# Interacting Dirac fermions and the rise of Pfaffians in graphene

Vadym Apalkov1 and Tapash Charkaborty2,3 1Department of Physics and Astronomy,
Georgia State University, Atlanta, Georgia 30303, USA;
2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada
R3T 2N2;
3Department of Physics, Brock University, St. Catharines, ON, Canada L2S 3A1

###### Abstract

Fractional Quantum Hall effect (FQHE) is a unique many-body phenomenon, which
was discovered in a two-dimensional electron system placed in a strong
perpendicular magnetic field. It is entirely due to the electron-electron
interactions within a given Landau level. For special filling factors of the
Landau level, a many-particle incompressible state with a finite collective
gap is formed. Among these states, when the Landau level is half filled, there
is a special FQHE state that is described by the Pfaffian function and the
state supports charged excitations that obey non-Abelian statistics. Such a
$1/2$-FQHE state can be realized only for a special profile of the electron-
electron potential. For example, for conventional electron systems the
$1/2$-FQHE state occurs only in the second Landau level, while in a graphene
monolayer, no $1/2$-FQHE state can be found in the first and the second Landau
levels. Another type of low-dimensional system is the bilayer graphene, which
consists of two graphene monolayers coupled through the inter-layer hopping.
The system is quasi-two-dimensional, which makes it possible to tune the
inter-electron interaction potential by applying either the bias voltage or
the magnetic field that is applied parallel to the bilayer. Interestingly, in
the bilayer graphene with AB staking, there is one Landau level per valley
where the $1/2$-FQHE state can indeed be present. The properties of that
$1/2$-FQHE state have a nonmonotonic dependence on the applied magnetic field
and this state can be even more stable than the one discovered in conventional
electron systems.

Fractional Quantum Hall Effect, Pfaffian state, non-abelian statistics,
graphene, bilayer graphene, fractional charge, half-filled Landau level,
strong magnetic field,

## Key points

  * •

In two-dimensional electron systems an externally applied magnetic field
results in the formation of highly discrete and degenerate Landau levels.

  * •

In a strong enough magnetic field, the discrete nature of the energy spectrum
of two-dimensional electrons results in the Integer Quantum Hall effect, which
happens when integer number of Landau levels are occupied.

  * •

For a partially filled Landau levels, inter-electron interactions result in
formation of incompressible liquids, which manifest themselves as the
Fractional Quantum Hall Effect.

  * •

The conventional FQHE corresponds to filling factors of type $p/q$, where $q$
is odd.

  * •

The unconventional FQHE at half-filling ($\nu=1/2$) of a given Landau level is
described by the Pfaffian function as the wave function of the ground state
and has excitations with the charge of $e/4$ and obey non-abelian statistics.

  * •

For conventional 2D systems, $1/2$-FQHE is realized only in the $n=1$ Landau
level.

  * •

For graphene monolayer, there is no $\nu=1/2$ incompressible liquid in any
Landau level.

  * •

For bilayer graphene there is a special Landau level in which the stability of
the $\nu=1/2$ incompressible state can be tuned by the magnitude and direction
of the magnetic field and the bias voltage.

##  I Introduction

A moving charge placed in an external magnetic field experiences a magnetic
force, which can change only the direction of the charge velocity but not its
magnitude. One of the manifestations of this property of the magnetic field is
the Hall effect. The Hall effect occurs when the current flows through a solid
and the magnetic field is applied perpendicular to the current. In this case
the voltage difference, which is called the Hall voltage, is generated across
the conductor in the direction transverse to both the current and the magnetic
field. This effect was discovered by Edwin Hall in 1879 Hall (1879). If the
electric current is in the $x$ direction with the current density of $j_{x}$
and magnetic field is in the $z$ direction, then there is a Hall electric
field generated in the $y$ direction. The strength of the Hall effect is
characterized by the Hall coefficient defined by the following expression
Ashcroft and Mermin (1976); Hurd (1972); Kittel (2005)

$R_{H}=\frac{E_{y}}{j_{x}B_{z}}.$ (1)

It is possible to show that, within the classical approach, the Hall constant
is related to the charge of the carriers, $q$, and their density, $n$,

$R_{H}=\frac{1}{nq}.$ (2)

One important property of this expression is that the Hall constant and
correspondingly the Hall voltage depends on the sign of carrier’s charge, $q$.
For example, for semiconductors, when the carriers are negatively charged
electrons or positively charged holes, the Hall constant can be negative or
positive depending on the type of the major carriers.

The above expression for the Hall coefficient corresponds to the classical
description of the electron dynamics in a magnetic field, or in the low
magnetic field limit. With increasing magnetic field, the electron dynamics in
the magnetic field becomes essentially quantum mechanical, which results in
quantization of the corresponding energy spectrum and formation of highly
degenerate and discrete Landau levels Landau and Lifshitz (1965). The energy
distance between the Landau levels is proportional to the magnetic field. Due
to discrete nature of the energy spectrum of carriers, the diagonal resistance
of a solid as a function of the magnetic field shows oscillations, which are
called the Shubnikov de Haas oscillations Seiler and Stephens (1991).

More interesting and unexpected phenomena occur in two-dimensional (2D)
electron systems, such as in heterojuctions, atomic monolayers, or in quantum
wells. If the magnetic field is applied perpendicular to the 2D layer, i.e.,
the $(x,y)$ plane, then for weak magnetic fields there is the classical Hall
effect, which can be also characterized by the Hall resistance $R_{xy}$

$R_{xy}=\frac{E_{y}}{j_{x}}=R_{H}B_{z}.$ (3)

Therefore in the classical regime, i.e., in a weak magnetic field, the Hall
resistance is proportional to the magnetic field, $B_{z}$. At the same time,
in stronger magnetic fields when the quantization of the in-plane electron
dynamics results in Landau levels, there is another unique effect, which is
the Integer Quantum Hall Effect (IQHE) Stone (1992); Klitzing et al. (1980);
Yoshioka (2002); Prange and Girvin (1987); Chakraborty and Pietiläinen (1995);
Klitzing (2017); von Klitzing et al. (2020). In this regime, the Hall
resistance takes only quantized values of the form

$R_{xy}=\frac{h}{e^{2}\nu},$ (4)

where $h$ is the Plank constant, $e$ is the electron charge, and $\nu$ is an
integer, $\nu=1,2,3,\ldots$. The integer number $\nu$ corresponds to the
number of completely filled Landau levels and whenever the Fermi energy is
between the two Landau levels $n$ and $n+1$, the Hall resistance is constant,
while the longitudinal resistance, $R_{xx}$, becomes zero. Strictly speaking,
to understand the Quantum Hall Effect we also need to consider broadening of
the Landau levels due to disorder and spatial localization of the in-gap
electron states between the Landau levels Stone (1992); Chakraborty and
Pietiläinen (1995). However the main condition for the IQHE is the existence
of the gaps in the energy spectrum which naturally occurs in 2D systems in a
strong magnetic field.

Figure 1:  Non-diagonal and diagonal resistivities, $\rho_{xy}$ and
$\rho_{xx}$, as a function of magnetic field, $B$. The sample is GaAs-
Al0.3Ga${}_{0}.7$As with electron density of $n=1.23\times 10^{11}$ cm-2 and
mobility $\mu=90000$ cm2/Vs (Adapted from Tsui et al. (1982)).

The IQHE was discovered experimentally in 1980 and two years later another
unexpected (and more spectacular) effect was observed in even stronger
magnetic fields, and in better quality samples. This effect is known as the
Fractional Quantum Hall Effect (FQHE) Tsui et al. (1982); Chakraborty and
Pietiläinen (1995). In this case the filling factor $\nu$ in Eq. (4) is
fractional and corresponds to partial occupation of a given Landau level.
Since all levels within the Landau level are degenerate, the electron-electron
interaction completely determines the properties of the system. At special
fractional occupations of the Landau level, the interaction generates the
incompressible ground states with gapped excitations, which finally results in
the FQHE. The primary filling factors where the FQHE occur are
$\nu=\frac{1}{3},\frac{2}{3},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{1}{7}$
etc. Prange and Girvin (1987); Chakraborty and Pietiläinen (1995); Halperin
and Jain (2020). In all these cases the denominator is an odd number, which is
due to the fermionic nature of electrons.

