}(1,r,-1-r)$ (58)

The first homotopy group is isomorphic to the quaternion group ${\mathbb{Q}}$
in Eq. (57)

$\displaystyle\pi_{1}(SU(2)/{\mathbb{Q}})\simeq{\mathbb{Q}}.$ (59)

The elements in Eq. (57) correspond to the ground state (${\bf
1}{\color[rgb]{0,0,0}{}_{2}}$), spin vortices of $2\pi$ rotation ($-{\bf
1}{\color[rgb]{0,0,0}{}_{2}}$), and spin vortices of
${\color[rgb]{0,0,0}\pm}\pi$ rotation about the $x,y,z$-axes ($\pm
i\sigma_{x,y,z}$).  These vortex configurations at the large distance can be
asymptotically written as

$\displaystyle A\sim O(\theta)A_{\theta=0}O^{T}(\theta),\quad O(\theta)\in
SO(3)$ (60)

with the azimuthal angle $\theta$. The concrete forms of $O(\theta)$’s are
given in Table 1. Corresponding $SU(2)$ elements $g(\theta)$ can be defined as
a double covering of $SO(3)$ with $g(2\pi)\in\pi_{1}(G/H)$. A spin vortex of
$2\pi$ rotation ($-{\bf 1}{\color[rgb]{0,0,0}{}_{2}}$) and a spin vortex of
$\pi$ rotation ($\pm i\sigma_{a}$) $(a=x,y,z)$ are given by

$\displaystyle g(\theta)=\exp\left(\pm i\frac{\theta}{2}\sigma\cdot{\bf
n}\right)=\cos\frac{\theta}{2}{\bf 1}_{2}\pm i\sigma\cdot{\bf
n}\sin\frac{\theta}{2},$ $\displaystyle g(\theta)=\exp\left(\pm
i\frac{\theta}{4}\sigma_{a}\right)=\cos\frac{\theta}{4}{\bf 1}_{2}\pm
i\sigma_{a}\sin\frac{\theta}{4},\quad$ (61)

respectively, with a unit three-vector ${\bf n}$, giving the elements of
$\pi_{1}(G/H)$ at $\theta=2\pi$: $g(2\pi)=-{\bf 1}_{2}$ and $g(2\pi)=\pm
i\sigma_{a}$, respectively. While the former is Abelian because
$[g(2\pi),H^{*}]=0$ (even though $g(2\pi)\neq g(0)$), the latter is non-
Abelian because $[g(2\pi),H^{*}]\neq 0$ with the universal covering
$H^{*}\simeq SU(2)$,333 Note that $[O(2\pi),H]=0$, and thus the criterion of
non-Abelian property should be considered in the universal covering group
$H^{*}$ but not in $H$.  and $H^{*}$ is topologically broken to
$K^{*}_{a}=\\{\pm{\bf 1}_{2},\pm i\sigma_{a}\\}$.

The elements in Eq. (57) are grouped to the conjugacy classes, Eq. (53),
consisting of five elements:

$\\{{\bf 1}_{2}\\},\\{-{\bf 1}_{2}\\},\\{\pm i\sigma_{x}\\},\\{\pm
i\sigma_{y}\\},\\{\pm i\sigma_{z}\\}.$ (62)

This follows for instance from
$(i\sigma_{y})^{-1}(i\sigma_{x})(i\sigma_{y})=-i\sigma_{x}$, impling that a
vortex corresponding to $i\sigma_{x}$ becomes $-i\sigma_{x}$ when it travels
around a vortex corresponding to $i\sigma_{y}$. Thus, the vortices
corresponding to $i\sigma_{x}$ and $-i\sigma_{x}$ are indistinguishable
although they are not the same. From $(i\sigma_{a})(-i\sigma_{a})={\bf 1}_{2}$
($a=x,y,z$), vortices belonging to the same conjugacy class are anti-particles
of each other, and thus they are similar to Majorana fermions discussed in the
last section.

In three spatial dimensions, this phenomenon of non-commutativity appears as
follows: when two vortex lines corresponding to $i\sigma_{x}$ and
$i\sigma_{y}$ collide in three spatial dimensions, the third vortex bridging
them is created Poenaru & Toulouse (1977); Mermin (1979).

$D_{4}$-BN liquid crystals. The order parameter of $D_{4}$ BNs is a set of two
unoriented indistinguishable rods of the same lengths orthogonal to each
other. In this case, the (unbroken) symmetry is a dihedral group $D_{4}$
keeping a square invariant:

$\displaystyle G/H=SO(3)/D_{4}\simeq SU(2)/D_{4}^{*},$ (63)

where $M^{*}$ denotes a universal covering group of $M$. The fundamental group
is given by

$\displaystyle\pi_{1}(SU(2)/D_{4}^{*})\simeq D_{4}^{*}$ (64)

with the sixteen elements

$\displaystyle\left\\{\pm\bm{1}_{2},\pm i\sigma_{x},\pm i\sigma_{y},\pm
i\sigma_{z},\pm C_{4},\pm{\color[rgb]{0,0,0}C_{4}^{-1}},\pm
i\sigma_{x}C_{4},\pm i\sigma_{x}{\color[rgb]{0,0,0}C_{4}^{-1}}\right\\}$ (65)

with $C_{4}\equiv e^{i{\pi\over 4}\sigma_{z}}=(1/\sqrt{2})({\bf
1}_{2}+i\sigma_{z})$. Note $(C_{4})^{2}=i\sigma_{z}$ and $(C_{4})^{4}=-{\bf
1}_{2}$. The conjugacy classes consist of the following seven elements

$\displaystyle\\{\bm{1}_{2}\\},\\{-\bm{1}_{2}\\},\\{\pm i\sigma_{x},\pm
i\sigma_{y}\\},\\{\pm i\sigma_{z}\\},$
$\displaystyle\\{C_{4},{\color[rgb]{0,0,0}C_{4}^{-1}}\\},\\{-C_{4},{\color[rgb]{0,0,0}-C_{4}^{-1}}\\},\\{\pm
i\sigma_{x}C_{4},\pm i\sigma_{x}{\color[rgb]{0,0,0}C_{4}^{-1}}\\}.\quad\quad.$
(66)

Since $-C_{4}=C_{4}^{3}$ ($-C_{4}^{-1}=C_{4}^{-3}$), vortices of these
elements have more energy than those of $C_{4}$ and $C_{4}^{-1}$.  This
symmetry breaking of $D_{4}$ nematics can be realized by a higher rank tensor
order parameter Mietke & Dunkel (2022), in which more generally $D_{n}$
nematic liquid crystals were also discussed.

