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\DOI{10.5802/crphys.232}
\datereceived{2024-10-18}
\daterevised{2025-01-10}
\dateaccepted{2025-01-13}
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\def\overbar{\bar}

\begin{noXML}

\CDRsetmeta{articletype}{congress}

\title{Interlayer phase coherence and composite fermions}

\alttitle{Coh\'erence de phase entre bicouche et fermions composites}

\author{\firstname{Thierry} \lastname{Jolicoeur}\CDRorcid{0000-0003-2510-0473}}
\address{Paris-Saclay University, CNRS, CEA, IPhT, France}
\email{thierry.jolicoeur@ipht.fr}

\begin{abstract}
The fractional quantum Hall effect (FQHE) realized in two-dimensional
electron systems is explained by the emergent composite fermions (CF)
out of ordinary electrons. It is possible to write down explicit
wavefunctions explaining many if not all FQHE states. In bilayer
systems there is a regime at integer filling of the lowest Landau level
that displays a spontaneous breakdown of the \textit{U}(1) relative phase
between the two layers. This can be seen as interlayer phase coherence
(ILC) in terms of \textit{electrons}. Recent experiments in double
layer samples of graphene have revealed the appearance of many FQHE
states unique to the bilayer case. We discuss extensions of the CF idea
in this situation as well as the possible existence of ILC of CFs.
\end{abstract}

\begin{altabstract}
L'effet Hall quantique fractionnaire (FQHE) r\'{e}alis\'{e} dans les
syst\`{e}mes \'{e}lectroniques bidimensionnels s'explique par
l'\'{e}mergence de fermions composites (CF) \`{a} partir
d'\'{e}lectrons ordinaires. Il est possible d'\'{e}crire des fonctions
d'onde explicites expliquant de nombreux, voire tous les \'{e}tats de
l'effet Hall quantique fractionnaire. Dans les syst\`{e}mes form\'es de
bicouches, il existe un r\'{e}gime de remplissage entier du niveau de
Landau le plus bas qui pr\'{e}sente une brisure spontan\'{e}e de la
phase relative \textit{U}(1) entre les deux couches. Cela peut \^{e}tre
consid\'{e}r\'{e} comme une coh\'{e}rence de phase entre les couches
(ILC) en termes d'\textit{\'{e}lectrons}. Des exp\'{e}riences
r\'ecentes dans des \'{e}chantillons \`{a} double couche de
graph\`{e}ne  ont r\'{e}v\'{e}l\'{e} l'apparition de nombreux \'{e}tats
FQHE propres au cas bicouche. Nous discutons des extensions de
l'id\'{e}e de CF dans cette situation ainsi que de l'existence possible
d'ILC de CFs.
\end{altabstract}

\keywords{\kwd{Quantum Hall effect}
\kwd{Composite fermions}
\kwd{Interlayer phase coherence}
\kwd{Bilayer graphene}}

\altkeywords{\kwd{Effet Hall quantique}
\kwd{Fermions composites}
\kwd{Coherence entre couches}
\kwd{Bicouche de graphene}}

\maketitle

\end{noXML}

\section{Introduction}
In the low-temperature regime quantum fluids may display physical
properties governed by spontaneous symmetry breaking. This is the case
of superfluids with the symmetry associated to the number of neutral
particles. This also happens in superconductors with breakdown of the
gauge symmetry of electrodynamics due to the condensation of Cooper
pairs. Another set of phenomenon is due to the development of strong
topological order known to happen in two-dimensional electrons gases in
a magnetic field. This is the realm of the fractional quantum Hall
effect  (FQHE)~\cite{Tsui1982,Laughlin}. In this short review we will
discuss the interplay of both phenomena in the context of graphene
bilayers. Indeed it has been possible recently to fabricate devices by
stacking monolayers of graphene separated by a barrier of hexagonal
boron nitride leading to an atomic scale physical separation between
the layers and essentially no tunneling. Such devices have revealed a
complex pattern of FQHE states and some of them may be described by
spontaneous interlayer phase coherence of emerging
entities~\cite{jainbook} called ``composite fermions''. These composite
fermions are in this case even more complex objects~\cite{CsathyJain}
than the ones appearing in single layer devices.

We will focus only on the approaches based on explicit wavefunctions
written in terms of electron coordinates that have been shown to be
extremely useful~\cite{jainbook,scarola,KamillaCF}.


\section{Fractional quantum Hall effect}
To set the stage for the wavefunctions describing FQHE we first give a
short introduction to Landau level physics and the integer quantum Hall
effect which does not involve in a crucial way electron--electron
interactions. The emerging dressed electrons called ``composite
fermions'' that describe the FQHE will essentially be ruled by this
regime. Numerous books are available, giving all necessary
details~\cite{jainbook,dassarma,heinonenbook,Prange}.

