# Theory of Competing Charge Density Wave, Kekule and Antiferromagnetic
Fractional Quantum Hall States in graphene Aligned with boron nitride

Suraj S. Hegde<EMAIL_ADDRESS>Institut für Theoretische physik,
Technische Universität Dresden, 01069 Dresden, Germany  Max-Planck institut
für physik komplexer systeme, Nöthnitzer straße 38,01187 Dresden, Germany
Inti Sodemann Villadiego<EMAIL_ADDRESS>Institut für Theoretische
physik, Universität Leipzig, Brüderstraße 16, 04103, Leipzig, Germany  Max-
Planck institut für physik komplexer systeme, Nöthnitzer straße 38,01187
Dresden, Germany

###### Abstract

We investigate spin and valley symmetry-broken fractional quantum Hall phases
within a formalism that naturally extends the paradigm of quantum Hall
ferromagnetism from integer to fractional quantum Hall states, allowing us to
construct detailed phase diagrams for a large class of multi-component states.
Motivated by recent experiments on graphene aligned with a boron nitride
substrate, we predict a sequence of transitions realized by increasing the
magnetic field, starting from a sub-lattice polarized state to a valley
coherent Kekule charge density wave state and further to an anti-ferromagnetic
phase. Moreover for filling fractions such as $\nu=\pm 1/3$, we predict that
the system undergoes a transition at low fields, that not only differ by the
spin-valley orientation of the fractionally filled flavors but also by their
intrinsic fractional quantum Hall nature. This transition is from a Laughlin-
like state to a two component Halperin-like state both with a charge density
wave order. Moreover for $\nu=\pm 1/3,\pm 2/3$, we predict a “canted Kekule
density phase”(CaKD) where the spinors of integer and fractionally occupied
components have different orientations in the valley Bloch sphere, in contrast
to the Kekule state for the integer quantum Hall state at neutrality where
both occupied components have the same orientation in the valley Bloch sphere.

##  I Introduction

The quantum Hall regime in graphene offers a fantastic arena to investigate
the interplay of broken symmetry, topology and fractionalization MacDonald
(1990); Nomura and MacDonald (2006); Feldman  _et al._ (2009); Bolotin  _et
al._ (2009); Goerbig (2011); Kharitonov (2012a); Young  _et al._ (2012);
Abanin  _et al._ (2013); Young  _et al._ (2014); Sodemann and MacDonald
(2014); Wu  _et al._ (2014, 2015); Roy  _et al._ (2014); Peterson and Nayak
(2014); Amet  _et al._ (2015); Knothe and Jolicoeur (2015); Balram  _et al._
(2015); DaSilva  _et al._ (2016); Lukose and Shankar (2016); Yan and Ting
(2017); Polshyn  _et al._ (2018); Zibrov  _et al._ (2018); Stepanov  _et al._
(2018); Chen  _et al._ (2019); Maiti and Sedrakyan (2019); Schmitz  _et al._
(2020); Kim  _et al._ (2021); Atteia  _et al._ (2021); Pierce  _et al._
(2021); Veyrat  _et al._ (2020), that continues to flourish thanks to a
combination of high quality samples, enlarged degeneracy from its valley
degree of freedom, and its exposed nature that allows new ways to measure and
control the states of interest. For example, recent experiments have developed
remarkable techniques to generate and detect long-range transmission of
collective excitations in broken-symmetry integer and fractional quantum Hall
states Young  _et al._ (2012); Wei  _et al._ (2017, 2018); Zibrov  _et al._
(2018); Stepanov  _et al._ (2018); Assouline  _et al._ (2021); Fu  _et al._
(2021); Pierce  _et al._ (2021); Zhou  _et al._ (2021), and recent theoretical
and scanning tunneling microscopy studies has provided detailed evidence of
valence-bond quantum Hall Kekule-type states Liu  _et al._ (2022); Coissard
_et al._ (2021); Li  _et al._ (2019); Das  _et al._ (2022).

Moreover, the possibility to align graphene with substrates, such as
hexagonal-boron-nitride (hBN), can lead to a substantially enriched space of
states. This is in fact one of the key purposes of this work, where we will
demonstrate that a delicate competition of a variety of broken symmetry
fractional quantum Hall states in the zeroth Landau level of graphene arise as
a consequence of the interplay of the hBN-induced sub-lattice symmetry
breaking and the tendency of the intrinsic interactions in graphene to
stabilize anti-ferromagnetic states at high fields, as summarized in Fig.1.

Another key purpose of our work is to elaborate on a framework introduced by
MacDonald and one of us in Ref. Sodemann and MacDonald, 2014, which
generalizes the Hartree-Fock theory of integer quantum Hall ferromagnets to a
vast class of experimentally relevant multi-component fractional quantum Hall
states 111For a closely related discussion see also Ref. Abanin  _et al._ ,
2013. This framework allows to make detailed quantitative predictions of
broken symmetry phase diagrams and quasi-particle excitation gaps, and, in
particular, it can be applied to determine the competing phases arising in the
model Hamiltonian introduced by Kharitonov Kharitonov (2012a), which has
proved to be a valuable tool for understanding the broken symmetry phases of
the zeroth Landau level of graphene Kharitonov (2012a, b); Sodemann and
MacDonald (2014); Wu  _et al._ (2014); Knothe and Jolicoeur (2015); Murthy
_et al._ (2016); Tikhonov  _et al._ (2016); Lian  _et al._ (2016); de Nova and
Zapata (2017); Lian and Goerbig (2017); Pientka  _et al._ (2017); Jolicoeur
and Pandey (2019). This model can be viewed as descending from the model
introduced by Aleiner, Kharzeev and Tsvelik Aleiner  _et al._ (2007) upon
projection onto the zeroth Landau level of graphene Kharitonov (2012a).

While we will develop analytical formulae for a class of fractional quantum
states with arbitrary fillings, we will focus specially on states realized at
fillings $\nu=\pm 1/3,\pm 2/3$ which are often the most robust in experiments.
At $\nu=\pm 1/3$ in particular we will describe a competition between a class
of Laughlin-like states and singlet Halperin-like states that not only differ
by the spin-valley orientation of the fractionally filled flavors but also by
their intrinsic fractional quantum Hall nature, and are analogous to spin
polarized and spin singlet states realized at total filling 2/3 in GaAs
Eisenstein  _et al._ (1990); Lay  _et al._ (1997); Cho  _et al._ (1998);
NiemelÃ€  _et al._ (2000); Feldman  _et al._ (2009, 2012); Peterson  _et al._
(2015); Zhang  _et al._ (2016). This makes graphene an attractive platform to
study these states, their edge modes and the interface between them, which due
to its non-chiral nature, is a candidate to realize parafermion modes by
coupling it to superconductors, which is an interesting platform for
topological quantum computationStern  _et al._ (2004); Bid  _et al._ (2010);
Gross  _et al._ (2012); Gurman  _et al._ (2012); Mong  _et al._ (2014); Ronen
_et al._ (2018); Lafont  _et al._ (2019); Wang  _et al._ (2021) .

Our paper is organized as follows: Sec.II discusses the framework that
generalizes the theory of broken symmetry quantum Hall ferromagnetism to
multi-component fractional quantum Hall states. The discussion here is kept
general so as to apply to multi-component quantum Hall systems beyond
graphene. Sec.III then reviews how to apply this formalism to Kharitonov model
of symmetry breaking interactions in the $n=0$ Landau level of graphene, where
$n=0$ corresponds to the orbital Landau level located at the Dirac point. Sec.
IV then applies the formalism to compute the phase diagram of the broken
symmetry states of the integer quantum Hall states at $\nu=0$ paying special
attention to the modifications brought in by the hBN-induced sublattice
potential. Sec. V generalizes these considerations to the phase diagrams of a
large class of general fractional quantum Hall states. Sec.VI presents our
prediction of a series of transitions between valley- and spin-broken order in
fractional quantum Hall states realised within the regime of experimental
parameters. In Sec.VII, we comment on the distinction in the universal
topological order between the Laughlin-like and Halperin-like states. Finally,
Sec.VIII presents conclusions and discussions.

Figure 1: Sequence of transitions within and between the valley-ordered phases
and spin-ordered phases estimated using the variation theory for SU(4)
symmetry breaking interaction in multi-component fractional quantum Hall
states of graphene for a total filling $\nu=2/3$ measured relative to charge
neutrality.

##  II Multi-component FQH states and variational theory of iso-spin symmetry-
breaking short range interactions.

###  II.1 Constructing Correlated states on Quantum Hall Ferromagnet vacua

We consider a multi-component quantum Hall system with $N$ internal flavors
and imagine that the Hamiltonian can be separated into a dominant part that is
$SU(N)$ invariant plus smaller symmetry breaking perturbations. It is then
natural to first understand many-body ground states of the $SU(N)$ invariant
part, which appear in degenerate sets forming irreducible representations.
Irreducible representations of $SU(N)$ arising from a system $N_{e}$ electrons
can be labeled by a set of $N$ integers satisfying:

$n_{1}\geq n_{2}...\geq n_{N}\geq 0,\ \sum_{i}n_{i}=N_{e}.$ (1)

These integers count the number of boxes in each row of a Young tableau
depiction of the representation Pfeifer (2003), and, more intuitively, they
simply count the invariant number of particles occupying each of the internal
flavors 222For the familiar case of $SU(2)$ the $S^{2}$ quantum number is
$S=(n_{1}-n_{2})/2$.. This leads to a generalization of the notion of filling
factor into an $N$-component filling vector which specifies the $SU(N)$
invariant occupations of each of the flavors and is defined as:

$(\nu_{1},\nu_{2},...,\nu_{N}),\ \nu_{i}\equiv\frac{n_{i}}{N_{\phi}},\
\sum_{i}\nu_{i}=\nu,$ (2)

where $N_{\phi}$ is the number of flux quanta. When the fillings $\nu_{i}$ are
integers the state corresponds to an integer quantum Hall ferromagnet (IQHFM)
which are well-known to be exact ground states of a large class of repulsive
$SU(N)$ invariant Hamiltonians projected into a Landau level. Moreover, IQHM
states also serves as ”perfect vacua on which one could construct correlated
FQH states. Namely, if we have an eigenstate of the Hamiltonian with with $K\
(\leq N)$ fractionally filled components and $N-K$ empty components, labeled
by $(\nu_{1},...,\nu_{K},0,....,0)$, then we can “glue” this state with any
IQHFM with $L\ (\leq N-K)$ integer filled components, to obtain a new
eigenstate of the form $(1,...,1,\nu_{1},...,\nu_{K},0,...,0)$. The $SU(N)$
invariant interaction energy, denoted by $V$, of this new state can be
obtained from that of the other states, and is given by Sodemann (2014) :

$V[\Psi_{L+K}]=V[\Psi_{K}]+V[\Psi_{L}]+\nu_{K}LN_{\phi}V_{H},$ (3)

$V_{H}=\frac{1}{2\pi l^{2}}\int d^{2}r\ v(r),$ (4)

where $\Psi_{K}$ denotes the $K$-component FQH state with filling
$\nu_{K}=\sum_{i=1}^{K}\nu_{i}$, $\Psi_{L}$ denotes the $L$-component IQHFM
state, and $\Psi_{K+L}$ denotes the glued state of these two. $v(r)$ is simply
the Hartree potential between the electrons in the fractionally and integer
occupied components, and would be absent if a neutralizing background was
present, as it is the case with Coulomb interactions. $l$ is the magnetic
length.