Spinor BECs. Next we provide examples of spinor BECs Kawaguchi & Ueda (2012).
The order parameter of spin-2 BECs is a traceless symmetric $3\times 3$ tensor
$A$ with complex components transforming under the symmetry $G=U(1)\times
SO(3)$ as

$\displaystyle A\to e^{i\theta}gAg^{T},\quad e^{i\theta}\in U(1),\quad g\in
SO(3).$ (67)

Phases of condensations with total angular momentum two are classified by
Mermin Mermin (1974). They are nematic, cyclic and ferromagnetic phases. All
phases are theoretically possible in the case of spin-2 BECs. The spin-2 BECs
are experimentally realized by 87Rb atoms for which the phase is around the
boundary between cyclic phase and ferromagnetic phase, see e.g. Ref. Tojo et
al. (2009). Here, we discuss nematic and cyclic phases which host non-Abelian
vortex anyons. In the next section, we discuss ${}^{3}P_{2}$ SF which is in
the nematic phase.

BN phases in spin-2 BEC. Song et al. (2007); Uchino et al. (2010); Kobayashi
et al. (2012); Borgh & Ruostekoski (2016). The nematic phase consists of three
degenerate phases: the UN, $D_{2}$\- and $D_{4}$-BN phases. The order
parameters of the UN, $D_{2}$\- and $D_{4}$-BN phases are given by

$\displaystyle{\rm UN}:$ $\displaystyle A\sim{\rm diag}(1,-1/2,-1/2),$
$\displaystyle D_{2}\mbox{-BN}:$ $\displaystyle A\sim{\rm diag}(1,r,-1-r),$
$\displaystyle D_{4}\mbox{-BN}:$ $\displaystyle A\sim{\rm diag}(1,-1,0),$ (68)

respectively, where $r\in{\mathbb{R}}$ and $-1<r<-1/2$. In the $D_{2}$-BN
phase, the limits $r\to-1/2,-1$ correspond to UN and $D_{4}$-BN phases,
respectively. The symmetry breaking patterns and the order parameter manifolds
are

$\displaystyle{\rm UN}:$ $\displaystyle{G\over H}=U(1)\times{SO(3)\over
O(2)}\simeq{U(1)\times{\mathbb{R}}P^{2}},$ $\displaystyle D_{2}\mbox{-BN}:$
$\displaystyle{G\over H}=U(1)\times{SO(3)\over D_{2}}\simeq
U(1)\times{SU(2)\over{\mathbb{Q}}},$ $\displaystyle D_{4}\mbox{-BN}:$
$\displaystyle{G\over H}={U(1)\times SO(3)\over D_{4}}\simeq{U(1)\times
SU(2)\over D_{4}^{*}}.$

These phases are continuously connected by the parameter $r$ interpreted as a
quasi-Nambu-Goldstone mode. The degeneracy is lifted by quantum effects and
either of these phases remains as the ground state Uchino et al. (2010).

The $D_{2}$-BN and $D_{4}$-BN admit non-Abelian vortices. The order parameters
of the UN and $D_{2}$-BN phases of spin-2 BECs are merely products of the
$U(1)$ phonon and the order parameters of the corresponding nematic liquids.
Thus, we concentrate on the $D_{4}$-BN phase hereafter. The fundamental group
of the $D_{4}$-BN phase is given by

$\displaystyle\pi_{1}\left(U(1)\times SU(2)\over
D_{4}^{\ast}\right)\cong\mathbb{Z}\times_{h}D_{4}^{\ast}$ (70)

where $\times_{h}$ is defined in Ref. Kobayashi et al. (2012). This consists
of the following sixteen elements

$\displaystyle\Big{\\{}(N,\pm\bm{1}_{2}),(N,\pm i\sigma_{x}),(N,\pm
i\sigma_{y}),(N,\pm i\sigma_{z}),$ $\displaystyle\quad\left(N+{1\over 2},\pm
C_{4}\right),\left(N+{1\over 2},\pm i\sigma_{x}C_{4}\right),$
$\displaystyle\quad\left(N+{1\over
2},\pm{\color[rgb]{0,0,0}C_{4}^{-1}}\right),\left(N+{1\over 2},\pm
i\sigma_{x}{\color[rgb]{0,0,0}C_{4}^{-1}}\right)\Big{\\}},$ (71)

where the first and second elements of a pair $(\cdot,\cdot)$ denote the
circulation $\kappa$ ($U(1)$ winding number) and an $SU(2)$ element,
respectively with $N\in{\mathbb{Z}}$. Here $C_{4}\equiv e^{i{\pi\over
4}\sigma_{z}}=(1/\sqrt{2})({\bf 1}_{2}+i\sigma_{z})$ satisfying
$C_{4}^{4}=-{\bf 1}_{2}$. The conjugacy classes of Eq. (71) are composed of
seven elements for each $N$:

