# SU(4) spin waves in the $\nu=\pm 1$ quantum Hall ferromagnet in graphene

Jonathan Atteia<EMAIL_ADDRESS>Laboratoire de Physique des
Solides, Université Paris Saclay, CNRS UMR 8502, F-91405 Orsay Cedex, France
Mark Oliver Goerbig<EMAIL_ADDRESS>Laboratoire de Physique des Solides, Université Paris Saclay, CNRS UMR 8502,
F-91405 Orsay Cedex, France

###### Abstract

We study generalized spin waves in graphene under a strong magnetic field when
the Landau-level filling factor is $\nu=\pm 1$. In this case, the ground state
is a particular SU(4) quantum Hall ferromagnet, in which not only the physical
spin is fully polarized but also the pseudo-spin associated with the valley
degree of freedom. The nature of the ground state and the spin-valley
polarization depend on explicit symmetry breaking terms that are also
reflected in the generalised spin-wave spectrum. In addition to pure spin
waves, one encounters valley-pseudo-spin waves as well as more exotic
entanglement waves that have a mixed spin-valley character. Most saliently,
the SU(4) symmetry-breaking terms do not only yield gaps in the spectra, but
under certain circumstances, namely in the case of residual ground-state
symmetries, render the originally quadratic (in the wave vector) spin-wave
dispersion linear.

##  I Introduction

Graphene, a one-atom-thick layer of carbon atoms arranged in a honeycomb
lattice, is the prototype of a large class of two-dimensional materials such
as transition metal dichalchogenoidsManzeli  _et al._ (2017), van der Waals
heterostructuresGeim and Grigorieva (2013) or twisted bilayersLu  _et al._
(2019) and multilayersSinha  _et al._ (2020) that present striking properties
such as topological, correlated or superconducting phases. It is the paradigm
of Dirac fermions in condensed matter since its dispersion is described by the
Dirac-Weyl equation in two dimensionsNovoselov  _et al._ (2005); Cayssol
(2013). These fermions come in two flavours with different chiralities,
represented here by the valley index, which acts as an effective ”pseudo-
spin”.

Upon the application of a magnetic field $B$ perpendicular to the graphene
plane, the relativistic character of the Dirac fermions is at the origin of an
anomalous quantum Hall effect. While the effect is still a consequence of the
quantization of the electrons’ energy into highly degenerate Landau levels
(LLs), the latter inherit from the $B=0$ system a twofold valley degeneracy,
in addition to the spin degeneracy, such that the low-energy Hamiltonian is
invariant under SU(4) spin-valley transformations. This SU(4) symmetry is
furthermore respected to leading order by the Coulomb interaction between the
electrons, which constitutes the dominant energy scale in partially filled LLs
due to the flatness of the latter. If only some spin-valley branches of a
specific LL are filled, all the electrons inside this LL choose to
spontaneously break the SU(4) symmetry and to be polarized in a certain spin
and pseudo-spin state. This marks the onset of SU(4) quantum Hall
ferromagnetismNomura and MacDonald (2006); Doretto and Smith (2007); Goerbig
(2011); Young  _et al._ (2012).

The physics inside a LL is thus dominated by the Coulomb interaction
$E_{C}=e^{2}/\varepsilon l_{B}=625\sqrt{B[T]}$\mathrm{K}$/\varepsilon$, where
$\varepsilon$ is the dielectric constant of the environment the graphene sheet
is embedded into, and $l_{B}=\sqrt{\hbar/eB}$ is the magnetic length. However,
at much smaller energies, explicit symmetry breaking terms become relevant,
such as the Zeeman term, short-range electron-electron interactions, electron-
phonon interactions or coupling to the substrateAlicea and Fisher (2006);
Abanin  _et al._ (2006); Sheng  _et al._ (2007); Nomura  _et al._ (2009);
Kharitonov (2012). These symmetry-breaking terms, which happen to be all on
the same order of magnitude, determine thus the spin-valley polarization of
the ground state. At half-filling of the $n=0$ LL ($\nu=0$) several phases
have been proposed such as a ferromagnetic (F), charge density wave (CDW),
Kekulé distortion (KD) and canted anti-ferromagnetic (CAF) phase as a function
of the symmetry breaking termsHerbut (2007a, b); Kharitonov (2012); Durić  _et
al._ (2014). Notice that there is experimental evidence for three of these
phasesYoung  _et al._ (2014); Li  _et al._ (2019); Veyrat  _et al._ (2020),
indicating that the nature of the SU(4) ferromagnetic ground state may be
sample and/or substrate dependent. At quarter filling $\nu=\pm 1$ – the signs
are related by particle-hole symmetry – the phase diagram has been obtained by
Lian et al. Lian and Goerbig (2017) using the same symmetry breaking terms as
KharitonovKharitonov (2012), and one obtains similar phases as in the $\nu=0$
case.

Spin waves are the lowest energy excitations in a ferromagnet. They have been
observed in a wide variety of materialsBerger (1996); Tsoi  _et al._ (2000);
Kajiwara  _et al._ (2010); An  _et al._ (2014); Hamadeh  _et al._ (2014);
Collet  _et al._ (2016); Wimmer  _et al._ (2019) and are promising platforms
for spintronicsWolf  _et al._ (2001); Chumak  _et al._ (2015). In a two-
dimensional electron gas (2DEG) in GaAs/AlGaAs heterostructures at filling
$\nu=1$, the first example of a quantum Hall ferromagnet, the ground state
consists of all spins pointing in the direction of the magnetic field, and the
spin waves correspond simply to the precession of the spins around their
ground state position. Generalized spin waves have also been extensively
studied and observed in bilayer 2DEGs where the layer index plays the role of
the pseudo-spin. When the distance $d$ is on the order of the magnetic length
$l_{B}$, quantum Hall ferromagnetism of the layer pseudo-spin is observed and
manifests itself in the form of a global phase coherence between electrons in
the two layersMacDonald  _et al._ (1990); Moon  _et al._ (1995). At $\nu=1$
(quarter filling of the $n=0$ LL), the ground state is an interlayer coherent
state where each electron is in a superposition of the two layers, and the
physical spin is fully polarized. This ground state can be viewed as a
condensate of electron-hole pairs which then possesses a gapless, linearly
dispersing superfluid modeWen and Zee (1992); Fertig (1989); MacDonald (2001);
Eisenstein and MacDonald (2004). This mode was observed experimentallySpielman
_et al._ (2001) using tunneling spectroscopy. Put differently, this superfluid
mode is associated with a U(1) symmetry of the ground state that corresponds
to the phase of the electron-hole superposition. At $\nu=2$ (half-filling of
the $n=0$ LL), one is confronted with a frustrated situation: a complete spin
polarization excludes a full pseudo-spin polarization, and vice versa.
Depending on the relative strength of the Zeeman and interlayer tunneling
term, the ground state can thus be a spin ferromagnet, a spin-singlet or an
intermediate phase with CAF orderYang (1999); Demler and Sarma (1999). The
dispersion of the modes at $\nu=2$ are presented in Ref. [Hama  _et al._ ,
2012]. The peculiarity of the CAF phase is that it possesses a U(1) symmetry
associated with the invariance under spin rotation around the $z$ axis. Such a
symmetry implies also a gapless linearly dispersing mode which was observed
experimentally by inelastic light scattering Pellegrini  _et al._ (1998) and
nuclear magnetic resonanceKumada  _et al._ (2006, 2007).

In graphene, due the SU(4) spin-valley symmetry, one can have valley pseudo-
spin waves in addition to spin waves, and what we call “entanglement” waves of
mixed spin-valley character. Recent experimentsStepanov  _et al._ (2018); Wei
_et al._ (2018); Zhou  _et al._ (2020); Assouline  _et al._ (2021) have
managed to electrically emit and detect spin wavesTakei  _et al._ (2016) using
local gates. This is a highly promising result in the prospective of probing
and controlling the spin degree of freedom in quantum-Hall systems. So far,
the observed threshold for the emission of a spin wave is equal to the size of
the Zeeman gap, a strong indication of the emission a pure spin wave. However,
Ref. [Wei  _et al._ , 2020] has suggested a setup susceptible to generate
valley waves at the edge located at the interface between two regions with
filling factors $(\nu_{1},\nu_{2})=(+1,-1)$. The full dispersion relation of
spin waves in graphene at $\nu=0$ has been studied in Refs. [Lambert and Côté,
2013] and [De Nova and Zapata, 2017], while the low-energy dispersion and gaps
of the KD and CAF state spin-waves was obtained using a non-linear sigma model
in Ref. [Wu  _et al._ , 2014], which showed the presence of gapless linearly
dispersing modes in these two phases. Ref. [Wei  _et al._ , 2020] has studied
the transmission of spin waves at a junction between regions with different
filling factors.

Motivated by these recent experiments considering interfaces between regions
at $\nu=1$, 0 and $-1$, we present in this paper a classification of the
dispersion relations and the associated gaps in the graphene quantum Hall
ferromagnet at $\nu=1(-1)$ when one sub-LL is empty (filled). We consider the
spin waves in the four phases introduced in Ref. [Lian and Goerbig, 2017] with
the addition of a “valley Zeeman” term. However, since this term does not
modify substantially the phases but rather the location of their phase
transitions, we consider only the dispersion in the phases of Ref. [Lian and
Goerbig, 2017]. At $\nu=-1$, there are three Goldstone mode corresponding to
flipping one electron from the filled sub-LL to each one of the three empty
sub-LLs. In the simple phases such as KD or CDW, the three modes correspond to
a pure spin wave, a pseudo-spin wave and an entanglement wave. We derive a
non-linear sigma model valid at long wave lengths generalized to the CP3 coset
space corresponding to the space of broken symmetries. In the absence of
explicit symmetry-breaking terms at low energies, all the dispersions are
gapless and quadratic in the wave vector, corresponding thus to true Goldstone
modes. In the presence of the symmetry breaking terms, some modes acquire a
gap, while others remain gapless but acquire a linear dispersion relation
until a certain momentum at which they recover their quadratic dispersion at
higher momentum. We find that this behavior originates from a residual
symmetry of the ground state. We also find that at several high-symmetry
points in the phase diagram, some originally gapped modes become gapless.

The paper is organized as follows. In Sec. II, we present the phase diagram
originally introduced in Ref. [Lian and Goerbig, 2017] using a different
labelling for the phases and also discuss the introduction of a valley Zeeman
term. In Sec. III, we present our non-linear sigma model using a Lagrangian
formalism, while in Sec. IV, we present our results for the dispersion
relation in the different regions of the phase diagram. In the conclusion
section, we present a summary of the various spin waves one encounters in each
phase, in view of their dispersion, i.e. whether they are quadratic and gapped
or linear and gapless.