##  II Fractional Quantum Hall Effect

First, we consider the conventional two-dimensional electron systems, such as
the ones grown in heterojuncions or in a quantum well, for which the low-
energy dispersion relation is parabolic. The Hamiltonian of such systems has
the following form

${\cal H}_{c}=\frac{\vec{p}^{2}}{2m^{*}},$ (5)

where $\vec{p}$ is the two-dimensional momentum and $m^{*}$ is the electron
effective mass. When the system is placed in a perpendicular external magnetic
field, the Hamiltonian becomes

${\cal H}_{c}=\frac{\mathbf{\pi}^{2}}{2m^{*}},$ (6)

where $\vec{\pi}=\vec{p}+e\vec{A}/c$ is the generalized momentum and $\vec{A}$
is the vector potential. The energy spectrum of the Hamiltonian (6) consists
of highly degenerate discrete Landau levels Landau and Lifshitz (1965) with
energy

$\epsilon_{n}=\hbar\omega_{B}\left(n+\frac{1}{2}\right),$ (7)

where $\omega_{B}=eB/m^{*}$ is the cyclotron frequency and $n=0,1,2,\ldots$ is
the Landau level index. We write the corresponding Landau eigenfunctions as
$\phi_{n,m}$, where the index $m$ labels the degenerate wavefunctions within a
given Landau level and is determined by the choice of the gauge. For example,
in the Landau gauge, $A_{x}=0$ and $A_{y}=Bx$, the index $m$ is the
$y$-component of the momentum, while in the symmetric gauge,
$\vec{A}=\frac{1}{2}\vec{B}\times\vec{r}$, the index $m$ is the $z$-component
of electron angular momentum Landau and Lifshitz (1965). In what follows, we
consider $m$ as the $z$ component of the angular momentum.

When a given Landau level is partially occupied by electrons, the properties
of the system is completely determined by the electron-electron interactions.
If the mixing of Landau levels due to the interactions is weak, then the
interaction properties of electrons within a single Landau level are
completely determined by the Haldane pseudopotentials $V^{(n)}_{m}$ Haldane
(1983). The Haldane pseudopotential $V^{(n)}_{m}$ is the energy of two
electrons with the relative angular momentum $m$ and they can be found from
the following expression Haldane (1987)

$V_{m}^{(n)}=\int_{0}^{\infty}\frac{dq}{2\pi}qV(q)\left[F_{n}(q)\right]^{2}L_{m}(q^{2}){\rm
e}^{-q^{2}},$ (8)

where $L_{m}(x)$ are the Laguerre polynomials, $V(q)=2\pi{\rm
e}^{2}/(\kappa\ell_{0}q)$ is the Coulomb interaction in the momentum space,
$\kappa$ is the dielectric constant, and $F_{n}(q)$ is the form factor of the
$n$-th Landau level. The form factor is totally determined by the structure of
the wavefunctions of the corresponding Landau level. Different two-dimensional
systems, such as graphene monolayer, transition-metal dichalcogenide
monolayer, graphene bilayer, etc., have different form factors $F_{n}(q)$, but
in all cases, the corresponding Haldane pseudopotentials are given by the same
expression (8). For conventional two-dimensional electron systems with Landau
wavefunctions $\phi_{n,m}$ the form factor is Haldane (1987)

$F_{n}(q)=L_{n}\left(q^{2}/2\right).$ (9)

If the electron system is fully spin polarized then the spatial component of
the many-particle wavefunction is antisymmetric with respect to the particle
exchange. In this case, only the Haldane pseudopotentials, $V_{m}^{(n)}$, with
odd values of $m$, $m=1,3,5,\ldots$, determine the properties of the system.
The nature of the ground state mainly depends on the ratio of the first few
pseudopotentials, $V_{1}^{(n)}/V_{3}^{(n)}$ and $V_{3}^{(n)}/V_{5}^{(n)}$.

At special values of the filling factor $\nu$, which is defined as
$\nu=N_{e}/N_{0}$, where $N_{e}$ is the number of electrons in a given Landau
level and $N_{0}$ is its degeneracy, the electron system becomes
incompressible with a finite excitation gap. This incompressibility is
entirely due to electron-electron interactions and is determined by the values
of the Haldane pseudopotentials. Only for these filling factors the FQHE can
be observed.

The conventional FQHE occurs at the filling factors of the form $p/q$, where
$q$ is odd, which is related to the antisymmetric property of a many-particle
electron wavefunction when two particles are interchanged, i.e.,
$\Psi(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3},\ldots)=-\Psi(\vec{r}_{2},\vec{r}_{1},\vec{r}_{3},\ldots)$.
A prominent example of this type of FQHE is the $1/q$ sequence, i.e., $1/3$,
$1/5$ etc. The corresponding many-particle wave function is well described by
the Laughlin function Laughlin (1983) of the form

$\Psi(z_{1},z_{2},z_{3},\ldots)=\prod_{i<j}(z_{i}-z_{j})^{q}\exp\left(-\sum_{i}\frac{z_{i}^{2}}{4\ell_{0}^{2}}\right),$
(10)

where the positions of electrons are described in terms of complex variable
$z=x-iy$. The Laughlin function (10) corresponds to the filling factor $1/q$
and is very close to the exact many-particle wave function of the system. The
accuracy of the trial wave functions, for instance the Laughlin functions is
tested through numerical calculations for a finite-size electron system. For a
finite-size system placed in an external magnetic field, the many-particle
basis is finite, which allows us to find both the ground state and the
excitation spectrum of the system. Then, the overlap of the trial function
with the exact ground state wave function of the many-particle system can be
determined and that can be used to test the accuracy of the trial
approximation. The incompressibility of the system is determined by the
magnitude of the gap in the collective excitations of the system. Another
interesting property of the $1/q$ FQHE, which is described by the Laughlin
function, is that the charged excitations have fractional charge $e^{*}=e/q$.
For example, for the $1/3$-FQHE the quasiparticles carry a fractional charge
$e^{*}=e/3$.

Our fundamental understanding of the origin of the odd-denominator FQHE is
from the brilliant work of Laughlin which requires that we consider a
incompressible fluid that has no long-range positional order. The
incompressibility implies that all the excited states have a non-zero energy
difference from the ground state. The ground state can be fully spin polarized
or also have spin degree of freedom Chakraborty and Zhang (1984); Chakraborty
et al. (1986); Apalkov et al. (2001) in the ground state and spin-reversed
excitations. It is worth emphasizing that despite the torrent of ideas
unleased by the Laughlin function, the origin of incompressibility in the
Laughlin state still remains unresolved Haldane (2011).

In addition to the conventional FQHE set of type $p/q$, where $q$ is odd,
another type of FQHE corresponding to the filling $5/2$ has been discovered
experimentally (Willett et al., 1987; Willett, 2013). This filling factor
implies that there are two completely occupied Landau levels, which correspond
to spin-up and spin-down Landau levels with index $n=0$, and the next Landau
level is half-filled. It means that in the $n=1$ Landau level the $1/2$-FQHE
occurs, i.e., the filling factor of the $n=1$ Landau level is $\nu=1/2$. This
unusual FQHE cannot be described by the Laughlin function (10). This is
because the many-particle wave function should be antisymmetric with respect
to the particle interchange, because the corresponding electrons are fermions
and $q$ must be an odd integer. But for $q=2$, the Laughlin function is
symmetric, and therefore describes a system of bosons. Another important
property of the $1/2$-FQHE is that the $1/2$ incompressible liquid is realized
only in higher Landau levels, for example, for $n=1$ but not for the $n=0$
level. Different wavefunctions have been proposed theoretically to describe
this state. Among them are the Pfaffian Moore and Read (1991), anti-Pfaffian
Levin et al. (2007), particle-hole symmetric Pfaffian Son (2015), and the
221-parton Wen (1991) wavefunctions. The $1/2$-FQHE is also sensitive to the
inter-Landau mixing Rezayi (2017) which modifies the electron-electron
interaction potential Peterson and Nayak (2013) and opens up the possibility
to observe the $1/2$-FQHE in higher Landau levels Kim et al. (2019) where the
inter-Landau mixing becomes strong. Below we consider the properties of the
$1/2$-FQHE, which is described only by the Pfaffian wavefunction.

###  II.1 Pfaffian function

Some properties of the FQHE states can be understood by looking at the
specially created composite objects, that are called the composite fermions.
The composite fermion is an electron with an even number of flux quanta
attached to it. These objects still have the same Fermi statistics as original
electrons, but they feel a different magnetic field. To understand their
properties, first, let us consider a fully filled Landau level. In this case,
there is one magnetic flux quantum per each electron and under this condition
we have the IQHE. Now let us consider a Landau level with the filling factor
of $\nu=1/3$, i.e., only $1/3$ states within a given Landau level are
occupied. For such a system, there are three magnetic flux quanta per electron
and this state corresponds to the $\nu=1/3$-FQHE. The composite fermions for
this system are electrons bound to two magnetic flux quanta each. Therefore,
for these composite fermions (one electron plus two flux quanta), there is one
magnetic flux quanta per fermion, which corresponds to a fully occupied Landau
level and is therefore, the IQHE of composite fermions. Another way to state
this is that, the FQHE for electrons is the IQHE for composite fermions. This
correspondence exists only for the FQHE of type of $p/q$, where $q$ is odd. As
an example, for $\nu=1/5$ the composite fermion is an electron plus four
magnetic flux quanta and the composite fermions all occupy the first Landau
level. It is important that the number of magnetic flux quanta attached to an
electron is even. Only in this case the composite particles have the same
Fermi statistics as electrons.