\subsection{Integer quantum Hall effect}
We consider a system of $N_{\mathrm{e}}$ electrons confined to two-dimensional
plane under a perpendicular uniform magnetic field. This is a physical
situation describing some electron gases in semiconductors,
heterostructures or quantum wells. A simple description of this
situation is given by the following Hamiltonian:
{\begin{eqnarray}
\mathcal{H}=\frac{1}{2m^*}(p+e\mathcal{A})^2,
 \label{defHam2D}
\end{eqnarray}}\unskip
where the vector potential can be taken in the so-called symmetric
gauge $\mathcal{A}=(\mathcal{B}\times r)/2$. The effective
mass $m^*$ is a material dependent parameter. Solving for eigenvalues
of the one-body Hamiltonian Equation~(\ref{defHam2D}) gives the Landau levels
that have energies $E_N=(N+1/2)\hbar\omega_{\mathrm{c}}$ with $N$ a positive
integer and $\omega_{\mathrm{c}}=eB/m^*$ the cyclotron frequency. There a
macroscopic degeneracy in this situation: for each value of $N$ there
are exactly $N_\phi$ states that have this peculiar energy where
$N_\phi$ is the number of flux quanta through the system
$N_\phi=B\times A/(h/e)$. The FQHE is strongest when all electrons lie
in the lowest Landau level (LLL) whose one-body eigenstates are given
by the following formula:
{\begin{eqnarray}
 \phi_M(r)=z^M \exp (-|z|^2/4\ell^2),
 \label{onebody}
\end{eqnarray}}\unskip
where we use the complex coordinate in the plane $z=x-\mathrm{i}y$ and we have
defined the magnetic length $\ell=\sqrt{\hbar/(eB)}$. The power $M$ is
a positive or zero integer (we omit the normalization). With the
special Landau level spectrum it is clear that even without considering
interactions there are special cases when one has exact filling of an
\textit{integer} number of Landau levels, say $p$. With the formula for
the degeneracy this happens when:
{\begin{eqnarray}
 \nu = \frac{N_{\mathrm{e}}}{N_\phi} = \frac{nh}{eB} = p,
 \label{fillfactor}
\end{eqnarray}}\unskip
where we have defined the filling factor $\nu$ which is the fraction of
occupied states, and $n$ is the areal density of electrons. A naive
thought is that in this integer case one has a magnetic-field induced
insulator due to the presence of cyclotron gap $\hbar\omega_{\mathrm{c}}$. This
gap lies between the highest occupied $p$th Landau level and the lowest
empty $p+1$th level at zero temperature. This is the physical picture
for the bulk sample. In a real finite-size sample the Landau levels are
bent when approaching the boundaries and as a consequence the Landau
levels will cross the Fermi energy at some points that we may call
``Fermi points'' at the real-space edges of the system. As a
consequence there is electric conduction with appearance of a Hall
voltage $V_{\mathrm{H}}$ perpendicular to an imposed current $I$ and its value is
given by:
{\begin{eqnarray}
 V_{\mathrm{H}} = \frac{1}{p}\frac{h}{e^2}I= \frac{1}{\nu}\frac{h}{e^2}I,
 \label{HallRes}
\end{eqnarray}}\unskip
so that the Hall resistance defined by $R_{\mathrm{H}}=V_{\mathrm{H}}/I$ is quantized in
units of $h/e^2$ and $I$ is the current intensity in the sample. The
presence of disorder in the sample leads to the appearance of plateaus
as as a function of the magnetic field where the Hall resistance stays
constant at the value Equation~(\ref{HallRes}). This phenomenon, the
so-called integer quantum Hall effect~\cite{Halperin1983}, does not
involve in any crucial way the electron--electron interactions. 

The closed-shell argument above suggests that for fractional filling no
similar quantization should develop since partial filling of a flat
band enlarged by disorder may host a Fermi liquid albeit with a large
density of states. But striking experiments found evidence for the
existence of a fractional quantum Hall effect when the filling factor
$\nu$ is some rational fraction and the phenomenology of current
transport is the same as in the integer case. The original discovery
involved a plateau in the Hall resistance $R_{\mathrm{H}}=3h/e^2$ corresponding to
filling factor $\nu=1/3$ in the LLL. To understand the fractional case
one has to dig deep into the many-body problem of interacting electrons
in the LLL. A generic many-body wavefunction is a sum of products of
one-body states Equation~(\ref{onebody}) and thus can be written as a
polynomial $\mathcal{P}$ in complex coordinates $z_i$ times a universal
(state-independent) Gaussian factor:
{\begin{eqnarray}
 \Psi(z_1,\ldots,z_N)=\mathcal{P}(z_1,\ldots,z_N)  \exp 
 \left(-\sum_i |z_i|^2/4\ell^2\right).
\end{eqnarray}}\unskip
In the spin-polarized case the polynomial $\mathcal{P}$ is
antisymmetric to comply with the Pauli principle. In the general case
it is hard to guess what are the relevant polynomials $\mathcal{P}$ but
in the realm of the FQHE the successfull approach leads to explicit
guess of the polynomials, the most famous case being the Laughlin
wavefunction. We conclude this section by the giving the expression of
$\mathcal{P}$ for the completely filled LLL. We first observe that the
one-body states Equation~(\ref{onebody}) has a probability distribution which
is a ring centered at $z=0$ (because of the choice of gauge) whose
radius grows with the exponent $M$. So fully filled LLL means that we
occupy all  one-body states from $M=0$ up to a maximal value given by
$M=N-1$ without any vacancy $z^0,z^1,z^2,\ldots,z^{N-1}$. This state is
exactly a Slater determinant:
{\begin{eqnarray}
 \Psi_{\nu=1}=\det(z_i^{j-1})=\prod_{i<j}(z_i-z_j),\quad i,j=1,\ldots,N,
 \label{VDM}
\end{eqnarray}}\unskip
where we have used the fact that the determinant is the so-called
Vandermonde determinant. From now on we will omit the Gaussian factor
from the one-body states since it is independent of the state under
consideration. If we draw the charge density of state $\Psi_{\nu=1}$ in
real space we find that it has the shape of a very flat pancake with
uniform density in its interior and going to zero very quickly at the
edge with a characteristic length $\ell$.

