# Mechanism of skyrmion condensation and pairing for twisted bi-layer graphene

Dian Jing1,2,3, Alexander Conkey Tyner4, Pallab Goswami1,4 1 Department of
Physics and Astronomy, Northwestern University, Evanston, IL 60208  2
Department of Chemical and Biological Engineering, Northwestern University,
Evanston, IL 60208  3 Integrated Science Program, Northwestern University,
Evanston, IL 60208  4 Graduate Program in Applied Physics, Northwestern
University, Evanston, IL 60208

###### Abstract

When quantum flavor Hall insulator phases of itinerant fermions are disordered
by strong quantum fluctuations, the condensation of skyrmion textures of order
parameter fields can lead to superconductivity. In this work, we address the
mechanism of skyrmion condensation by considering the scattering between
(2+1)-dimensional, Weyl fermions and hedgehog type tunneling configurations of
order parameters that violate the skyrmion-number conservation law. We show
the quantized, flavor Hall conductivity ($\sigma^{f}_{xy}$) controls the
degeneracy of topologically protected, fermion zero-modes, localized on
hedgehogs, and the overlap between zero-mode eigenfunctions or ’t Hooft vertex
determines the nature of pairing. We demonstrate the quantum-disordered,
flavor Hall insulators with $\sigma^{f}_{xy}=2N$ lead to different types of
charge $2Ne^{-}$ superconductivity. Some implications for the competition
among flavor Hall insulators, the charge $2e^{-}$ paired states in BCS and
pair-density-wave channels, and the composite, charge $4e^{-}$ superconductors
for twisted bilayer graphene are outlined.

_Introduction_ : Skyrmions are smooth, topologically non-trivial, textures of
three-component, unit vector field $\hat{\boldsymbol{\Omega}}(\mathbf{x})$ of
$O(3)$ non-linear sigma model (NLSM) at two spatial dimensions. Rajaraman
Since $\hat{\boldsymbol{\Omega}}(\mathbf{x})$ is equivalent to a two-sphere
($S^{2}$), the skyrmion configurations are classified by the second homotopy
group: $\pi_{2}(S^{2})=\mathbb{Z}$. Inside the ordered phase of NLSM, the
skyrmion requires a finite creation energy $E_{sk}=4\pi\rho_{s}|W_{sk}|$,
where $\rho_{s}$ is the spin-stiffness of the NLSM, and $W_{sk}$ is the
integer-valued, winding number. If $\hat{\boldsymbol{\Omega}}(\mathbf{x})$ is
coupled to (2+1)-dimensional Weyl fermions as a mass type order parameter, the
skyrmions can support induced “fermion numbers”, indicating the presence of
fluctuating competing orders inside the skyrmion core. Wilczek1 ; Jaroszewicz1
; Jaroszewicz2 ; Wilczek2 ; Abanov ; TanakaHu ; FisherSenthil ; Grover ;
YaoLee ; FuSachdev ; RoyHerbut ; Herbut1 ; Moon ; Chakravarty1 ; GoswamiSi2 ;
GoswamiSi3 ; GoswamiSi4  When the NLSM is disordered by quantum fluctuations,
the vanishing of excitation gap can allow the condensation of skyrmions.
Hence, the quantum-disordered, anti-ferromagnet (AFM) and quantum spin Hall
(QSH) states can support nucleation of spin-singlet, valence bond solids (VBS)
FisherSenthil  and $s$-wave superconductivity, Grover ; Herbut1 ; Moon ;
Chakravarty1  respectively.