$\displaystyle{\rm(I)}$ $\displaystyle\\{(N,\bm{1}_{2})\\},$
$\displaystyle{\rm(II)}$ $\displaystyle\\{(N,-\bm{1}_{2})\\},$
$\displaystyle{\rm(III)}$ $\displaystyle\\{(N,\pm i\sigma_{x}),(N,\pm
i\sigma_{y})\\},$ $\displaystyle{\rm(IV)}$ $\displaystyle\\{(N,\pm
i\sigma_{z})\\},$ $\displaystyle{\rm(V)}$
$\displaystyle\left\\{\left(N+{1\over 2},C_{4}\right),\left(N+{1\over
2},{\color[rgb]{0,0,0}C_{4}^{-1}}\right)\right\\},$ $\displaystyle{\rm(VI)}$
$\displaystyle\left\\{\left(N+{1\over 2},-C_{4}\right),\left(N+{1\over
2},{\color[rgb]{0,0,0}-C_{4}^{-1}}\right)\right\\},$ $\displaystyle{\rm(VII)}$
$\displaystyle\left\\{\left(N+{1\over 2},\pm
i\sigma_{x}C_{4}\right),\left(N+{1\over 2},\pm
i\sigma_{x}{\color[rgb]{0,0,0}C_{4}^{-1}}\right)\right\\}.$

They describe (I) integer vortices ($N=0$ corresponds to the vacuum), (II)
spin vortices of $2\pi$ rotation, (III),(IV) spin vortices of $\pi$ rotation
around the $x$, $y$, and $z$ axes, and (V) – (VII) non-Abelian HQVs.

In Sec. 5, we see that ${}^{3}P_{2}$ SFs are in the $D_{4}$-BN phase in a
strong magnetic field. There, a singly quantized vortex $(1,{\bf 1})$ splits
into two HQVs $(1/2,C_{4})$ and $(1/2,C_{4}^{-1})$ both in the conjugacy class
(V)444 Topologically a decay into a pair $(1/2,-C_{4})$ and
$(1/2,-C_{4}^{-1})$ in the conjugacy class (VI) is also possible, but
energetically disfavored.  . When they fuse, they go back to the singly
quantized vortex $(1,{\bf 1})$. However, one of them, e.g., $(1/2,C_{4}^{-1})$
can transform to the other $(1/2,C_{4})$ once some other vortex passes through
between them because they belong to the same conjugacy class. If they fuse
after that, they become $(1,i\sigma_{z})$ because $C_{4}^{2}=i\sigma_{z}$
which is a composite belonging to (IV), of a singly quantized vortex and a
spin vortex of $\pi$ rotation.

Cyclic phase in spin-2 BEC. Semenoff & Zhou (2007); Kobayashi et al. (2009,
2012). The order parameter of the cyclic phase is

$\displaystyle A\sim{\rm diag}(1,e^{2\pi i/3},e^{4\pi i/3}),$ (73)

yielding the symmetry breaking to the tetrahedral group $H\simeq T$ and the
order parameter manifold, given by

$\displaystyle{G\over H}={U(1)\times SO(3)\over T}\simeq{U(1)\times SU(2)\over
T^{*}}$ (74)

with the universal covering group $T^{*}$ of $T$. The fundamental group in the
cyclic phase is given by

$\displaystyle\pi_{1}\left({U(1)\times SU(2)\over
T^{\ast}}\right)\cong\mathbb{Z}\times_{h}T^{\ast}$ (75)

with the 24 elements

$\displaystyle\Big{\\{}$ $\displaystyle(N,\pm\bm{1}_{2}),(N,\pm
i\sigma_{x}),(N,\pm i\sigma_{y}),(N,\pm i\sigma_{z}),$
$\displaystyle\left(N+\frac{1}{3},\pm C_{3}\right),\left(N+\frac{1}{3},\pm
i\sigma_{x}C_{3}\right),$ $\displaystyle\left(N+\frac{1}{3},\pm
i\sigma_{y}C_{3}\right),\left(N+\frac{1}{3},\pm i\sigma_{z}C_{3}\right),$
$\displaystyle\left(N-\frac{1}{3},\pm C_{3}^{2}\right),\left(N-\frac{1}{3},\pm
i\sigma_{x}C_{3}^{2}\right),$ $\displaystyle\left(N-\frac{1}{3},\pm
i\sigma_{y}C_{3}^{2}\right),\left(N-\frac{1}{3},\pm
i\sigma_{z}C_{3}^{2}\right)\Big{\\}}$ (76)

with the same notation with Eq. (71). Here $C_{3}\equiv(1/2)({\bf
1}_{2}+i\sigma_{x}+i\sigma_{y}+i\sigma_{z})$ satisfying $(C_{3})^{3}=-{\bf
1}_{2}$. The conjugacy classes of Eq. (76) are composed of the following seven
elements for each $N$ Semenoff & Zhou (2007):

$\displaystyle{\rm(I)}$ $\displaystyle\\{(N,\bm{1}_{2})\\},$
$\displaystyle{\rm(II)}$ $\displaystyle\\{(N,-\bm{1}_{2})\\},$
$\displaystyle{\rm(III)}$ $\displaystyle\\{(N,\pm i\sigma_{x}),(N,\pm
i\sigma_{y}),(N,\pm i\sigma_{z})\\},$ $\displaystyle{\rm(IV)}$
$\displaystyle\Big{\\{}\left(N+\frac{i}{3},C_{3}\right),\left(N+\frac{1}{3},-i\sigma_{x}C_{3}\right),$
$\displaystyle\quad\left(N+\frac{1}{3},-i\sigma_{y}C_{3}\right),\left(N+\frac{1}{3},-i\sigma_{z}C_{3}\right)\Big{\\}},$
$\displaystyle{\rm(V)}$
$\displaystyle\Big{\\{}\left(N+\frac{i}{3},-C_{3}\right),\left(N+\frac{1}{3},i\sigma_{x}C_{3}\right),$
$\displaystyle\quad\left(N+\frac{i}{3},i\sigma_{y}C_{3}\right),\left(N+\frac{1}{3},i\sigma_{z}C_{3}\right)\Big{\\}},$
$\displaystyle{\rm(VI)}$
$\displaystyle\Big{\\{}\left(N-\frac{1}{3},C_{3}^{2}\right),\left(N-\frac{1}{3},i\sigma_{x}C_{3}^{2}\right),$
$\displaystyle\quad\left(N-\frac{1}{3},i\sigma_{y}C_{3}^{2}\right),\left(N-\frac{1}{3},i\sigma_{z}C_{3}^{2}\right)\Big{\\}},$
$\displaystyle{\rm(VII)}$
$\displaystyle\Big{\\{}\left(N-\frac{1}{3},-C_{3}^{2}\right),\left(N-\frac{1}{3},-i\sigma_{x}C_{3}^{2}\right),$
$\displaystyle\quad\left(N-\frac{1}{3},-i\sigma_{y}C_{3}^{2}\right),\left(N-\frac{1}{3},-i\sigma_{z}C_{3}^{2}\right)\Big{\\}}.$