\subsection{Laughlin wavefunction}
To explain the properties of the quantum state of electrons at
$\nu=1/3$ Laughlin proposed the following candidate wavefunction:
{\begin{eqnarray}
 \Psi^{(3)}=\prod_{i<j}(z_i-z_j)^3 \exp \left(-\sum_i |z_i|^2/4\ell^2\right).
\end{eqnarray}}\unskip
If we compute the charge density by some means we find also a very flat
pancake  as in state Equation~(\ref{VDM}) but with a mean density three
times lower, corresponding to a uniform electronic state with filling
factor 1/3 (this would be $1/m$ if we were to put power $m$ to the
Vandermonde factor). This wavefunction has very good trial energy but
is not an exact eigenstate of the Coulomb interacting electrons. So its
theoretical status is not immediately clear. Another ansatz was the
idea of starting from the Slater determinant of $\nu=1$ and fill only
one out of three orbitals: $z^0,z^3,z^6,\ldots.$ Since this is still a
determinant it can be easily computed giving
$\prod_{i<j}(z_i^3-z_j^3)$. This is not a good candidate for the FQHE
state at $\nu=1/3$. While it has the correct filling factor by
construction it has higher energy than the Laughlin state because it
lacks the big correlation hole around each electron due to the 
$(z_i-z_j)^3$ factor in the Laughlin wavefunction. Also this state
called the Tao-Thouless state does not have a uniform density in real
space and does not lead to the correct phenomenology of the FQHE at
$\nu=1/3$, contrary to the Laughlin  state.


To clarify the status of the Laughlin state we now show to solve by
elementary means any two-body problem in the LLL. The interacting
Hamiltonian is:
{\begin{eqnarray}
 \mathcal{H}_2=\frac{1}{2m^*}(p_1+e\mathcal{A}_{r_1})^2+
 \frac{1}{2m^*}(p_2+e\mathcal{A}_{r_2})^2 + V(r_1-r_2),
\end{eqnarray}}\unskip
where the positions of the electrons are $r_1,r_2$ and $V$ is the
interaction potential. In the context of the FQHE it is given by the
Coulomb interaction $V(r)=e^2/r$. We now introduce the center of mass
and relative coordinates $R=(r_1+r_2)/2$, $r=r_1-r_2$. The fact that
the vector potential is linear in the coordinates leads to a
simplification:
{\begin{eqnarray}
 \mathcal{H}_2=\frac{1}{2M}(\mathcal{P}+2e\mathcal{A}_{R})^2+
 \frac{1}{2\mu}(p_r+(e/2)\mathcal{A}_{r})^2 + V(r),
\end{eqnarray}}\unskip
so the center of mass as well as the relative particle will live in
their own separate Landau levels. If we focus on the relative particle
we can take as a basis the one-body states of Equation~(\ref{onebody}) that
are eigenstates of the angular momentum. If we consider a potential
$V(r)$ which is rotationally invariant it will be automatically
diagonal in such a basis. The eigenenergies are thus given by
$V_m=\langle\phi_m|V(r)|\phi_m\rangle$. The numbers $V_m$ are often
called pseudopotentials and are given by a simple integral. In the
Coulomb case the pseudopotentials decrease as $1/\sqrt{m}$. For
spinless electrons only the odd values of $m$ matter due to
wavefunction antisymmetry. As a consequence, the full many-body
interacting Hamiltonian can be written as:
{\begin{eqnarray}
 \mathcal{H}=\sum_{i<j}\, \sum_{m=0}^\infty \, V_m\,
 \, \mathcal{P}^{(m)}_{ij}
 \label{generic}
\end{eqnarray}}\unskip
where we have defined $\mathcal{P}^{(m)}_{ij}$ the projector onto the
state of relative angular momentum $m$ for the pair $i,j$ of particles.




If now we retain only the $m=1$ projector in the Hamiltonian
Equation~(\ref{generic}) then we note that the Laughlin state is an
exact zero-energy eigenstate. Indeed since for all pairs of particles
there is an overall factor $z^3$ with $z$ the relative particle of pair
$i,j$ it means that it has no weight onto $m=1$. For the experimentally
relevant case of Coulomb interaction the $V_{m=1}$ pseudopotential is
indeed the strongest which gives some weight to the relevance of the
Laughlin state beyond the fact of having a good energy. The
parametrization of Equation~(\ref{generic}) suggests a way to
interpolate between the Coulomb case and the hard-core limit involving
only the projector $\mathcal{P}_1$ by varying the set of discrete
pseudopotentials $V_m$. This idea was successfully implemented by F. D.
M. Haldane (see his contribution in book~\cite{Prange}) showing that
the Laughlin state is smoothly connected to the Coulomb ground state
and thus these two states share the same physics.


As a polynomial in the electronic coordinates the Laughlin ansatz gives
us a state which is already fully factorized.  This is a mathematical
statement concerning the polynomial appearing in the many-body
wavefunction. It does not mean that the quantum state is factorizable,
the Laughlin state is \textit{not} a Slater determinant. Indeed it is
known to be a case of strong entanglement, a property which does not
exist in Slater determinant states. The zeros of this polynomial are
located exactly at the positions of the electrons, a very special
feature which is not true in general.  Let us rewrite the Laughlin
polynomial in the following way:
{\begin{eqnarray}
 \prod_{i<j}(z_i-z_j)^3=\prod_{i<j}(z_i-z_j)\times \prod_{i< j}(z_i-z_j)^2
 =\det(z_i^{j-1}) \times \prod_{i< j}(z_i-z_j)^2.
\end{eqnarray}}\unskip
Here we have explcitly factored out one power of the Vandermonde
determinant Equation~(\ref{VDM}). We observe that  if we pick  one electron 
and drag it around the closest neighbor we see that the phase of the
wavefunction changes by $6\uppi$ while antisymmetry due to the Pauli
principle requires only a $2\uppi$ turn. We interpret this property by
saying that there are exactly two vortices bound to each electron in
this state of matter. The vortex attachment is directly due to the
extra factor $\prod_{i< j}(z_i-z_j)^2$. While this reveals a
fundamental property of the FQHE state, it does not lead immediately to
a recipe to construct other candidate wavefunctions for filling factors
beyond $\nu=1/3$. We now show in the next section how another rewriting
of the Laughlin polynomial leads naturally to the so-called composite
fermion construction.