The analysis of such competing orders is often guided by the construction of
$SO(5)$ NLSMs that treat spin-triplet and spin-singlet orders on an equal
footing. After integrating out the gapped fermion fields, one obtains a non-
Abelian, Berry’s phase for the five-component vector field Abanov ; TanakaHu ;
FisherSenthil ; Grover ; YaoLee ; FuSachdev ; RoyHerbut ; Herbut1 ; Moon ;
Chakravarty1 ; GoswamiSi2 , which is also known as the Wess-Zumino-Witten
(WZW) term. The relationship between $SO(5)$ WZW model and the unconventional,
continuous quantum phase transitions between two ordered states Senthiletal2
is being actively studied by many groups. Chalker1 ; CWang ; ZWang

Owing to the discovery of superconductivity in twisted bilayer graphene (TBLG)
in the vicinity of correlated insulating states, the analysis of competing
particle-hole and particle-particle orders of Weyl fermions has become an
important, physically relevant problem. Cao1 ; Taniguchi ; Efetov1 ; Efetov2 ;
Saito ; Kerelsky ; Choi ; Jiang ; Xie ; Wong ; Zondiner ; LiuWang ; Cao2  The
interactions between fermion fields and bosonic collective modes, such as
phonons and smooth order parameter fluctuations are being vigorously
investigated, as potential candidates for pairing mechanism. Wu ; Isobe ; Lian
; You ; Classen ; Chubukov ; Kozil ; Fernandes2  Recently, the skyrmion
condensation has also been considered for addressing the competition between
correlated insulators and adjacent superconducting states. Zaletel1 ; Christos
; Zaletel2 ; Khalaf  Based on the low energy theories of $4-$ and $8-$ flavors
of Weyl fermions, different types of level-$1$ Zaletel1  and level-$2$
Christos  $SO(5)$ WZW models have been proposed for the integer filling
fractions $\nu=2$, and $0$, respectively. The topological pairing mechanism
can provide critical, non-perturbative insights into the nature of competing
orders and emergent symmetries, which are generally inaccessible through
perturbation theories about topologically trivial configurations of collective
modes. This motivates us to ask the following question.

How do the skyrmions condense and determine the nature of competing orders?
Due to the vanishing of $E_{sk}$, all topologically distinct ground states,
labeled by different $W_{sk}$’s can become energetically degenerate. Such
infinite degeneracy of ground states is removed by hedgehog or monopole type
tunneling configurations of order parameter in Euclidean space-time. They
violate the skyrmion-number conservation law and lead to a superposition of
different skyrmion configurations as the true ground state. Duncan ;
MurthySachdev ; Read  Therefore, the mechanism of condensation and the nature
of superconducting states cannot be clearly understood, without considering
interactions between Weyl fermions and hedgehogs.

Furthermore, for $4N$-flavors of Weyl fermions with $N>1$, a spin-triplet
order has many competing spin-singlet, mass orders, which anti-commute with
each other. Naturally, there are multiple candidates for level-$N$ $SO(5)$ WZW
theories, and they cannot provide an unbiased description of competing spin-
singlet orders. Such issues for the $N=2$ case were identified in Refs.
GoswamiSi2, ; GoswamiSi3, , while addressing the competition between AFM, VBS,
and Kondo-singlets. Ref. GoswamiSi3,  has showed the space-time dependent,
Weyl operator in the presence of hedgehog configurations of AFM order
parameter possesses topologically protected fermion zero-modes. Many
similarities between this problem and (3+1)-dimensional quantum chromodynamics
Shuryak ; Diakonov  were pointed out. The overlap between zero-modes was shown
to cause an effective $2N$-fermion interaction or ’t Hooft vertex (TV). 'T
Hooft  For $N=1$, TV is a frequency-momentum dependent fermion billinear,
describing VBS order. For $N\geq 2$, the quartic TV corresponds to a
composite, spin-singlet order, which cannot be described by VBS or Kondo-
singlet type fermion bilinears.

The primary goal of our current work is to address the nature of paired states
arising from the condensation of skyrmion textures of quantum flavor Hall
(QFH) insulators for $4N$-flavors of Weyl fermions, by explicitly considering
the role of hedgehogs. For the clarity of presentation, we will only consider
triplet orders for one of the $SU(2)$ sub-groups of $SU(4N)$. In response to
the externally applied electric fields, uniformly ordered, QFH states exhibit
quantized Hall conductivity $\sigma^{f}_{xy}=2N$ for the spin or valley type
flavor currents. For TBLG, we will mainly focus on $\nu=0$. Next we set up our
notations for the effective theory of Weyl fermions.