A unique situation occurs for the Landau level that is exactly half filled.
For such a system, the composite fermion picture dictates that there are two
magnetic flux quanta associated with each electron. In that case, a composite
fermion comprise of an electron bound to two magnetic flux quanta and in a
mean-field picture, the system of composite fermions does not experience a net
magnetic field Halperin et al. (1993). The properties of this composite
fermion system are determined by the inter-fermion interaction, i.e., by the
Haldane pseudopotentials. For the $n=0$ Landau level, the composite fermions
forms a gapless composite Fermi liquid. A more interesting situation happens
for the $n=1$ Landau level, i.e., for the total filling factor $\nu=5/2$. In
this case, the theoretical and experimental results suggest that the system
forms a gapped state which is due to pairing of the composite fermions, i.e.,
by formation of Cooper pairs, similar to what we see in the supercoductor
systems. The pairing symmetry can then be different and below we consider only
one type of symmetry, i.e., when the Cooper pairs of composite fermions have
angular momentum $-1$. It was proposed that the corresponding ground state of
the system expressed in terms of the coordinates of original electrons is
described by the Pfaffian Moore and Read (1991); Greiter et al. (1991, 1992);
Nayak et al. (2008) function, which has the following form

$\Psi_{\mbox{Pf}}=\mbox{Pf}\left[\frac{1}{z_{i}-z_{j}}\right]\prod_{i<j}(z_{i}-z_{j})^{2}\exp\left(-\sum_{i}\frac{z_{i}^{2}}{4\ell_{0}^{2}}\right),$
(11)

where the Pfaffian is defined as Moore and Read (1991); Greiter (2011)

${\rm
Pf}\,\left[\mathbf{M}\right]=\frac{1}{2^{N/2}\left(N/2\right)!}\sum_{\sigma\in
S_{N}}{\rm sgn}\,\sigma\prod_{l=1}^{N/2}M_{\sigma(2l-1)\sigma(2l)},$ (12)

for an $N\times N$ antisymmetric matrix $\mathbf{M}$ whose elements are
$M_{ij}$, where $N$ is an even number. Here $S_{N}$ is the group of
permutations of $N$ objects.

Historically, the concept of Pfaffians arose from a very surprising
mathematical result: Consider the determinant of a $N\times N$ antisymmetric
matrix $\mathbf{M}=[M_{ij}]$ written as $\det[M_{ij}]_{1\leq i,j\leq N}$ in
which $M_{ij}=-M_{ji}$ for $i,j=1,2,\ldots,N$. If $\mathbf{M}=-\mathbf{M}^{T}$
is an antisymmetric matrix of order $N$ then
$\det[\mathbf{M}]=(-1)^{N}\det[\mathbf{M}]$ so that if the order is odd, then
the determinant vanishes. However, when $N=2m$, then the determinant is the
square of a polynomial in coefficients of the given determinant. This result
was first discovered by the British mathematician Arthur Cayley in 1847 Cayley
(1847); Halton (1966), who named this polynomial Pfaffian in honor of the
great German mathematician Johann Pfaff, who discovered it in 1815, in the
course of his work on differential equations.

For $N=2$,

$\det\left|\begin{array}[]{cc}0&M_{12}\\\
-M_{12}&0\end{array}\right|=\left(M_{12}\right)^{2},$ (13)

and the Pfaffian is ${\rm Pf}\left[\mathbf{M}\right]=M_{12}$. Similarly, for
$N=4$,

$\det\left|\begin{array}[]{cccc}0&M_{12}&M_{13}&M_{14}\\\
-M_{12}&0&M_{23}&M_{24}\\\ -M_{13}&-M_{23}&0&M_{34}\\\
-M_{14}&-M_{24}&-M_{34}&0\end{array}\right|=\left(M_{12}M_{34}-M_{13}M_{24}+M_{14}M_{23}\right)^{2},$
(14)

and the corresponding Pfaffian is

${\rm Pf}\,\left[\mathbf{M}\right]=M_{12}M_{34}-M_{13}M_{24}+M_{14}M_{23}.$
(15)

There are three terms in the above expression. For a general antisymmetric
matrix $N\times N$, the number of terms in the pfaffian is $(N-1)!!$. The
Pfaffians are sometimes referred to as triangular determinant or half-
determinant Crilly (2016),

${\rm
Pf}\,\left[\mathbf{M}\right]=\left.\begin{array}[]{cccc}|M_{12}&M_{13}&M_{14}\\\
&M_{23}&M_{24}\\\ &&M_{34}\end{array}\right|.$ (16)

Moving on, one can obtain the following relation between the determinant of
the matrix $\mathbf{M}$ and its Pfaffian Cayley (1847); Halton (1966),

$\rm{det}\left[\mathbf{M}\right]=\left({\rm
Pf}\,\left[\mathbf{M}\right]\right)^{2}.$ (17)

This relation can be generalized for a special partially antisymmetric matrix
of the form Cayley (1847); Bajdich et al. (2008)

$\mathrm{det}\left[\begin{array}[]{ccccc}0&b_{12}&b_{13}&\ldots&b_{1N}\\\
-a_{12}&0&a_{23}&\ldots&a_{2N}\\\ -a_{13}&-a_{23}&0&\ldots&a_{3N}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
-a_{1N}&-a_{2N}&-a_{3N}&\ldots&0\end{array}\right]=\mathrm{Pf}\left[\begin{array}[]{ccccc}0&a_{12}&a_{13}&\ldots&a_{1N}\\\
-a_{12}&0&a_{23}&\ldots&a_{2N}\\\ -a_{13}&-a_{23}&0&\ldots&a_{3N}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
-a_{1N}&-a_{2N}&-a_{3N}&\ldots&0\end{array}\right]\times\mathrm{Pf}\left[\begin{array}[]{ccccc}0&b_{12}&b_{13}&\ldots&b_{1N}\\\
-b_{12}&0&a_{23}&\ldots&a_{2N}\\\ -b_{13}&-a_{23}&0&\ldots&a_{3N}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
-b_{1N}&-a_{2N}&-a_{3N}&\ldots&0\end{array}\right]$ (18)

Other useful relations, involving pfaffians of $N\times N$ matrices
$\mathbf{M}$ and $\mathbf{A}$, are the following

$\displaystyle{\rm Pf}\,\left[\mathbf{M}^{T}\right]=(-1)^{N/2}{\rm
Pf}\,\left[\mathbf{M}\right]$ (19) $\displaystyle{\rm
Pf}\,\left[\lambda\mathbf{M}\right]=\lambda^{N/2}{\rm
Pf}\,\left[\mathbf{M}\right]$ (20) $\displaystyle{\rm
Pf}\,\left[\mathbf{A}\mathbf{M}\mathbf{A}^{T}\right]={\rm
det}\left[\mathbf{A}\right]{\rm Pf}\,\left[\mathbf{M}\right].$ (21)
$\displaystyle{\rm Pf}\left[\begin{array}[]{cc}0&\mathbf{M}\\\
-\mathbf{M}^{\mathrm{T}}&0\end{array}\right]=(-1)^{N(N-1)/2}\mathrm{det}\left[\mathbf{M}\right]$
(24) $\displaystyle{\rm Pf}\left[\begin{array}[]{cc}\mathbf{M}_{1}&0\\\
0&\mathbf{M}_{2}\end{array}\right]=\mathrm{Pf}\left[\mathbf{M}_{1}\right]\mathrm{Pf}\left[\mathbf{M}_{2}\right]$
(27)

There is also a special expression for the pfaffian of skew-symmetric
tridiagonal matrix,

$\mathrm{Pf}\left[\begin{array}[]{ccccccc}0&a_{1}&0&0&&&\\\ -a_{1}&0&0&0&&&\\\
0&0&0&a_{2}&&&\\\ 0&0&-a_{2}&0&&&\\\ &&&&\ddots&&\\\ &&&&&0&a_{N/2}\\\
&&&&&-a_{N/2}&0\end{array}\right]=a_{1}a_{2}\ldots a_{N/2}$ (28)

Pfaffians actually describe a pairing state. In fact, the famous Bardeen-
Cooper-Schrieffer (BCS) wave function that is the wavefunction for spin-
singlet pairs can be expressed in terms of the Pfaffians or a determinant
Bajdich et al. (2008); Bouchaud et al. (1988).