The construction described above is based on adding correlated particles to
the IQHFM. It is also possible to construct a different set of states by
adding correlated holes. Namely, for any eigenstate with fillings
$(\nu_{1},...,\nu_{K},0,....,0)$ one can obtain other correlated states in
which these particles are removed from an IQHFM with $L\ (\geq K)$ integer
filled components, leading to a new state with $L-K$ integer filled components
and the remaining fractionally filled as follows
$(1,...1,1-\nu_{K},\cdots,1-\nu_{1},0,\cdots,0)$. The energy of these states
can be shown to be related as followsSodemann (2014) :

$\displaystyle V[\Psi_{L-K}]=$ $\displaystyle V[\Psi_{K}]$
$\displaystyle+N_{\phi}(L-\nu)\bigg{(}LV_{H}-\int\frac{d^{2}q}{(2\pi)^{2}}v(q)|F(q)|^{2}\bigg{)},$
(5)

where $|F(q)|^{2}=\exp(-q^{2}l^{2}/2)$ is the squared density form factor of
the lowest Landau level and $v(q)$ is the Fourier transform of the Hartree
potential $v(r)$.

###  II.2 Hard-core states and variational theory of short range SU(N)
breaking interactions

The above two rules allow to construct a large set of new FQHE states and
determine their energy from knowledge of FQHE states constructed on vacuum. An
important set of FQHE states in vacuum are those we will refer to as “hard-
core” states. A hard-core state is one in which the wave-function vanishes
when any two-particles approach each other regardless of whether they have the
same $SU(N)$ pseudo-spin flavor. Clearly this property endows these hard-core
states with energetically desirable correlations for repulsive interactions,
and this set includes many celebrated wave-functions like the Laughlin states,
many of the Halperin multicomponent states, the Jain states, the Moore-Read
states and others.

An important property of the set of any hard-core states is that they
necessarily have total filling factors restricted to $\nu\leq 1$ Sodemann and
MacDonald (2014); Sodemann (2014). This can be seen easily by noting that any
wave-function which vanishes whenever two particles approach each other must
necessarily be proportional to the Laughlin-Jastrow-like function
$\prod_{i<j}(z_{i}-z_{j})$ with the indices $i,j$ running over all the
particles times another analytic polynomial of the coordinates. But the
wavefunction associated with the Laughlin-Jastrow-like factor has filling
factor 1, and since multiplication by an analytic polynomial always reduces
the density of the wave-function, it follows that their fillings are
restricted to $\nu\leq 1$.

Another immediate property of hard-core states is that they are zero
eigenstates of any generic delta function interaction even if it breaks the
$SU(N)$ symmetry. To illustrate this, let us consider the following $SU(N)$
symmetry breaking Hamiltonian:

$\displaystyle
H_{a}=\sum_{i<j,\alpha\beta}g_{\alpha\beta}\mathcal{T}_{\alpha}^{i}\mathcal{T}_{\beta}^{j}\delta^{(2)}({\bf
r}_{i}-{\bf r}_{j})-\sum_{i\alpha}h_{\alpha}\mathcal{T}_{\alpha}^{i},$ (6)

here $\mathcal{T}_{\alpha}$ are generators of $SU(N)$ and $g_{\alpha\beta}$ is
a symmetric matrix, and $h_{\alpha}$ denote the strength of uniform single-
particle symmetry-breaking terms. It is easy to show that the hard-core states
are exact ground states of the above Hamiltonian, simply because they have
zero probability for any two particles to coincide at the same point in space
regardless of their $SU(N)$ flavor, and therefore they are annihilated by the
delta function in Eq.6. Moreover the $SU(N)$ flavors of the occupied states
can be chosen to diagonalize the single-particle matrix
$h_{\alpha}\mathcal{T}_{\alpha}$, so that they are eigenstates of the full
Hamiltonian in Eq.6.

Now the states with $\nu>1$ that follow from the two previously discussed
rules are not necessarily exact eigenstates of this Hamiltonian. Let us
imagine, however, that the $SU(N)$ invariant part of the Hamiltonian is much
stronger than the symmetry breaking terms in Eq.6, and therefore that the
orbital two-particle correlation functions of these states are sufficiently
rigid so as to remain well approximated by the ideal $SU(N)$ invariant states,
and that the role of the terms in Eq.6 is to select the ground state within
the $SU(N)$ degenerate manifold. This assumption allows to construct a
variational mean field theory of these states somewhat analogous to Hartree-
Fock theory, albeit extended to states that are far from being approximated as
Slater determinants, namely, the FQH states generated by the two previously
discussed rules. Specifically, consider an IQHFM vacuum occupying $L$ flavors,
which can be parametrized by a projector of the form:

$P_{i}=\sum_{k=1}^{L}\ket{\chi_{k}}\bra{\chi_{k}},$ (7)

and consider a hard-core FQH state of $K$ flavors, and to which we associate a
density matrix of fractional fillings associated as follows:

$P_{f}=\sum_{k=1}^{K}\nu_{k}\ket{\chi_{L+k}}\bra{\chi_{L+k}}.$ (8)

In writing $P_{i}$ and $P_{f}$ we are assuming that all the spinors
$\ket{\chi_{k}},\ket{\chi_{L+k}}$ are orthogonal and normalized. Now, it can
be shown Sodemann and MacDonald (2014); Sodemann (2014) that the expectation
value of the symmetry breaking Hamiltonian in Eq.6 for the “glued” state of
the above IQHFM and FQH states can be written as:

$\displaystyle\frac{\langle
H_{a}\rangle}{N_{\phi}}=\frac{1}{2}\text{Tr}(P_{i}H_{i}^{HF})+\text{Tr}(P_{f}H_{i}^{HF})$
$\displaystyle-\sum_{\alpha}h_{\alpha}\text{Tr}((P_{i}+P_{f})\mathcal{T}_{\alpha}).$
(9)

where $H_{i}^{HF}$ is the mean-field Hartree-Fock Hamiltonian arising from
interactions with the particles occupying integer filled components of the
IQHFM, and it is given by:

$\displaystyle
H_{i}^{HF}=\sum_{\alpha,\beta}u_{\alpha\beta}[\text{Tr}(P_{i}\mathcal{T}_{\alpha})\mathcal{T}_{\beta}-\mathcal{T}_{\alpha}P_{i}\mathcal{T}_{\beta}]$
(10)

where $u_{\alpha,\beta}=\frac{g_{\alpha,\beta}}{2\pi l^{2}}$. Notice that when
the fractionally filled components are empty (i.e. $P_{f}=0$), the energy in
Eq.(II.2) reduces to the Hartree-Fock energy of the integer quantum Hall
ferromagnetic states (compare e.g. with Eqs.(44)-(48) of Ref. Kharitonov,
2012a). Thus we see that our description is a natural extension of the theory
of integer quantum Hall ferromagnets to the case of multicomponent fractional
quantum Hall states.

Therefore the task is reduced to minimizing the energy from Eq. (II.2) as a
function of the spinors $\ket{\chi_{k}}$ parametrizing the IQHFM from Eqs.(7)
and (8). The above equations are obtained from the assumption of hard-core
repulsion and the following property of completely filled spinors

$\hat{\rho}_{m}(r)\ket{\Psi}=\frac{1}{2\pi\ell^{2}}\ket{\Psi},$ (11)

where
$\hat{\rho}_{m}(r)=P_{LLL}\sum_{i}(\delta(r-r_{i})\ket{\chi_{m}}_{ii}\bra{\chi_{m}})P_{LLL}$
is the density projected to the mth completely filled spinor. We would like to
also mention in passing, that while the above framework captures a large
variety of multi-component states, other multi-component states that deviate
from this picture, to this date have not been understood theoretically, have
been found in numerical studies Wu  _et al._ (2015); Le and Jolicoeur (2021).

##  III Weakly broken SU(4) FQH states in graphene on h-BN

In this section we apply the formalism of Sec.II to the specific case of SU(4)
symmetry breaking interactions known to occur in FQHEs of graphene. The
integer and fractional quantum Hall states of graphene under high magnetic
fields have been considered in detail in various works Kharitonov (2012a);
Abanin  _et al._ (2013); Sodemann and MacDonald (2014) and in particular
recent experiments have developed ingenious ways to probe the pseudo-spin
order in these systems by exploiting the coupling of their order parameters to
non-local transport phenomenaStepanov  _et al._ (2018); Zibrov  _et al._
(2018); Wei  _et al._ (2018); Polshyn  _et al._ (2018); Zhou  _et al._ (2021).