_Flavor symmetry of Weyl fermions_ : The low energy theory of many two-
dimensional systems can be described by multiple species of linearly
dispersing Weyl fermions. The Weyl points with chirality $+1$ ($-1$) act as
unit strength vortices (anti-vortices) in momentum space. In this work, we
denote them as the right- and the left- handed Weyl points, respectively,
which come in pairs, due to the fermion doubling theorem. On a general ground,
we will consider $2N$ pairs of right- and left- handed fermions, where the
factor of $2$ ($N$) counts for the spin (layer or other internal) degrees of
freedom. We will describe the two-component, right-handed and left-handed
fermion fields by
$R^{T}_{i,s}(x_{0},\boldsymbol{x})=(R_{A,i,s,}(x_{0},\boldsymbol{x}),R_{B,i,s}(x_{0},\boldsymbol{x}))$,
and
$L^{T}_{i,s}(x_{0},\boldsymbol{x})=(L_{A,i,s}(x_{0},\boldsymbol{x}),L_{B,i,s}(x_{0},\boldsymbol{x}))$,
respectively, where $s=\uparrow,\downarrow$ is the spin index, $i=1,...,N$ is
the collective index for layer/internal degrees of freedom, and $A/B$
corresponds to the sub-lattice (SL) index. We have absorbed the Fermi velocity
$v_{F}$ on the imaginary time to write $x_{0}=-iv_{F}t$. After performing the
SL transformation $L_{i,s}\to\tau_{2}L_{i,s}$ and combining all degrees of
freedom into an $8N$-component spinor
$\psi^{\dagger}_{8N}=(R^{\dagger}_{1,\uparrow},R^{\dagger}_{1,\downarrow},...,R^{\dagger}_{2N,\uparrow},R^{\dagger}_{2N,\downarrow},L^{\dagger}_{1,\uparrow},L^{\dagger}_{1\downarrow},...,L^{\dagger}_{2N,\uparrow},L^{\dagger}_{2N\downarrow})$,
the Euclidean action of $4N$ flavors of Weyl fermions can be written in a
manifestly flavor-symmetric form

$S_{0}=\int d^{3}x\;\bar{\psi}_{8N}\;\mathbb{1}_{4N\times
4N}\otimes\left[\;\sum_{\mu=0}^{2}\;\gamma_{\mu}\;\partial_{\mu}\right]\;\psi_{8N},$
(1)

where $\mathbb{1}_{4N\times 4N}$ is the $4N\times 4N$ identity matrix,
$\bar{\psi}_{8N}=\psi^{\dagger}_{8N}\mathbb{1}_{4N\times
4N}\otimes\gamma_{0}$, $\gamma_{0}=\tau_{3}$, $\gamma_{1}=\tau_{2}$,
$\gamma_{2}=-\tau_{1}$. The $2\times 2$ identity matrix $\tau_{0}$ and the
Pauli matrices $\tau_{j}$’s with $j=1,2,3$ act on the sub-lattice (SL) index.
Notice in the Euclidean frequency-momentum space, the Weyl points appear as
the unit strength, hedgehog singularities.

The action is invariant under (i) the global $U(1)$ transformation:
$\psi_{8N}\to\exp[i\theta\;\mathbb{1}_{4N\times 4N}\otimes\tau_{0}]\psi_{8N}$,
and (ii) the global, flavor symmetry transformation: $\psi_{8N}\to
U\psi_{8N}$, with $U\in SU(4N)$. Hence, the free fermion action possesses
$U(1)\times SU(4N)=U(4N)$ symmetry, and we can define the conserved total
number current $J^{0}_{\mu}=\bar{\psi}_{8N}\mathbb{1}_{4N\times
4N}\otimes\gamma_{\mu}\psi_{8N}$, and the flavor currents
$J^{i}_{\mu}=\bar{\psi}_{8N}\hat{\Lambda}_{i}\otimes\gamma_{\mu}\psi_{8N}$,
where $\hat{\Lambda}_{i}$ with $i=1,2,..,(4N)^{2}-1$ are the $4N\times 4N$,
Hermitian generators of $SU(4N)$ group. For the $N=1$ model of spinful mono-
layer graphene (MLG), QFH order can describe the QSH or the quantum valley
Hall (QVH) states. The $N=2$ model corresponds to the simplest low energy
theory of nearly-flat bands of TBLG. Bistritzer ; Gail ; Santos ; Po  For
TBLG, the QFH mass can correspond to the QSH, the QVH, or the quantum mini-
valley Hall (QMVH) states. Additional Weyl points can also arise from the
effects of trigonal warping. Geim  The $SU(4N)$ generators can be expressed as
suitable linear combinations of direct products of $\log_{2}(4N)$ sets of
identity and Pauli matrices operating on spin and valley type degrees of
freedom. While describing MLG and TBLG, we will reserve $\sigma_{\lambda}$,
$\eta_{\lambda}$, $\rho_{\lambda}$ respectively for the spin, the valley ($L$
vs. $R$), and the mini-valley ($R_{1}$ vs $R_{2}$ within a Moiŕe Brillouin
zone) sectors. Next, we consider the skyrmion textures of QFH orders.