##  II QHFM ground state

In a single particle picture, flat Landau levels (LLs) are formed in graphene
under a magnetic field with energies $E_{\lambda
n}=\lambda\hbar\omega_{c}\sqrt{n}$ where $\lambda=\pm$ is the band index, $n$
is the LL index, $\omega_{c}=\sqrt{2}v/l_{B}$ is the cyclotron energy, and $v$
is the Fermi velocity of graphene. For a sufficiently strong magnetic field,
the low-energy physics of a quantum Hall ferromagnet in the $n=0$ LL is
dominated by the Coulomb interaction

$\displaystyle\hat{V}_{C}=\frac{1}{2}\sum_{\mathbf{q}\neq
0}v(\mathbf{q})\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),$ (1)

in terms of the Coulomb potential multiplied by the lowest Landau level (LLL)
form factor,

$\displaystyle v(\mathbf{q})=\frac{1}{\mathcal{A}}\frac{2\pi
e^{2}}{\varepsilon|\mathbf{q}|}|\mathcal{F}_{0}(\mathbf{q})|^{2},$ (2)

where $\mathcal{A}$ is the area of the sample and
$\mathcal{F}_{0}(\mathbf{q})$ is the form factor of the LLL (see eg. Ref.
[Goerbig, 2011]). Furthermore, $\bar{\rho}(\mathbf{q})$ represents the density
operator in momentum space projected into the LLL. This Hamiltonian is
approximately SU(4) invariant under spin-valley rotations. The exchange terms
favors a completely antisymmetric orbital wavefunction to minimize the Coulomb
replusion, which then favors a completely symmetric spin-valley spinor. At
filling $\nu=-1$, there is thus one electron per orbital site and the uniform
ground state is described by the Slater determinant

$\displaystyle|\psi_{0}\rangle=\prod_{m}\left(\sum_{\mu}F_{\mu}c^{\dagger}_{m,\mu}\right)|0\rangle$
(3)

where $\mu=\\{\sigma,\xi\\}$ runs over the spin
($\sigma\in\\{\uparrow,\downarrow\\}$) and valley ($\xi\in\\{K,K^{\prime}\\}$)
indices, $m$ is the Landau site index and $F$ is a normalized four-component
spinor which describes the QHFM ground state.

###  II.1 Parametrization of the spinor

The Coulomb Hamiltonian is SU(4) symmetric, while the broken symmetry ground
state is invariant under SU(3)$\otimes$U(1) rotations corresponding to
rotations between the three empty sub-LL and the relative phase between the
empty and filled sub-LL. The coset space is thus $CP^{3}=U(4)/U(3)\otimes
U(1)$ which has 6 real dimensionsYang  _et al._ (2006). A general spinor
describing the broken symmetry ground state is thus parametrized by 6 angles.
In order to describe the spinor $F$, we express it as a Schmidt decomposition
in the basis
$\\{|K\uparrow\rangle,|K\downarrow\rangle,|K^{\prime}\uparrow\rangle,|K^{\prime}\downarrow\rangle\\}$
asDouçot  _et al._ (2008); Lian and Goerbig (2017)

$\displaystyle|F\rangle=\cos\frac{\alpha}{2}|\mathbf{n}\rangle|\mathbf{s}\rangle+e^{i\beta}\sin\frac{\alpha}{2}|-\mathbf{n}\rangle|-\mathbf{s}\rangle,$
(4)

where
$|\mathbf{n}\rangle|\mathbf{s}\rangle=|\mathbf{n}\rangle\otimes|\mathbf{s}\rangle$
is the tensor product of the spinors

$\displaystyle|\mathbf{n}\rangle$
$\displaystyle=\begin{pmatrix}\cos\frac{\theta_{P}}{2}\\\
\sin\frac{\theta_{P}}{2}e^{i\varphi_{P}}\end{pmatrix},$ (5)
$\displaystyle|\mathbf{s}\rangle$
$\displaystyle=\begin{pmatrix}\cos\frac{\theta_{S}}{2}\\\
\sin\frac{\theta_{S}}{2}e^{i\varphi_{S}}\end{pmatrix},$ (6)

acting in valley and spin spaces respectively. We have
$\bm{\sigma}\cdot\mathbf{s}|\pm\mathbf{s}\rangle=\pm|\pm\mathbf{s}\rangle$ and
$\bm{\tau}\cdot\mathbf{n}|\pm\mathbf{n}\rangle=\pm|\pm\mathbf{n}\rangle$,
where

$\displaystyle\mathbf{s},\mathbf{n}=\begin{pmatrix}\sin\theta_{S,P}\cos\varphi_{S,P}\\\
\sin\theta_{S,P}\sin\varphi_{S,P}\\\ \cos\theta_{S,P}\end{pmatrix}$ (7)

are the unit vectors on the spin and pseudo-spin Bloch spheres, respectively,
with $\theta_{S},\theta_{P}\in[0,\pi]$ and
$\varphi_{S},\varphi_{P}\in[0,2\pi]$. The angles $\alpha\in[0,\pi]$ and
$\beta_{1}\in[0,2\pi]$ are the angles of the ”entanglement” Bloch sphere of
the particleDouçot  _et al._ (2008). The spinors $|-\mathbf{s}\rangle$ and
$|-\mathbf{n}\rangle$ are obtained from $|\mathbf{s}\rangle$ and
$|\mathbf{n}\rangle$ by the replacement $\theta\rightarrow\pi-\theta$ and
$\varphi\rightarrow\varphi+\pi$ such that we have
$\langle\mathbf{s}|-\mathbf{s}\rangle=\langle\mathbf{n}|-\mathbf{n}\rangle=0$.

When $\theta_{P}=0(\pi)$, the vector $\mathbf{n}$ lies at the north (south)
pole of the pseudo-spin Bloch sphere corresponding to a polarization in valley
$K(K^{\prime})$. Analogously, for $\theta_{S}=0(\pi)$, the vector $\mathbf{n}$
lies at the north (south) pole of the spin Bloch sphere corresponding to spin
up (down) polarization. Finally, this parametrization includes the possibility
of “entanglement” between the spin and the pseudo-spin. In fact, this
decomposition of the spinors does not correspond to real entanglement between
two particles because here it is the spin and pseudo-spin of the same particle
which is “entangled”, and the Schmidt decomposition can be viewed as a
decomposition of SU(4) spinors in the basis of SU(2)$\otimes$SU(2) spinors.
Because of this reminiscence and the relevance of the spin and pseudospin
magnetizations in experimental measurements, we will refer loosely to the
angle $\alpha$ as entanglement angle for simplicity.

###  II.2 Symmetry breaking terms

Inspired by earlier worksKharitonov (2012); Nomura  _et al._ (2009); Lian and
Goerbig (2017); Atteia  _et al._ (2021) that focus on short-range electron-
electronAlicea and Fisher (2006) and electron-phononKharitonov (2012)
interactions at the lattice scale, we consider the local anisotropic
Hamiltonian

$\displaystyle H_{A}=\frac{1}{2}$ $\displaystyle\int
d^{2}r\left\\{U_{\perp}[P_{x}^{2}(\mathbf{r})+P_{y}^{2}(\mathbf{r})]+U_{z}P_{z}^{2}(\mathbf{r})\right\\}$
$\displaystyle-$ $\displaystyle\int
d^{2}r\left\\{\Delta_{Z}S_{z}(\mathbf{r})+\Delta_{P}P_{z}(\mathbf{r})\right\\},$
(8)

where

$\displaystyle\mathbf{P}(\mathbf{r})$
$\displaystyle=\Psi^{\dagger}(\mathbf{r})(\sigma_{0}\otimes\bm{\tau})\Psi(\mathbf{r}),$
(9) $\displaystyle\mathbf{S}(\mathbf{r})$
$\displaystyle=\Psi^{\dagger}(\mathbf{r})(\bm{\sigma}\otimes\tau_{0})\Psi(\mathbf{r})$
(10)

are the local spin and pseudo-spin densities, respectively, in terms of the
vectors $\bm{\sigma}$ and $\bm{\tau}$ of Pauli matrices vectors acting in spin
and pseudo-spin spaces, respectively, while $\sigma_{0}$ and $\tau_{0}$ are
the identity matrices. In the following, we neglect the identity and consider
$\bm{\sigma}\equiv\bm{\sigma}\otimes\tau_{0}$ and
$\bm{\tau}\equiv\sigma_{0}\otimes\bm{\tau}$. The potentials $U_{\perp}$ and
$U_{z}$ correspond to local interactions that act when two electrons are at
the same position, and they act only in valley space thus favoring in-plane or
out-of-plane pseudo-spin polarizations. The relative values of $\Delta_{Z}$,
$\Delta_{P}$, $U_{z}$ and $U_{\perp}$ determine thus the spin or pseudo-spin
polarization of the ground state.

The first term in Eq. (8) represents the electrons’ interaction with ”frozen”
in-plane phononsNomura  _et al._ (2009) and is estimated to be of the order of
$U_{\perp}\sim 2.0B[(T)]K$. This term creates a Kekulé-like distortion. The
term $U_{z}$ originates from short-range Hubbard type interactionsAlicea and
Fisher (2006) and intervalley scattering which originate from the SU(4)
symmetry breaking the in Coulomb interactionGoerbig  _et al._ (2006). Out-of-
plane phonons also contribute to $U_{z}$ and is estimated to be of the order
of $\sim 0.5B[(T)]K$. The Zeeman coupling $\Delta_{Z}=g\mu_{B}B$ is of the
order of $\sim 1.2B[(T)]K$. Finally, $\Delta_{P}$ corresponds to a staggered
potential on the A and B sublattice which generates a mass term in the Dirac
equation and can be generated by the interaction with a substrate, eg
hexagonal Boron-Nitride (hBN)Hunt  _et al._ (2013); Amet  _et al._ (2013). Due
to the locking of the sublattice and valley indices in the $n=0$ LL, this term
is analogous to a Zeeman term acting in pseudo-spin space, we thereby dub it
”valley Zeeman” term. This terms favors a polarization in one valley and thus
on one sublattice. The energies $U_{\perp}$ and $U_{z}$ are proportional to
the perpendicular magnetic fieldLi  _et al._ (2019) while $\Delta_{z}$ is
proportional to the total magnetic field. Moreover, $\Delta_{P}$ is an
intrinsic effect and thus independent of the magnetic field. Notice that these
energy scales are all on the same order of magnitude and are likely to be
strongly sample-dependent. We thus consider them, here, as tunable parameters
that determine the phase diagram of the QHFM ground states as well as that of
the skyrmions formed on top of these states.

Applying the Hartree-Fock approximation, the energy of the anisotropic energy
$E_{A}=\langle F|H_{A}|F\rangle$ can be expressed asLian and Goerbig (2017)

$\displaystyle
E_{A}[F]=\frac{N_{\phi}}{2}\left[u_{\perp}\left(M_{P_{x}}^{2}+M_{P_{y}}^{2}\right)+u_{z}M_{P_{z}}^{2}\right]$
(11) $\displaystyle-
N_{\phi}\left[\Delta_{Z}M_{S_{z}}+\Delta_{P}M_{P_{Z}}\right],$ (12)

where $N_{\phi}=A/(2\pi l_{B}^{2})$ is the number of flux quanta threading the
area $A$ of the sample and

$\displaystyle\mathbf{M_{P}}$ $\displaystyle=\langle
F|\bm{\tau}|F\rangle=\mathbf{n}\cos\alpha$ (13) $\displaystyle\mathbf{M_{S}}$
$\displaystyle=\langle F|\bm{\sigma}|F\rangle=\mathbf{s}\cos\alpha$ (14)

are the spin and pseudo-spin magnetization respectively. The parameters
$u_{\perp,z}$ are obtained as

$\displaystyle
u_{\perp,z}=\mathcal{V}^{H}_{\perp,z}-\mathcal{V}^{F}_{\perp,z}$ (15)

where $\mathcal{V}^{H}_{\perp,z}$ and $\mathcal{V}^{F}_{\perp,z}$ are the
Hartree and Fock potentials, respectively, associated with the potentials
$U_{\perp,z}$. For a $\delta(\mathbf{r})$ interaction, at $\nu=\pm 1$, the
Hartree and Fock potentials are identical and thus cancel each otherLian and
Goerbig (2017). We thus postulate a slightly non-local interaction.