\subsection{Composite fermion wavefunctions}\label{JCF}

Experiments revealed also the appearance of the FQHE for fractions
other than 1/3. The most prominent set of such states appears for
filling factors $\nu=p/(2p\pm 1)$ with $p$ integer, asking for an
explanation in terms of wavefunctions beyond the  Laughlin state. One
such state is given by the so-called composite fermion construction
that we describe now. Much of our intuition of electron systems is
based on Slater determinants and occupied/empty orbitals so the first
step is to rephrase the successfull Laughlin state in this language. We
write the  correlation factor of the Laughlin state in the following
way:
{\begin{eqnarray}
 \prod_{i<j}(z_i-z_j)^3=\prod_{i<j}(z_i-z_j)\times \prod_{i\neq j}(z_i-z_j)
 =\det(z_i^{j-1}) \times \prod_{i\neq j}(z_i-z_j).
\end{eqnarray}}\unskip
Distributing the last among the columns of the determinant we arrive at
the following identity:
{\begin{eqnarray}
 \prod_{i<j}(z_i-z_j)^3
 =\det\left(z_i^{j-1} \prod_{ k\neq i}(z_i-z_k)\right).
\end{eqnarray}}\unskip
This can be interpreted as a Slater determinant for a filled Landau level
provided one replaces the one-body wavefunctions by correlated one-body
wavefunctions:
{\begin{eqnarray}
 z^m \rightarrow z^m \prod_j (z-z_j),
 \label{Jastrow}
\end{eqnarray}}\unskip
where the product over $j$ is over all other particles. The extra
correlation factor is called a Jastrow factor in many-body physics. If
we decide to adopt such correlated orbitals instead of the one-body
states we can view the Laughlin state at $\nu=1/3$ as a completely
filled Landau level with $\nu^*=1$. We guess then that excited states
may involve higher-lying Landau levels provided one uses modified
orbitals including the correlation factor Equation~(\ref{Jastrow}). In Landau
levels other than the lowest the one-body eigenstates involve the
complex conjugate $z^*$ in addition of the complex coordinate $z$. Let
us call $\phi_{N,m}(z,z^*)$ such a state in the $N$th Landau level. We
add Laughlin-style correlations by making the product as above:
{\begin{eqnarray}
 \phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)  \prod_j (z-z_j).
\end{eqnarray}}\unskip
Such a state however does not live in the LLL due to the appearance of
$z^*$ factors. Since there is overwhelming evidence for FQHE in the LLL
only we want to project such states into the LLL and then play the game
of making Slater determinants. The operation of projection onto the LLL
amounts to putting all $z^*$  to the left-hand side of the formula and
next replacing them by the operator $2\partial/\partial z$. The result
is then entirely in the LLL. In the $N=1$ Landau level this
manipulation amounts to the substitution:
{\begin{eqnarray}
  z^m \rightarrow z^m \frac{\partial}{\partial z}\prod_j (z-z_j),
\end{eqnarray}}\unskip
and going to even higher Landau levels simply add more derivatives
acting onto the correlation factor. Since the correlation factors map
$\nu=1/3$ onto $\nu^*=1$ one can say that the composite fermions feel a
reduced magnetic field $B^*=B-2n\phi_0$ by using the definition
Equation~(\ref{fillfactor}) of the filling factor. This immediately suggests
that there will be an IQHE for the composite fermions when $\nu^*=p$
with $p$ filled Landau levels of the CFs. This translates in filling
factors for electrons as $\nu=p/(2p+1)$. One can then write down Slater
determinants made of correlated orbitals to describe such states. This
procedure gives us explicit trial wavefunctions whose energies can be
computed by a simple Metropolis sampling. Many if not all properties
extracted from these CF wavefunctions are in excellent agreement with
the known experimental data and also in agreement with numbers obtained
from exact diagonalization of systems with a (very) small number of
electrons. The series of FQHE states at $\nu=p/(2p+1)$ is prominent in
high-quality samples and is called the Jain series of states. It is
observed  from $p=1$ (the Laughlin state) up to at least $p=10$. It is
important to note that these states have \textit{no} variational
parameters and nevertheless offer a very accurate description of FQHE
states (for detailed comparisons see e.g.~\cite{jainbook}). The
heuristic mapping $B\rightarrow B^*$ also correctly predict that when
$B^*=0$ the CFs form some kind of Fermi sea which is gapless as is
observed in electron gases at\unskip\break $\nu=1/2$.

With the notion of effective magnetic field $B^*$ we note that it may
be negative, leading to states with $\nu^*=-p$ hence $\nu=p/(2p-1)$ as
observed in experiments. It is also easy to generalize the CF
construction to fractions descending from the parent state $\nu=1/5$.
Indeed the Laughlin state can accommodate any odd power of the Jastrow
factor for spin polarized fermions (even power for bosons). The same
line of reasoning leads to series of states with $\nu=p/(4p+1)$ and
$\nu=p/(4p-1)$, again many of such states are observed in nature. When
the filling factors becomes low the FQHE are competing with a crystal
state made out of electrons, called the Wigner crystal which does not
have the same striking properties as the FQHE states.