They describe (I) integer vortices ($N=0$ corresponds to the vacuum), (II)
spin vortices of $2\pi$ rotation, (III) spin vortices of $\pi$ rotation around
the $x,y,z$ axes, and (IV) – (VII) 1/3-quantum vortices.

Other examples. Another example in condensed matter physics can be found in
vortex lines in the dipole-free A-phase of SF 3He Balachandran et al. (1984);
Salomaa & Volovik (1987). An example in high energy physics can be found in
color flux tubes (“non-Abelian” vortices) in high density QCD matter Fujimoto
& Nitta (2021a, b, c); Eto & Nitta (2021).

###  4.3 Fusion rules for non-Abelian vortex anyons

Two non-Abelian vortices belonging to the same conjugacy class are
indistinguishable even if they are not identical. For instance, a vortex
$a\in\pi_{1}(G/H)$ and a vortex $b=cac^{-1}$ with $c\in\pi_{1}(G/H)$ are not
the same and thus $a$ and anti-vortex of $b$ cannot pair annihilate.
Nevertheless they are indistinguishable with a help of $c$. This leads a class
of non-Abelian statistics Brekke et al. (1993); Lo & Preskill (1993); Lee
(1994); Brekke et al. (1997). Such non-Abelian anyons are called non-Abelian
vortex anyons.

In order to discuss non-Abelian vortex anyons, first we define anyons $\\{{\bf
1},\sigma,\tau\\}$ appropriately by conjugacy classes of non-Abelian vortices.
Then, the fusion rule (3) for non-Abelian vortex anyons can be written as
Mawson et al. (2019)

$\displaystyle\tau\otimes\tau=N^{\bf 1}_{\tau\tau}{\bf 1}\oplus
N^{\sigma}_{\tau\tau}\sigma\oplus N^{\tau}_{\tau\tau}\tau,$
$\displaystyle\tau\otimes\sigma=\tau,\quad\sigma\otimes\sigma={\bf 1},\quad
x\otimes{\bf 1}=x,$ (78)

with $x\in\\{{\bf 1},\sigma,\tau\\}$.

| I0 | II0 | III0 | ${\rm IV}_{0}$ | ${\rm V}_{0}$ | ${\rm VI}_{-1}$ | ${\rm VII}_{-1}$
---|---|---|---|---|---|---|---
${\bf 1}={\rm I}_{0}$ | I0 | II0 | III0 | ${\rm IV}_{0}$ | ${\rm V}_{0}$ | ${\rm VI}_{-1}$ | ${\rm VII}_{-1}$
$\sigma={\rm II}_{0}$ | II0 | I0 | III0 | ${\rm V}_{0}$ | ${\rm IV}_{0}$ | ${\rm VII}_{-1}$ | ${\rm VI}_{-1}$
$\tau={\rm III}_{0}$ | III0 | III0 | $6{\rm I}_{0}\oplus 6{\rm II}_{0}\oplus 4{\rm III}_{0}$ | $3{\rm IV}_{0}\oplus 3{\rm V}_{0}$ | $3{\rm IV}_{0}\oplus 3{\rm V}_{0}$ | $3{\rm VI}_{-1}\oplus 3{\rm VII}_{-1}$ | $3{\rm VI}_{-1}\oplus 3{\rm VII}_{-1}$
${\rm IV}_{0}$ | ${\rm IV}_{0}$ | ${\rm V}_{0}$ | $3{\rm IV}_{0}\oplus 3{\rm V}_{0}$ | $3{\rm VI}_{0}\oplus{\rm VII}_{0}$ | ${\rm VI}_{0}\oplus 3{\rm VII}_{0}$ | $4{\rm I}_{0}\oplus 3{\rm III}_{0}$ | $4{\rm II}_{0}\oplus 2{\rm III}_{0}$
${\rm V}_{0}$ | ${\rm V}_{0}$ | ${\rm IV}_{0}$ | $3{\rm IV}_{0}\oplus 3{\rm V}_{0}$ | ${\rm VI}_{0}\oplus 3{\rm VII}_{0}$ | $3{\rm VI}_{0}\oplus{\rm VII}_{0}$ | $4{\rm II}_{0}\oplus 2{\rm III}_{0}$ | $4{\rm I}_{0}\oplus 2{\rm III}_{0}$
${\rm VI}_{-1}$ | ${\rm VI}_{-1}$ | ${\rm VII}_{-1}$ | $3{\rm VI}_{-1}\oplus 3{\rm VII}_{-1}$ | $4{\rm I}_{0}\oplus 2{\rm III}_{0}$ | $4{\rm II}_{0}\oplus 2{\rm III}_{0}$ | $3{\rm IV}_{-1}\oplus{\rm V}_{-1}$ | ${\rm IV}_{-1}\oplus 3{\rm V}_{-1}$
${\rm VII}_{-1}$ | ${\rm VII}_{-1}$ | ${\rm VI}_{-1}$ | $3{\rm IV}_{-1}\oplus 3{\rm VII}_{-1}$ | $4{\rm II}_{0}\oplus 2{\rm III}_{0}$ | $4{\rm I}_{0}\oplus 2{\rm III}_{0}$ | ${\rm IV}_{-1}\oplus 3{\rm V}_{-1}$ | $3{\rm IV}_{-1}\oplus 2{\rm V}_{-1}$