As a function of the angles, we obtain the expression

$\displaystyle E_{A}[F]=$ $\displaystyle
N_{\phi}\left[\frac{1}{2}\cos^{2}\alpha(u_{\perp}\sin^{2}\theta_{P}+u_{z}\cos^{2}\theta_{P})\right.$
$\displaystyle\left.-\Delta_{P}\cos\alpha\cos\theta_{S}-\Delta_{Z}\cos\alpha\cos\theta_{S}\right].$
(16)

The phase diagram is obtained by minimizing Eq. (16). We first consider the
phase diagram without the valley Zeeman term $\Delta_{P}$ in Sec. II.3, while
we show its effect in Sec. II.4.

###  II.3 Phase Diagram without valley Zeeman term

The phase diagram of the QHFM at $\nu=\pm 1$ without the valley Zeeman term
was calculated by Lian et alLian and Goerbig (2017). Here, we briefly review
the different phases in order to discuss the spin waves associated with each
ground state. There is a $\mathbb{Z}_{2}$ redundancy in the parametrization of
the spinors (see appendix of Ref. [Lian and Goerbig, 2017]) such that without
loss of generality we can assume $\alpha\in\left[0,\pi/2\right]$. Using this
fact we can see that the anisotropic energy is minimized for
$\cos\theta_{S}=1$ everywhere.

Figure 1: (a) Phase diagram of the QHFM ground state composed of four phase :
charge density wave (CDW), Kekulé distortion (KD), anti-ferrimagnetic (AFI)
and canted anti-ferromagnetic (CAF). (b)-(e) Spin magnetization on the A and B
sublattices of the different phase. (b) CDW (c) KD, (d) AFI and (e) CAF.

Minimizing Eq. (16), we find the four phases shown in Fig. (1) which can be
separated in two types : for $u_{\perp}>u_{z}$, an easy-axis pseudo-spin
polarization is favored, which is the case of the charge density wave (CDW)
and anti-ferrimagnetic (AFI) phases, while for $u_{z}>u_{\perp}$, an easy-
plane polarization is favored, namely, the Kekulé distortion (KD) and canted
anti-ferromagnetic (CAF) phase. In addition to that, the phases can present
entanglement ($\alpha\neq 0$) or not ($\alpha=\\{0,\pi\\}$). The CDW and KD
phases are not entangled and they have maximal spin and pseudo-spin
magnetizations, they are thereby ferromagnetic phases. The AFI and CAF phases
are entangled, such that their spin and pseudo-spin magnetizations are
reduced. These phases are realized in the regions of positive $u_{\perp}$ and
$u_{z}$ because entanglement allows to reduce the pseudo-spin magnetization
thus making a compromise between the spin and pseudo-spin magnetizations. In
the limit of vanishing Zeeman term (compared to $u_{\perp}$ and $u_{z}$),
these two phases are maximally entangled become both anti-ferromagnetic. We
mention that, as opposed to the $\nu=0$ case, at $\nu=\pm 1$, the spin and
pseudo-spin can be maximal at the same time. Thus the CDW and KD phases are
pseudo-spin polarized and spin ferromagnetic, whereas at $\nu=0$, the phases
can be either be spin polarized and pseudo-spin unpolarized (F), pseudo-spin
polarized and spin unpolarized (KD and CDW) or entangled (CAF). Notice that in
Ref. [Lian and Goerbig, 2017], these phases were named after their valley
pseudo-spin magnetization : the CDW (AFI) phases are associated with an
unentangled (entangled) easy-axis pseudo-spin order, while the KD (CAF) comes
along with an unentangled (entangled) easy-plane pseudo-spin magnetization.

In order to characterize the different phases, we focus on experimentally
measurable quantities such as the spin magnetization and electronic density on
the A and B sublattices

$\displaystyle\rho_{A,B}$ $\displaystyle=\frac{1}{2}\langle
F|(\tau_{0}\pm\tau_{z})|F\rangle,$ (17) $\displaystyle\mathbf{M_{S}}_{A,B}$
$\displaystyle=\frac{1}{2}\langle
F|\bm{\sigma}(\tau_{0}\pm\tau_{z})|F\rangle,$ (18)

respectively.

The spinor of the CDW phase is

$\displaystyle|F\rangle=|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle,$ (19)

where $\mathbf{n}_{z}=(1,0)^{T}$ and $\mathbf{s}_{z}=(1,0)^{T}$ correspond to
a spin and pseudo-spin both polarized at the north of their respective Bloch
spheres, such that the electrons have spin-up and are polarized in valley $K$
or $K^{\prime}$ corresponding thus to a ferromagnetic phase restricted to a
single sublattice. The sublattice polarization is given by $\rho_{A}=1$ and
$\rho_{B}=0$ or $\rho_{A}=0$ and $\rho_{B}=1$ and there is thus a spontaneous
$\mathbb{Z}_{2}$ sublattice symmetry breaking. The spin magnetizations on
sublattices A and B are $\mathbf{M_{S_{A}}}=\mathbf{s}_{z}$ and
$\mathbf{M_{S_{B}}}=0$.

The spinor of the KD phase is given by

$\displaystyle|F\rangle=|\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle,$
(20)

where $|\mathbf{n}_{\perp}\rangle=\frac{1}{\sqrt{2}}(1,e^{i\varphi})^{T}$
points to a position at the equator of the pseudo-spin Bloch sphere and
corresponds thus to a superposition of the two valleys. The angle $\varphi$
corresponds to the orientation of the pseudo-spin magnetization in the $xy$
plane. There is thus a residual $U(1)$ symmetry corresponding to the angle
$\varphi$. Both sublattices are equally populated such that
$\rho_{A}=\rho_{B}=1/2$ and
$\mathbf{M_{S_{A}}}=\mathbf{M_{S_{B}}}=\frac{1}{2}\mathbf{s}_{z}$.

The spinor of the AFI phase has the expression

$\displaystyle|F\rangle$
$\displaystyle=\cos\frac{\alpha_{1}}{2}|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{1}}{2}|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle,$
(21)

with

$\displaystyle\cos\alpha_{1}=\frac{\Delta_{Z}}{u_{z}}.$ (22)

This phase corresponds thus to an entangled phase which in turn reduces the
amplitude of the spin magnetization in order to minimize the anisotropic
energy. The spin magnetization on the A and B sublattices are
$\mathbf{M_{S_{A}}}=\frac{1}{2}(1+\cos\alpha_{1})\mathbf{s}_{z}$ and
$\mathbf{M_{S_{B}}}=\frac{1}{2}(-1+\cos\alpha_{1})\mathbf{s}_{z}$ such that
the spin magnetization on each sublattice points along the $z$ direction but
there is an imbalance between the spin magnetization in sublattices A and B.
For $u_{z}=\Delta_{Z}$ ($\alpha_{1}=0$), namely at the CDW-AFI transition, we
recover the CDW phase, while for $u_{z}\gg\Delta_{Z}$
($\alpha_{1}\rightarrow\pi/2$), we have a maximally entangled phase with
$\mathbf{M_{S_{A}}}=-\mathbf{M_{S_{B}}}=\frac{1}{2}\mathbf{s}_{z}$ which is
anti-ferromagnetic, as we would expect in the limit of a vanishing Zeeman
effect.

The spinor of the CAF phase has the expression

$\displaystyle|F\rangle$
$\displaystyle=\cos\frac{\alpha_{2}}{2}|\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{2}}{2}|-\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle,$
(23)

with

$\displaystyle\cos\alpha_{2}=\frac{\Delta_{Z}}{u_{\perp}}.$ (24)

This phase has its pseudo-spin polarized in the $xy$ plane of the Bloch sphere
and presents entanglement analogously to the AFI phase. Both sublattices are
populated equally $\rho_{A}=\rho_{B}=1/2$. The spin magnetization on the A and
B sublattices forms a canted anti-ferromagnetic pattern with
$\mathbf{M_{S_{A,B}}}=(\pm\sin\alpha_{2}\cos(\beta-\varphi),\pm\sin\alpha_{2}\sin(\beta-\varphi),\cos\alpha_{2})$
such that the $z$ component of the magnetization is identical on both
sublattices, but there is a canting of the spin in the $xy$ plane with
opposite orientation on the sublattices. At the transition with the KD phase
($\Delta_{Z}=u_{\perp}\rightarrow\alpha_{2}=0$), we recover a ferromagnetic
phase with equal weight on the K and K’ valleys, while in the fully entangled
limit ($u_{\perp}\gg\Delta_{z}\rightarrow\alpha_{2}=\pi/2$), we obtain an
anti-ferromagnetic phase with spins pointing in the $xy$ plane.

###  II.4 Phase diagram with valley Zeeman

Figure 2: (a)Phase diagram of the QHFM ground state with the valley Zeeman
term $\Delta_{P}$ such that $\Delta_{P}=\Delta_{Z}$. The KD and CAF phases are
modified compared to the case without the valley Zeeman term and are turned to
a canted KD phase (CDW) and a different CAF phase (CAF’).

Experimentally, graphene is generally placed on top of a substrate. In the
case of hBN, a potential difference is generated between the A and B sites of
graphene and yields a valley-dependent potential due to the valley-sublattice
equivalence in the LLL of graphene. Such a term favors a polarization on one
sublattice and thus in one valley, analougously to a Zeeman term in valley
space. The evolution of the phase diagram in the presence of the valley Zeeman
term is shown in Fig. 2. The phases CDW and AFI are not modified by the valley
Zeeman term because their pseudo-spin is already polarized in one valley.
However, the presence of the valley Zeeman breaks the $Z_{2}$ symmetry between
the two valleys by favoring one valley corresponding to the sublattice with
smallest on-site potential. However, the KD and CAF phases are modified such
that their pseudo-spin polarization is now canted towards the north pole of
the Bloch sphere (or the south pole if the staggered potential is reversed).
The KD phase becomes a canted KD phase with spinor

$\displaystyle|F\rangle=|\mathbf{n}\rangle|\mathbf{s}_{z}\rangle,$ (25)

with

$\displaystyle\cos\theta_{P}=\frac{\Delta_{P}}{(u_{z}-u_{\perp})}.$ (26)

There is thus a continuous phase transition between the CDW and CKD phase
transition located at $u_{z}-u_{\perp}=\Delta_{P}$, where the pseudo-spin is
progressively canted relative to the $z$ direction. For
$u_{z}-u_{\perp}\gg\Delta_{P}$, we recover the KD phase. The CDW occupied thus
a larger portion of the phase diagram compared to the $\Delta_{P}=0$ case (see
Fig. 1).