The Pfaffian function (11) which determines the $\nu=1/2$ state is defined
only for even number of particles, which illustrates its fundamental property
as a collective state of the Cooper pairs, i.e., the bound state of two
electrons. Interestingly, while the pfaffian itself, i.e., the first factor in
Eq. (11), is singular when two electrons are at the same position
($z_{i}=z_{j}$), when multiplied by the second factor, the pfaffian function
(11) becomes regular. In other words, the Pfaffian state has a nonzero
amplitude at the coincidence of two particles. As an example, for four
electrons the pfaffian function takes the following form

$\displaystyle\Psi_{\mbox{Pf}}=\left[\frac{1}{z_{1}-z_{2}}\frac{1}{z_{3}-z_{4}}-\frac{1}{z_{1}-z_{3}}\frac{1}{z_{2}-z_{4}}+\frac{1}{z_{1}-z_{4}}\frac{1}{z_{2}-z_{3}}\right]\times$
$\displaystyle(z_{1}-z_{2})^{2}(z_{1}-z_{3})^{2}(z_{1}-z_{4})^{2}(z_{2}-z_{3})^{2}(z_{2}-z_{4})^{2}(z_{3}-z_{4})^{2}\exp\left(-\sum_{i}\frac{z_{i}^{2}}{4\ell_{0}^{2}}\right).$
(29)

One can see that the highest power of any $z_{i}$, e.g., $z_{1}$, in the above
expression is 5. In general, for the number of electrons $N_{e}$, the highest
power is $2N_{e}-3$. In the thermodynamic limit, it gives the required filling
factor $1/2$.

The Pfaffian state has a very unique property in terms of the charged
excitations. In fact, the function that describes creation of two positively
charged holes has the following form

$\Psi_{\mbox{Pf}}=\mbox{Pf}\left[\frac{(z_{i}-\eta_{1})(z_{j}-\eta_{2})+(z_{j}-\eta_{1})(z_{i}-\eta_{2})}{z_{i}-z_{j}}\right]\prod_{i<j}(z_{i}-z_{j})^{2}\exp\left(-\sum_{i}\frac{z_{i}^{2}}{4\ell_{0}^{2}}\right).$
(30)

Here $\eta_{1}$ and $\eta_{2}$ are the coordinates of the holes. The charge of
these excitations is fractional, $e^{*}=e/4$, and they obey the ‘non-Abelian’
statistics Nayak et al. (2008); Stern and Halperin (2006); Stern (2008). The
non-Abelian statistics means that when two excitations exchange their
positions, it changes not only the phase of the wave function but also
introduces an unitary transformation of the wave function itself. In general,
these unitary transformations do not commute and the system of these
excitations is called non-Abelian.

The charged excitations of the Pfaffian state also carry the signature of
Majorana fermions (MFs) Read and Green (2000); Ivanov (2001). The Majorana
fermion is a special type of ‘particle’, which can be seen as its own anti-
particle. If $\gamma_{i}$ is an operator corresponding to the Majorana fermion
$i$, then it satisfies the relations

$\displaystyle\left\\{\gamma_{i},\gamma_{i}\right\\}=2\delta_{ij}$ (31)
$\displaystyle\gamma_{i}^{\dagger}=\gamma_{i},$ (32)

where $\left\\{\gamma_{i},\gamma_{i}\right\\}$ is the anti-commutator of
$\gamma_{i}$ and $\gamma_{j}$. The first relation determines that these
particles are fermions and the second relation tells us that the particle is
its own anti-particle. The Majorana particle can be formally expressed as a
combination of the creation, $c_{i}^{\dagger}$, and annihilation, $c_{i}$,
operators of regular fermions,

$\gamma_{i}=c_{i}+c_{i}^{\dagger}.$ (33)

It means that the Majorana operator creates simultaneously a particle, e.g.,
an electron with the negative charge $-e$, and an anti-particle, e.g., a hole
with positive charge $e$. In conventional systems such an operation is
impossible, but in systems such as the superconductors which support Cooper
pairs and for which the electric charge is conserved mod 2, the Majorana
fermions are possible. Superposition of two Majorana fermions correspond to a
fermionic state where the constituent MFs are spatially separated. This
intrinsic non-local nature of the fermionic state makes the state being
protected from local perturbations. This property and the associated non-
abelian exchange statistics makes the MFs well suited for fault-tolerant
quantum computing Kitaev (2003); Ayukaryana et al. (2021); Das Sarma et al.
(2005).

Many numerical studies of the $\nu=1/2$ state have been performed in the
spherical geometry (see, e.g., Fano et al. (1986)). In this geometry, the
electrons are placed on a surface of a sphere and the magnetic field is
created by a magnetic monopole placed at the center of the sphere. The
strength of the magnetic field is characterized by the parameter $S$, where
$2S$ is the number of magnetic fluxes through the sphere in units of the flux
quantum. At the same time the parameter $S$ is equal to the angular momentum
of single-particle states. All single-particle states have the same energy and
form the Landau level, where different states are distinguished by the $z$
component of the angular momentum. Therefore, in spherical geometry, the total
number of states in the Landau level is $2S+1$. In a many-electron system the
electron-electron interaction is introduced through the Haldane
pseudopotentials and the number of electrons, $N_{e}$ determines the filling
factor of the Landau level. Here the $\nu=1/2$ FQHE corresponds to the
following condition: $2S=2N_{e}-3$. Although in this case the ratio of the
number of electrons and the number of single-particle states is not 1/2, i.e.,
$N_{e}/(2S+1)=N_{e}/(2N_{e}-2)\neq 1/2$, for this value of $S$, the system has
a gap and the ground state wave function is close to the Pfaffian function. In
the thermodynamic limit, i.e., for $N\rightarrow\infty$, there is a correct
identity $N_{e}/(2S+1)=N_{e}/(2N_{e}-2)\rightarrow 1/2$.

The position of an electron in spherical geometry can be described by either
the two angles, $\phi$ and $\theta$, or two components of a spinor,
$u=\cos\theta\sin\phi$ and $v$. In terms of the variables $u$ and $v$, the
Pfaffian function takes the following form

$\Psi_{\mbox{Pf}}=\mbox{Pf}\left[\frac{1}{u_{i}v_{j}-u_{j}v_{i}}\right]\prod_{i<j}(u_{i}v_{j}-u_{j}v_{i})^{2}.$
(34)

In the spherical geometry the Pfaffian function is an exact wave function of
the ground state for the system with three-particle interaction defined as

$H_{int}=\frac{e^{2}}{\kappa\ell_{0}}\sum_{i<j<k}P_{ijk}(3S-3),$ (35)

where $P_{ijk}$ is the three-particle projection operator onto the state with
the total angular momentum $L$.

In a planar geometry, the Pfaffian function is the exact ground state wave
function of the system with the three-body interaction Greiter et al. (1991,
1992) given by the following expression

$H_{3B}=V_{0}\sum_{i<j<k}^{N_{el}}S_{ijk}\nabla_{i}^{2}\delta(i-j)\delta(i-k),$
(36)

where $\delta(i-j)=\delta(\vec{r}_{i}-\vec{r}_{j})$ is the $\delta$ function,
$\nabla_{i}$ is the partial derivative with respect to $\vec{r}_{i}$, and
$S_{ijk}$ denotes symmetrization over permutations within $(ijk)$:
$S_{123}=f_{123}+f_{231}+f_{312}$, and $f$ is symmetric in its first two
indices. To understand why for the three-body interaction potential (36) the
energy of the Pfaffian state is zero, we need to look at the structure of the
Pfaffian wave function. For example, by looking at the four-particle system
(see Eq. (29)) we can see that the terms with $z_{1}$ are of the type:
$(z_{1}-z_{2})(z_{1}-z_{3})^{2}(z_{1}-z_{4})^{2}\ldots$. Therefore, they have
one factor with linear dependence on $z_{1}$, $(z_{1}-z_{2})$, and all other
factors have quadratic dependence on $z_{1}$,
$(z_{1}-z_{3})^{2}(z_{1}-z_{4})^{2}\ldots$. Now let us consider the following
term in the three-particle Hamiltonian (36):
$\nabla_{1}^{2}\delta(1-2)\delta(1-3)$. Then to find the contribution to the
energy due to that term, we need to calculate the second derivative of
$(z_{1}-z_{2})(z_{1}-z_{3})^{2}(z_{1}-z_{4})^{2}\ldots$ with respect to
$z_{1}$ and then set $z_{1}$ equals to $z_{2}$ and $z_{3}$, which finally
gives us zero. Analyzing different terms both in the Hamiltonian (36) and the
Pfaffian wave function, we can conclude that the energy of the Pfaffian state
is zero and it is an eigenfunction of the three-particle Hamiltonian (36).

For the real two-body interaction, whether the Pfaffian function is the ground
state wave function or not is determined by the values of Haldane
pseudopotentials, mainly by the values of $V_{1}/V_{5}$ and $V_{3}/V_{5}$. The
diagram, which shows under what values of these parameters the half-filled
Landau level becomes incompressible and is described by the Pfaffian function,
is shown in Fig. 2 Storni et al. (2010). The diagram depicts the contour plot
of the gap of the system, where for the compressible state the gap is zero.
The gap also characterizes the stability of the corresponding $1/2$-FQHE
state. Here the blue region corresponds to the compressible state, which does
not support the $1/2$-FQHE. The maximum overlap of the ground state wave
function with the Pfaffian function is marked by a solid black line.