Table 2: Fusion rule for vortex anyons in a cyclic spin-2 BEC Mawson et al.
(2019). The subscript on the conjugacy class denotes the $U(1)$ winding number
$N$.

| I0 | II0 | III0 | IV0 | V0 | VI0 | VII0
---|---|---|---|---|---|---|---
${\bf 1}={\rm I}_{0}$ | I0 | II0 | III0 | IV0 | V0 | VI0 | VII0
$\sigma={\rm II}_{0}$ | II0 | I0 | III0 | IV0 | VI0 | V0 | VII0
$\tau_{1}={\rm III}_{0}$ | III0 | III0 | $4{\rm I}_{0}\oplus 4{\rm II}_{0}\oplus 4{\rm IV}_{0}$ | 2III0 | 2VII0 | 2VII0 | $4{\rm V}_{0}\oplus 4{\rm VI}_{0}$
$\tau_{2}={\rm IV}_{0}$ | IV0 | IV0 | 2III0 | $2{\rm I}_{0}\oplus 2{\rm II}_{0}$ | ${\rm V}_{0}\oplus{\rm VI}_{0}$ | ${\rm V}_{0}\oplus{\rm VI}_{0}$ | 2VII0
${\rm V}_{0}$ | ${\rm V}_{0}$ | ${\rm VI}_{0}$ | $2{\rm VII}_{0}$ | ${\rm V}_{0}\oplus{\rm VI}_{0}$ | $2{\rm I}_{1}\oplus{\rm IV}_{1}$ | $2{\rm II}_{1}\oplus{\rm IV}_{1}$ | $2{\rm III}_{1}$
${\rm VI}_{0}$ | ${\rm VI}_{0}$ | ${\rm V}_{0}$ | $2{\rm VII}_{0}$ | ${\rm V}_{0}\oplus{\rm VI}_{0}$ | $2{\rm II}_{1}\oplus{\rm IV}_{1}$ | $2{\rm I}_{1}\oplus{\rm IV}_{1}$ | $2{\rm III}_{1}$
${\rm VII}_{0}$ | ${\rm VII}_{0}$ | ${\rm VII}_{0}$ | $4{\rm IV}_{0}\oplus 4{\rm V}_{0}$ | $2{\rm VII}_{0}$ | $2{\rm III}_{1}$ | $2{\rm III}_{1}$ | $4{\rm I}_{1}\oplus 4{\rm II}_{1}\oplus 4{\rm IV}_{1}$

Table 3: Fusion rule for for vortex anyons in a $D_{4}$-BN spin-2 BEC Mawson
et al. (2019). The subscript on the conjugacy class denotes the $U(1)$ winding
number $N$.

Non-Abelian vortex anyons in $D_{2}$-BN. Let us discuss the simplest case. For
$D_{2}$-BN liquid crystals or $D_{2}$-BN phases of spin-2 BECs, the conjugacy
class is given in Eq. (62). We then define

$\displaystyle{\bf 1}=\\{+{\bf 1}_{2}\\},\quad\sigma=\\{-{\bf
1}_{2}\\},\quad\tau=\\{\pm i\sigma_{a}\\}$ (79)

with $a$ being either $x$, $y$, or $z$. The coefficient $N^{c}_{ab}$ counts
how many ways anyons $a$ and $b$ fuse to an anyon $c$. The relations
$(-i\sigma_{a})(+i\sigma_{a})=(+i\sigma_{a})(-i\sigma_{a})=+{\bf 1}_{2}$ and
$(-i\sigma_{a})(-i\sigma_{a})=(+i\sigma_{a})(+i\sigma_{a})=-{\bf 1}_{2}$ lead
$N^{\bf 1}_{\tau\tau}=2$ and $N^{\sigma}_{\tau\tau}=2$, respectively.
Furthermore, $\tau$ anyons do not fuse to $\tau$ in this case and then
$N^{\tau}_{\tau\tau}=0$. We thus reach

$\displaystyle\tau\otimes\tau=2{\bf 1}_{2}\oplus 2\sigma,$
$\displaystyle\tau\otimes\sigma=\tau,\quad\sigma\otimes\sigma={\bf 1},\quad
x\otimes{\bf 1}=x,$ (80)

with $x\in\\{{\bf 1},\sigma,\tau\\}$. Since two $\tau$-anyons fuse to two
different anyons ${\bf 1}$ and $\sigma$, $\tau$ anyons are non-Abelian anyons
Comparing this fusion rule with that of the Ising anyons in Eq. (4), we find
that these non-Abelian vortex anyons are similar to the Ising anyons. In fact,
a $\tau$ aynon coincide with its anti-particle, and so similar to a Majorana
fermion.

Gathering all $\tau_{a}=\\{\pm{\color[rgb]{0,0,0}i}\sigma_{a}\\}$ ($a=x,y,z$)
together, we obtain the following fusion formula from the relation
$(i\sigma_{a})(i\sigma_{b})=-\delta_{ab}+\epsilon_{abc}(-i\sigma_{c})$:

$\displaystyle\tau_{a}\otimes\tau_{b}=2\delta_{ab}{\bf 1}\oplus
2\delta_{ab}\sigma\oplus 2\epsilon_{abc}\tau_{c}$
$\displaystyle\tau_{a}\otimes\sigma=\tau_{a},\quad\sigma\otimes\sigma={\bf
1},\quad x\otimes{\bf 1}=x,$ (81)

with $x\in\\{{\bf
1},\sigma,{\color[rgb]{0,0,0}\tau_{x},\tau_{y},\tau_{z}}\\}$. In this case,
there are the three non-Abelian anyons $\tau_{a}$ coupled to each other.