We focus on the zero Landau level of graphene, which has N=4 flavors arising
from two valleys and two spins. The long-range part of the Coulomb
interactions projected into these Landau levels are viewed as the dominant
SU(4) invariant part of the Hamiltonian. In addition there are short-distance
lattice scale corrections to the Coulomb interactions Kharitonov (2012a);
Aleiner  _et al._ (2007) that can be modeled as sub-lattice and valley
dependence contact interactions 333For counterparts of these interactions in
bilayer graphene see e.g. Refs. Lemonik  _et al._ (2010, 2012). Including
these terms is crucial because they play the role of the leading symmetry
breaking terms in the Landau level, and therefore determine the specific
broken symmetry ground states. We also imagine that the alignment of graphene
with an hBN substrate gives rise to a single-particle valley splitting with
strength $\Delta$, as relevant for experiments in Ref.Zibrov  _et al._ , 2018.
Together with the ordinary Zeeman term and short distance interactions
considered in previous studies Kharitonov (2012a); Sodemann and MacDonald
(2014); Sodemann (2014) the Hamiltonian in Eq. 6 can be written as:

$\displaystyle
H_{a}=\sum_{i<j,\alpha}(g_{\perp}(\tau_{x}^{i}\tau_{x}^{j}+\tau_{y}^{i}\tau_{y}^{j})+g_{z}\tau_{z}^{i}\tau_{z}^{j})\delta^{(2)}({\bf
r}_{i}-{\bf r}_{j})$
$\displaystyle-\sum_{i}(h\sigma_{z}^{i}+\Delta\tau_{z}^{i})$ (12)

where $\tau_{i}$ and $\sigma_{i}$ are Pauli matrices for the valley iso-spin
and intrinsic spin respectively, $g_{\perp,z}$ are the strengths of the delta-
function interactions, $h$ is the spin-Zeeman term and $\Delta$ is the valley-
Zeeman term. The valley-Zeeman term comes from sublattices A and B having
different energies, originating from the alignment of graphene with an hBN
substrate, as seen in penetration capacitance measurements Zibrov  _et al._
(2018). These terms break the SU(4) weakly in the sense that
$g_{\perp},g_{z}<e^{2}/\epsilon\ell$ and $h,\Delta<e^{2}/\epsilon\ell$. The
values of $g_{\perp},g_{z}$ are dependant on the magnetic field component
perpendicular to the graphene plane $B_{\perp}$ ($g_{\perp,z}\propto
B_{\perp}$), whereas the Zeeman term $h$ is determined by the total magnetic
field. From the above short-range interaction terms one can define two energy
scales:

$u_{z,\perp}=\frac{g_{z,\perp}}{2\pi l^{2}}$ (13)

Now, let us obtain the contribution of these anisotropy energies in order to
determine the different phases realised in the parameter space, following the
formalism described in Sec. II.2. The many-body states we consider are the
ones which are constructed from the hardcore states which have a filling of
the form: $(1,1,\nu_{3},\nu_{4})$ 444There are more possibilities of the form
$(1,\nu_{2},\nu_{3},0)$, but as discussed in Ref.Sodemann and MacDonald, 2014,
these states are disfavored much more strongly by the short range interactions
and tend to be stabilized at very small magnetic fields, and therefore we will
not consider them here. This is also supported by recent detailed numerical
studies Le and Jolicoeur (2021). The variational coherent states associated
with the completely filled flavors are denoted by
$\\{\ket{\chi_{1}},\ket{\chi_{2}}\\}$. The ones of the partially filled
flavors will be denoted by $\\{\ket{\chi_{3}},\ket{\chi_{4}}\\}$. The weighed
projectors onto these states are denoted by
$P_{i}=\ket{\chi_{1}}\bra{\chi_{1}}+\ket{\chi_{2}}\bra{\chi_{2}}$ and
$P_{f}=\nu_{3}\ket{\chi_{3}}\bra{\chi_{3}}+\nu_{4}\ket{\chi_{4}}\bra{\chi_{4}}$.
The expectation value of the anisotropy energy per flux quantum is given
bySodemann and MacDonald (2014); Sodemann (2014)

$\displaystyle\frac{\langle
H_{a}\rangle}{N_{\phi}}=\frac{1}{2}\text{Tr}(P_{i}H_{i}^{HF})+\text{Tr}(P_{f}H_{i}^{HF})$
$\displaystyle-\frac{h}{2}\text{Tr}(P_{i}\sigma_{z})-\frac{\Delta}{2}\text{Tr}(P_{i}\tau_{z})$
$\displaystyle
H_{i}^{HF}=\sum_{\alpha}u_{\alpha}[\text{Tr}(P_{i}\tau_{\alpha})\tau_{\alpha}-\tau_{\alpha}P_{i}\tau_{\alpha}]-h\sigma_{z}-\Delta\tau_{z}$
(14)

In the subsequent sections we will minimize the above variational energy with
respect to the SU(4) orientation of the spinors
$\\{\ket{\chi_{1}},\ket{\chi_{2}},\ket{\chi_{3}},\ket{\chi_{4}}\\}$, for
various cases.

##  IV Phase diagram for IQH states at neutrality with valley-Zeeman term

Figure 2: Phase diagram of $(1,1,0,0)$ integer Quantum Hall ferromagnetic
state in graphene in the presence of valley-Zeeman term $\Delta$ (sub-lattice
staggered potential).

In this section we will consider the symmetry broken phases and phase diagram
of the integer quantum Hall effect of graphene at neutrality. We sometimes
will denote the total filling by $\tilde{\nu}$, such that
$\tilde{\nu}\in[0,4]$. Neutrality corresponds to $\tilde{\nu}=2$, and this
relates to the other frequently used notation of filling $\nu\in(-2,2)$ as
follows: $\tilde{\nu}=2+\nu$.

The phase diagram of $\tilde{\nu}=2$ ($\nu=0$) quantum Hall state in graphene
at neutrality has been studied extensivelyKharitonov (2012a); Abanin  _et al._
(2013); Wu  _et al._ (2014); Zibrov  _et al._ (2018). Here we will review
relevant aspects of it in order set-up the notation and to include the valley-
Zeeman term induced by the hBN substrate in the same manner as in Ref. Zibrov
_et al._ , 2018. This state also serves as the vacuum for creating
fractionally filled phases that we discuss in the next section. Charge
neutrality corresponds to occupation of two components
$(\nu_{1},\nu_{2},\nu_{3},\nu_{4})=(1,1,0,0)$.

Within the theory described in Sec.III, the possible ground states are
parametrized by the projector:
$P_{i}=\ket{\chi_{1}}\bra{\chi_{1}}+\ket{\chi_{2}}\bra{\chi_{2}}$. We will
also further restrict to states in which the occupied spinors do not have
valley-spin entanglementKharitonov (2012a); Sodemann and MacDonald (2014),
namely states for which the SU(4) spinors can be taken as a tensor product of
the state in the valley space times one in the spin space 555An interesting
study found spin-valley entanglement near the egde of the integer quantum Hall
ferromagnets Knothe and Jolicoeur (2015), but here we are focusing on bulk
ground states.. Restriction to this sub-set is parametrised by the projector
$P_{i}$:

$P_{i}=P_{n_{1}}\otimes P_{s_{1}}+P_{n_{2}}\otimes P_{s_{2}},$ (15)

where $\vec{n}_{1},\vec{n}_{2}$ are the unit vectors in the Bloch sphere of
the valley iso-spin and $\vec{s}_{1},\vec{s}_{2}$ are the unit vectors in the
spin space. $P_{n_{j}}$ is the valley-spin density matrix given by

$P_{nj}=\ket{\vec{n}_{j}}\bra{\vec{n}_{j}}=\frac{1}{2}(1+\vec{n}_{j}\cdot\vec{\tau})$
(16)

and of similar structure for the spins $\vec{s}_{1},\vec{s}_{2}$. In addition
one needs to enforce that the two projectors in the right hand side of Eq.(15)
are orthogonal. This is achieved by either making the spin vectors anti-
parallel while leaving their valley orientations arbitrary, and we refer to
this states “valley ordered”, or by the converse choice in which the valley
vectors are anti-aligned and the spin vectors are arbitrary, and we refer to
the latter as “spin ordered” states. The valley ordered states can be written
as:

$\displaystyle\chi_{1}=\ket{\vec{n}_{1}}\otimes\ket{\vec{s}},\quad\chi_{2}=\ket{\vec{n}_{2}}\otimes\ket{-\vec{s}},$
$\displaystyle P_{n}=P_{n_{1}}\otimes P_{s}+P_{n_{2}}\otimes P_{-s}$ (17)

These contain states that correspond to charge density wave (CDW) and Kekule-
distortion (KD) phases. The spin-ordered states are:

$\displaystyle\chi_{a}=\ket{\vec{n}}\otimes\ket{\vec{s}_{1}},\quad\chi_{b}=\ket{-\vec{n}}\otimes\ket{\vec{s}_{2}},$
$\displaystyle P_{s}=P_{n}\otimes P_{s_{1}}+P_{-n}\otimes P_{s_{2}}$ (18)

These contain states such as ferromagnetic (FM) phase and various anti-
ferromagnetic (AFM) phases. From Eq.(III) the anisotropy energy of the valley-
ordered states parametrized in Eq.(17) can be found to be:

$E_{va}=u_{\perp}(n_{1x}n_{2x}+n_{1y}n_{2y})+u_{z}n_{1z}n_{2z}-\Delta(n_{1z}+n_{2z})$
(19)

Similarly, the anisotropy energy of the spin-ordered states parametrised in
Eq. (18) is found to be:

$E_{sa}=-u_{\perp}(1+\vec{s}_{1}.\vec{s}_{2})-u_{z}-h(s_{1z}+s_{2z})$ (20)

State | Valley | Spin | Energy
---|---|---|---
CDW | $n_{1z}=n_{2z}=1$ | $\vec{s}_{1}=-\vec{s}_{2}$ | $u_{z}-2\Delta$
Kekule | $n_{1z}=n_{2z}=\frac{\Delta}{u_{\perp}-u_{z}}$ | $\vec{s}_{1}=-\vec{s}_{2}$ | $u_{\perp}-\frac{\Delta^{2}}{u_{z}-u_{\perp}}$
CaAFM | $\vec{n}_{1}=-\vec{n}_{2}$ | $s_{1z}=s_{2z}=\frac{h}{2u_{\perp}}$ | $-u_{z}-\frac{h^{2}}{2|u_{\perp}|}$
FM | $\vec{n}_{1}=-\vec{n}_{2}$ | $\vec{s}_{1}=\vec{s}_{2}$ | $-2u_{\perp}-u_{z}-2h$
Table 1: Spin-valley orientations of integer quantum Hall ferromagnets at
neutrality and energies with Zeeman and valley-Zeeman coupling.