The thick lines with dots correspond to the conventional electron system with
parabolic energy dispersion and different thickness of the 2D layer. Here the
thickness of the layer changes the Haldane pseudopotentials, resulting in the
corresponding lines in the diagram. The results of Fig. 2 clearly show that,
for the conventional system, the state in the half-filled $n=0$ Landau level
is compressible, while that for the half-filled $n=1$ Landau level is
incompressible and the overlap of the ground state wave function with the
Pfaffian function is large. The half-filled $n=1$ Landau level corresponds to
the total filling factor of $5/2$.

The diagram shown in Fig. 2 can be also used to analyze the properties of
half-filled Landau levels for other systems, such as graphene monolayer and
graphene bilayer. The Haldane pseudopotentials for these systems is different
from the ones for the conventional system.

The effective electron-electron interaction potential also depends on the
Landau level mixing. This mixing can be important for systems with small
Landau level gaps, such as the ZnO quantum wells Luo and Chakraborty (2017).
The strength of mixing is determined by the ratio of the Coulomb interaction
and the Landau level gap. When this ratio is varied, it was shown by numerical
analysis Luo and Chakraborty (2017) that, in the ZnO quantum well systems the
topological transitions can be observed for the half-filled Landau level.
Further, in ZnO systems a stable Pfaffian state have been predicted at filling
factors $5/2$ and $7/2$ Luo and Chakraborty (2017).

Figure 2:  Contour plot of the FQHE gap as a function of $V_{1}/V_{5}$ and
$V_{3}/V_{5}$. The black line marks the maximum of the overlap of the ground
state wave function and the Pfaffian function. The lines with dots depict the
effect of the finite width of a 2D layer. Here the red line corresponds to
filling factor $\nu=5/2$, which is the same as $\nu=1/2$ at the $n=1$ Landau
level, while the blue line corresponds to $\nu=1/2$ at the $n=0$ Landau level.
The dots denote width $w/\ell_{0}=0,1,2,3,4$ (from right to left). The
compression region is shaded blue. Reproduced with permission from Storni et
al. (2010).

##  III Monolayer and bilayer graphene in a strong magnetic field

###  III.1 Graphene monolayer

The graphene is a monolayer of carbon atoms Wallace (1947); Novoselov et al.
(2004), which form the 2D honeycomb crystal structure with two sublatices, say
A and B [see Fig. 3(a)] Abergel et al. (2010). The corresponding electron band
structure has two valleys at two inequivalent points,
$K=(2\pi/a)(\frac{1}{3},\frac{1}{\sqrt{3}})$ and
$K^{\prime}=(2\pi/a)(\frac{2}{3},0)$, in the reciprocal space [see Fig. 3(b)].
Here $a=0.246$ nm is the lattice constant. The unique property of graphene is
that the low-energy dispersion at each valley is determined by the massless
Hamiltonian of the Dirac type (Dirac fermions) Castro Neto et al. (2009); Geim
and Novoselov (2007); Abergel et al. (2010)

${\cal H}_{\xi}=\xi v_{\rm F}\left(\begin{array}[]{cc}0&p_{-}\\\
p_{+}&0\end{array}\right),$ (37)

where $\xi$ is the valley index, which is 1 at the $K$ valley and -1 at the
$K^{\prime}$ valley, $p_{-}=p_{x}-{\rm i}p_{y}$, $p_{+}=p_{x}+{\rm i}p_{y}$,
and $v_{\rm F}\approx 10^{6}$ m/s is the Fermi velocity. The corresponding
eigenfunctions of the Hamiltonian (37) have two components due to two
sublattices, $A$ and $B$, of the graphene honeycomb lattice. The wave
functions are expressed as $(\psi_{A},\psi_{B})^{T}$ for valley $K$ and
$(\psi_{B},\psi_{A})^{T}$ for valley $K^{\prime}$, where $\psi_{A}$ and
$\psi_{B}$ correspond to sublattices $A$ and $B$, respectively.

Figure 3:  (a) Honeycomb crystal structure of graphene monolayer. Two
equivalent sublattices A and B are marked by open and filled dots. (b) The
first Brillouin zone of monolayer graphene. Two valley $K$ and $K^{\prime}$
are also shown.

The Landau level of electrons in graphene can be found from Hamiltonian (37)
by replacing the electron momentum $\vec{p}$ with the generalized momentum
$\vec{\pi}=\vec{p}+e\vec{A}/c$,

${\cal H}_{\xi}=\xi v_{\rm F}\left(\begin{array}[]{cc}0&\pi_{-}\\\
\pi_{+}&0\end{array}\right).$ (38)

Then the eigenfunctions of the Hamiltonian (38) have the following form

$\Psi^{K}_{n,m}=\left(\begin{array}[]{c}\psi_{A}\\\
\psi_{B}\end{array}\right)=C_{n}\left(\begin{array}[]{c}{\rm sgn}(n){\rm
i}^{|n|-1}\phi_{|n|-1,m}\\\ {\rm i}^{|n|}\phi_{|n|,m}\end{array}\right),$ (39)

for the valley $K$ ($\xi=1$) and

$\Psi^{K^{\prime}}_{n,m}=\left(\begin{array}[]{c}\psi_{B}\\\
\psi_{A}\end{array}\right)=C_{n}\left(\begin{array}[]{c}{\rm sgn}(n){\rm
i}^{|n|-1}\phi_{|n|-1,m}\\\ {\rm i}^{|n|}\phi_{|n|,m}\end{array}\right),$ (40)

for the valley $K^{\prime}$ ($\xi=-1$). Here $C_{n}=1$ for $n=0$ and
$C_{n}=1/\sqrt{2}$ for $n\neq 0$ and

$\mbox{sgn}(n)=\left\\{\begin{array}[]{cc}0&(n=0)\\\ 1&(n>0)\\\
-1&(n<0).\end{array}\right.,$ (41)

where the positive and negative values of $n$ correspond to the conduction and
valence bands, respectively. The corresponding Landau energy spectrum takes
the form McClure (1956); Castro Neto et al. (2009); Goerbig (2011)

$\varepsilon_{n}=\hbar\omega_{B}\mbox{sgn}(n)\sqrt{|n|},$ (42)

where $\omega_{B}=\sqrt{2}v_{\rm F}/\ell_{0}$ and $\ell_{0}=\sqrt{\hbar/eB}$
is the magnetic length.

Summarizing, we can say that there are two types of Landau levels in graphene.
The first one corresponds to $n=0$. In this case the wave function is

$\Psi^{K}_{n=0,m}=\left(\begin{array}[]{c}0\\\ \phi_{0,m}\end{array}\right).$
(43)

This wave function consists of only the $\phi_{0,m}$ Landau function of a
conventional system. The interacting electron system at $n=0$ graphene Landau
level is therefore identical to the interacting electron system in $n=0$
Landau level of the conventional system.

The second class of graphene Landau levels correspond to $n\neq 0$. In this
case the wave function is

$\Psi^{K}_{n,m}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}{\rm sgn}(n){\rm
i}^{|n|-1}\phi_{|n|-1,m}\\\ {\rm i}^{|n|}\phi_{|n|,m}\end{array}\right).$ (44)

Now, the graphene wave function is the mixture of $n$ and $n-1$ Landau wave
functions of a conventional electron system.

Figure 4:  The first few Landau levels of graphene monolyaer as a function of
perpendicular magnetic field. The Landau levels are labeled with integer index
$n$, where positive and negative $n$ correspond to the conduction and valence
bands, respectively. The red and blue lines mark the Landau levels for which
the FQHE can be observed.

For Dirac fermions in graphene, the unique feature of the Landau levels (42)
is their square-root dependence on both the magnetic field $B$ and the Landau
level index $n$. This behavior is different from that in conventional
semiconductor 2D systems with the parabolic energy dispersion discussed above,
where the Landau levels have linear dependence on both the magnetic field and
the Landau level index [see Eq. (7)]. The Landau level of graphene are shown
in Fig. 4 as a function of the magnetic field. Here the positive and negative
Landau level indices $n$ correspond to the conduction and valence bands,
respectively. These unique properties of the Landau levels of Dirac fermions
in graphene result in the unconventional Quantum Hall Effect in graphene with
quantum Hall plateaus at filling factors $4\left(n+\frac{1}{2}\right)$
Novoselov et al. (2004); Zhang et al. (2005).

###  III.2 Bilayer graphene

The bilayer graphene consists of two coupled graphene monolayers McCann and
Fal’ko (2006); McCann and Koshino (2013), which can be in two possible main
stackings: (i) AA stacking and (ii) Bernal (AB) stacking, which are shown
schematically in Fig. 5.