The CF construction also gives a very simple picture of excited states.
Since the CF are filling an integer of pseudo LLs a first type of
excitations consists of promoting a CF from the topmost filled pseudo
LL to the lowest empty pseudo LL. This is a neutral excitation with no
change of the number  of electrons or the number of flux quanta. One
may expect that such an excited state has an energy cost given by the
effective cyclotron energy for the CFs. There are also charged excited
states obtained by making a hole in the topmost filled pseudo LL: such
a state is called a quasihole. By reducing the applied magnetic flux
one can also create a situation with only one electron promoted to the
next pseudo LL. This is then the quasielectron\unskip\break state.

In the CF picture one still has to explain why the Hall resistance is
quantized as $R_{xy}=(1/\nu)(h/e^2)$ with the $\nu$ the
\textit{electron} filling while there are $p$ filled pseudo Landau
levels of CF in the series $\nu=p/(2p+1)$. The explanation is that
while the CF contains a charged electron it also binds two vortices. 
Indeed in the construction of the CF states the Jastrow factor squared
is always present and in line with the case of the Laughlin state we
interpret this factor by saying that there are two vortices bound to
each eletron. The CF vortex carries two units of flux $\phi_0$ and if
such a vortex crosses a Hall bar it will induce a voltage drop
$e=-\mathrm{d}\Phi/\mathrm{d}t=2(h/e)(I/e)$ where $I/e$ is the number of CF per unit time
crossing the Hall bar. So there is an additional contribution to the
Hall voltage: $V_{\mathrm{H}}=(1/p+2)h/e^2 I$. This completes the explanation of
transport phenomenology in the FQHE regime which is mapped onto that of
the IQHE.

Finally we mention that the composite fermions may undergo a pairing
instability. This was proposed by Moore and Read~\cite{MR} who  
introduced yet another intriguing explicit wavefunction called the
Pfaffian:
{\begin{eqnarray}
 \Psi_{\mathrm{Pf}}=\mathrm{Pf}\left(\frac{1}{z_i-z_j}\right)
 \prod_{i<j}(z_i-z_j)^2,
 \label{MR}
\end{eqnarray}}\unskip
where the symbol Pf stands for the Pfaffian of the square matrix
$1/(z_i-z_j)$. If we compute the determinant of an antisymmetric matrix
we find that it is the square of a polynomial of the matrix elements.
This polynomial is called the Pfaffian of the matrix. It appears when
one projects a paired state like the BCS wavefunction onto a state with
fixed number of particles. So the appearance of this peculiar  factor
is indicative of the paired nature of the Pfaffian state. Indeed the
Pfaffian state is a $p$-wave paired state of composite fermions. The
filling factor of the state can be easily computed and is $\nu=1/2$.
This state is not an exact eigenstate of the Coulomb interaction and is
a trial state competing with the Fermi sea of composite fermions that
exists at the same filling factor. In the LLL in semiconductors the
Fermi sea has lower energy and the $\nu=1/2$ state is compressible but
it may be that this state becomes the ground state in the next Landau
level where the effective Coulomb interaction is different from the LLL
so it is a candidate for the FQHE state observed for $\nu=2+1/2=5/2$ in
some very clean devices. This state supports excitations with
non-Abelian statistics which are very interesting quasiparticles. We
will not discuss in more details this important FQHE topic.


\section{Interlayer phase coherence}
\subsection{The role of spin}

We now discuss the modifications of previous ideas when we consider the
spin degree of freedom of charge carriers. The first remark is that the
$\nu=1$ state should be written as:
{\begin{eqnarray}
 \Psi^{(1)}=\prod_{i<j}(z_i-z_j)
 |\uparrow\cdots\uparrow\mathop{\rangle}
 \label{spinnu1}
\end{eqnarray}}\unskip
since by construction it is fully polarized. More general states will
not be a simple product of a spin part times an orbital part. If we
consider the many-body problem of electrons interacting by the Coulomb
potential we note that it has full $SU(2)$ spin rotation symmetry. This
rotation symmetry will be broken down to $U(1)$ by the Zeeman coupling
of the external field to the total spin. The state (\ref{spinnu1}) has
a spin projection onto the $z$ axis equal to $S^z_{\mathrm{tot}}=+N/2$ and by
rotational symmetry of the Hamiltonian it is also member of a multiplet
of total spin $S_{\mathrm{tot}}=S^z_{\mathrm{tot}}$. This multiplet is exactly degenerate
without Zeeman effect.
{\begin{eqnarray}
 \Psi^{(1)}=\prod_{k}c^\dagger_{k\uparrow}|0 \mathop{\rangle}
 \label{spinnu1-2nd}
\end{eqnarray}}\unskip
The $S^z_{\mathrm{tot}}=0$ member of this spin multiplet is obtained
by acting repeatedly with the spin lowering operator:
{\begin{eqnarray}
 |S^z=0\mathop{\rangle} =(S^-_{\mathrm{tot}})^{N/2}\,\,\prod_{k}c^\dagger_{k\uparrow}|0 \mathop{\rangle} .
 \label{spinzero}
\end{eqnarray}}\unskip
In this formula the spin operator acts only on the spin degrees of
freedom but does not change the orbital part of the state. If we write
the state in first quantization we decide to call $z_i$ the coordinates
of $\uparrow$ spins and $w_k$ of $\downarrow$ spins and the orbital
part of the $S^z_{\mathrm{tot}}=0$ state is then:
{\begin{eqnarray}
 \Psi^{(1)}=\prod_{i<j}(z_i-z_j)\prod_{k<l}(w_k-w_l)
 \prod_{i,k}(z_i-w_k) .
\end{eqnarray}}\unskip
The full wavefunction with the spin part is the antisymmetrized product
of this orbital factor and of the zero spin state 
$|\uparrow\cdots\uparrow\downarrow\cdots\downarrow\mathop{\rangle}$. Halperin has
proposed a generalization of the Laughlin wavefunction appropriate to
states involving spin:
{\begin{eqnarray}
 \Psi^{(mmn)}=\prod_{i<j}(z_i-z_j)^m\prod_{k<l}(w_k-w_l)^m
 \prod_{i,k}(z_i-w_k)^n .
 \label{Hal}
\end{eqnarray}}\unskip
Again this only the orbital part---it has to be supplemented by the
spin part  and antisymmetrized. Evaluation of spin-independent
observables like the  Coulomb energy only involve the orbital part so
we simply omit the spin part for clarity. The filling factor of the
trial state $mmn$ has to be computed and is found to be $\nu=2/(m+n)$
where the filling factor refers to the total filling including both
species.  In general it is not an eigenstate of total spin. Special
cases include $m=n+1$ which is a singlet $S_{\mathrm{tot}}=0$ and $m=n$ states
that are ferromagnetic states as the state Equation~(\ref{spinnu1}). For
example the $(332)$ Halperin state describes the singlet state
$\nu=2/5$ that is a FQHE state appearing in samples with small Zeeman
effect.