Minimization of these energy functionals leads to four phases listed in Table
1 and to the corresponding phase diagram shown in Fig.2.

##  V Phase diagram of FQHE states in graphene with valley-Zeeman term

State | Valley iso-spin | Spin | Energy
---|---|---|---
CDW | $n_{1z}=-n_{3z}=n_{2z}=-n_{4z}=1$ | $\vec{s}_{1}=-\vec{s}_{2}=\vec{s}_{3}=-\vec{s}_{4}$ | $u_{z}(1-2\nu)-2\nu u_{\perp}-\Delta(2-\nu)-h(\nu_{3}-\nu_{4})$
CantedKD |  $\vec{n}_{1}=-\vec{n}_{3}$, $\vec{n}_{2}=-\vec{n}_{4}$, | $\vec{s}_{1}=-\vec{s}_{2}=\vec{s}_{3}=-\vec{s}_{4}$ | $(1-\nu)u_{\perp}\sqrt{\frac{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})-(1-\nu_{4})^{2}\Delta^{2}}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})-(1-\nu_{3})^{2}\Delta^{2}}}\bigg{(}1-\frac{\Delta^{2}(1-\nu_{3})^{2}}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})}\bigg{)}$
| $n_{3z}\neq n_{4z}$ |  | $-\frac{(1-\nu_{3})(1-\nu_{4})u_{z}\Delta^{2}}{(1-\nu)(u_{z}^{2}-u_{\perp}^{2})}-2\nu u_{\perp}-\nu u_{z}$
CoAFM | $\vec{n}_{1}=-\vec{n}_{2}=\vec{n}_{3}=-\vec{n}_{4}$ | $s_{1z}=-s_{2z}=-s_{3z}=s_{4z}$ | $-u_{z}-2\nu u_{\perp}-h(\nu_{3}-\nu_{4})-\Delta(\nu_{3}-\nu_{4})$
CaAFM | $\vec{n}_{1}=-\vec{n}_{2}=\vec{n}_{3}=-\vec{n}_{4}$ |  $\vec{s}_{1}=-\vec{s}_{3}$,$\vec{s}_{2}=-\vec{s}_{4}=1$, | $|u_{\perp}|(1+\nu)-\frac{|u_{\perp}|(1-\nu)}{2}\bigg{(}\frac{1-\nu_{4}}{1-\nu_{3}}+\frac{1-\nu_{3}}{1-\nu_{4}}\bigg{)}-\frac{h^{2}}{2|u_{\perp}|}\frac{(1-\nu_{3})(1-\nu_{4})}{(1-\nu)}$
|  | $s_{3z}\neq s_{4z}$ | $-u_{z}-\Delta(\nu_{3}-\nu_{4})$
FM | $\vec{n}_{1}=-\vec{n}_{2}=\vec{n}_{3}=-\vec{n}_{4}$ | $s_{1z}=s_{2z}=s_{3z}=s_{4z}=1$ | $-u_{z}-2u_{\perp}-h(2-\nu)-\Delta(\nu_{3}-\nu_{4})$
Table 2:  Orientations of valley iso-spin and spin for different phases at
fractional filling $(1,1,\nu_{3},\nu_{4})$ and the corresponding energies. For
the expressions of $n_{3z},n_{4z}$ see B. The expression for $s_{3z},s_{4z}$
are given in Ref.Sodemann and MacDonald, 2014.  Figure 3: Phase diagrams in
the presence of valley-Zeeman term $\Delta$ (sub-lattice staggered potential).
The phase boundaries of $(1,1,2/3,0)$ state are indicated by dark lines and
phase boundaries of $(1,1,1/3,1/3)$ indicated by dashed lines.

In this section, we will obtain the phase diagram for a class of fractional
quantum Hall phases in the presence of valley-Zeeman term. We will explicitly
consider the states with filling vector $(1,1,\nu_{3},\nu_{4})$. Now we need
to specify both the fully filled spinors $\\{\ket{\chi_{1}},\ket{\chi_{2}}\\}$
and fractionally filled spinors $\\{\ket{\chi_{3}},\ket{\chi_{4}}\\}$. Again
after taking the occupied spinors to be in valley-spin tensor product states
and enforcing their orthogonality, the states that minimise the anisotropy
energy in Eq.(III) fall into two classes: spin-ordered and valley-ordered
states. The spin-ordered states have the form:

$\displaystyle\ket{\chi_{1}}=\ket{\vec{n},\vec{s}_{3}},\quad\ket{\chi_{2}}=\ket{-\vec{n},\vec{s}_{4}}$
$\displaystyle\ket{\chi_{3}}=\ket{\vec{n},-\vec{s}_{3}},\quad\ket{\chi_{4}}=\ket{-\vec{n},-\vec{s}_{4}}$

And their anisotropy energy can be obtained from Eq.(III) as:

$\displaystyle E_{so}$
$\displaystyle=-u_{\perp}(1-\nu)\vec{s}_{3}\cdot\vec{s_{4}}-u_{z}-u_{\perp}(1+\nu)$
$\displaystyle-h((1-\nu_{3})s_{3z}+(1-\nu_{4})s_{4z})-\Delta(\nu_{3}-\nu_{4}).$
(21)

The valley-ordered states have the spinors:

$\displaystyle\ket{\chi_{1}}=\ket{\vec{n}_{3},\vec{s}},\quad\ket{\chi_{2}}=\ket{\vec{n}_{4},-\vec{s}}$
$\displaystyle\ket{\chi_{3}}=\ket{-\vec{n}_{3},\vec{s}},\quad\ket{\chi_{4}}=\ket{-\vec{n}_{4},-\vec{s}}$

And their anisotropy energy is obtained from Eq. (III) as:

$\displaystyle E_{vo}$ $\displaystyle=$ $\displaystyle
u_{\perp}(1-\nu)u_{\perp}{\vec{n}}_{3\perp}\cdot{\vec{n}}_{4\perp}+(1-\nu)u_{z}n_{3z}n_{4z}-2\nu
u_{\perp}$ $\displaystyle-$ $\displaystyle\nu
u_{z}-\Delta((1-\nu_{3})n_{3z}+(1-\nu_{4})n_{4z})-h(\nu_{3}-\nu_{4}),$ (22)

where $\nu=\nu_{3}+\nu_{4}$ and
${\vec{n}}_{3\perp}\cdot{\vec{n}}_{4\perp}=n_{3x}n_{4x}+n_{3y}n_{4y}$. Notice
that in the anisotropy energy of valley ordered states with fractionally
filled components the occupied spinors have different couplings to the
sublattice symmetry breaking strength ($\Delta$), except in the special case
when $\nu_{3}=\nu_{4}$. This special case includes the integer quantum Hall
ferromagnetic states at neutrality ($\nu_{3}=\nu_{4}=0$). Therefore
generically we expect that the fractional quantum Hall Kekule states will have
spinors with different orientations in the valley Bloch sphere, and we dub
these states “Canted Kekule States”. For the specific case of fractional
fillings $\nu_{3}\neq\nu_{4}$, within the anti-ferromagnetic order, there is a
competition between a canted AFM (CaAFM)(where the spinors have different
orientation in spin Bloch sphere) and a collinear AFM (CoAFM).

Minimisation of the energy functionals lead to five distinct phases as shown
in Table. 2 and phase diagram for different cases are given in Figs. 3,4,5. In
appendix A we describe a mapping from states $(1,\nu,0,0)$ to
$(1,1,\nu_{3},\nu_{4})$ that allows to compute the energies and phase diagrams
for other states from the results presented here. Fig. 3 shows the difference
between the phase boundaries in the two cases $\nu_{3}=2/3,\nu_{4}=0$ and
$\nu_{3}=\nu_{4}=1/3$, in the absence of Zeeman term $h$. The phase diagrams
in the presence of Zeeman term $h$ are shown in Figs.4,5. One can see the
stark difference between phase boundaries and phases realised for
$(1,1,2/3,0)$ and $(1,1,1/3,1/3)$. In the following section, we highlight the
differences in these phase diagrams and their relevance for experiments.

Figure 4: Phase diagrams in the presence of both valley-Zeeman and spin-Zeeman
terms. The Zeeman and valley-Zeeman field ratio $h/\Delta=0.1$. In the
$(1,1,2/3,0)$ state there is a small sliver of region of stability of the
CaAFM state. In the $(1,1,1/3,1/3)$ state, the boundary of the CaAFM state
with the CoAFM state is pushed to infinity.

##  VI Isospin and valley-/spin-order transitions and relation to experiments.

We will now describe the range of parameters and the phase diagrams that are
accessible in experiments by tuning the magnetic field for the case of
graphene aligned with an hBN substrate. Recent experiments have reported the
detection of such transitions using magnon transmission Zibrov  _et al._
(2018); Zhou  _et al._ (2021). In order to relate to these experiments, we
need to express the relation between the experimentally tunable parameters and
the parameters of the model.

The Zeeman term has a linear dependence on the total applied magnetic field
$h=2\mu_{B}B$, $\mu_{B}$ is the Bohr magneton,
$B=\sqrt{B_{\parallel}^{2}+B_{\perp}^{2}}$ (we take the g-factor $g=2$). We
parametrize the coefficients of the anisotropic delta-function interactions,
$u_{\perp,z}$, as follows $(u_{\perp},u_{z})=g(\cos(\theta),\sin(\theta))$.
Typical values for these parameters, obtained from experiments in Refs.Zibrov
_et al._ , 2018; Zhou  _et al._ , 2021 are listed in Table 3.