For bilayer graphene with AA stacking, there is a interlayer coupling between
the Landau levels of two layers with the same Landau level indices. Such
coupling changes the energies of the Landau levels of graphene monolayers, but
does not affect the wavefunctions of the layers. Therefore, the bilayer
graphene Haldane pseudopotentials, which characterize the electron-electron
interaction properties, are completely identical to the corresponding
pseudoptentials of monolayer graphene.

Figure 5:  (a) Bilayer graphene with AA stacking. Here the interlayer hopping
couples sublattice A (B) in the top layer with sublattice A (B) in the bottom
layer. (b) Bilayer graphene with AB staking. For such staking, sublattice A
(B) in the top layer is coupled to sublattice B (A) in the bottom layer.

For bilayer graphene with Bernal (AB) stacking, the interlayer coupling
strongly modifies the properties of the Landau level in the system. The
corresponding Hamiltonian for valley $\xi=\pm 1$ has the form McCann and
Fal’ko (2006)

${\cal H}_{\xi}^{(AB)}=\xi\left(\begin{array}[]{cccc}\frac{U}{2}&v_{\rm
F}\pi_{-}&0&0\\\ v_{\rm F}\pi_{+}&\frac{U}{2}&\xi\gamma_{1}&0\\\
0&\xi\gamma_{1}&-\frac{U}{2}&v_{\rm F}\pi_{-}\\\ 0&0&v_{\rm
F}\pi_{+}&-\frac{U}{2}\end{array}\right).$ (45)

where $U$ is the inter-layer bias voltage and $\gamma_{1}\approx 400$ meV is
the interlayer hopping integral. The corresponding wave functions have the
structure of $(\psi_{A_{1}},\psi_{B_{1}},\psi_{A_{2}},\psi_{B_{2}})^{T}$,
where $A_{1}$, $B_{1}$ correspond to the lower monolayer and $A_{2}$, $B_{2}$
correspond to the upper monolayer. From Hamiltonian (45), the Landau level
wave functions are

$\Psi^{\rm(bi)}_{n,m}=\left(\begin{array}[]{c}\xi C_{1}\phi_{n-1,m}\\\
C_{2}\phi_{n,m}\\\ C_{3}\phi_{n,m}\\\ \xi
C_{4}\phi_{n+1,m}\end{array}\right),$ (46)

where $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$ are constants. The wave functions
in the bilayer graphene with AB stacking are therefore the mixtures of the
conventional Landau wavefunctions with indices $n-1$, $n$, and $n+1$.

In Eq. (46) the Landau index $n$ can take the following values:
$n=-1,0,1,\ldots$. Here we assume that if the index of the Landau wave
function, $\phi_{n,m}$, is negative then the function is identically zero,
which means that $\phi_{-2,m}\equiv 0$ and $\phi_{-1,m}\equiv 0$. Hence, for
$n=-1$, the wave function (46) is just
$\Psi^{\rm(bi)}_{-1,m}=(0,0,0,\phi_{0,m})$, i.e., the coefficients $C_{1}$,
$C_{2}$, $C_{3}$ are zero. Therefore, there is only one energy level
corresponding to $n=-1$. For $n=0$, the wave function (46) has zero
coefficient $C_{1}$ and, correspondingly, there are only three energy levels.

For $n>0$, there are four eigenvalues of the Hamiltonian (45), corresponding
to four Landau levels in a bilayer graphene at a given valley $\xi=\pm 1$. The
eigenvalue equation, which determines the Landau levels is written as Pereira
et al. (2007)

$\left[\left(\varepsilon+\xi u\right)^{2}-2n\right]\left[\left(\varepsilon-\xi
u\right)^{2}-2(n+1)\right]=\tilde{\gamma}_{1}^{2}\left[\varepsilon^{2}-u^{2}\right],$
(47)

where $\varepsilon$ is the energy of the Landau level in units of
$\epsilon_{B}$, $\epsilon_{B}=\hbar v_{\rm F}/\ell_{0}$,
$\tilde{\gamma}_{1}=\gamma_{1}/\epsilon_{B}$, and $u=U/\epsilon_{B}$.

Figure 6:  A few lowest Landau levels of bilayer graphene as a function of the
magnetic field. The bilayer graphne has the AB staking. The Landau levels of
both $K$ and $K^{\prime}$ are shown. The red lines show special Landau levels,
at which the $\nu=1/2$ Pfaffian incompressible state can be realized. The bias
voltage is 5 meV and interlayer hopping integral is $\gamma_{1}=400$ meV.

The four Landau levels determined by Eq. (47) have two levels with negative
energy (valence band) and two levels with positive energy (conduction band).
It is then convenient to label these levels as follows: for a given value of
$n$ and a given valley $\xi$ we label the levels as $n^{(\xi)}_{i}$, where
$i=-2,-1,1,2$ in the ascending order and the negative and positive values of
$i$ correspond to the valence and conduction bands, respectively. Although for
$n=0$ there are only three Landau levels and for $n=-1$ there is only one
Landau level, it is convenient to combine them in a single set of $n=0$ Landau
levels and label them as $0^{(\xi)}_{i}$, where $i=-2,-1,1,2$. Also, the
Landau levels of different valleys are related as follows
$\epsilon(n^{(\xi)}_{i})=-\epsilon(n^{(-\xi)}_{-i})$.

With the known wave functions (46) of the bilayer graphene Landau levels, the
form factor in Eq. (8) for Haldane pseudopotentials can be obtained from the
following expression

$F_{n}(q)=|C_{1}|^{2}L_{n-1}(q^{2}/2)+\left(|C_{2}|^{2}+|C_{3}|^{2}\right)L_{n}(q^{2}/2)+|C_{4}|^{2}L_{n+1}(q^{2}/2).$
(48)

There are two special Landau levels of bilayer graphene with almost zero
energy Apalkov and Chakraborty (2011). The first one corresponds to $n=-1$
with the energy of $\varepsilon=-\xi u$ and the corresponding wave function,

$\Psi^{\rm(bi)}_{0^{(+)}_{1},m}=\Psi^{\rm(bi)}_{0^{(-)}_{-1},m}=\left(\begin{array}[]{c}0\\\
0\\\ 0\\\ \phi_{0,m}\end{array}\right).$ (49)

The wave function consists of only the $n=0$ conventional Landau level wave
function. Therefore, the FQHE in this bilayer Landau level is exactly the same
as the one in the $0$-th conventional Landau level. For example, we can say
that there is no FQHE in the half-filled $0^{(+)}_{1}$ Landau level.

In order to find the half-filled level with possible Pfaffian function as the
ground state, the most interesting bilayer Landau level is $0^{(+)}_{-1}$ in
the $K$ valley and $0^{(-)}_{1}$ in the $K^{\prime}$ valley Apalkov and
Chakraborty (2011). For small values of $U$, the energy of this state is
almost zero, $\varepsilon\approx 0$. The wave function of $0^{(+)}_{-1}$
Landau level has the form

$\Psi^{\rm(bi)}_{0_{-1},m}=\frac{1}{\sqrt{\tilde{\gamma}_{1}^{2}+2}}\left(\begin{array}[]{c}0\\\
\sqrt{2}\phi_{0,m}\\\ 0\\\
\tilde{\gamma}_{1}\phi_{1,m}\end{array}\right)=\frac{1}{\sqrt{\gamma_{1}^{2}+2\epsilon_{B}^{2}}}\left(\begin{array}[]{c}0\\\
\sqrt{2}\epsilon_{B}\phi_{0,m}\\\ 0\\\
\gamma_{1}\phi_{1,m}\end{array}\right).$ (50)

The wave function is the mixture of $\phi_{0,m}$ and $\phi_{1,m}$ states. For
a relatively small magnetic field, $\epsilon_{B}\ll\gamma_{1}$, the wave
function becomes $(0,0,0,\psi_{1,m})^{T}$, i.e., it is completely identical to
the $n=1$ conventional Landau level. In a large magnetic field,
$\epsilon_{B}\gg\gamma_{1}$, the $0^{(+)}_{-1}$ Landau wave function becomes
$(0,0,\psi_{0,m},0)^{T}$, i.e., it is the same as $n=0$ conventional Landau
level. The interesting aspect of this is that, by varying the magnetic field
the electron-electron interactions within the $0^{(+)}_{-1}$ Landau level
transform from $n=1$ to the conventional $n=0$ case.