_Skyrmions of QFH orders_ : The $SU(2N)$ symmetry preserving, QSH mass term is
given by
$\mathcal{O}_{QSH}=\mathbf{\Omega}(x)\cdot\bar{\psi}\mathbb{1}_{2N\times
2N}\otimes\boldsymbol{\sigma}\otimes\gamma_{0}\psi$, where
$\mathbf{\Omega}(x)$ is a three-component, vector field. A uniform order
parameter field determines a global spin quantization axis along
$\mathbf{\Omega}=\mathbf{\Omega}_{0}$ and the total $U(1)$ spin-current
$J^{s}_{\mu}=\hat{\mathbf{\Omega}}_{0}\cdot\bar{\psi}\mathbb{1}_{2N\times
2N}\otimes\boldsymbol{\sigma}\otimes\gamma_{\mu}\psi$. In response to the
externally applied electro-magnetic and spin gauge fields, the QSH phase
supports cross-correlated, charge and spin Hall currents,
$J^{0}_{\alpha}=\frac{\sigma_{xy}^{s}}{2\pi}\;\epsilon_{\alpha\mu\nu}F^{0}_{\mu\nu}$,
and
$J^{s}_{\alpha}=\frac{\sigma_{xy}^{s}}{2\pi}\;\epsilon_{\alpha\mu\nu}F^{s}_{\mu\nu}$
where $\sigma^{s}_{xy}=2N$ is the quantized spin-Hall conductivity, and the
Abelian field strength tensors are
$F^{i}_{\mu\nu}=\partial_{\mu}A^{i}_{\nu}-\partial_{\nu}A^{i}_{\mu}$. Similar
results for the cross-correlated charge and flavor quantum Hall currents can
be obtained for QVH and QMVH mass orders in TBLG, respectively defined as
$\mathcal{O}_{QVH}=\mathbf{\Omega}(x)\cdot\bar{\psi}\boldsymbol{\eta}\otimes\rho_{0}\otimes\sigma_{0}\otimes\gamma_{0}\psi$,
and
$\mathcal{O}_{QMVH}=\mathbf{\Omega}(x)\cdot\bar{\psi}\eta_{0}\otimes\boldsymbol{\rho}\otimes\sigma_{0}\otimes\gamma_{0}\psi$.

The static skyrmion configurations of QSH order parameter are described by

$\hat{\mathbf{\Omega}}(\rho,\varphi)=\left[\sin\theta\cos(n\varphi),\sin\theta\sin(n\varphi),\cos\theta\right],$
(2)

where $\varphi=\tan^{-1}(x_{2}/x_{1})$, and $\theta$ is a function of radial
variable $\rho=\sqrt{x^{2}_{1}+x^{2}_{2}}$, such that $\theta(\rho\to 0)\to 0$
and $\theta(\rho\to\infty)\to\pi$. The winding number
$W_{sk}=\frac{1}{4\pi}\;\int
d^{2}x\;\hat{\mathbf{\Omega}}\cdot(\partial_{\nu}\hat{\mathbf{\Omega}}\times\partial_{\lambda}\hat{\mathbf{\Omega}})\;=n$,
with $n=\pm 1,\pm 2,..$. The Belavin-Polyakov solutions of skyrmions, Polyakov
which minimize the energy of static NLSMs, correspond to
$\cos(\theta)=(R^{n}-\rho^{n})/(R^{n}+\rho^{n})$, where $R$ is the size of
skyrmion core.