We note that at a given filling factor there are several Halperin state
that are competing. For example at $\nu_{\mathrm{tot}}=1/3$ one may construct
the $(333)$ state which has Laughlin correlations irrespective of the
particle index and one can also build the $(551)$ state which may
become relevant when repulsion is weaker between the two components.


\subsection{Quantum Hall bilayers}

There are several physical situations where the electrons have a
pseudospin index. This happens notably in many-valleys semiconductors.
Monolayer graphene has two valleys and thus an extra pseudospin with
two values in addition to the real spin. Some semiconductors like Si
have up to six valleys. It may happens that the Coulomb interaction is
independent of these extra degrees of freedom. This is approximately
the case of monolayer graphene. We discuss now the case of engineered
systems where two spatially separated layers of two-dimensional
electron gases are close enough so that there are sizable Coulomb
interactions. The layer index is then a pseudospin and the Coulomb
interaction is different inside a given layer and between layers:
{\begin{eqnarray}
 V_{\uparrow\uparrow}=V_{\downarrow\downarrow}=e^2/r, \quad
  V_{\uparrow\downarrow}=e^2/\sqrt{r^2+d^2},
\end{eqnarray}}\unskip
where $d$ is the distance between layers. With this interaction the
problem does not have the full $SU(2)$ rotation symmetry in pseudospin
space but only the $U(1)$ rotation around the $z$ axis which is the
conservation law of the difference of particle numbers in the two
layers. Let us concentrate on the case $\nu=1$ first. In the limit
$d\rightarrow 0$ we are back to the symmetric situation discusses in
the previous section and the ground state is a ferromagnetic spin
multiplet. If we now tune $d$ small the members of the multiplet will
no longer be degenerate. Since electron--electron interactions are
weaker when they are in separate layers it means that the $S^z_{\mathrm{tot}}=0$
state will have lower energy than all other states in the multiplet.


\subsection{Symmetry breaking and the phase}

If we increase the number of electrons at fixed interlayer distance $d$
we discover that the multiplet of states $S^z_{\mathrm{tot}}=-N/2,\ldots,+N/2$
becomes degenerate as $N\rightarrow\infty$. This emergent degeneracy is
the hallmark of broken symmetry. Since these states differ by the
transfer of electron between layer it is the XY symmetry associated to
the relative phase between layers which is broken. As in situations
involving broken symmetry the physics become transparent once we use a
function which breaks explicitly the symmetry. A simple choice is:
{\begin{eqnarray}
 \Psi_x=\prod_k c^\dagger_{k, x}|0\mathop{\rangle}
 =\prod_k (c^\dagger_{k, \uparrow}+c^\dagger_{k, \downarrow})|0 \mathop{\rangle} ,
\end{eqnarray}}\unskip
where $k$ labels the one-body states of the LLL. This state has
$\langle S^z_{\mathrm{tot}}\rangle=0$ even though it is not  an eigenstate of
$S^z_{\mathrm{tot}}$. Expansion of the second formula reveals that it has weight
over all members of the multiplet. This state is the analog of the BCS
wavefunction for broken particle number. Its generalization to an
arbitrary relative phase  $\phi$ between layers is then:
{\begin{eqnarray}
 \Psi(\phi)=\prod_k (c^\dagger_{k, \uparrow}+\mathrm{e}^{\mathrm{i}\phi}
 c^\dagger_{k, \downarrow})|0 \mathop{\rangle}
 \label{psiphi}
\end{eqnarray}}\unskip
This state gives equal weight to the two layers and this broken
symmetry is aptly called interlayer phase coherence. In the magnetic
language we are dealing with XY symmetry breaking and the relative
phase $\phi$ is the XY order parameter. This phenomenon has been
observed in semiconductor devices~\cite{Eisenstein} for total filling
factor unity and $d \lesssim\ell$. If the layer separation is too large
then we have two essentially decoupled layers each hosting a $\nu=1/2$
CF Fermi sea. There is at least one transition between the fully
decoupled regime at large separation and the ILC phase at small
separation. As expected in a system with a phase associated with
symmetry breaking there should supercurrents in states with a gradient
in space of this phase. However since the phase corresponds to the
relative phase between the layers it implies that the supercurrent
consists of opposite flows of charge carriers in opposite layers so
with zero total current.