Figure 5: Phase diagram of $(1,1,2/3,0)$ state with Zeeman and valley-Zeeman
field ratio $h/\Delta=1/3$. As the Zeeman field increases the region of
stability of CaAFM phase enlargens, but still occupies a narrow strip in the
phase diagram.

The total energy of a state will contain in addition to the anisotropy
energies listed in Tables 2, an SU(4) invariant long-range part of the Coulomb
interaction, and thus can be written as:

$\frac{E}{N_{\phi}}=E_{\text{ani}}+\frac{e^{2}}{\epsilon\ell_{1T}}C(\nu)\sqrt{B_{\perp}(T)},$
(23)

Here $\ell_{1T}$ is the magnetic length at 1 Tesla and $E_{\text{ani}}$ is the
corresponding anisotropy energy of the state in question that can be read from
Table 2. In the above we imagine that electrons interact via the ideal Coulomb
interaction $e^{2}/\epsilon r$ (i.e. not screened by gates). The screening
constant, $\epsilon$, accounts both for the screening from the dielectric
substrate as well as the self-screening of graphene, which when accounted at
the RPA level leads Sodemann and Fogler (2012) to
$\epsilon=\epsilon_{hBN}+\frac{\pi}{2}\alpha$ (with
$\alpha=c/(137v_{F})\approx 2.2$). For fully hBN encapsulated samples we take
$\epsilon_{hBN}=4.4$. $C$ in Eq. (23) is a dimensionless constant measuring
the ideal Coulomb energy per flux quantum of the FQHE state in consideration,
and which we extract from previous exact diagonalization studies Xie  _et al._
(1989); NiemelÃ€  _et al._ (2000); Výborný  _et al._ (2009); Davenport and
Simon (2012).

While the Coulomb energy will not affect the SU(4) spinor orientation of a FQH
State, it can play a crucial role determining the competition among different
FQH states with different fillings of the flavors. To illustrate this we
discuss in detail the competition of the Laughlin-like state and a singlet
state at $\nu=2/3$, which has lower Coulomb energy. Their competition has been
widely studied specially in two-component systems in the pastNayak and Wilczek
(1995a, b); McDonald and Haldane (1996); Lay  _et al._ (1997); Wu  _et al._
(2012). The full four-component filling vectors of the states in question are:

Laughlin-like: $\displaystyle(1,1,\frac{2}{3},0)$ vs Singlet-like:
$\displaystyle(1,1,\frac{1}{3},\frac{1}{3})$

Their Coulomb energy difference per flux quantum obtained from exact
diagonalisation studies Xie  _et al._ (1989); NiemelÃ€  _et al._ (2000);
Výborný  _et al._ (2009); Davenport and Simon (2012):

$\displaystyle\frac{\delta E^{\rm Coul}_{2/3}}{N_{\Phi}}\approx
0.006\frac{e^{2}}{\epsilon\ell_{1T}}\sqrt{B_{\perp}(T)}$ (24)

Parameters | Values in experiment at 1T Zibrov  _et al._ (2018); Zhou  _et al._ (2021)
---|---
Valley-Zeeman $\Delta$ | 3.7meV
Zeeman $h$ | 0.115meV
$(u_{z},u_{\perp})=g(\cos\theta,\sin\theta)$ |  $g$=3.87meV
$\theta$ | $123\deg$
$e^{2}/(\epsilon\ell_{1T})$ | 7.1meV
Table 3: List experimental parameters.

We estimate from the experiments in Ref.Zhou  _et al._ , 2021; Zibrov  _et
al._ , 2018 that by tuning the perpendicular field the system follows the line
in the $(u_{z},u_{\perp})$ space shown in Fig.6. Below we describe our key
results and predictions.

Figure 6: (Top) Schematics for different phases and the transitions at total
filling $\nu=2/3$ (see also Fig.1). (Left) Phase diagram as a function of
experimental parameters, the valley-Zeeman term $\Delta$ (hBN potential) and
the perpendicular magnetic field $B_{\perp}$. The horizontal dotted line
indicates a typical value realized in the experiments of Ref. Zhou  _et al._ ,
2021. (Right) The phase diagram schematic indicating the locus of the system
in the $(u_{z},u_{\perp})$ phase diagram under experimental tuning of magnetic
field as shown by dotted line(red). The phase diagram is in the limit of
Zeeman field $h$ much smaller than $\Delta$($h_{1T}\sim 0.1meV,\Delta\sim
3meV$). The region of CaAFM(2/3,0) has been scaled up for visibility.

###  VI.1 Halperin CDW to Laughlin-like CDW iso-spin transition.

At low fields, the system is in a CDW state favored by the hBN induced valley
Zeeman term $\Delta$ (sublattice staggered potential). However, there is a
competition between two distinct CDW states with different fractional
occupations. Namely, one CDW state of Laughlin-like character, with filling
vector $(1,1,2/3,0)$, and the singlet-like CDW with fillings $(1,1,1/3,1/3)$.
These CDW states have the same valley/sublattice polarization but different
spin polarization (their anisotropy energy difference depends only on the
Zeeman term). The singlet-like CDW is spin unpolarized and is the ground state
at low fields where the Coulomb energy is dominant, while the Laughlin-like
CDW becomes the ground state for fields above $\sim 1.6T$ (see Figs. 1 and 6).
This transition is analogous of those in GaAs Eisenstein  _et al._ (1990); Lay
_et al._ (1997); Cho  _et al._ (1998); NiemelÃ€  _et al._ (2000); Feldman  _et
al._ (2009, 2012); Peterson  _et al._ (2015); Zhang  _et al._ (2016) and it
features and interesting interplay of symmetry enriched fractionalization as
we further discuss in Sec.VII.

We emphasize that while this transition should be present in the system, it
has so far not been reported in experiments Polshyn  _et al._ (2018); Zhou
_et al._ (2021). A possible reason for this might be that it is realized at
small fields where disorder effects can be more important. We hope that future
studies can focus on this field range to shed more light on this transition.

###  VI.2 The intermediate CaKD phase and transition to CoAFM.

One of the key predictions from our analysis is the occurrence of an
intermediate CaKD phase between the CDW and AFM phases, as seen in Fig.6.
Notice that unlike the CaKD state realized for the integer quantum Hall
ferromagnet at neutrality (Sec. IV), in the fractionally filled case the
spinors have different orientations in the valley Bloch sphere. This arises
from the unequal coupling of the valley-Zeeman term to the fully filled and
fractionally filled spinors, leading to unequal canting angles of the spinors
in the valley Bloch sphere. This is why we have termed this state a “CaKD”
(Canted Kekule Distortion) to emphasize this difference from the KD state at
neutrality where the two occupied spinors have a common orientation in the
valley Bloch sphere and thus a valley ferromagnetic-like character. Recent
tunneling experiments have demonstrated clear and tantalizing evidence for the
presence of these states Liu  _et al._ (2022); Coissard  _et al._ (2021).

We estimate the transition from CDW to CaKD phase to occur around $B\sim 6T$.
This transition should be “dark” within Magnon transmission experiments since
there is no change of the spin order, explaining why it was not reported in
Refs. Zibrov  _et al._ (2018); Zhou  _et al._ (2021). However, more recently,
remarkably beautiful scanning tunneling spectroscopy (STS) experiments have in
fact detected a transition from the CDW states to a KD state at neutrality Liu
_et al._ (2022); Coissard  _et al._ (2021), and therefore these experiments
have a great potential to detect the corresponding transition in the
fractionally filled case that we predict here. We note in passing that the
interesting STS study of Ref. Li  _et al._ , 2019 also reported the
observation of the Kekule state at neutrality in samples of multilayer
graphene without the hBN substrate. Here the top layer was viewed as decoupled
from the rest, but there was no independent control of the magnetic field and
the density of this layer.

As the applied field is further increased the CaKD Laughlin-like $(1,1,2/3,0)$
state eventually transitions into a CoAFM phase. We estimate the field of the
transition to be $B\sim 8T$. We interpret this as the transition from a
valley-ordered phase to an anti-ferromagnetic that has been detected through
magnon transmission experiments Zhou  _et al._ (2021).

There is one additional phase, the CaAFM (Canted AFM) which in the case of the
Laughlin-like filling $(1,1,2/3,0)$, occupies a very small region in the
parameter space, as in seen in Fig. 4(cf. Also see Fig.5). Its region of
stability is given bySodemann and MacDonald (2014):

$\frac{(1-\nu_{3})(1-\nu_{4})}{(1-\nu)(\nu_{3}-\nu_{4})}\geq\frac{u_{\perp}}{h}\geq\frac{(1-\nu_{3})(1-\nu_{4})}{(1-\nu)(2-\nu)}.$
(25)

Because of the smallness of this region, we believe that the experimentally
accessible line of parameters for the filling $\tilde{\nu}=2+2/3$ does not
cross this phase (see Fig. 6).

##  VII On the universal topological distinction between the Laughlin-like and
singlet-like states.

All the transitions occurring among the states with filling vector (1,1,2/3,0)
that were discussed previously (see Figs.1 and 6 for a summary) can be
understood as a re-orientation of the valley-spin degrees of freedom of the
state. However the transition between the (1,1,1/3,1/3) and (1,1,2/3,0)
occurring around $B_{\perp}\sim 1.6T$ is of a fundamentally different kind
where a change of the more intrinsic nature of the correlations occurs. The
nature of this transition has been the subject of dedicated study of FQH
researchers for decades specially in the context of two-component FQH
systemsNayak and Wilczek (1995a, b); McDonald and Haldane (1996); Lay  _et
al._ (1997); Wu  _et al._ (2012). In this section we would like to briefly
review the theoretical underpinnings on why this transition is special. We
believe that graphene is an unprecedented physical platform that could open
new windows to investigate this amazing physics. For simplicity only in this
section we will concentrate on these states viewed as two component states,
namely as $(2/3,0)$ and $(1/3,1/3)$.