The transition between the CDW and AFI phase is also modified because the cost
to entangle the easy-axis phase implies a non-zero weight on the valley
$K^{\prime}$. Thereby, the transition occurs at
$u_{z}=(\Delta_{P}+\Delta_{Z})$ and the entanglement angle in the AFI phase
$\alpha_{1}$ is now given by

$\displaystyle\cos\alpha_{1}=\frac{\Delta_{Z}+\Delta_{P}}{u_{z}}.$ (27)

Finally, the CAF phase is also modified into a different CAF phase such that
the spinor reads

$\displaystyle|F\rangle$
$\displaystyle=\cos\frac{\alpha_{2}}{2}|\mathbf{n}\rangle|\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{2}}{2}|-\mathbf{n}\rangle|-\mathbf{s}_{z}\rangle,$
(28)

where

$\displaystyle\cos\alpha_{2}=\frac{\Delta_{Z}}{u_{\perp}}\quad\text{and}\quad\cos\theta_{P}=\frac{\Delta_{Z}}{\Delta_{P}}\frac{u_{\perp}}{(u_{z}-u_{\perp})}.$
(29)

Once again, the AFI phase is favored in a larger part of the phase diagram and
the transition between the AFI and CAF phases is located at
$u_{z}=u_{\perp}(\Delta_{P}/\Delta_{Z}+1)$. The four phase transitions meet at
the point $(u_{\perp},u_{z})=(\Delta_{Z},\Delta_{Z}+\Delta_{P})$.

##  III Non-linear sigma model

In order to find the dispersion relations of the Goldstone modes, we derive an
effective Lagrangian which describes the low-energy (long-wavelength)
excitations of the ground state. In the SU(4) invariant limit (in the absence
of symmetry breaking terms), this Lagrangian consists of a non-linear sigma
model describing the fields associated with the broken symmetries. The
collective modes of this Lagrangian are the different Goldstone modes. In the
presence of the symmetry breaking terms, the Goldstone modes acquire a mass
gap.

###  III.1 Broken symmetries and their generators

Figure 3: Four sub-LLs of the $n=0$ LL and the three associated spin wave
modes corresponding to the mixing of the filled sub-LL described by the spinor
$|F\rangle$ with each of the three empty sub-LLs described by the spinors
$|C_{i}\rangle$.

At filling factor $\nu=\pm 1$, the spontaneous symmetry breaking mechanism
corresponds to filling one sub-LL out of the four with any SU(4) spin-valley
orientation (in the absence of symmetry breaking term). Explicitely, this
symmetry breaking mechanism corresponds to

$\displaystyle SU(4)\rightarrow SU(3)\otimes U(1),$ (30)

where SU(4) is the original symmetry of the Hamiltonian which in composed of
15 generators and SU(3)$\otimes$U(1) is the residual symmetry the ground state
which is invariant under tranformations that mixes the 3 empty sublevels
corresponding to 8 generators times the relative U(1) phase between the empty
and the occupied sub-LLs. According to Refs. [Arovas  _et al._ , 1999] and
[Yang  _et al._ , 2006], there are thus $15-8-1=6$ generators associated with
the broken symmetries. For simplicity, we label these generators ”broken
generators”. The corresponding coset space of the non-linear sigma model is
the complex projective space $CP^{3}=U(4)/[U(3)\otimes U(1)]$ which has six
dimensionsYang  _et al._ (2006).

In order to find an explicit expression for the broken generators, we consider
for simplicity the CDW ground state
$|F\rangle=|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle=|K\uparrow\rangle$ to
be the filled sub-LL in the basis
$\mathcal{A}=\\{|F\rangle,|C_{1}\rangle,|C_{2}\rangle,|C_{3}\rangle\\}=\\{|K\uparrow\rangle,|K\downarrow\rangle,|K^{\prime}\uparrow\rangle,|K^{\prime}\downarrow\rangle\\}$
as shown in Fig. 3. The spinors $|C_{i}\rangle$ define the empty sub-LLs of
the basis $\mathcal{A}$. In this basis, we are able to define the six broken
generators

$\begin{aligned} \Gamma^{1}_{x}&=\frac{1}{2}\sigma_{x}P_{+n_{z}}\\\
\Gamma^{2}_{x}&=\frac{1}{2}\tau_{x}P_{+s_{z}}\\\
\Gamma^{3}_{x}&=\frac{1}{4}(\sigma_{x}\tau_{x}-\sigma_{y}\tau_{y})\end{aligned}\quad\begin{aligned}
\Gamma^{1}_{y}&=\frac{1}{2}\sigma_{y}P_{+n_{z}}\\\
\Gamma^{2}_{y}&=\frac{1}{2}\tau_{y}P_{+s_{z}}\\\
\Gamma^{3}_{y}&=\frac{1}{4}(\sigma_{x}\tau_{y}+\sigma_{y}\tau_{x}),\end{aligned}$
(31)

where $P_{+s_{z}}=\frac{1}{2}(1+\sigma_{z})$ and
$P_{+n_{z}}=\frac{1}{2}(1+\tau_{z})$ are the projectors over the spin up and
valley $K$, respectively. Here, the matrices $\bm{\sigma}$ and $\bm{\tau}$ are
the usual Pauli matrices acting in the spin and pseudo-spin spaces,
respectively. Explicitely, the $\Gamma_{x}$ operators are

$\displaystyle\Gamma^{1}_{x}$ $\displaystyle=\begin{pmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix}\quad\Gamma^{2}_{x}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 0&0&0&0\end{pmatrix}$ $\displaystyle\Gamma^{3}_{x}$
$\displaystyle=\begin{pmatrix}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\
1&0&0&0\end{pmatrix}.$ (32)

The matrices $\Gamma^{1}_{x,y}$ mix $|F\rangle$ and $|C_{1}\rangle$, the
matrices $\Gamma^{2}_{x,y}$ mix $|F\rangle$ and $|C_{2}\rangle$ while the
matrices $\Gamma^{3}_{x,y}$ mix $|F\rangle$ and $|C_{3}\rangle$. We have thus
three sets of canonically conjugate matrices such that for each mode $a$

$\displaystyle[\Gamma_{\mu}^{a},\Gamma_{\nu}^{a}]$
$\displaystyle=i\varepsilon_{\mu\nu\lambda}\Gamma_{\lambda}^{a}$ (33)
$\displaystyle\\{\Gamma_{\mu}^{a},\Gamma_{\nu}^{a}\\}$
$\displaystyle=\frac{i}{2}\delta_{\mu\nu},$ (34)

where $\mu,\nu,\lambda\in\\{x,y,z\\}$, $a\in\\{1,2,3\\}$,
$\varepsilon_{\mu\nu\lambda}$ is the three-dimensional Levi-Civita tensor,
$\delta_{\mu\nu}$ is the identity matrix and we have introduced the additional
matrices

$\displaystyle\Gamma^{1}_{z}=\frac{1}{2}\sigma_{z}P_{+n_{z}},\quad\Gamma^{2}_{z}=\frac{1}{2}\tau_{z}P_{+s_{z}},\quad\Gamma^{3}_{z}=\frac{1}{4}(\sigma_{z}+\tau_{z}),$
(35)

to complete the algebra. To study the spin waves for another phase, we simply
rotate the spinors and the generators by a SU(4) unitary transformation $U$

$\displaystyle|\tilde{F}\rangle=U|F\rangle,$ (36a)
$\displaystyle|\tilde{C}_{i}\rangle=U|C_{i}\rangle,$ (36b)
$\displaystyle\tilde{\Gamma}_{\mu}^{a}=U\Gamma_{\mu}^{a}U^{\dagger}.$ (36c)

An important object that characterizes the spin waves in a (anti-)ferromagnet
is the matrix of the commutators of the broken generators over the ground
state

$\displaystyle M_{\mu\nu}^{ab}=\langle
F|[\Gamma_{\mu}^{a},\Gamma_{\nu}^{b}]|F\rangle,$ (37)

with $\mu,\nu\in\\{x,y\\}$. We find that it is independent of the basis and
defines the number and dispersion of the Goldstone modes associated with the
number of broken symmetryNielsen and Chadha (1976); Watanabe and Brauner
(2011); Hidaka (2013) (in the absence of explicit symmetry breaking terms). We
find that this matrix has the expression for any phase

$\displaystyle\langle
F|[\Gamma_{\mu}^{a},\Gamma_{\nu}^{b}]|F\rangle=\frac{i}{2}\varepsilon_{\mu\nu}\delta_{ab}.$
(38)

where $\varepsilon_{\mu\nu}$ is the two-dimensional Levi-Civita tensor for
$\mu,\nu\in\\{x,y\\}$. According to the general theory of Refs. [Watanabe and
Brauner, 2011] and [Hidaka, 2013], the number of quadratic spin waves is equal
to $\text{Rank}[M]/2=3$ while no linearly dispersing modes are found which is
in agreement with Refs. [Arovas  _et al._ , 1999] and [Yang  _et al._ , 2006],
where the number of Goldstone modes is shown to be half the number of the
broken symmetries because half of the fields are conjugate to the other half.
We thus expect three quadratically dispersing modes in the absence of symmetry
breaking terms. However, we show below that in some cases, the Goldstone modes
become linear at small wavevectors due to spin-valley anisotropic terms that
explicitely break the residual symmetry to yet lower ones.

###  III.2 Lagrangian

The effective low-energy Lagrangian is obtained analogously to Ref. [Moon  _et
al._ , 1995] by constructing a coherent state

$\displaystyle|\psi[\pi]\rangle=e^{i\sum_{\mathbf{r}_{i}}O(\mathbf{r}_{i},t)}|\psi_{0}\rangle,$
(39)

where $|\psi_{0}\rangle$ is the second quantized QHFM ground state (3) and

$\displaystyle
O(\mathbf{r}_{i},t)=\pi_{\mu}^{a}(\mathbf{r}_{i},t)\Gamma_{\mu}^{a}(\mathbf{r}_{i}),$
(40)

where $\pi_{\mu}^{a}(\mathbf{r}_{i},t)$ are six real fields associated with
the broken generators $\Gamma_{\mu}^{a}(\mathbf{r}_{i})$ acting at the Landau
site $\mathbf{r}_{i}$ and we have assumed summation over repeated indices.
They correspond to generalized local spin-valley rotations and thus describe
the quantum state $|\psi[\pi]\rangle$ with spin-valley textures.

The total Lagrangian $\mathcal{L}$ is the sum of the kinetic term
$\mathcal{L}_{K}$, the Coulomb term $\mathcal{L}_{C}$ and the symmetry
breaking $\mathcal{L}_{SB}$ terms

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{K}-\mathcal{L}_{C}-\mathcal{L}_{SB},$ (41)
$\displaystyle\mathcal{L}_{K}$
$\displaystyle=\langle\psi[\pi]|i\partial_{t}|\psi[\pi]\rangle,$ (42)
$\displaystyle\mathcal{L}_{C}$
$\displaystyle=\langle\psi[\pi]|H_{C}|\psi[\pi]\rangle,$ (43)
$\displaystyle\mathcal{L}_{SB}$
$\displaystyle=\langle\psi[\pi]|H_{A}|\psi[\pi]\rangle-\langle\psi_{0}|H_{A}|\psi_{0}\rangle.$
(44)

In order to derive the effective non-linear sigma model at low-energy, we
follow closely Refs. [Arovas  _et al._ , 1999], [Yang  _et al._ , 2006] and
[Kharitonov, 2012].