Non-Abelian vortex anyons in spin-2 BECs.

Non-Abelian vortex anyons in spin-2 BECs were studied in Mawson et al. (2019).
For the cyclic phase in a spin-2 BEC, $\\{{\bf 1},\sigma,\tau\\}$ are defined
by the conjugacy classes (I), (II) and (III) with $N=0$ in Eq. (LABEL:eq:conj-
cyclic). Then, we have (see Table 3) $N^{\bf
1}_{\tau\tau}=6,\;N^{\sigma}_{\tau\tau}=6,\;N^{\tau}_{\tau\tau}=4$:

$\displaystyle\tau\otimes\tau=6{\bf 1}\oplus 6\sigma\oplus 4\tau,$
$\displaystyle\tau\otimes\sigma=\tau,\quad\sigma\otimes\sigma={\bf 1},\quad
x\otimes{\bf 1}=x,$ (82)

with $x\in\\{{\bf 1},\sigma,\tau\\}$.  The non-Abelian 1/3-quantum vortices in
(IV)–(VII) are also non-Abelian anyons but one cannot restrict to the $N=0$
sector, and needs infinite numbers of anyons for a closed algebra.

The fusion rule for vortex anyons in the $D_{4}$-BN phase of a spin-2 BEC was
also obtained by the conjugacy classes (I)–(IV) in Eq. (LABEL:eq:conj-nematic)
Mawson et al. (2019). In this case, $\tau$ in the cyclic phase is split into
$\tau_{1}={\rm III}_{0}$ and $\tau_{2}={\rm IV}_{0}$. Then, we have $N^{\bf
1}_{\tau_{1}\tau_{1}}=4,\;N^{\sigma}_{\tau_{1}\tau_{1}}=4,\;N^{\tau_{1}}_{\tau_{1}\tau_{1}}=4,N^{\bf
1}_{\tau_{2}\tau_{2}}=2,\;N^{\sigma}_{\tau_{2}\tau_{2}}=2$ with the other
components zero:

$\displaystyle\tau_{1}\otimes\tau_{1}=4{\bf 1}\oplus 4\sigma\oplus 4\tau_{1},$
$\displaystyle\tau_{2}\otimes\tau_{2}=2{\bf 1}\oplus 2\sigma,$
$\displaystyle\tau_{i}\otimes\sigma=\tau_{i},\quad\sigma\otimes\sigma={\bf
1},\quad x\otimes{\bf 1}=x,$ (83)

with $i=1,2$ and $x\in\\{{\bf 1},\sigma,\tau_{i}\\}$. See Table 3. The non-
Abelian HQVs in (V)–(VII) are also non-Abelian anyons, but again infinite
numbers of anyons are necessary for a closed algebra. The same fusion rule in
Eq. (83) should hold for a $D_{4}$-BN liquid crystal, as seen in Eq. (66).

Including Bogoliubov modes, the algebra of vortex anyons is extended to a
quantum double Koornwinder et al. (1999); Mawson et al. (2019). An application
to quantum computation was also proposed in Ref. Génetay Johansen & Simula
(2022).

##  5 ${}^{3}P_{2}$ topological SFs

Finally we discuss the two-fold non-Abelian object in a novel topological SF
called a ${}^{3}P_{2}$ SF: The two-fold non-Abelian nature is attributed to
the fermionic part (Sec. 3) and the vortex part (Sec. 4). The ${}^{3}P_{2}$ SF
is a condensate of spin-triplet ($S=1$) $p$-wave ($L=2$) Cooper pairs with
total angular momentum $J=2$. In the following, we briefly review
${}^{3}P_{2}$ SFs in Sec. 5.1. In Sec. 5.2, the stability of a pair of HQVs
compared with a singly quantized vortex is discussed. In Sec. 5.3, we show
that a Majorana fermion, a non-Abelian Ising anyon, exists in the core of each
HQV.

###  5.1 Overview of ${}^{3}P_{2}$ topological SFs

Study of ${}^{3}P_{2}$ SFs was initiated since 1970s Hoffberg et al. (1970);
Tamagaki (1970); Takatsuka & Tamagaki (1971); Takatsuka (1972); Richardson
(1972); Sedrakian & Clark (2018). On the basis of the analysis of the phase-
shifts from nucleon-nucleon scattering in a free space, spin-singlet $s$-wave
${}^{1}S_{0}$ symmetry is a dominant attractive channel in the inner crust
region ($\rho\lesssim 0.5\rho_{0}$, where $\rho_{0}=0.17\mathrm{fm}^{-3}$ is
called the nuclear density) Tamagaki (1970). The critical temperature was
estimated as $10^{8}-10^{10}$ K, which is two orders of magnitude smaller than
the Fermi energy. For further increase in density
$0.7\rho_{0}\lesssim\rho\lesssim 3\rho_{0}$ in the inner core region, the
${}^{1}S_{0}$ superfluidity is suppressed, and instead the attractive channel
of the ${}^{3}P_{2}$ grows. Because of the attractively strong spin-orbit
force, the ${}^{3}P_{2}$ Cooper pair channels with high angular momentum
become more attractive than other ${}^{3}P$ pairing symmetry in contrast to
the atomic physics where the lower total angular momentum state is favored as
in the SF 3He-B phase. The anisotropic form of the Cooper pair is thought to
affect the Cooling rate of nuetron stars. Extreme conditions of neutron stars
are not only high density, but also rapid rotation, and a strong magnetic
field. Especially, neutron stars accompanied by magnetic field ranging from
$10^{15}$ – $10^{18}$ G are called magnetars. Rich structures of exotic
condensates and vortices explained below may be the origin of the observed
pulser glitches, that is, the sudden increase of the angular momentum of
neutron stars. Among them, the existence of non-Abelian HQVs in high magnetic
fields is responsible for the scaling law of the glitches without any fitting
parameters Marmorini et al. (2020).