Both the Laughlin like-state $(2/3,0)$ and the singlet-like state $(1/3,1/3)$
possess anyon excitations with same fractional Abelian statistics, namely
under exchange of two of them the many-body wave-function acquires phases
quantized in units of $\pi/3$. In fact, as detailed in Appendix C, these two
states have the same “topological order” Wen (2007). This means that if all
the symmetries of the problem are ignored, then it is possible to find an
adiabatic path smoothly deforming a Hamiltonian with ground state (2/3,0) into
a Hamiltonian with ground state (1/3,1/3) without encountering a phase
transition. As we will see however, such path does not exist if certain
symmetries are enforced, and therefore these states possess distinct “symmetry
enriched topological order”.

But which symmetries need to be enforced to distinguish these states as
distinct phases of matter? These states have anyons with charged quantized in
units of $e/3$ and charge conservation is not enough to distinguish these two
states from a universal point of view. It turns out, however, that these two
states have different topological “spin vectors” Wen and Zee (1992); Nayak and
Wilczek (1995a, b); McDonald and Haldane (1996) (see Appendix C for details).
Therefore, interestingly, space isotropy and its associated rotational
symmetry, should be enough to distinguish these as topologically distinct
states, even if the separate conservation of particle number of the two
components is broken (e.g. layer number conservation or spin conservation
depending on the context).

Moreover these Laughlin-like and singlet-like states can be also be
distinguished by the properties of their edges. The unpolarised state displays
spin-charge separation at the edge in having counter-propagating spin and
charge edge modesWu  _et al._ (2012). Interestingly the presence of separate
particle/spin conservation symmetries in the two components leads to the
appearance of neutral upstream modes at the interface of Laughlin-like and
singlet-like phasesMacDonald (1990); Kane  _et al._ (1994); Kane and Fisher
(1995); Moore and Haldane (1997); Hu  _et al._ (2008); Wang  _et al._ (2013).
There are proposals for using these modes as building elements for topological
quantum computation applications Mong  _et al._ (2014). These neutral modes
have been extensively probed in experiments as well Stern  _et al._ (2004);
Bid  _et al._ (2010); Gross  _et al._ (2012); Gurman  _et al._ (2012); Ronen
_et al._ (2018); Lafont  _et al._ (2019); Wang  _et al._ (2021). We hope that
our study can help guide future efforts to further investigate the fascinating
physics of these states in graphene.

##  VIII Summary and Discussion

We have elaborated on a theory that generalizes the theory of integer quantum
Hall ferromagnets to the fractionally filled states and allows to
quantitatively predict the patterns of symmetry breaking in multi-component
fractional quantum Hall systems, such as graphene. We have in particular
exploited this framework to investigate the role of the staggered sub-lattice
potential that results from the alignment with a boron nitride substrate on
the fractional quantum Hall states in graphene. As a case study, we have made
specific predictions for the states realized at a total filling of $\nu=2/3$
above charge neutrality. Within the typical parameter regimes accessed in the
experiments, we predict a rich sequence of phase transitions as a function of
perpendicular magnetic field, starting from a singlet-like CDW state, then to
a Laughlin-like CDW state, then to a Canted Kekule (CaKD) Laughlin-like state
and finally to a collinear Anti-ferromagnet (CoAFM) Laughlin-like state.

One of our key predictions is the presence of a Canted Kekule (CaKD) Laughlin-
like state, in which the occupied spinors point in different directions in the
valley Bloch sphere, in contrast to the Kekule state (KD) realized at the
integer filling corresponding to neutrality, where the spinors point into a
common direction in the Bloch sphere (this state has been detected in very
recent experiments Liu  _et al._ (2022); Coissard  _et al._ (2021)). Another
important prediction is the existence of a transition around $B\sim 1.6T$ from
a singlet-like CDW phase to a Laughlin-like CDW state. This transition is
hitherto unobserved in graphene, which could offer a new platform to
investigate the interesting physicsNayak and Wilczek (1995a, b); McDonald and
Haldane (1996); Crépel  _et al._ (2019); Liang  _et al._ (2019) of the
competing singlet-like and Laughlin like states at $2/3$.

We would like to also comment on some of the challenges in relating theory and
experiment, which arise partly because some of the orders described here are
hard to detect. Recently an ingenious set of experimental methods has detected
the spin-ordering transitions through non-local magnon excitation/transmission
devices Stepanov  _et al._ (2018); Wei  _et al._ (2018); Zhou  _et al._ (2020,
2021), but these experiments make hard to detect when the transition involves
primarily some change of the valley ordering. However, it has been proposed
Wei  _et al._ (2021) that in graphene aligned with hBN, the interface of the
$\nu=1$ and $\nu=-1$ could be used as a tunable valley-wave source. This could
open up avenues to probe various low-energy excitations in the broken
valley-/spin-order phases on FQHEs of graphene.

Another interesting avenue for experimentally probing these phases, that has
been developed very recently, is the use of scanning tunneling
spectroscopy(STS) to detect and distinguish valley-symmetry broken phases and
their transitions Liu  _et al._ (2022); Coissard  _et al._ (2021). It is
possible that in the future these probes could allow to pinpoint the presence
of the Canted Kekule state that we have found. They are also very powerful for
imaging in detail the defects, quasi-particles, domain walls and other
interesting textures that could appear in the different competing phases
Feldman  _et al._ (2016); Ding  _et al._ (2017); Papić  _et al._ (2018); Tam
_et al._ (2020).

###### Acknowledgements.

We thank Haoxin Zhou and Andrea Young for stimulating discussions, and Allan
H. MacDonald for previous collaborations that were key stepping stones for
this work.

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##  Appendix A  Phase diagram with filling $(1,\nu,0,0)$

We want to obtain the phase diagram with fractional multicomponent fillings
such as $(1,\nu,0,0)$ and in the presence of valley-Zeeman term. Here we
consider only states that are valley ordered. The treatment of spin-active
states are not altered in the presence of the valley-Zeeman term and is the
same as given in previous works Sodemann and MacDonald (2014); Kharitonov
(2012a).

The weighed projection operators (density matrices) for the fully filled and
partially filled spinors are given by:

$\displaystyle P_{i}=\ket{\chi_{1}}\bra{\chi_{1}}$ (26) $\displaystyle
P_{f}=\nu\ket{\chi_{f}}\bra{\chi_{f}}.$ (27)

For the fillings we consider here $(1,\nu)$, the first component $\chi_{1}$ is
fully filled and the second spinor is partially filled at the fraction $\nu$.

We make the following choices for the spinors

$\displaystyle\ket{\chi_{1}}=\ket{n_{1}}\otimes\ket{\uparrow}$ (28)
$\displaystyle P_{1}=P_{n_{1}}\otimes P_{s}$ (29)

$\displaystyle\ket{\chi_{f}}=\ket{n_{2}}\otimes\ket{\downarrow}$ (30)
$\displaystyle P_{f}=\nu P_{n_{2}}\otimes P_{-s}=\nu P_{2}$ (31)

Using these we obtain

$E_{a}=\nu u_{\perp}(n_{1x}n_{2x}+n_{1y}n_{2y})+\nu
u_{z}n_{1z}n_{2z}-\Delta(n_{1z}+\nu n_{2z})$ (32)

Now let us consider the value of $E_{a}$ in different phases. In the CDW
phase,

$n_{1z}=n_{2z};\quad\epsilon_{CDW}=\nu u_{z}-\Delta(1+\nu)$ (33)

Away from the CDW phase, valley symmetry breaking term tends to cant the
valley-isospins by different angles, which would destabilise the KD phase.

Determination of the minimal energy valley-ordered state through the method of
Lagrange multipliers:

We need to obtain the minimum of the energy given in Eq. 32 for the valley
ordered state under the constraint for the valley spinor
$n_{\perp}^{2}+n_{z}^{2}=1$, let us consider the Lagrangian function:

$\displaystyle\mathcal{L}=\nu u_{\perp}n_{1x}n_{2x}+\nu
u_{z}n_{1z}n_{2z}-\Delta(n_{1z}+\nu n_{2}z)$ (34)
$\displaystyle-\frac{\lambda_{1}}{2}(n_{1x}^{2}+n_{1z}^{2}-1)-\frac{\lambda_{2}}{2}(n_{2x}^{2}+n_{2z}^{2}-1),$
(35)

where $\lambda_{1},\lambda_{2}$ are the Lagrangian multipliers.

The optimum is obtained from:

$\displaystyle\frac{\partial\mathcal{L}}{\partial n_{1x}}$ $\displaystyle=$
$\displaystyle\nu u_{\perp}n_{2x}-\lambda_{1}n_{1x}=0$ (36)
$\displaystyle\frac{\partial\mathcal{L}}{\partial n_{2x}}$ $\displaystyle=$
$\displaystyle\nu u_{\perp}n_{1x}-\lambda_{2}n_{2x}=0$ (37)
$\displaystyle\frac{\partial\mathcal{L}}{\partial n_{1z}}$ $\displaystyle=$
$\displaystyle\nu u_{z}n_{2z}-\lambda_{1}n_{1z}-\Delta=0$ (38)
$\displaystyle\frac{\partial\mathcal{L}}{\partial n_{2z}}$ $\displaystyle=$
$\displaystyle\nu u_{z}n_{1z}-\lambda_{2}n_{2z}-\Delta\nu=0$ (39)

The equations for x-components give a constrain for the Lagrangian multipliers
as

$\displaystyle\frac{n_{1x}}{n_{2x}}=\frac{\nu
u_{\perp}}{\lambda_{1}}\quad\frac{n_{1x}}{n_{2x}}=\frac{\lambda_{2}}{\nu
u_{\perp}}$ (40)
$\displaystyle\implies\lambda_{1}\lambda_{2}=\nu^{2}u_{\perp}^{2}.$ (41)

Also multiplying the first and the third equations by $n_{1x}$ and $n_{1z}$
respectively and adding them, we get

$\nu u_{\perp}n_{1x}n_{2x}+\nu u_{z}n_{1z}n_{2z}-\Delta
n_{1z}-\lambda_{1}(n_{1x}^{2}+n_{1z}^{2})=0.$ (42)

Using the constraint $n_{1x}^{2}+n_{1z}^{2}=1$, we get

$\lambda_{1}=\nu u_{\perp}n_{1x}n_{2x}+\nu u_{z}n_{1z}n_{2z}-\Delta n_{1z}.$
(43)

Similarly using the second and the fourth equations for the optimum
conditions, we obtain the expression for $\lambda_{2}$:

$\lambda_{2}=\nu u_{\perp}n_{1x}n_{2x}+\nu u_{z}n_{1z}n_{2z}-\Delta\nu
n_{2z}.$ (44)

The z-components of the spinors can be obtained by solving the following
equations

$\displaystyle\lambda_{1}n_{1z}-\nu u_{z}n_{2z}=-\Delta$ (45)
$\displaystyle\nu u_{z}n_{1z}-\lambda_{2}n_{2z}=\nu\Delta.$ (46)

$n_{1z}=\frac{\lambda_{2}\Delta+\nu^{2}u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})},\quad
n_{2z}=\frac{\lambda_{1}\nu\Delta+\nu
u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}.$ (47)

The Lagrangian multipliers can be determined from the above equations for
$n_{ix},n_{iz}$, $\lambda_{1}\lambda_{2}=\nu^{2}u_{\perp}^{2}$ and the
constraints $n_{ix}^{2}+n_{iz}^{2}=1$.