Let us now perform a particle-hole transformation on only one spin
species of the state Equation~(\ref{psiphi}) and call 
$d_{k,\uparrow},d_{k,\uparrow}^\dagger$ the associated
creation/annihilation operators. Then the state with definite phase
Equation~(\ref{psiphi})  can be written as:
{\begin{eqnarray}
  \Psi(\phi)=\prod_k (1+\mathrm{e}^{\mathrm{i}\phi}
 c^\dagger_{k, \downarrow}d_{k, \uparrow})|\overbar{0}\mathop{\rangle}
 \label{excphi}
\end{eqnarray}}\unskip
where we have defined the new vacuum $|\overbar{0}\mathop{\rangle}=\prod_k
c^\dagger_{k, \uparrow}|0\mathop{\rangle}$. This new writing show that the ILC
state at total filling factor $\nu=1$ can be aptly called an exciton
condensate where the two members of the exciton pairs reside in
different layers.


There are two configurations of currents that can be used to reveal the
ILC. The first one is the drag configuration in which a current is
imposed only one layer (the ``drive'' layer) and may then measure the
Hall voltage across the drive layer giving then a measurement of
$R_{xy}^{\mathrm{drive}}$ and measure the Hall voltage across the other layer
with no drive current (the ``drag'' layer) giving access to
$R_{xy}^{\mathrm{drag}}$. If we consider the Halperin (111) wavefunction we note
that driving an electron in one layer is accompanied by one vortex in
the other layer so ILC has the special value:
{\begin{eqnarray}
 R_{xy}^{\mathrm{drive}} = R_{xy}^{\mathrm{drag}} = \frac{h}{e^2}
\end{eqnarray}}\unskip
The other configuration is the counterflow set-up in which the current
of the drive layer is injected at the end of the Hall bar backwards in
the top layer inducing a regime in which the currents flow in opposite
directions in the two layers. If now we tune the magnetic field to
reach $\nu_{\mathrm{tot}}=1$ electrons and holes are locked together as in
Equation~(\ref{psiphi}) and thus form a neutral entity that does not feel the
Lorenz force. So the Hall voltage in both layers goes to zero right at
$\nu_{\mathrm{tot}}=1$ a striking evidence of ILC~\cite{Eisenstein}.


\section{Graphene bilayers in the modern era}

It has been feasible recently to build devices with stacking of two
atomically thin monolayers of graphene. Such systems can reach a regime
where $d/\ell\approx 0.1$ which was out of reach of previous set-ups.
Combined with high electronic mobility experiments have revealed
numerous FQHE states~\cite{NatPhysKim,NatPhysDean} that do not match
the fillings of one-component states discussed in Section~\ref{JCF}.
One can first ask whether there is an extension of the CF wavefunctions
giving trial wavefunctions for these states. Another more intriguing
question is whether one can observe ILC involving as basic building
blocks CF instead of electrons.

\subsection{${}^2_1CF$ wavefunctions}

The Halperin family of states Equation~(\ref{Hal}) suggests a simple way to
construct CF states~\cite{scarola}. We take a CF state in each layer,
make a product and add a correlation factor between the two layers:
{\begin{eqnarray}
 \Psi=\Psi_{\overbar\nu}(z_i)\times\Psi_{\overbar\nu}(w_k)\times\prod_{i,k}(z_i-w_k)^m .
\end{eqnarray}}\unskip
Now the exponent $m$ can be even or odd since there is no restriction
from Fermi statistics. One may expect that the relevant values of $m$ 
are smaller for larger separations between layers. This state has 
filling factors $\nu_{1,2}$ for each layer:
{\begin{eqnarray}
 \frac{1}{\nu_{1,2}}=\frac{1}{\overbar\nu}+m
 = \frac{1}{\nu^*}+2p+m
\end{eqnarray}}\unskip
The CF that are formed in such states are called ${}^{2p}_mCF$ where
there are $2p$ vortices attached to the electron in the same layer  and
$m$ interlayer vortices, according to the interpretation of Jastrow
factor in the CF language.

By increasing the distance $d$ between the layers we expect a weakening
of correlations and transitions~\cite{scarola} between competing states
at fixed filling factor. For example at $\nu_{\mathrm{tot}}=1/3$ one may have a
ground state described by the $(333)$ state at small separations which
is a state of ${}^{2}_3CF$. For larger separation we construct the
$(551)$ at the same filling which is now a state of ${}^{4}_1CF$. For
very large separations there should should no interlayer repulsion and
so each layer should form a Fermi sea of ${}^{6}CF$ (note that at such
a low filling factor $\nu=1/6$ a Wigner crystal may form instead of a
FQHE state). By considering compressible states one may enlarge the
familty of trial states. For example there is the Fermi sea of
${}^{4}_2CF$ which is also competing at $\nu_{\mathrm{tot}}=1/3$ that would lead
to a compressible intermediate phase sandwiched between $(333)$ and
$(551)$ states if we guess that the ordering of phase follow the power
of the interlayer repulsion factor. At filling factor $\nu_{\mathrm{tot}}=1/2$
one may consider the Halperin state $(331)$ which may be realized in
GaAs/GaAlAs devices~\cite{Suen92,Eisenstein12}.

By similar reasoning we expect that for $\nu_{\mathrm{tot}}=2/5$ by increasing
the separation we find the $(332)$ state which is a ${}^{2}_2CF$, then
a Fermi sea of ${}^{4}_1CF$, then the state $(550)$ which is a product
of two Laughlin states at filling 1/5, an incompressible state of
${}^{4}CF$.

It is also plausible~\cite{faugno} that other non-FQHE state enter the
competition like crystal states of electrons or of composite fermions.
Only partial theoretical analysis of this complicated situtation is
available.