Figure 8: (a) Phase diagram without magnetic field based on the GL theory. (b)
Gap structure of the nematic state. (c) Phase diagram with magnetic field in
the weak coupling limit. (d) The momentum resolved local density state at the
surface perpendicular to the $\hat{z}$ axis. The magnetic field is applied
parallel (perpendicular) to the surface [top (bottom) panel]. Figures are
adapted from Ref. Mizushima et al. (2017) ©2022 American Physical Society.

In terms of the symmetry point of view, ${}^{3}P_{2}$ SFs spontaneously break
the global gauge symmetry $U(1)_{\varphi}$ and the simultaneous rotational
symmetry in the spin-momentum space $SO(3)_{J}$ which originally exist in the
normal state inside neutron stars. The order parameter of ${}^{3}P_{2}$ SFs is
spanned by 3 by 3 traceless symmetric tensor $A_{\mu i}$, where the 2 by 2 gap
function in momentum space can be given by $\hat{\Delta}(\bm{k})=A_{\mu
i}\bar{k}_{i}\hat{\sigma}_{\mu}i\hat{\sigma}_{y}$. For later convenience, we
introduce a representation based on the total angular momentum along the
quantization axis, $M=-2,\cdots,2$:

$\displaystyle A_{\mu i}=\sum_{M=-2}^{2}\gamma_{M}[\Gamma_{M}]_{\mu i},$ (84)

where $\gamma_{M}$ is a complex wave function of the Cooper pair in the
angular momentum sector $M$. A basis set $\Gamma_{M}$ is given for the
quantization axis $\hat{w}$ by

$\displaystyle[\Gamma_{\pm 2}]_{\mu i}$ $\displaystyle=\frac{(\hat{u}\pm
i\hat{v})_{\mu}(\hat{u}\pm i\hat{v})_{i}}{2},$ (85) $\displaystyle[\Gamma_{\pm
1}]_{\mu i}$ $\displaystyle=\mp\frac{(\hat{u}\pm
i\hat{v})_{\mu}\hat{w}_{i}+\hat{w}_{\mu}(\hat{u}\pm i\hat{v})_{i}}{2},$ (86)
$\displaystyle[\Gamma_{0}]_{\mu i}$
$\displaystyle=\frac{-\hat{u}_{\mu}\hat{u}_{i}-\hat{v}_{\mu}\hat{v}_{i}+2\hat{w}_{\mu}\hat{w}_{i}}{\sqrt{6}},$
(87)

where $\hat{u}$, $\hat{v}$, and $\hat{w}$ constitute the orthonormal triad.

From a phenomenological aspect, the Ginzburg-Landau (GL) energy functional
invariant under the $SO(3)_{J}$ and $U(1)_{\varphi}$ symmetry was obtained for
${}^{3}P_{2}$ SFs as Richardson (1972); Fujita & Tsuneto (1972); Sauls &
Serene (1978); Muzikar et al. (1980); Sauls et al. (1982)

$\displaystyle\mathcal{F}$
$\displaystyle=\alpha\mathrm{tr}[AA^{*}]+\beta_{1}|\mathrm{tr}A^{2}|^{2}$
$\displaystyle\quad+\beta_{2}(\mathrm{tr}[AA^{*}])^{2}+\beta_{3}\mathrm{tr}[A^{2}A^{*2}].$
(88)

It should be remarked that spin-2 spinor BECs ($L=0,S=2$) and $d$-wave
superconductivty ($L=2,S=0$) have similar order parameter structures. The GL
functional for the ${}^{3}P_{2}$ Cooper pairs was minimized by Sauls and
Serene by using the correspondence to the $L=2$ GL functional which was solved
by Mermin Sauls & Serene (1978); Mermin (1974). The ground state solutions are
classified into three types of phases determined by $\beta_{i=1,2,3}$, as
shown in Fig. 8(a): nematic phases with time reversal symmetry and the cyclic
and ferromagnetic phases as non-unitary state. The microscopic derivation of
the GL parameters clarifies that the ground state is the nematic phases in the
weak coupling limit Sauls & Serene (1978); Sauls (1980). The order parameter
tensor of a nematic phase is given by $A_{\mu
i}=\Delta[\hat{u}_{\mu}\hat{u}_{i}+r\hat{v}_{\mu}\hat{v}_{i}-(1+r)\hat{w}_{\mu}\hat{w}_{i}]$
with a real number $r\in[-1,-1/2]$ [see also Eq. (68)]. The nematic phase has
a continuous degeneracy which is lifted by either a magnetic field or sixth-
order terms in the GL free energy. For the GL parameters derived from the
microscopic model, the UN phase for $r=-1/2$ is favored at zero magnetic field
while $D_{2}$-BN ($-1<r<-1/2$) and $D_{4}$-BN $r=-1$ phases are favored for
moderate and strong magnetic fields, respectively Masuda & Nitta (2016),
relevant for magnetars. The schematic images of the gap structures for the UN
and $D_{4}$-BN states are shown in Fig. 8(b), where the arrows account for the
momentum-dependent direction of the $d$-vector, defined by
$d_{\mu}(\bm{k})=\sum_{i}A_{\mu i}k_{i}$. We will discuss other possible
states later in the context of the fermionic topology.