$\displaystyle n_{1x}^{2}+n_{1z}^{2}=1$ $\displaystyle n_{1x}=$
$\displaystyle\frac{\nu u_{\perp}}{\lambda_{1}}n_{2x},\quad
n_{2x}^{2}=1-n_{2z}^{2}$ $\displaystyle\bigg{(}\frac{\nu
u_{\perp}}{\lambda_{1}}n_{2x}$
$\displaystyle\bigg{)}^{2}+\bigg{(}\frac{\lambda_{2}\Delta+\nu^{2}u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}\bigg{)}^{2}=1$
$\displaystyle\frac{\nu^{2}u_{\perp}^{2}}{\lambda_{1}^{2}}\bigg{(}1-$
$\displaystyle\bigg{(}\frac{\lambda_{1}\nu\Delta+\nu
u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}\bigg{)}^{2}\bigg{)}+\bigg{(}\frac{\lambda_{2}\Delta+\nu^{2}u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}\bigg{)}^{2}=1$

After some algebra, we obtain,

$\lambda_{2}=\pm\nu^{2}u_{\perp}\sqrt{\frac{u_{z}^{2}-u_{\perp}^{2}-\Delta^{2}}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})-\Delta^{2}}}$
(48)

Given this one can determine the energy minimum. The energy function is given
by

$E_{a}=\nu u_{p}(n_{1x}n_{2x}+n_{1y}n_{2y})+\nu
u_{z}n_{1z}n_{2z}-\Delta(n_{1z}+\nu n_{2z})$ (49)

From Eq.43 and Eq.44, we get

$E_{a}=\lambda_{1}-\nu\Delta n_{2z}=\lambda_{2}-\Delta n_{1z}$ (50)

using the above expression for $\lambda_{2}$, we get

$\displaystyle
E_{a}^{*}=-\nu^{2}u_{\perp}\sqrt{\frac{u_{z}^{2}-u_{\perp}^{2}-\Delta^{2}}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})-\Delta^{2}}}\bigg{(}1$
$\displaystyle-$
$\displaystyle\frac{\Delta^{2}}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}\bigg{)}$
(51) $\displaystyle-$
$\displaystyle\frac{u_{z}\Delta^{2}}{u_{z}^{2}-u_{\perp}^{2}}$

one can check that for $\nu=1$, this equation reduces to the minimum energy
expression at neutrality.

Phase boundary between CDW and FM phases: In the CDW phase, the valley spinor
is ordered such that $n_{1z}=n_{2z}=1$. Therefore the equations obtained from
the optimisation reduce to

$\displaystyle\lambda_{1}n_{1z}-\nu u_{z}n_{2z}=-\Delta\implies\lambda_{1}=\nu
u_{z}-\Delta$ (52) $\displaystyle\nu
u_{z}-\lambda_{2}n_{2z}=\nu\Delta\implies\lambda_{2}=\nu u_{z}-\nu\Delta$ (53)
$\displaystyle
n_{1z}=1\implies\frac{\lambda_{2}\Delta+\nu^{2}u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}=1$
(54) $\displaystyle n_{2z}=1\implies\frac{\lambda_{1}\nu\Delta+\nu
u_{z}\Delta}{\nu^{2}(u_{z}^{2}-u_{\perp}^{2})}=1$ (55)

Solving these equations, one obtains the hyperbola(Fig. 7)

$\bigg{(}u_{z}-\frac{\Delta(1+\nu)}{2\nu}\bigg{)}^{2}-u_{\perp}^{2}=\bigg{(}\frac{\Delta(1-\nu)}{2\nu}\bigg{)}^{2}$
(56)

The intercept of the lower branch of the hyperbola with the $u_{z}$ axis is
given by

$\displaystyle u_{z0}=\Delta$ (57)

Figure 7: The hyperbola of instability of CDW phase and the phase boundary
obtain by comparing the CDW and FM energies

Also, comparing the energy of the CDW state with the FM state

$\displaystyle E_{CDW}=\nu u_{z}-\Delta(1+\nu)$ (58) $\displaystyle
E_{FM}=-\nu(2u_{\perp}+u_{z})-\Delta(1-\nu)$ (59)

. Comparing these two the phase boundary is given by (Fig. 7)

$u_{z}=-u_{\perp}+\Delta$ (60)

Phase boundary between caKD and AFM: The phase phase boundary can be obtained
by comparing the above expression for $E_{a}^{*}$ in Eq.51 with the energy for
AFM $\epsilon_{AFM}=\nu u_{\perp}-\Delta(1-\nu)$. Analytically, this does not
give any simple expression for the phase boundary. Nevertheless, one can see
that the expression for $\lambda_{2}$ is real only for
$\nu^{2}(u_{z}^{2}-u_{\perp}^{2})>\Delta^{2}$ and also
$u_{z}^{2}-u_{\perp}^{2}>\Delta^{2}$, beyond, which it becomes complex.
Therefore, the minimal energy obtained is valid only below the (upper branch
of )hyperbola

$u_{z}^{2}-u_{\perp}^{2}=\Delta^{2}$ (61)

and also above the hyperbola
$u_{z}^{2}-u_{\perp}^{2}=\frac{\Delta^{2}}{\nu^{2}}$. $\lambda_{2}$ will again
be real for $\nu^{2}(u_{\perp}^{2}-u_{z}^{2})>\Delta^{2}$ and
$u_{\perp}^{2}-u_{z}^{2}>\Delta^{2}$. Here we have considered only states that
are valley ordered. The energies of spin-ordered states are not affected by
the valley-Zeeman term.

### Mapping from $(1,1,\nu_{3},\nu_{4})$ to $(1,\nu,0,0)$

The energy of the valley-ordered states for $(1,1,\nu_{3},\nu_{4})$ can be
mapped to the energy of the valley-ordered states of $(1,\nu,0,0)$. The energy
for the latter is given by

$E_{a}=\nu u_{\perp}n_{1x}n_{2x}+\nu u_{z}n_{1z}n_{2z}-\Delta(n_{1z}+\nu
n_{2z})$ (62)

Upto constants, the energy for the $(1,1,\nu_{3},\nu_{4})$ state is

$\displaystyle
E_{a}=u_{\perp}(1-\nu)(n_{3x}n_{4x}+u_{z}(1-\nu)n_{3z}n_{4z}-2\nu
u_{\perp}-\nu u_{z}$
$\displaystyle-\Delta((1-\nu_{3})n_{3z}+(1-\nu_{4})n_{4z})$

Now lets define the following scaled quantities:

$\displaystyle\bar{\nu}=\frac{1-\nu_{4}}{1-\nu_{3}};\quad\bar{u}_{\perp}$
$\displaystyle=\frac{1-\nu}{1-\nu_{4}}u_{\perp};\quad\bar{u}_{z}=\frac{1-\nu}{1-\nu_{4}}u_{z};$
(63) $\displaystyle\bar{\epsilon}_{a}$ $\displaystyle=\frac{E_{a}}{1-\nu_{3}}$
(64)

Upon this rescaling, the energy for $(1,1,\nu_{3},\nu_{4})$ is now:

$\bar{\epsilon}_{a}=\bar{\nu}\bar{u}_{\perp}n_{1x}n_{2x}+\bar{\nu}\bar{u_{z}}n_{1z}n_{2z}-\Delta(n_{1z}+\bar{\nu}n_{2z})$
(65)

This exactly matches the expression for $(1,\nu,0,0)$ but with scaled
quantities.

For the case of $\nu_{3}=\nu_{4}$, $\bar{\nu}=1$. Therefore the phase diagram
of $(1,1,\nu_{3},\nu_{3})$ is identical to the one for fully filled state and
for $\nu_{4}=0$, the phase diagram resembles the one for $(1,\nu,0,0)$ state.

##  Appendix B Phase diagram with filling $(1,1,\nu_{3},\nu_{4})$

Here we will consider the valley-ordered states of type
$(1,1,\nu_{3},\nu_{4})$ in the presence of the valley-Zeeman term to obtain
the anisotropy energies and phase boundaries. The anisotropy energy for the
$(1,1,\nu_{3},\nu_{4})$ valley ordered state is

$\displaystyle
E_{a}=(1-\nu)u_{\perp}n_{3x}n_{4x}+(1-\nu)u_{z}n_{3z}n_{4z}-2\nu u_{\perp}-\nu
u_{z}$ $\displaystyle-\Delta((1-\nu_{3})n_{3z}+(1-\nu_{4})n_{4z}),$

where $\nu=\nu_{3}+\nu_{4}$ and we have put $n_{3y}=n_{4y}=0$ for simplicity.
The CDW phase is characterised by $n_{3z}=n_{4z}=1$. The energy corresponding
to that phase is

$E_{a}=(1-2\nu)u_{z}-2\nu u_{\perp}-\Delta(2-\nu)$ (66)

.