Deep inside the ordered phase, the induced fermion current can be computed by
employing the gradient expansion scheme, which is controlled by the amplitude
$|\mathbf{\Omega}|$. The spin-singlet, total number current is found to be

$J^{0}_{\mu}=\sigma^{s}_{xy}J^{sk}_{\mu}=2NJ^{sk}_{\mu},$ (3)

where
$J^{sk}_{\mu}=\frac{1}{4\pi}\;\epsilon_{\mu\nu\lambda}\hat{\mathbf{\Omega}}\cdot(\partial_{\nu}\hat{\mathbf{\Omega}}\times\partial_{\lambda}\hat{\mathbf{\Omega}})$
is the skyrmion current density. Since the ordered phase does not allow
singular tunneling events or hedgehog configurations, both $J^{sk}_{\mu}$ and
$J^{0}_{\mu}$ satisfy the continuity equation. Hence, the induced fermion
number $\langle\bar{\psi}\gamma_{0}\psi\rangle=2N\int
d^{2}xJ^{sk}_{0}=2N\;W_{sk}$, is determined by the quantized, spin Hall
conductivity, and $W_{sk}$ acts as the generator of flavor-singlet, $U(1)$
symmetry.

As the $SU(2N)$ flavor symmetry remains unbroken, the spin-singlet $SU(2N)$
flavor currents
$J^{0,l}_{\mu}=\bar{\psi}\hat{\lambda}_{l}\otimes\sigma_{0}\otimes\gamma_{\mu}\psi$,
with $l=1,..,(2N)^{2}-1$ remain identically conserved. When the QSH order is
destroyed, the condensation of skyrmions is expected to give rise to a spin-
singlet, paired state, breaking global $U(1)$ symmetry. Similarly, the valley-
(mini-valley-) singlet paired states can arise from the quantum-disordered QVH
(QMVH) phase. _However, all charge $2e^{-}$ fermion bilinears break $SU(2N)$
flavor symmetry, when $N>1$_ (see Appendix A). _Can the skyrmion condensation
determine the pattern of flavor-symmetry-breaking (FSB) by paired states_? To
answer this question, we consider the role of hedgehog configurations, which
serve as the source and sink of $J^{sk}_{\mu}$.

_Hedgehogs and fermion zero modes_ : These tunneling singularities in
Euclidean space-time are also classified according to the second homotopy
group $\Pi_{2}(S^{2})=\mathbb{Z}$. The topological invariant or hedgehog
charge is given by $q=\frac{1}{8\pi}\int
d^{2}S_{a}\epsilon_{abc}\;\epsilon_{\alpha\beta\lambda}\;\hat{\Omega}_{\alpha}\partial_{b}\hat{\Omega}_{\beta}\partial_{c}\hat{\Omega}_{\lambda}$,
where the integrals are performed over a sphere surrounding the singularity.
Arafune  The minimal or unit strength ($q=\pm 1$), radial (anti-)hedgehog
configurations correspond to $\hat{\Omega}_{\mu}=\pm\hat{x}_{\mu}$. More
general hedgehogs with charge $q_{h}=l$ can be described by

$\displaystyle\mathbf{\Omega}=\left[F_{1,l}(\rho)\cos(l\varphi),F_{1,l}(\rho)\sin(l\varphi),F_{2,l}(x_{0})\right],$
(4)

where $F_{1,l}$ and $F_{2,l}$ are two profile functions with the asymptotic
properties: (i) $F_{1,l}(\rho\to 0)\sim\rho^{l}$, and
$F_{1,l}(\rho\to\infty)\sim c_{1}$, and (ii) $F_{2,l}(x_{0}\to 0)\sim x_{0}$
and $F_{2,l}(x_{0}\to\pm\infty)\sim\pm c_{2}$, with $c_{1}$ and $c_{2}$ being
two constants. Rotating these configurations by an angle $\phi$ about a
constant unit vector $\hat{\mathbf{n}}$, we can find various topologically
equivalent configurations
$\hat{\mathbf{\Omega}}^{\prime}=\mathcal{R}(\hat{\mathbf{n}},\phi)\hat{\mathbf{x}}$.
where $\mathcal{R}(\hat{\mathbf{n}},\phi)$ is a rotation matrix.