The microscopic model of ${}^{3}P_{2}$ SFs in terms of the fermion degrees of
freedom was constructed by Richardson Richardson (1972) and Tamagaki and
Takatsuka Tamagaki (1970); Takatsuka & Tamagaki (1971); Takatsuka (1972): The
starting microscopic Hamiltonian is composed of the one-body term $H_{1}$ and
the interaction term $H_{2}$ given, respectively, in the following forms:

$\displaystyle H_{1}$
$\displaystyle=\int\mathrm{d}\bm{r}\vec{\psi}^{\dagger}(\bm{r})[\hat{h}_{N}(-i\bm{\nabla})]\vec{\psi}(\bm{r}),$
(89) $\displaystyle H_{2}$
$\displaystyle=-\int\mathrm{d}\bm{r}\sum_{\alpha\beta=x,y,z}\dfrac{g}{2}T_{\alpha\beta}^{\dagger}(\bm{r})T_{\alpha\beta}(\bm{r}).$
(90)

In the first line, $[\vec{\psi}(\bm{r})]_{\sigma}=\psi_{\sigma}(\bm{r})$ and
$[\hat{h}_{N}(-i\bm{\nabla})]_{\sigma,\sigma^{\prime}}$ is the sum of the
kinetic energy
$h_{0}(-i\bm{\nabla})\delta_{\sigma,\sigma^{\prime}}=(-\nabla^{2}/2m-\mu)\delta_{\sigma,\sigma^{\prime}}$
measured from the chemical potential $\mu$, and the Zeeman energy
$-\bm{B}\cdot\hat{\bm{\sigma}}$ for a magnetic field $\bm{B}=B\bm{n}$. The
${}^{3}P_{2}$ force with coupling strength $g$ $>$ $0$ is represented by
$H_{2}$ with the pair-annihilation operator $T$ defined by
$T_{\alpha\beta}(\bm{r})$ $=$
$\sum_{\sigma\sigma^{\prime}}[t_{\alpha\beta,\sigma\sigma^{\prime}}(-i\bar{\bm{\nabla}})\psi_{\sigma^{\prime}}(\bm{r})]\psi_{\sigma}(\bm{r})$,
where $\bar{\bm{\nabla}}\equiv k_{\mathrm{F}}^{-1}\bm{\nabla}$. We have
introduced a spin-momentum coupling in the pair force via the 2 $\times$ 2
matrix in spin space $\hat{t}_{\alpha\beta}$ defined by
$\hat{t}_{\alpha\beta}(-i\bar{\bm{\nabla}})$ $=$
$i\hat{\sigma}_{y}\left\\{\left[\hat{\sigma}_{\alpha}(-i\bar{\nabla}_{\beta})+\hat{\sigma}_{\beta}(-i\bar{\nabla}_{\alpha})\right]/2\sqrt{2}-\delta_{\alpha\beta}\hat{\bm{\sigma}}\cdot(-i\bar{\bm{\nabla}})/3\sqrt{2}\right\\}$.
Within the mean field approximation, the BdG equation is derived as,

$\displaystyle\check{\mathcal{H}}(-i\bm{\nabla},\bm{R})\vec{u}_{\nu}(\bm{R})=\epsilon_{\nu}\vec{u}_{\nu}(\bm{R}),$
(91)
$\displaystyle\check{\mathcal{H}}(-i\bm{\nabla},\bm{R})=\begin{pmatrix}\hat{h}_{N}(-i\bm{\nabla})&\hat{\Delta}(-i\bm{\nabla},\bm{R})\\\
-[\hat{\Delta}(-i\bm{\nabla},\bm{R})]^{*}&-[\hat{h}_{N}(-i\bm{\nabla})]^{*}\end{pmatrix},$
(92)

where the gap matrix is given by
$\hat{\Delta}(-i\bm{\nabla},\bm{R})=\sum_{\alpha,\beta}\frac{i\hat{\sigma}_{\alpha}\hat{\sigma}_{y}}{2k_{\mathrm{F}}}\\{2A_{\alpha\beta}(\bm{R})(-i\nabla_{\beta})+[(-i\nabla_{\beta})A_{\alpha\beta}(\bm{R})]\\}$.
This model was also used to determine the aforementioned GL parameters in the
weak coupling limit and was directly investigated within a quasiclassical
approximation recently in Ref. Mizushima et al. (2017). Even in the presence
of the magnetic field, the quasiclassical approximation where the size of the
Fermi surface is treated as an infinite one, predicts that a nematic state is
the most stable with parameter $r$ determined by the strength of the magnetic
field as in the GL theory. The $T$-$B$ phase diagram is shown in Fig. 8(c). As
explained later, the ferromagnetic state appears near $T_{c}$ when the finite-
size correction of the Fermi surface is taken into account. The microscopic
analysis reveals the existence of the tricritical point on the phase boundary
between the $D_{4}$-BN and $D_{2}$-BN states in the $T$-$B$ phase diagram
shown in Fig. 8(c): at temperatures below the tricritical point, the phase
transition becomes discontinuous Mizushima et al. (2017). The existence of the
tricritical point was later confirmed in Ref. Mizushima et al. (2020) also in
the GL theory with higher order terms (up to the eighth order Yasui,
Chatterjee, Kobayashi & Nitta (2019)). In addition to such drastic change in
critical phenomena, the advantage of the microscopic model lies in studies of
the fermionic topology of the superfluidity. In the following, we explain the
fermionic topology of the possible phases of the ${}^{3}P_{2}$ SFs.

Nematic state. In the weak coupling limit without a magnetic field, the UN
state is the ground state. The order parameter tensors of the UN state and the
BN state under a magnetic field are already explained above. On the basis of
the symmetry of the BdG Hamiltonian, the nematic state is revealed as a
topological SF with time reversal symmetry (a class DIII in the classification
of topological insulators and SCs). The analysis of the BdG equation in the
presence of the boundary clarifies the existence of the gapless surface bound
Majorana fermion, the hallmark character of the topological states. The
surface Majorana fermion has an Ising spin character Chung & Zhang (2009);
Nagato et al. (2009); Volovik (2010); Mizushima & Machida (2011), that is, the
only external field coupled to the Ising spin gives a mass gap to the Majorana
fermion as shown in Fig. 8(d). The magnetic field direction giving the gap to
the Majorana fermion is the direction perpendicular to the surface, denoted by
$\bm{n}_{\perp}$. This is because the surface Majorana fermion is protected
not only by the time reversal symmetry but also another key symmetry, which is