The formation of KD phase is obstructed by the valley-Zeeman term. One needs
to find out the valley-spinor configuration that minimises the anisotropy
energy. We shall follow the method of Lagrangian multipliers

Lagrangian multipliers for finding the valley-optimal states The Lagrangian
function is given by

$\displaystyle\mathcal{L}$
$\displaystyle=u_{\perp}(1-\nu)n_{3x}n_{4x}+u_{z}(1-\nu)n_{3z}n_{4z}$
$\displaystyle-2\nu u_{\perp}-\nu
u_{z}-\Delta((1-\nu_{3})n_{3z}+(1-\nu_{4})n_{4z})$
$\displaystyle-\frac{\lambda_{1}}{2}(n_{3x}^{2}+n_{3z}^{2}-1)-\frac{\lambda_{2}}{2}(n_{4x}^{2}+n_{4z}^{2}-1).$

The optimum is obtained from:

$\displaystyle\frac{\partial\mathcal{L}}{\partial
n_{3x}}=(1-\nu)u_{\perp}n_{4x}-\lambda_{1}n_{3x}=0$ (67)
$\displaystyle\frac{\partial\mathcal{L}}{\partial
n_{4x}}=(1-\nu)u_{\perp}n_{3x}-\lambda_{2}n_{4x}=0.$ (68)
$\displaystyle\frac{\partial\mathcal{L}}{\partial
n_{3z}}=(1-\nu)u_{z}n_{4z}-\lambda_{1}n_{3z}-\Delta(1-\nu_{3})=0$ (69)
$\displaystyle\frac{\partial\mathcal{L}}{\partial
n_{4z}}=(1-\nu)u_{z}n_{3z}-\lambda_{2}n_{4z}-\Delta(1-\nu_{4})=0$ (70)

The equations for x-components give a constrain for the Lagrangian multipliers
as

$\displaystyle\frac{n_{3x}}{n_{4x}}=\frac{(1-\nu)u_{\perp}}{\lambda_{1}}\quad\frac{n_{3x}}{n_{4x}}=\frac{\lambda_{2}}{(1-\nu)u_{\perp}}$
(71) $\displaystyle\implies\lambda_{1}\lambda_{2}=(1-\nu)^{2}u_{\perp}^{2}.$
(72)

Multiplying the first and the third equations by $n_{3x}$ and $n_{3z}$
respectively and adding them, we get

$\displaystyle(1-\nu)u_{\perp}n_{3x}n_{4x}+(1-\nu)u_{z}n_{3z}n_{4z}$
$\displaystyle-\Delta(1-\nu_{3})n_{3z}$ $\displaystyle-$
$\displaystyle\lambda_{1}(n_{3x}^{2}+n_{3z}^{2})=0.$

Using the constraint $n_{3x}^{2}+n_{3z}^{2}=1$, we get

$\lambda_{1}=(1-\nu)u_{\perp}n_{3x}n_{4x}+(1-\nu)u_{z}n_{3z}n_{4z}-\Delta(1-\nu_{3})n_{3z}.$
(73)

Similarly using the second and the fourth equations for the optimum
conditions, we obtain the expression for $\lambda_{2}$:

$\lambda_{2}=(1-\nu)u_{\perp}n_{3x}n_{4x}+(1-\nu)u_{z}n_{3z}n_{4z}-\Delta(1-\nu_{4})n_{4z}.$
(74)

The z-components of the spinors can be obtained by solving the following
equations

$\displaystyle-\lambda_{1}n_{3z}+(1-\nu)u_{z}n_{4z}=\Delta(1-\nu_{3})$ (75)
$\displaystyle(1-\nu)u_{z}n_{3z}-\lambda_{2}n_{4z}=\Delta(1-\nu_{4}).$ (76)

$\displaystyle
n_{3z}=\frac{\lambda_{2}\Delta(1-\nu_{3})+(1-\nu)(1-\nu_{4})u_{z}\Delta}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})}$
$\displaystyle
n_{4z}=\frac{\lambda_{1}(1-\nu_{4})\Delta+(1-\nu)(1-\nu_{3})u_{z}\Delta}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})}.$
(77)

The Lagrangian multiplier $\lambda_{2}$ is obtained to be

$\lambda_{2}=\pm(1-\nu)u_{\perp}\sqrt{\frac{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})-(1-\nu_{4})^{2}\Delta^{2}}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})-(1-\nu_{3})^{2}\Delta^{2}}}$
(78)

This is real for
$(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})>(1-\nu_{3})^{2}\Delta^{2}$ and the phase
boundary is given by:

$u_{z}^{2}-u_{\perp}^{2}=\frac{(1-\nu_{3})^{2}}{(1-\nu)^{2}}\Delta^{2}$ (79)

There is another boundary beyond which $\lambda_{2}$ is a complex number:

$u_{z}^{2}-u_{\perp}^{2}=\frac{(1-\nu_{4})^{2}}{(1-\nu)^{2}}\Delta^{2}$ (80)

For $\nu_{3}=\nu_{4}$, the boundaries are the same and are separated for the
case of $\nu_{3}>\nu_{4}$.

####  B.0.1 Boundary of stability of CDW phase

$\displaystyle
n_{3z}=\frac{\lambda_{2}\Delta(1-\nu_{3})+(1-\nu)(1-\nu_{4})u_{z}\Delta}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})}$
(81) $\displaystyle
n_{4z}=\frac{\lambda_{1}(1-\nu_{4})\Delta+(1-\nu)(1-\nu_{3})u_{z}\Delta}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})}.$
(82)

In the CDW phase, $n_{3z}=n_{4z}=1$.

$\displaystyle
n_{3z}=\frac{\lambda_{2}\Delta(1-\nu_{3})+(1-\nu)(1-\nu_{4})u_{z}\Delta}{(1-\nu)^{2}(u_{z}^{2}-u_{\perp}^{2})}=1.$
(83)

The Lagrangian multiplier $\lambda_{2}$ is given by

$\lambda_{2}=(1-\nu)u_{z}-\Delta(1-\nu_{4})$ (84)

Plugging these in to the equation for $n_{3z}=1$and simplifying, one gets

$u_{z}=\sqrt{u_{\perp}^{2}+\frac{\Delta^{2}(\nu^{2}-4\nu_{3}\nu_{4})}{4(1-\nu)^{2}}}+\frac{\Delta}{2}\frac{2-\nu}{1-\nu}$
(85)

####  B.0.2 Boundary between FM and CDW phases.

The energy of the FM phase is given by

$\epsilon_{FM}=-u_{z}-2u_{\perp}-\Delta(\nu_{3}-\nu_{4})$ (86)

The energy of the CDW phase is

$\epsilon_{CDW}=(1-2\nu)u_{z}-2\nu u_{\perp}-\Delta(2-\nu)$ (87)

The boundary between the phase is obtained by equating the above two
expressions:

$u_{z}=-u_{\perp}+\Delta\frac{(1-\nu_{3})}{(1-\nu)}$ (88)

The transition to FM phase from CDW phase appears before the boundary of
stability of CDW phase.

##  Appendix C Topological orders of Multicomponent states.

On considering the Chern-Simons description of the Laughlin-like and singlet
multi-component states at $2/3$ filling, there is a subtle distinction in the
intrinsic nature of the states. This distinction is seen particularly when the
states are on a manifold with intrinsic curvature such as a sphere. The Chern-
Simons theory for the multi-component FQHEs on a curved manifold is given by
Wen (1995); Fradkin (2013).

$\displaystyle\mathcal{L}=-\frac{1}{4\pi}K_{IJ}\epsilon_{\mu\nu\lambda}a_{I}^{\mu}\partial^{\nu}a_{J}^{\lambda}-\frac{e}{2\pi}A_{\mu}t_{I}\epsilon_{\mu\nu\lambda}\partial^{\nu}a^{\lambda}_{I}$
(89)
$\displaystyle+s_{I}\omega^{i}\epsilon_{i\nu\lambda}\partial_{\nu}a^{\lambda}_{I}$
(90)

Here, $t_{I}$ is the charge vector and $s_{I}$ is the spin vector indicating
the charge and spin in each flavors/components rspectively. $K_{IJ}$ is a
$2\times 2$ symmetric matrix(considering 2-component states only here.)

A given FQH state is completely characterised by the triplet
$(K,\vec{t},\vec{s})$ in the presence of a compact curved manifold. Otherwise,
it suffices to specify only $(K_{IJ},\vec{t})$. The $(K,\vec{t},\vec{s})$ are
unique upto $SL(2,Z)$ transfromations:

$K\rightarrow WKW^{T};\quad\vec{t}\rightarrow W\vec{t};\quad\vec{s}\rightarrow
W\vec{s}$ (91)

The K-matrix for the Laughlin-like state is given byWen (1995):

$K=\begin{pmatrix}1&2\\\ 2&1\end{pmatrix};\quad t^{T}=(1,1);\quad
s^{T}=(1/2,-1/2).$ (92)

The state corresponding to this description is known to be obtained from Jain
constructionWen (1995). This is equivalent to a state obtained from a
$\nu=-1/3$ state of holes on a $\nu=1$ quantum Hall state of electrons, for
which the K-matrix is given by:

$K=\begin{pmatrix}1&0\\\ 0&-3\end{pmatrix};\quad t^{T}=(1,-1);\quad
s^{T}=(1/2,-3/2).$ (93)

The above two states are related by the transformation

$W=\begin{pmatrix}1&0\\\ -2&1\end{pmatrix}.$ (94)

On the other hand, the singlet state is described by:

$K=\begin{pmatrix}1&2\\\ 2&1\end{pmatrix};\quad t^{T}=(1,1);\quad
s^{T}=(1/2,1/2).$ (95)

This is not equivalent to the Laughlin-like state by any $SL(2,Z)$
transformation, because of their different topological spins.