####  III.2.1 Kinetic term

In the continuum limit, the kinetic term can be expressed as

$\displaystyle\mathcal{L}_{K}=\rho_{0}\int
d^{2}rZ^{\dagger}(\mathbf{r},t)i\partial_{t}Z(\mathbf{r},t),$ (45)

in terms of the spinor field

$\displaystyle Z(\mathbf{r},t)=e^{iO(\mathbf{r},t)}|F\rangle,$ (46)

where $|F\rangle$ is the ground state spinor corresponding to Eq. (3).
Expanding $O(\mathbf{r},t)$ up to second order in the $\pi$ fields, with the
help of Eq. (38), we obtain

$\displaystyle\mathcal{L}_{K}$ $\displaystyle=\frac{\rho_{0}}{2}\int
d^{2}r\varepsilon_{\mu\nu}\pi_{\mu}^{a}\partial_{t}\pi_{\nu}^{a}$ (47)
$\displaystyle=\frac{\rho_{0}}{2}\int
d^{2}r\bm{\mathcal{A}}^{a}[\bm{\pi}]\cdot\partial_{t}\bm{\pi}^{a},$ (48)

where $\rho_{0}=(2\pi l_{B}^{2})^{-1}$ is the electron density, and
$\bm{\mathcal{A}}^{a}[\bm{\pi}]=(-\pi_{y}^{a},\pi_{x}^{a},0)$ is the Berry
connection associated with the mode $a$.

####  III.2.2 Gradient term

To lowest order in the spatial derivatives, the energy associated with the
Coulomb Hamiltonian gives rises to a gradient termArovas  _et al._ (1999);
Yang  _et al._ (2006); Kharitonov (2012)

$\displaystyle\mathcal{L}_{\text{C}}$ $\displaystyle=\rho_{s}\int
d^{2}r\text{Tr}\left[\bm{\nabla}P\bm{\nabla}P\right]$ (49)
$\displaystyle=2\rho_{s}\int
d^{2}r\partial_{j}Z^{\dagger}(1-ZZ^{\dagger})\partial_{j}Z,$ (50)

where

$\displaystyle P(\mathbf{r},t)=ZZ^{\dagger}$ (51)

is the (space-time dependent) order parameter of the ferromagnet and

$\displaystyle\rho_{s}=\frac{1}{16\sqrt{2\pi}}\frac{e^{2}}{\varepsilon l_{B}}$
(52)

is the spin stiffness. This gradient term corresponds to the cost in exchange
energy associated with the misalignment of neighboring spins.

The matrix $P$ is a projectorKharitonov (2012) that obeys $P^{2}=P$,
$P^{\dagger}=P$ and $\text{Tr}[P]=1$. Up to second order in the $\pi$-fields,
the gradient term is given by

$\displaystyle\mathcal{L}_{C}=\frac{\rho_{s}}{2}\int
d^{2}r(\bm{\nabla}\pi_{\mu}^{a})^{2},$ (53)

where we have used the property that $\langle
F|\Gamma_{\mu}^{a}\Gamma_{\nu}^{b}|F\rangle=\frac{1}{4}\delta_{ab}(\delta_{\mu\nu}+i\varepsilon_{\mu\nu})$.
We recover thus the usual non-linear sigma model term extended to the six
fields in the $CP^{3}$ space.

####  III.2.3 Anisotropic terms

Finally, the symmetry breaking terms correspond to the anisotropic energy
$E_{A}[Z]$ of the slowly varying field $Z$ minus the anisotropic energy of the
ground state such that we consider only the excess energy corresponding to the
spin wave

$\displaystyle\mathcal{L}_{A}=E_{A}[Z]-E_{A}[F],$ (54)

where $E_{A}[F]$ is given by Eq. (12) and

$\displaystyle E_{A}[Z]=\rho_{0}\int
d^{2}r\Big{\\{}\sum_{i}u_{i}M_{P_{i}}^{2}[Z]-\Delta_{Z}M_{S_{z}}[Z]\Big{\\}},$
(55)

with $i\in\\{x,y,z\\}$, $u_{x}=u_{y}=u_{\perp}$, and

$\displaystyle\mathbf{M_{P}}[Z]$ $\displaystyle=\langle Z|\bm{\tau}|Z\rangle$
(56) $\displaystyle\mathbf{M_{S}}[Z]$ $\displaystyle=\langle
Z|\bm{\sigma}|Z\rangle$ (57)

are the spin and pseudo-spin magnetizations analogous to (14) generalized to
the field $Z$. We can express the anisotropic Lagrangian in a more compact way

$\displaystyle\mathcal{L}_{A}=\rho_{0}\int
d^{2}r\sum_{i}u_{i}t_{i}-\Delta_{Z}s_{z},$ (58)

with

$\displaystyle t_{i}$ $\displaystyle=\langle Z|\tau_{i}|Z\rangle^{2}-\langle
F|\tau_{i}|F\rangle^{2}$ (59) $\displaystyle s_{z}$ $\displaystyle=\langle
Z|\sigma_{z}|Z\rangle-\langle F|\sigma_{z}|F\rangle.$ (60)

We now expand the pseudo-spin magnetization up to second order in the
$\pi$-fields

$\displaystyle\langle Z|\tau_{i}|Z\rangle$ $\displaystyle=\langle
F|e^{-iO}\tau_{i}e^{iO}|F\rangle$ $\displaystyle=\langle
F|\tau_{i}|F\rangle-i\pi_{\mu}^{a}\langle
F|[\Gamma_{\mu}^{a},\tau_{i}]|F\rangle$
$\displaystyle-\frac{1}{2}\pi_{\mu}^{a}\langle
F|[\Gamma_{\mu}^{a},[\Gamma_{\nu}^{b},\tau_{i}]]|F\rangle\pi_{\nu}^{b},$ (61)

and we have a similar expression for the spin magnetization. Upon squaring,
the pseudo-spin anisotropy has a linear and a quadratic term in the
$\pi$-fields

$\displaystyle
t_{i}=R_{\mu}^{0a}\pi_{\mu}^{a}+\pi_{\mu}^{a}R_{i,\mu\nu}^{ab}\pi_{\nu}^{b},$
(62)

with

$\displaystyle R_{i\mu}^{0a}=$ $\displaystyle-2i\langle
F|\tau_{i}|F\rangle\langle F|[\Gamma_{\mu}^{a},\tau_{i}]|F\rangle$ (63)
$\displaystyle R_{i,\mu\nu}^{ab}=$ $\displaystyle-\langle
F|[\Gamma^{a}_{\mu},\tau_{i}]|F\rangle\langle
F|[\Gamma^{b}_{\nu},\tau_{i}]|F\rangle$ $\displaystyle-\langle
F|\tau_{i}|F\rangle\langle
F|[\Gamma_{\mu}^{a},[\Gamma^{b}_{\nu},\tau_{i}]]|F\rangle.$ (64)

The Zeeman term is linear in the spin magnetization such that we have

$\displaystyle
s_{z}=R_{Z\mu}^{0a}\pi_{\mu}^{a}+\pi_{\mu}^{a}R_{Z,\mu\nu}^{ab}\pi_{\nu}^{b},$
(65)

where

$\displaystyle R_{Z\mu}^{0a}$ $\displaystyle=-i\langle
F|[\Gamma_{\mu}^{a},\sigma_{z}]|F\rangle$ (66) $\displaystyle
R_{Z,\mu\nu}^{ab}$ $\displaystyle=-\frac{1}{2}\langle
F|[\Gamma_{\mu}^{a},[\Gamma^{b}_{\nu},\sigma_{z}]]|F\rangle$ (67)

For every state $|F\rangle$, the linear terms cancel each other

$\displaystyle\sum_{i}u_{i}R_{i\mu}^{0a}-\Delta_{Z}R_{Z\mu}^{0a}=0$ (68)

for all $\mu$ and $a$. The anisotropic Lagrangian can thus be written as

$\displaystyle\mathcal{L}_{A}=\int d^{2}r\bm{\pi}^{T}\mathcal{R}\bm{\pi}$ (69)

where $\bm{\pi}=(\pi_{\mu}^{a})$ is the six-component vector made of the
$\pi$-fields and

$\displaystyle\mathcal{R}_{\mu\nu}^{ab}=\sum_{i}u_{i}R_{i\mu\nu}^{ab}-\Delta_{Z}R_{Z\mu\nu}^{ab}$
(70)

is a $6\times 6$ matrix in the basis $\\{\mu,a\\}$ that we call the anisotropy
matrix.

We now consider the effective action $\mathcal{S}=\int dt\mathcal{L}$ and
Fourier transform the kinetic and gradient Lagrangians (47) and (53) in space
and time

$\displaystyle\mathcal{S}=\int d\omega
d^{2}k\bm{\pi}^{T}(\mathbf{k},\omega)\mathcal{M}\bm{\pi}(-\mathbf{k},-\omega),$
(71)

with

$\displaystyle\mathcal{M}_{\mu\nu}^{ab}=\left(\frac{\rho_{0}}{2}i\omega\varepsilon_{\mu\nu}-\frac{\rho_{s}}{2}\mathbf{k}^{2}\delta_{\mu\nu}\right)\delta_{ab}-\rho_{0}\mathcal{R}_{\mu\nu}^{ab}.$
(72)

The dispersion relations of the collective mode are obtained by minimizing the
action, $\delta\mathcal{S}/\delta\mathbf{\pi}(\mathbf{k},\omega)=0$, which
gives the equation

$\displaystyle\mathcal{M}(\mathbf{k},\omega)\bm{\pi}(\mathbf{k},\omega)=0.$
(73)

Because the matrix $\mathcal{M}(\mathbf{k},\omega)$ is hermitian, the
frequencies always come in pairs $\pm\omega(\mathbf{k})$. However, we only
consider the three positive eigenfrequencies $\omega_{\alpha}$($\mathbf{k}$),
which correspond to the physically relevant modes, and discard the negative-
energy solutions. The corresponding fields $\bm{\pi}$ are obtained by finding
the null space of $\mathcal{M}$. The resulting spinor is thus given by

$\displaystyle|Z_{\alpha}\rangle=\left(\mathbb{1}+i\pi_{\mu,\alpha}^{a}\Gamma_{\mu}^{a}-\frac{1}{2}\pi_{\mu,\alpha}^{a}\pi_{\nu,\alpha}^{b}\Gamma_{\mu}^{a}\Gamma_{\nu}^{b}\right)|F\rangle,$
(74)

where $\pi_{\mu\alpha}^{a}$ is the eigenstate corresponding to the frequency
$\omega_{\alpha}$. When the matrix is block-diagonal
$\mathcal{M}_{\mu\nu}^{ab}\propto\delta_{ab}$, the different modes are
decoupled and the eigenstate labels are identical to the mode label
$\alpha=a$. This is the case for the CDW and KD phases.