##  IV Pfaffian states in graphene

###  IV.1 Graphene monolayer

The electron-electron interactions and correspondingly the nature of the
ground state within a given Landau level are determined by the Haldane
pseudopotentials, which are defined by the corresponding form factor $F(q)$.
For graphene monolayer there are two types of Landau levels: (i) the $n=0$
Landau level and (ii) the $n\neq 0$ levels. For the $n=0$ Landau level, the
form factor is given by

$F_{0}(q)=L_{0}(q^{2}/2)=1.$ (51)

This is exactly the same form factor as the one for the $n=0$ conventional
Landau level, for which there is no $\nu=1/2$ FQHE state. We can then conclude
that in the $n=0$ graphene Landau level, there is no incompressible
$\nu=1/2$-FQHE state with the Pfaffian function as a ground state wave
function. At the same time, all other FHQE states, such as $p/q$ with odd $q$,
have exactly the same properties, i.e., the same ground state wave functions
and the same excitation gaps, as the ones for the $n=0$ conventional Landau
level. This suggests that, as long as there is no inter-Landau level mixing,
the electron systems in the $n=0$ graphene Landau level are identical to the
systems in the $n=0$ conventional Landau level.

On the other hand, for the $n\neq 0$ graphene Landau level, the form factor is

$F_{n}(q)=\frac{1}{2}\left(L_{n-1}(q^{2}/2)+L_{n}(q^{2}/2)\right).$ (52)

For example, for $n=1$, $L_{0}(q^{2}/2)=1$ and $L_{0}(q^{2}/2)=1-q^{2}/2$ and
the form factor is $F_{n}(q)=1-q^{2}/4$. In this case the form factor is the
mixture of the form factors of $n$ and $n-1$ conventional Landau levels. The
electron-electron interactions in this case are completely different from
those in conventional systems. One of these differences is related to
stability, i.e., the magnitude of the excitation gaps of the conventional FQHE
with the filling factor $p/q$, where $q$ is odd. This means that, out of all
the graphene Landau levels, including the $n=0$ Landau level, the largest FQHE
gaps are realized in the $n=1$ Landau level Apalkov and Chakraborty (2006);
Chakraborty and Apalkov (2014). This is different from conventional systems,
where the conventional FQHE states have the largest gaps for the $n=0$ Landau
level.

Our extensive numerical analyses have established that there is no
incompressible $\nu=1/2$ state of the Pfaffian type for $n=0$ and $n=1$
graphene Landau levels Apalkov and Chakraborty (2006). Here, the excitation
gap is small and the overlap of the ground state wave function with the
Pfaffian function is also small. This analysis has been done without
considering inter-Landau level mixing. This mixing can alter the properties of
the half-filled Landau level, resulting in the formation of an incompressible
state. For example, for conventional systems it was shown that the inter-
Landau mixing favors the anti-Pfaffian state over other type of ground states
of the half-filled $n=1$ Landau level Rezayi (2017). Also, both for
conventional and graphene systems, the Landau level mixing crucially modifies
for the $1/2$-FQHE state the Haldane pseudoptentials $V_{1}$, $V_{3}$, and
$V_{5}$ Peterson and Nayak (2013). The Landau level mixing is also more
pronounced at higher Landau levels. It should be noted that, recent
experimental results Kim et al. (2019) indeed suggest that the incompressible
$\nu=1/2$ state may be realized in monolayer graphene, albeit in higher $n=3$
Landau level. Here, the inter-Landau mixing can be important and it can be the
reason for the incompressible $\nu=1/2$ state being present in the higher
Landau level in graphene. Further, the even-denominator incompressible states
have been observed experimentally in low Landau levels in graphene within some
range of the magnetic fields Zibrov et al. (2018), but the origin of those
states is yet to be clearly determined and it can perhaps be related to
multicomponent fractional quantum Hall states.

###  IV.2 Bilayer graphene

For the general bilayer Landau level, the form factor is given by the
expression (53) below. Due to the bilayer nature of the system, there are
extra parameters which can control the form factor and correspondingly the
interaction strength. These parameters are the bias voltage and the direction
of the magnetic field, i.e., the component of the magnetic field that is
parallel to the bilayer. As we have mentioned above, in bilayer graphene there
are two ’special’ Landau levels $0_{-1}^{(+)}$ ($K$ valley) and $0_{1}^{(-)}$
($K^{\prime}$ valley) for which the Pfaffian function can be the ground state
of the half-filled Landau level Apalkov and Chakraborty (2011); Chakraborty
and Apalkov (2013).

Figure 7:  Panel (a): Overlap of the ground state wave function of $\nu=1/2$
state with the Pfaffian state as a function of the magnetic field. Panel (b):
the collective excitation gap of the $\nu=1/2$ state as a function of the
magnetic field. The data are shown for the Landau level $0_{-1}^{(+)}$ of the
bilayer graphene and for two values of the interlayer hopping integral,
$\gamma_{1}$. Reproduced with permission from Apalkov and Chakraborty (2011).

Extensive studies of the finite-size system of bilayer graphene have been
performed by us in the spherical geometry Haldane (1983); Haldane and Rezayi
(1985); Fano et al. (1986). Those studies have revealed that for all bilayer
Landau levels, except the ones $0_{-1}^{(+)}$ and $0_{1}^{(-)}$, the overlap
of the $\nu=\frac{1}{2}$ ground state with the Pfaffian state is relatively
small (less than $0.5$). Therefore, for those Landau levels, the
$\nu=\frac{1}{2}$ system is gapless and compressible. A different situation
occurs for the Landau levels $0_{-1}^{(+)}$ and $0_{1}^{(-)}$. These levels
belong to the $K$ and $K^{\prime}$ valleys, respectively, and have exactly the
same interaction properties. Hence, it is enough to consider only one of these
levels, e.g., $0_{-1}^{(+)}$, as for the other Landau level, $0_{1}^{(-)}$,
the results are exactly the same.

For the $0_{-1}^{(+)}$ Landau level the form factor is

$F_{0_{-1}}(q)=\left[\frac{\gamma_{1}^{2}}{\gamma_{1}^{2}+2\epsilon_{B}^{2}}\right]L_{1}(q^{2}/2)+\left[\frac{2\epsilon_{B}^{2}}{\gamma_{1}^{2}+2\epsilon_{B}^{2}}\right]L_{0}(q^{2}/2).$
(53)

The strength of magnetic field is determined by the parameter $\epsilon_{B}$.
With increasing magnetic field, i.e., with increasing $\epsilon_{B}$, the form
factor of the $0_{-1}^{(+)}$ level goes through the following main regions:
(i) for small $B$, $\epsilon_{B}\ll\gamma_{1}$, the form factor is
$L_{1}(q^{2}/2)$ and identical to the one of the $n=1$ conventional Landau
level; (ii) at $\epsilon_{B}=\gamma_{1}/\sqrt{2}$, the form factor is
$\frac{1}{2}[L_{0}(q^{2}/2)+L_{1}(q^{2}/2)]$ and is the same as the one of the
$n=1$ Landau level of monolayer graphene; (iii) at
$\epsilon_{B}\gg\gamma_{1}$, the form factor is $L_{0}(q^{2}/2)$ and is the
same as the one of the $n=0$ conventional Landau level.

Since the $\nu=1/2$ incompressible state is realized only in the $n=1$
conventional Landau level, then for bilayer graphene, the $\nu=1/2$ Pfaffian
state should be observed only for relatively small values of the magnetic
field. At the same time, it so happens that as a function of the magnetic
field, the characteristics of the $\nu=1/2$ system is not monotonic. In Fig. 7
(a) the overlap of the $\nu=1/2$ wave function with the Pfaffian function is
shown as a function of the magnetic field. The calculations were done in a
spherical geometry for $N_{e}=14$ electron system. The overlap has a maximum
at a finite magnetic field. The position of the maximum depends on the
interlayer hopping integral, $\gamma$. For example, for $\gamma=400$ meV, the
maximum overlap occurs for $\approx 10$ T. The profile of the overlap is also
correlated with the gap of the collective excitation of the system [see Fig.
7( b)]. The energy gap also has a maximum at a finite magnetic field. Since in
a small magnetic field, $B\rightarrow 0$, the $0_{-1}^{(+)}$ bilayer Landau
system is equivalent to $n=1$ conventional Landau systems, we are allowed to
conclude that the stability of the Pfaffian $\nu=\frac{1}{2}$ state in bilayer
graphene can be increased compared to that of the conventional systems.

The position of the maximum in Fig. 7 can be analyzed further by looking at
some dimensionless quantities. For instance, in dimensionless units the
maximum is achieved at $\gamma_{1}/\epsilon_{B}\approx 4.9$. This means that,
in terms of the magnetic field, the position of the maximum of the overlap is
approximately proportional to $\gamma_{1}$, which is also seen in Fig. 7.

Using the phase diagram shown in Fig. 2, which characterizes the stability of
the $\nu=1/2$ state described by the Pfaffian function, we can illustrate how
the interaction parameters in the $0_{-1}^{(+)}$ Landau level change with the
magnetic field [see Fig. 8]. For a small magnetic field, the ground state of
the system is well described by the Pfaffian function, while the overlap and
the corresponding excitation gap reach their maximum at intermediate values of
the magnetic field and finally, for a large magnetic field, $>100$ T, the
electron system becomes compressible.