\subsection{Observed states}
By analogy with monolayer electronic gases one expects to find a
prominent series of FQHE states involving ${}^2_1CF$~\cite{scarola}
which means $p=1$ and $m=1$ hence $\nu_{1,2}=n/(3n+1)$ where $n$ is the
number of filled pseudo-Landau levels of CFs. Indeed the states
$\nu_{1,2}=1/4 \ (n=1), 2/7 \ (n=2), 3/10 \ (n=3)$ are observed as 
incompressible states. The negative-flux  series is also observed for 
$\nu_{1,2}=1/2 \ (n=-1), 2/5 \ (n=-2), 3/8 \ (n=-3)$. These two series
of states should converge to a compressible state for $\nu_{1,2}=1/3$.
Some of these fractions have simple wavefunctions: this is the case of
the $1/4$ state which involves two Laughlin states with one Jastrow
power  between the layers. The wavefunction is the $(331)$ state in the
Halperin family Equation~(\ref{Hal}). But these series of states do not
exhaust the observations made in graphene bilayers. 

Some additional fractions beyond the principal $n/(3n\pm1)$ series
include states with fractional $n$ and odd denominator such as 
$\nu_{1,2}=3/14$ corresponding to $n=3/5$ or $\nu_{1,2}=2/9$ with
$n=2/3$. It may be that such corresponds to FQHE states of composite
fermions as observed in one-component systems for fractions like
$\nu=4/11$. They may fall in the general hierarchical scheme of FQHE.

Another set of states appear for $\nu_{1,2}=1/3$ and $2/3$. These
states  do not manifest any drag Hall resistance so they are decoupled
FQHE states likely $(330)$ for $1/3$ and its particle-hole partner for 
$2/3$. The state observed at $\nu_{1,2}=1/6$ may be a Halperin state
$(333)$ provided interlayer correlations are strong enough which is not
completely consistent with uncorrelated states nearby at 1/3. As we
observed in the case of multicomponent states at a given filling factor
there are several competing states when we vary the strength of
interlayer coupling.

Finally there is evidence for states with half-integer filling of
${}^2_1CF$. This includes $\nu_{1,2}=1/5\, (n=1/2), 3/11 \ (n=3/2),
5/17 \ (n=5/2)$ and with negative flux $\nu_{1,2}=3/7 \ (n=-3/2),  5/3
\ (n=-5/2)$. If we take into account the ILC phenomenology as well as
the CF construction it is natural to conjecture that these states may
display ILC of ${}^2_1CF$.

It is now a theoretical challenge to write down explicit wavefunctions
capturing these states that combine the formation of emerging
quasiparticles (the CFs) and their condensation in a broken symmetry
state.


\section{Conclusions}
Recent experiments have given evidence for a fascinating interplay of
topological order with the creation of a new type of composite
fermions, the ${}^2_1CF$ entities. These CF quasiparticles may form
Landau levels leading to incompressible states that are a
generalization of the series of FQHE states already observed in
single-layer one-component systems. However these experiments also
revealed the appearence of other FQHE states whose description may
involve interlayer phase coherence of composite fermions. Such states
await detailed theoretical explanation. The technological advances in
the manipulation of layered structures has opened a whole new field of
investigation of correlated quantum states and the transitions between
them.

Of course there are many more examples of multicomponent quantum Hall
systems that we have not addressed in this short review. The
experiments we have briefly discussed in this
review~\cite{NatPhysDean,NatPhysKim} have a thin insulating barrier of
hexagonal boron nitride between two graphene monolayers. The barrier is
thin enough to allow strong Coulomb interactions between the two
electron gases but nevertheless thick enough to suppress tunneling. It
is feasible to fabricate samples with nonzero tunneling. This leads a
new set of physical properties. Notably the spin texture of charge
excitations is changed and can be manipulated by tilting the applied
magnetic field away from the direction perpendicular to the layers. The
physics of this situation has been investigated in samples involving
two semiconductor quantum wells coupled by tunnel effect. Each one-body
quantum state in a well is then combined with its partner in the
opposite well giving rise to eigenstates that are
symmetric--antisymmetric (SAS) doublets with tunnel energy splitting
$\Delta_{\mathrm{SAS}}$. It is thus feasible to investigate such a system as a
function of the applied magnetic field and $\Delta_{\mathrm{SAS}}$. Such
experiments have revealed the competition between one-component and
two-component quantum Hall states as well as between single-layer and
bilayer Wigner crystals~\cite{ManoharanShayegan,ManoharanSST}.

In the graphene world one can also study pure graphene bilayers where
there is chemical bonding between two monolayers. The case of Bernal
stacking (AB) has been investigated in detail in the quantum Hall
regime. This is a very special case since the central Landau level has
an orbital degeneracy between levels with $N=0$ and $N=1$ character
where $N$ is the Landau level index. This is in addition  to the spin
and valley degeneracies which are also present as in the case of
monolayer graphene. The central level orbital degeneracy can be
adjusted by applying an electric bias between the two layers leading to
a tunable quantum Hall system which can interpolate between $N=0$ and
$N=1$ FQHE physics. One expects that some if not all FQHE states can be
described by appropriately generalized CF wavefunctions. Recent
investigations~\cite{JunZhu} have revealed many FQHE states with
\textit{even} denominators that are outside the scope of CF
wavefunctions but likely to belong to the Pfaffian family of FQHE
states or its particle-hole partner dubbed the ``AntiPfaffian''.

It is likely that progress in sample creation or fabrication will lead
to more insights into these remarkable states of matter and more
guidance for in-depth theoretical studies.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgments}
I thank Carlos S\'a de Melo and Yvan Castin for giving me the
opportunity to present this review at the workshop ``Open questions in
the quantum many-body problem'' held in Paris in July 2024 at Institut
Henri Poincar\'e.

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