###  III.3 Change of ground state

The general analysis of the previous sections has been performed by
considering the ground state spinor
$|F\rangle=|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$. To consider a
different ground state, we perform the unitary rotation given by Eqs. (36).
The spinor $Z$ is thus transformed as

$\displaystyle\tilde{Z}=UZ=e^{i\tilde{\pi}_{\mu}^{a}\tilde{\Gamma}_{\mu}^{a}}|\tilde{F}\rangle,$
(75)

where we have introduced the the fields $\tilde{\pi}_{\mu}^{a}$ which
correspond now to the modes $a$ associated with the broken generators
$\tilde{\Gamma}_{\mu}^{a}$. However, for simplicity, we will keep the notation
$\pi_{\mu}^{a}$ in every basis and assume that the $\pi$-fields correspond to
the modes in the corresponding basis.

The kinetic and gradient terms are independent of the basis because the SU(4)
transformation matrix $U$ is global
$\mathcal{L}_{K}[\tilde{Z}]=\mathcal{L}_{K}[Z]$ and
$\mathcal{L}_{C}[\tilde{Z}]=\mathcal{L}_{C}[Z]$. However, the symmetry
breaking terms are basis dependent. The spin and pseudo-spin magnetization
read

$\displaystyle\langle\tilde{Z}|\bm{\tau}|\tilde{Z}\rangle$
$\displaystyle=\langle Z|\mathbf{P}|Z\rangle$ (76)
$\displaystyle\langle\tilde{Z}|\sigma_{z}|\tilde{Z}\rangle$
$\displaystyle=\langle Z|S_{z}|Z\rangle,$ (77)

such that instead of computing the commutators in Eq. (61) using the
transformed matrices $\tilde{\Gamma}_{\mu}^{a}$, we simply replace the
matrices $\bm{\tau}$ and $\sigma_{z}$ by

$\displaystyle\mathbf{P}$ $\displaystyle=U^{\dagger}\bm{\tau}U$ (78)
$\displaystyle S_{z}$ $\displaystyle=U^{\dagger}\sigma_{z}U,$ (79)

such that the pseudo-spin magnetization reads

$\displaystyle\langle\tilde{Z}|\tau_{i}|\tilde{Z}\rangle$
$\displaystyle=\langle F|P_{i}|F\rangle-i\pi_{\mu}^{a}\langle
F|[\Gamma_{\mu}^{a},P_{i}]|F\rangle$
$\displaystyle-\frac{1}{2}\pi_{\mu}^{a}\langle
F|[\Gamma_{\mu}^{a},[\Gamma_{\nu}^{b},P_{i}]]|F\rangle\pi_{\nu}^{b},$ (80)

where $|F\rangle=|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$ and the
matrices $\Gamma_{\mu}^{a}$ are given by Eqs. (31). We have a similar
expression for the spin magnetization in the transformed basis. Thus instead
of computing the transformed matrices and spinors in the new basis, we simply
express the matrices $\bm{\tau}$ and $\sigma_{z}$ in the basis
$\tilde{\mathcal{A}}$. Thus the anisotropic Lagrangian reads

$\displaystyle\mathcal{L}_{A}[\tilde{Z}]=\int
d^{2}r\bm{\pi}\tilde{\mathcal{R}}\bm{\pi},$ (81)

where

$\displaystyle\tilde{\mathcal{R}}_{\mu\nu}^{ab}=\sum_{i}u_{i}\tilde{R}_{i\mu\nu}^{ab}-\Delta_{Z}\tilde{R}_{Z\mu\nu}^{ab},$
(82)

and the matrices $\tilde{R}_{i\mu\nu}^{ab}$ and $\tilde{R}_{Z\mu\nu}^{ab}$ are
obtained from Eqs. (64) and (67) by the replacements $\tau_{i}\rightarrow
P_{i}$ and $\sigma_{z}\rightarrow S_{z}$.

##  IV Dispersion relations

Using the formalism developped in the previous section, we now diagonalize the
matrix (72) to find the dispersion relations of the three different modes and
their associated gaps. We only consider the four phases of Sec. II.3 without
the valley Zeeman term since they are not substantially modified upon its
introduction.

###  IV.1 Charge density wave phase

In the charge density wave, the ground state spinor and the empty sub-LL
$|C_{a}\rangle$ defining the three mode $a$ have the expression

$\displaystyle|F\rangle=|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle=(1,0,0,0)^{T}$
(83a)
$\displaystyle|C_{1}\rangle=|\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle=(0,1,0,0)^{T}$
(83b)
$\displaystyle|C_{2}\rangle=|-\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle=(0,0,1,0)^{T}$
(83c)
$\displaystyle|C_{3}\rangle=|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle=(0,0,0,1)^{T}$
(83d)

in the basis
$\\{|K\uparrow\rangle,|K\downarrow\rangle,|K^{\prime}\uparrow\rangle,|K^{\prime}\downarrow\rangle\\}$.
We have chosen here a ground state polarized in valley $K$, but one can also
choose a polarization in valley $K^{\prime}$ by the replacement
$\mathbf{n}_{z}\rightarrow-\mathbf{n}_{z}$. The mode $a=1$ which mixes
$|F\rangle$ and $|C_{1}\rangle$ corresponds to a pure spin wave such that the
pseudo-spin remains unaffected. The mode $a=2$ mixes $|F\rangle$ and
$|C_{2}\rangle$ and corresponds to a pseudo-spin wave where the spin remains
unaffected. The mode $a=3$ corresponds to an entanglement wave in which
inverses both the spin and pseudo-spin such that the spinor $Z$ is in a
superposition of $|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$ and
$|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle$.

Figure 4: Dispersion relation of the three modes in the KD phase for
$u_{z}=-\Delta_{Z}$ and $u_{\perp}=2\Delta_{Z}$. The three modes are gapped
and quadratically dispersing.

The anisotropy matrix $\mathcal{R}$ is block diagonal
$\mathcal{R}^{ab}_{\mu\nu}\propto\delta_{ab}$ such that the three modes are
decoupled. We find the dispersion relations $\omega_{a}(\mathbf{k})$
corresponding to the three modes $a={1,2,3}$

$\displaystyle\omega_{1}(\mathbf{k})$
$\displaystyle=2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+\Delta_{Z}$ (84)
$\displaystyle\omega_{2}(\mathbf{k})$
$\displaystyle=2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+u_{\perp}-u_{z}$ (85)
$\displaystyle\omega_{3}(\mathbf{k})$
$\displaystyle=2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+\Delta_{Z}-u_{z}.$ (86)

The three modes have a quadratic dispersion and a mass term proportional to
the anisotropic energy terms. The CDW region is defined by $u_{\perp}>u_{z}$
and $u_{z}<\Delta_{Z}$ such that the three modes have a positive gap in the
region. The three eigenmodes have the same expression for each mode such that
the spinor with wavevector $\mathbf{k}$ corresponding to mode $a$ reads

$\displaystyle
Z_{\mathbf{k}a}(\mathbf{r},t)=\left(1-\frac{\pi_{0}^{2}}{8}\right)|F\rangle+i\frac{\pi_{0}}{2}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega_{a}t)}|C_{a}\rangle$
(87)

where $\pi_{0}\ll 1$ is the magnitude of the wave which is a scalar. As shown
in Fig. 5, The spin wave corresponds thus to a small weight on the spinor
$|C_{a}\rangle$ with the phase oscillating at frequency $\omega_{a}$ and
wavevector $\mathbf{k}$.

Figure 5: Bloch spheres corresponding to each modes in the CDW phase. (a) Spin
Bloch sphere of the pure spin mode 1. (b) Pseudo-spin Bloch sphere of the
pseudo-spin mode 2. (c) Entanglement Bloch sphere corresponding to the
entanglement mode 3. The black arrow indicates the ground state polarization,
while the red arrow corresponds to the magnetization at a point in space in
the presence of a spin wave. The red arrow rotates periodically around the
ground state polarization according to Eq. (87).

The first mode corresponds to a pure spin wave, its gap is unaffected by the
anisotropic term and depends only on the Zeeman term. Because the pseudo-spin
remains unaffected by the spin wave and is polarized in one valley, the spins
live only on one sublattice (we choose sublattice A here for illustration) and
spin the magnetization is

$\displaystyle\mathbf{M_{S_{A}}}=\begin{pmatrix}\pi_{0}\cos(\mathbf{k}\cdot\mathbf{r}-\omega_{1}t)\\\
\pi_{0}\sin(\mathbf{k}\cdot\mathbf{r}-\omega_{1}t)\\\
1-\frac{1}{2}\pi_{0}^{2}\end{pmatrix},\quad\mathbf{M_{S_{B}}}=0.$ (88)

The spin wave consist thus of the spins of sublattice A precessing around the
axis $z$ at frequency $\omega_{1}$ as shown in Fig. 6.(a).

(a) (b) (c)

Figure 6: Three modes of the CDW phase. (a) ”Snapshot” of the pure spin wave
mode $a=1$ seen from the top with wavevector $\mathbf{k}$ along the axis $y$.
We observe the precession of the spins around the $z$ axis of the spins in the
A sublattice. (b) Sublattice polarization of the pseudo-spin wave mode $a=2$.
We observe a small electronic density on sublattice B. The dynamic part of the
field is encoded in the relative phase of the superposition between valley $K$
and $K^{\prime}$. The spin magnetization is proportional to the sublattice
density and points along $\mathbf{s}_{z}$. (c) Spin magnetization on the A and
B sublattices of the entanglement mode $a=3$, there is a small spin
magnetization on the B sublattice with opposite direction as on sublattice A.

The second mode corresponds to a pseudo-spin wave for which the gap depend
only on the pseudo-spin anisotropic terms and not on the Zeeman term. In the
CDW region, we have chosen for simplicity a polarization in the valley $K$ (a
similar treatment can be done if the polarization is in valley $K^{\prime}$)
and thus the pseudo-spin points towards the north pole of the Bloch sphere.
Because the pseudo-spin magnetization points along the $z$ direction, the
anisotropic energy of the ground state depends only on $u_{z}$. The presence
of a pseudo-spin wave introduces a pseudo-spin magnetization in the $xy$ plane
of the Bloch sphere, such that the magnetization out-of-plane anisotropic
energy $u_{z}$ is reduced, while there is a cost in in-plane anisotropic
energy $u_{\perp}$, hence the gap is proportional to $(u_{\perp}-u_{z})$. The
pseudo-spin magnetization is given by

$\displaystyle\mathbf{M_{P}}=\begin{pmatrix}\pi_{0}\cos(\mathbf{k}\cdot\mathbf{r}-\omega_{2}t)\\\
\pi_{0}\sin(\mathbf{k}\cdot\mathbf{r}-\omega_{2}t)\\\
1-\frac{1}{2}\pi_{0}^{2}\end{pmatrix}.$ (89)

This expression for the pseudo-spin is analogous to the spin magnetization
(88) of the pure spin wave. It is now the pseudo-spin that precesses around
the $z$ axis, such that it corresponds to a superposition of the valley $K$
and $K^{\prime}$ with a relative phase oscillating at frequency $\omega_{2}$.
However, the electronic density imbalance of the sublattice, which corresponds
to the $z$ component of the pseudo-spin magnetization
($M_{P_{z}}=\rho_{A}-\rho_{B}$) remains uniform

$\displaystyle\rho_{A}=1-\frac{\pi_{0}^{2}}{4}\quad\rho_{B}=\frac{\pi_{0}^{2}}{4}$
(90)

as shown in Fig. 6.(b). We observe thus a small electronic density on the
sublattice $B$. Because the spinors $|F\rangle$ and $|C_{2}\rangle$ both have
spins pointing along the $z$ direction, the spin magnetization on sublattices
A and B is simply proportional to the electronic density,
$\mathbf{M_{S_{A}}}=\rho_{A}\mathbf{s}_{z}$ and
$\mathbf{M_{S_{B}}}=\rho_{B}\mathbf{s}_{z}$. The total spin magnetization is
thus $\mathbf{M_{S}}=\mathbf{s}_{z}$.