Another parameter that can be used to control the stability of the Pfaffian
ground state, is the magnetic field that is parallel to the 2D layer
Chakraborty and Apalkov (2013). The reason why the parallel magnetic field
changes the wave functions and the interaction potential is due to the bilayer
nature of the system. In this case the system has extra dynamics in the $z$
direction, which is described in the model as an inter-layer hopping. The
parallel component of the magnetic field can be introduced into the
Hamiltonian of bilayer graphene through a Peierls substitution as an extra
position-dependent phase factor in the inter-layer hopping integral. The
details of that study can be found elsewhere Chakraborty and Apalkov (2013).

Figure 8:  Trajectory of the inter-electron interaction with varying magnetic
field. The trajectory is shown by a solid red line in the plane
$(V_{1}/V_{5})-(V_{3}/V_{5})$ for the Landau level $0_{-1}^{(+)}$ of the
bilayer graphene. The initial point of the trajectory (at $B=0$) corresponds
to the conventional system at $n=1$ Landau level, while the final point (at
$B=\infty$) corresponds to the conventional system at the $n=0$ LL. The
hatched region illustrates the compressible $\nu=1/2$ state, while the blank
region corresponds to the incompressible $\nu=1/2$ state following the Ref.
Storni et al. (2010) and Fig. 2. The crossing of the boundary between the
compressible and incompressible states occurs at $B\sim 100$ Tesla for the
interlayer hopping integral $\gamma_{1}=400$ meV. The blue dashed line shows
the region of large overlap of the ground state wave function with the
Pfaffian function. Reproduced with permission from Apalkov and Chakraborty
(2011).

The effect of the parallel component of the magnetic field is illustrated in
Fig. 9, where the parameters $V_{1}/V_{5}$ and $V_{3}/V_{5}$ are shown as a
function of the perpendicular component of the magnetic field and for
different values of its parallel component. The most stable Pfaffian state is
illustrated by the dashed lines. The results demonstrate that with increasing
parallel component of the magnetic field the values of pseudopotentials, which
correspond to the most stable Pfaffian state, are realized for smaller values
of the perpendicular component of the magnetic field. Therefore, the parallel
component of the magnetic field can be used to tune the stability of the
Pfaffian state and the magnetic field where it can be realized.

Figure 9:  Ratios of pseudopotentials at two values of angular momentum
$V_{1}/V_{5}$ [panel (a)] and $V_{3}/V_{5}$ [panel (b)] are shown as the
function of perpendicular component of magnetic field and for different
parallel components of magnetic field, $B_{\parallel}=5$ T, 50 T, and 100 T.
The data are shown for Landau level $0_{-1}^{(+)}$ of bilayer graphene. The
hatched regions correspond to the values of pseudopotentials for which the
large overlap of the ground state wave function with the Pfaffian function and
large excitation gap of incompressible ground state are achieved. Reproduced
with permission from Chakraborty and Apalkov (2013).

###  IV.3 Finding the Pfaffians

Even-denominator fractional quantum Hall effect has been observed
experimentally in bilayer graphene Zibrov et al. (2017); Li et al. (2017). It
was found that the corresponding states were spin-polarized and it was
suggested that they are Pfaffian or anti-Pfaffian states corresponding to the
half-filled Landau level. Such states were observed at high energy bilayer
Landau levels that are a mixture of the $n=1$ and $n=0$ conventional Landau
levels. It was noted that the FQHE was observed only at the filling factor
$\nu=3/2$, but not at $\nu=1/2$ and $5/2$ Li et al. (2017). This observation
illustrates the high sensitivity of the half-filled incompressible states to
the structure of the electron wave functions which determine the electron-
electron interaction potentials. Also the measured activation gap of the half-
filled incompressible state was several times larger than the largest gaps
measured in the GaAs systems Zibrov et al. (2017), which also illustrates that
the half-filled FHQE state is much more stable in bilayer graphene then in
conventional electron systems. It was observed that the stability of the
experimentally found $1/2$-FQHE states is sensitive to both the magnitude of
the magnetic field and the perpendicular component of the electric field. With
variation of these parameters, transitions from the compressible to the
incompressible states have been observed Li et al. (2017). Such a behavior is
in complete accord with our theoretical predictions elaborated in the previous
section.

##  V Conclusion

Two-dimensional electron systems placed in a strong magnetic field can support
charged excitations with fractional charge and non-Abelian statistics. These
excitations are possible only due to the nature of electron-electron
interactions of a special type. The example of such a system is an electron
gas in a given Landau level that is only half filled. Under a special profile
of the interaction potential, the ground state of the $\nu=1/2$-Landau level
is described by the Pfaffian function. Here the profile of interaction
potential is determined by the form factor of the corresponding Landau level.
As a result, for the conventional electron systems, the incompressible
$\nu=1/2$-FQHE state is realized only in the $n=1$ Landau level, but for the
graphene monolayer which has the relativistic-like low-energy dispersion (that
of the Dirac fermions), there is no incompressible $\nu=1/2$ Pfaffian state in
any Landau level. This theoretical conclusion is based on the properties of
electrons within a given Landau level, without considering the admixture of
other Landau levels. The Landau level mixing, which is especially relevant for
higher Landau levels, can modify the properties of the half-filled Landau
states [see Rezayi (2017); Rezayi and Simon (2011); Simon and Rezayi (2013);
Pakrouski et al. (2015); Peterson and Nayak (2013); Luo and Chakraborty
(2017). It can also open up the possibility to observe the $\nu=1/2$-FQHE in
higher Landau levels in graphene Kim et al. (2019).

In this context, bilayer graphene is truly unique because for bilayer graphene
with AB staking, there are two special Landau levels: one in the $K$ valley
and another one in the $K^{\prime}$ valley, for which, there is a range of
magnetic fields where the $\nu=1/2$ ground state is determined by the Pfaffian
function. Stability of such a ground state, i.e., its collective excitation
gap, depends on the magnetic field and for a finite magnetic field, the
$\nu=1/2$-FQHE state becomes even more stable than the corresponding state in
a conventional electron system. Another important property of bilayer graphene
is that there are external parameters, such as the bias voltage and the in-
plane magnetic field, that can change the inter-electron interaction strength
within a given Landau level and correspondingly change the properties of the
$\nu=1/2$ Pfaffian state.

In addition to the unique properties of the $\nu=1/2$ Pfaffian state, the
system with half filling of a given Landau level has another interesting
feature. In particular, when the projection on a given Landau level is
considered, i.e., only the states within this Landau level are taken into
account with inter-landau level mixing, then the system itself has a particle-
hole symmetry. It means that it can be described either as the $\nu=1/2$
particle (electron) system or the $\nu=1/2$ hole system. The Pfaffian
function, which describes the $\nu=1/2$ particle system, is not invariant
under the exchange of particles and holes. When the particle-hole symmetry
operation is applied to the Pfaffian function it generates the Pfaffian
conjugated function, which is known as the anti-Pfaffian Levin et al. (2007);
Lee et al. (2007). The anti-Pfaffian function is given by

$\Psi_{\mbox{aPf}}=\mbox{Pf}\left[\frac{z_{i}-z_{j}}{(z_{i}^{*}-z_{j}^{*})^{2}}\right]\prod_{i<j}(z_{i}-z_{j})^{2}\exp\left(-\sum_{i}\frac{z_{i}^{2}}{4\ell_{0}^{2}}\right).$
(54)

In addition to the Pfaffian and anti-Pfaffian functions, another type of
function have been proposed in the literature. It is called the PH-Pfaffian
and is symmetric with respect to the particle-hole symmetry operator Son
(2015). The PH-Pfaffian is given by the following expression

$\Psi_{\mbox{PH-
Pf}}=\mbox{Pf}\left[\frac{1}{z_{i}^{*}-z_{j}^{*}}\right]\prod_{i<j}(z_{i}-z_{j})^{2}\exp\left(-\sum_{i}\frac{z_{i}^{2}}{4\ell_{0}^{2}}\right).$
(55)

All these functions, Pfaffian, anti-Pfaffian, and PH-Pfaffian, are of the
superconductor-type with Cooper pairing but having different pairing
symmetries. The corresponding charged excitations are non-Abelian anyons.
Discussion of the conditions under which different Pfaffian-type functions are
to be realized and their properties can be found in the literature Simon and
Rezayi (2013); Rezayi and Simon (2011); Zaletel et al. (2015); Rezayi (2017);
Sodemann and MacDonald (2013); Pakrouski et al. (2015); Peterson and Nayak
(2013); Wójs et al. (2010); Bonderson et al. (2011); Antonić et al. (2018);
Hsin et al. (2020); Rezayi et al. (2021). Confirmation of the presence of
Pfaffian functions in half-filled FQHE will provide another open door to our
understanding of this distinctive many-body phenomena that have enthralled us
for more than four decades. Additionally, it would be equally exciting to
watch that the work of Cayley and Pfaff finally see the light of day through
bilayer graphene.

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