The spinors of the third mode cannot be expressed as a tensor product of a
spin and a valley spinors. Thereby, this mode is an entanglement mode which
mixes the sub-LLs $|K\uparrow\rangle$ and $|K^{\prime}\downarrow\rangle$. It
corresponds to the electron being mainly polarized on sublatice A with spin up
with a small polarization on sublatice B with spin down with the relative
phase oscillating at frequency $\omega_{3}$. Analogously to the pseudo-spin
wave, the pseudo-spin magnetization along the $z$ direction is reduced
($M_{P_{z}}=1-\pi_{0}^{2}/2$) such that there is a gain in anisotropic energy
$u_{z}$. However, there is a cost in Zeeman energy, and the gap is
proportional to $\Delta_{Z}-u_{z}$. The sublattice polarizarion is identical
to the pseudo-spin wave but the spin magnetization is

$\displaystyle\mathbf{M_{S_{A}}}=\left(1-\frac{\pi_{0}^{2}}{4}\right)\mathbf{s_{z}}\quad\mathbf{M_{S_{B}}}=-\frac{\pi_{0}^{2}}{4}\mathbf{s_{z}}$
(91)

such that the total spin is reduced similarly to the spin wave.

b) c) a)

Figure 7: Size of the gap of a) the pseudo-spin and b) the entanglement waves
as a function of $u_{\perp}$ and $u_{z}$ in the CDW region. We observe that
the pseudo-spin gap $\Delta_{2}$ vanishes at the boundary with the KD phase,
and the entanglement gap $\Delta_{3}$ vanishes at the boundary with the AFI
entangled phase. c) ”Phase diagram” of the spin mode with the lowest gap. We
can see that the pseudo-spin and entanglement modes have the lowest energy
near the phase boundaries, whereas the spin mode dominates elsewhere.

Figs. 7.(a) and (b) show the size of the gaps $\Delta_{a}$ of the pseudo-spin
and entanglement modes in units of $\Delta_{Z}$. We can see that the size of
the gap decrease as we get closer to the boundaries and eventually vanish at
the boundaries.

The gap of the pseudo-spin wave vanishes at the boundary with the KD phase
defined by $u_{\perp}=u_{z}$. At this line, as one can see from Eq. (8), the
SU(2) pseudo-spin symmetry is restored and there is thus no preferred
orientation of the pseudo-spin. There is no cost in anisotropic energy for the
creation of a pseudo-spin wave. The pseudo-spin wave becomes thus a true
Goldstone mode where the spontaneously broken symmetry is the SU(2) pseudo-
spin rotation symmetry.

The gap of the entanglement wave vanishes at the boundary $u_{z}=\Delta_{Z}$
with the anti-ferrimagnetic phase which is an entangled phase. This comes from
the fact that the spin and pseudo-spin magnetizations along $z$ of the wave
are identical $M_{S_{z}}=M_{P_{z}}=(1-\pi_{0}^{2}/2)$ because we have a small
imbalance over the state $|K^{\prime}\downarrow\rangle$ with opposite spin and
pseudo-spin that of $|K\uparrow\rangle$. In addition, there is no spin and
pseudo-spin magnetization in the $xy$ plane
$M_{S_{x},P_{x}}=M_{S_{y},P_{y}}=0$. Thus, at the transition line
$u_{z}=\Delta_{Z}$, up to second order in $\pi_{0}$, the anisotropic energy
term

$\displaystyle
E_{A}[Z]=\frac{u_{z}}{2}M_{P_{Z}}^{2}-\Delta_{Z}M_{S_{z}}=\frac{u_{z}}{2}-\Delta_{Z}=E_{A}[F],$
(92)

which is independent of the amplitude $\pi_{0}$. Thereby, for small
amplitudes, the spin and pseudo-spin magnetizations cancel each other at the
transition line. This symmetry between the spin and pseudo-spin magnetization
will be explored further in Sec. IV.3.

###  IV.2 Kekulé distortion phase

In the KD phase, we apply the unitary transformation

$\displaystyle U_{KD}=e^{i\frac{\pi}{4}\mathbf{n}\cdot\bm{\tau}},$ (93)

with $\mathbf{n}=(\sin\varphi,-\cos\varphi,0)$ to the spinors (83) of the CDW
phase such that we have the spinors in the KD phase

$\displaystyle|\tilde{F}\rangle=|\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle=\frac{1}{\sqrt{2}}(1,0,e^{i\varphi},0)^{T}$
(94a)
$\displaystyle|\tilde{C}_{1}\rangle=|\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle=\frac{1}{\sqrt{2}}(0,1,0,e^{i\varphi})^{T}$
(94b)
$\displaystyle|\tilde{C}_{2}\rangle=|-\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle=\frac{1}{\sqrt{2}}(-e^{-i\varphi},0,1,0)^{T}$
(94c)
$\displaystyle|\tilde{C}_{3}\rangle=|-\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle=\frac{1}{\sqrt{2}}(0,-e^{-i\varphi},0,1)^{T},$
(94d)

where we have a U(1) pseudo-spin symmetry in the $xy$ plane of the Bloch
sphere. Similarly to the analysis for the CDW phase, the mode 1 is a pure spin
wave where the pseudo-spin is unaffected, the mode 2 is a pseudo-spin wave,
while the mode 3 is an entanglement mode.

Figure 8: Bloch spheres corresponding to each modes in the KD phase in the
same way as Fig. 5. (a) Spin Bloch sphere of the spin mode 1. (b) Pseudo-spin
Bloch sphere of the pseudo-spin mode 2. The ground state has a U(1) symmetry
for rotations around the $z$ axis. (c) Entanglement Bloch sphere corresponding
to the entanglement mode 3. For the spin and entanglement modes, the red arrow
rotates periodically around the ground state polarization according to Eq.
(87). At low-energy, the pseudo-spin mode is restricted to the equator of the
pseudo-spin Bloch sphere, which costs no anisotropic energy, while at higher
energy, it acquires an element along the $z$ direction.

The anisotropy matrix $\mathcal{R}$ is again block diagonal
$\mathcal{R}^{ab}_{\mu\nu}\propto\delta_{ab}$ such that the three modes are
decoupled. We find the dispersion relations $\omega_{a}(\mathbf{k})$
corresponding to the three modes $a={1,2,3}$,

$\displaystyle\omega_{1}(\mathbf{k})$
$\displaystyle=2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+\Delta_{Z}$ (95)
$\displaystyle\omega_{2}(\mathbf{k})$
$\displaystyle=|\mathbf{k}|l_{B}\sqrt{2\pi\rho_{s}}\sqrt{2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+u_{z}-u_{\perp}}$
(96) $\displaystyle\omega_{3}(\mathbf{k})$
$\displaystyle=2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+\Delta_{Z}-u_{\perp}.$ (97)

Figure 9: Dispersion relation of the three modes in the KD phase for
$u_{z}=2\Delta_{Z}$ and $u_{\perp}=-2\Delta_{Z}$. We observe that the pseudo-
spin mode is gapless, linear at low momentum $|\mathbf{k}\ll k_{0}$ and
becomes quadratic at higher momentum. the two other modes are gapped and
quadratic.

The dispersion of the three modes is shown in Fig. 9. Analogously to the CDW
case, the mode 1 corresponds to a spin mode, the mode 2 to a pseudo-spin mode
and the mode 3 to an entanglement mode.

The spin and entanglement modes are quite similar to the modes observed for
the CDW phase, they are quadratic gapped modes with a gap proportional to the
Zeeman coupling for the spin wave and a gap equal to $\Delta_{Z}-u_{\perp}$
for the entanglement wave, which corresponds to flipping both the spin and the
pseudo-spin. This gap is always positive since in the KD phase, we have
$u_{\perp}<\Delta_{Z}$. The space-time dependent spinor corresponding to these
two mode has the same expression as (87) with the basis spinors given by Eqs.
(94). The pure spin wave has the spins of each sublattice oscillating at
frequency $\omega_{1}$ with equal weight on both sublattices

$\displaystyle\mathbf{M_{S_{A}}}=\mathbf{M_{S_{B}}}=\frac{1}{2}\begin{pmatrix}\pi_{0}\cos(\mathbf{k}\cdot\mathbf{r}-\omega_{1}t)\\\
\pi_{0}\sin(\mathbf{k}\cdot\mathbf{r}-\omega_{1}t)\\\
1-\frac{1}{2}\pi_{0}^{2}\end{pmatrix}.$ (98)

Finally, the second mode looks different, it has a gapless linear dispersion
at low-momentum $\mathbf{k}^{2}\ll u_{z}-u_{\perp}$ while we recover a
quadratic dispersion relation at high momentum. The transition between these
two regimes occurs at a momentum of $k_{0}=\sqrt{u_{z}-u_{\perp}}$. Similarly
to the pseudo-spin mode in the CDW phase (85), the energy $u_{z}-u_{\perp}$
corresponds to the energy necessary to bring one pseudo-spin out of the plane,
namely there is a cost in out-of-plane anisotropic energy $u_{z}$ but a gain
in in-plane anisotropic energy $u_{\perp}$. This energy is always positive in
the KD region since $u_{z}>u_{\perp}$. Thereby, at low momentum, there is not
enough energy to bring one pseudo-spin out of the plane. The model corresponds
thus to an $XY$ model where the pseudo-spin is restricted to the equator of
the Bloch sphere and this mode is analogous to the linearly dispersing
superfluid mode in Helium and in bilayer 2DEGsFertig (1989); Moon  _et al._
(1995); MacDonald (2001). Its gaplessness originates from the U(1) symmetry of
the ground state : there is no cost in anisotropic energy cost for rotating a
pseudo-spin in the $xy$ plane. When the energy is larger than
$u_{z}-u_{\perp}$, there is now enough energy to bring the pseudo-spin out of
the plane and we recover the usual quadratic dispersion relation associated
with the fact that the two generators are now canonically conjugate.

###  IV.3 Anti-ferrimagnetic phase

The unitary matrix that tranforms the CDW spinors (83) into the entangled
spinors of the AFI phase is given by

$\displaystyle
U_{\text{AFI}}=e^{i\frac{\alpha_{1}}{2}\sigma_{x}\mathbf{m}\cdot\bm{\tau}},$
(99)

where $\mathbf{m}=(\sin\beta,-\cos\beta,0)$ and $\alpha$ is given by Eq. (22).
The basis spinors of the AFI phase are

$\displaystyle|\tilde{F}\rangle$
$\displaystyle=\cos\frac{\alpha_{1}}{2}|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{1}}{2}|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle$
(100a) $\displaystyle|\tilde{C}_{1}\rangle$
$\displaystyle=\cos\frac{\alpha_{1}}{2}|\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{1}}{2}|-\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$
(100b) $\displaystyle|\tilde{C}_{2}\rangle$
$\displaystyle=-\sin\frac{\alpha_{1}}{2}e^{-i\beta}|\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle+\cos\frac{\alpha_{1}}{2}|-\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$
(100c) $\displaystyle|\tilde{C}_{3}\rangle$
$\displaystyle=-\sin\frac{\alpha_{1}}{2}e^{-i\beta}|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle+\cos\frac{\alpha_{1}}{2}|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle.$
(100d)

We can see that the modes 1 and 2 involves the four basis spinors
$|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$,
$|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle$,
$|-\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$ and
$|\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle$ and one cannot factor the
spinors in order to have a definite spin or pseudo-spin mode. We find that
these two modes are coupled and their dispersions are given by

$\displaystyle\omega_{1,2}$
$\displaystyle=\pm\left(\frac{u_{\perp}}{2}-u_{z}\right)\cos\alpha_{1}$
$\displaystyle+\sqrt{2\pi\rho_{s}(\mathbf{k}l_{B})^{2}[2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+2u_{\perp}]+\frac{u_{\perp}^{2}}{4}\cos^{2}\alpha_{1}},$
(101)

which are both positive due to the gap term inside the square root. The gaps
$\Delta_{\alpha}$ of the modes $\alpha=1$ and $\alpha=2$ are

$\displaystyle\Delta_{1}$ $\displaystyle=u_{z}\cos\alpha_{1}=\Delta_{Z}$ (102)
$\displaystyle\Delta_{2}$
$\displaystyle=\left(u_{\perp}-u_{z}\right)\cos\alpha_{1}.$ (103)

Figure 10: Dispersion relation of the three modes in the AFI phase for
$u_{z}=2\Delta_{Z}$ and $u_{\perp}=6\Delta_{Z}$. The modes $a=1$ and $a=2$ are
coupled and form the $\alpha=1$ and $\alpha=2$ which are quadratically
dispersing while the entanglement mode is linear and gapless.

For $\alpha=0$, namely at the boundary with the CDW phase, the spinors (100)
simplify to the CDW spinors and we recover the pseudo-spin mode with gap
$u_{\perp}-u_{z}$ and the spin mode with gap $\Delta_{Z}=u_{z}$.

The dispersion for the entanglement mode $a=3$ is given by

$\displaystyle\omega_{3}(\mathbf{k})=\sqrt{2\pi\rho_{s}}|\mathbf{k}|l_{B}\sqrt{2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+u_{z}(1-\cos^{2}\alpha_{1})}.$
(104)

We can see that for $\cos\alpha_{1}=1$ ($\Delta_{Z}=2u_{z}$), namely at the
transition with the CDW phase, we obtain a gapless quadratic dispersion. When
$\Delta_{Z}<2u_{z}$, we have a linear dispersion at low momentum which
transforms into a quadratic dispersion around momentum
$k_{0}=\sqrt{2u_{z}(1-\cos^{2}\alpha_{1})}$. This mode is analogous to the
pseudo-spin mode in the KD phase. The linearity at low-momentum originates
from the U(1) symmetry of the ground state associated with the parameter
$\beta$ in Eqs. (100a) and (100d). The spinors $|\tilde{F}\rangle$ and
$|C_{3}\rangle$ are both in a superposition of the states
$|\mathbf{n}_{z}\rangle|\mathbf{s}_{z}\rangle$ and
$|-\mathbf{n}_{z}\rangle|-\mathbf{s}_{z}\rangle$ as shown in Fig. 11. It costs
thus no anisotropic energy to move the ground state (black arrow in Fig. 11)
around the parallel of the Bloch sphere at which lie both the black and red
arrows. At higher momentum, there is enough energy to bring the entanglement
mode out of this latitude and restore the symmetry between the $xy$ direction
and the $z$ direction.

###  IV.4 Canted anti-ferromagnetic phase

Figure 11: Entanglement Bloch spheres corresponding to the entanglement mode
in (a) the AFI phase and (b) the CAF phase. The spinor $|F\rangle$ indicated
by the black arrow corresponds to the ground state, while the spinor
$|C_{3}\rangle$ is located at opposite direction of the Bloch sphere. The
ground states possesses a U(1) symmetry associated with the angle $\beta$
corresponding to the latitude indicated by the circle at the tip of the black
and red arrows. At low-energy, the entanglement wave correponds to a small
deviation at equi-latitude indicated by the red arrow.

The unitary matrix that tranforms the CDW spinors (83) into the entangled
spinors of the canted anti-ferromagnetic phase is the product of the matrices
(93) and (99) of the KD and AFI phase

$\displaystyle
U_{\text{CAF}}=e^{i\frac{\pi}{4}\mathbf{n}\cdot\bm{\tau}}e^{i\frac{\alpha_{2}}{2}\sigma_{x}\mathbf{m}\cdot\bm{\tau}},$
(105)

where $\mathbf{n}=(\sin\varphi,-\cos\varphi,0)$,
$\mathbf{m}=(\sin\beta,-\cos\beta,0)$ and $\alpha_{2}$ is given by Eq. (24).

Figure 12: Dispersion relation of the three modes in the CAF region for
$u_{z}=12\Delta_{Z}$ and $u_{\perp}=2\Delta_{Z}$. We observe two gapless modes
: the entanglement mode and the mode $\alpha=2$ which originates from the
gapless mode of the KD region.

The basis spinors of the AFI phase are

$\displaystyle|\tilde{F}\rangle$
$\displaystyle=\cos\frac{\alpha_{2}}{2}|\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{2}}{2}|-\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle$
(106a) $\displaystyle|\tilde{C}_{1}\rangle$
$\displaystyle=\cos\frac{\alpha_{2}}{2}|\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle+e^{i\beta}\sin\frac{\alpha_{2}}{2}|-\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle$
(106b) $\displaystyle|\tilde{C}_{2}\rangle$
$\displaystyle=-\sin\frac{\alpha_{2}}{2}e^{-i\beta}|\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle+\cos\frac{\alpha_{2}}{2}|-\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle$
(106c) $\displaystyle|\tilde{C}_{3}\rangle$
$\displaystyle=-\sin\frac{\alpha_{2}}{2}e^{-i\beta}|\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle+\cos\frac{\alpha_{2}}{2}|-\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle$
(106d)

The modes $a=1$ and $a=2$ are also coupled and we don’t present their explicit
expression here since it is too lengthy. We find the corresponding gaps

$\displaystyle\Delta_{1}$ $\displaystyle=\Delta_{Z}$ (107)
$\displaystyle\Delta_{2}$ $\displaystyle=0,$ (108)

such that one mode is gapless with a linear dispersion relation at low-energy
as can be seen in Fig. 12 and one mode has a pure Zeeman gap. We can see that
the modes $\alpha=1$ and $\alpha=2$ originate from an anti-crossing around
momentum $|\mathbf{k}|l_{B}\approx 0.03$ between a linear mode and a gapped
quadratic mode, which are the descendants of the spin and the gapless pseudo-
spin modes of the KD phase. The mode 1 becomes quadratic at higher energy.

Once again, the mode 3 is decoupled from the others, and corresponds thus to
an entanglement mode with dispersion

$\displaystyle\omega_{3}(\mathbf{k})=\sqrt{2\pi\rho_{s}}|\mathbf{k}|l_{B}\sqrt{2\pi\rho_{s}(\mathbf{k}l_{B})^{2}+u_{\perp}(1-\cos^{2}\alpha_{2})}.$
(109)

This mode is the analog of the entanglement mode in the AFI phase except that
we are in the basis
$\\{|\mathbf{n}_{\perp}\rangle|\mathbf{s}_{z}\rangle,|-\mathbf{n}_{\perp}\rangle|-\mathbf{s}_{z}\rangle\\}$
as shown in Fig. 11. The gaplessness and linearity originates also from the
U(1) symmetry associated with the angle $\beta$ in Eqs. (106a) and (106d).

##  V Conclusion

Figure 13: Summary of the low-energy dispersion relation in the four phases.
The indices 1,2 and 3 refer to the spin, pseudo-spin and entanglement modes
respectively, except in the CAF and AFI region where the spin and pseudo-spin
modes are coupled. In the schematic expression of the dispersion relations, we
have set $\rho_{s}/\rho_{0}=2\pi\rho_{s}l_{B}^{2}\equiv 1$. In the CDW region,
the three modes are gapped. In the KD region, there are two gapped modes and
one gapless linear modes, the pseudo-spin mode. In the AFI region, the
entanglement mode is gapless while the two other modes are gapped. Finally, in
the CAF region, there are two gapless modes, the entanglement and the coupled
mode $\alpha=2$, the descendant of the pseudo-spin mode.

To conclude, we have presented the dispersions of the different types of spin
waves, namely pure spin, valley, and entanglement waves, in graphene at
filling factor $\nu=\pm 1$. We have considered the four different possible
ground states presented by Lian et alLian and Goerbig (2017) based on the
anisotropic terms $u_{\perp}$ and $u_{z}$ originally introduced by
KharitonovKharitonov (2012). We have introduced a non-linear sigma model based
on a Lagrangian formalism which describes the long wavelength space-time
dependent spin-valley rotations. The presence of small explicit symmetry-
breaking terms generally opens a gap in the dispersion relation of the
different types of spin waves. However, we have found that in each phase,
except in the CDW region, there remain one or two gapless modes with a linear
dispersion relation at low momentum. The fact that these modes remain gapless
originates from a residual symmetry of the ground state, which is present even
when the symmetry breaking terms are introduced. These modes recover a
quadratic dispersion relation at higher energies when the symmetry between the
different directions of oscillation is restored. The summary of our findings
for the presence or absence of a gap for the three modes in each region is
presented in Fig. 13.

Our study, along with the expression for the gaps at $\nu=0$ for the KD and
CAF phase presented in Ref. [Wu  _et al._ , 2014] opens the way to an analysis
of the scattering of spin waves at interfaces between regions with different
filling factor taking into account the different types of spin wave (spin,
pseudo-spin or entanglement). Depending on the steepness of the scattering
region, we expect a different scattering process and emit the possibility that
one wave type in the $\nu=\pm 1$ region might be changed in the scattering
process, or be in a superposition of different types, since the type of spin
waves are different at $\nu=0$. The scattering mechanism should also depend on
the phase the region at $\nu=0$ is in.

###### Acknowledgements.

We would like to thank Alexandre Assouline, Preden Roulleau, Rebeca Ribeiro
Palau, and François Parmentier for stimulating discussions. We acknowledge
financial support from Agence Nationale de la Recherche (ANR project
“GraphSkyrm”) under Grant No. ANR-17-CE30-0029.

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