Due to the non-conservation of skyrmion currents, the continuity equation for
the total number current is modified to

$\displaystyle\partial_{\mu}J^{0}_{\mu}=2N\sum_{i}\;\delta^{3}(x-x_{i})q_{i}.$
(5)

To satisfy the overall neutrality condition (i.e., $\sum_{i}q_{i}=0$), we must
have an equal number of hedgehogs and anti-hedgehogs. A fully self-consistent
treatment of fermion-hedgehog interactions lies beyond the scope of current
work. We will only discuss the topological structure of fermion determinant in
the presence of fixed hedgehog configurations.

Let us consider the coupling between hedgehogs of QSH order parameter and one
species of four-component Weyl fermions with sub-lattice and spin index. Due
to the $SU(2N)$ flavor symmetry, the results for all species of Weyl fermions
will be identical.To facilitate our analysis of fermion-hedgehog scattering
and induced pairing, we introduce the Nambu spinor
$\Psi^{T}_{N}=(\psi^{T},\bar{\psi}\gamma_{1}\otimes\sigma_{2})/\sqrt{2}$ and
$\bar{\Psi}_{N}=(\psi^{T}\gamma_{1}\otimes\sigma_{2},\bar{\psi})/\sqrt{2}$,
leading to

$\displaystyle S=\int d^{3}x\;\bar{\Psi}_{N}\mathcal{D}_{N}\Psi_{N}=\int
d^{3}x\;\bar{\Psi}_{N}\left(\begin{array}[]{cc}0&\mathcal{D}^{\dagger}\\\
\mathcal{D}&0\end{array}\right)\Psi_{N},$ (8)

where
$\mathcal{D}=\gamma_{\mu}\partial_{\mu}+\boldsymbol{\sigma}\cdot\boldsymbol{\Omega}$.
In this basis, all charge $2e^{-}$, fermion bilinears will be block diagonal
operators (see Appendix A). While $\mathcal{D}$ and $\mathcal{D}^{\dagger}$
are not Hermitian or anti-Hermitian operators, $\mathcal{D}_{N}$ is an
Hermitian operator. Therefore, the fermion fields $\Psi_{N}$ and
$\bar{\Psi}_{N}$ can be expanded in the eigenbasis of $\mathcal{D_{N}}$. The
eigenvalues follow from the squared operator

$\mathcal{D}^{2}_{N}=\left(\begin{array}[]{cc}\mathcal{D}^{\dagger}\mathcal{D}&0\\\
0&\mathcal{D}\mathcal{D}^{\dagger}\end{array}\right).$ (9)

Since $\\{\mathcal{D}_{N},\alpha_{3}\\}=0$, where $\alpha_{3}$ is the diagonal
Pauli matrix, operating on the particle-hole index, the non-zero eigenvalues
of $\mathcal{D}_{N}$ come in pairs.

However, the zero-modes possess definite chirality, corresponding to
$\alpha_{3}=+1$ (right/annihilation channel) or $-1$ (left/creation channel).
The four-component, right zero-mode $\phi_{R}$ (column vector) satisfies
$\mathcal{D}\phi_{R}=\mathcal{D}^{\dagger}\mathcal{D}\phi_{R}=0$ and the four-
component, left zero-mode $\phi_{L}$ (row vector) obeys
$\mathcal{D}^{\dagger}\phi_{L}^{\dagger}=\mathcal{D}\mathcal{D}^{\dagger}\phi_{L}^{\dagger}=0$.
The difference between the number of right zero-modes $n_{R}$ and the number
of left zero-modes $n_{L}$ is always -$|l|$ for hedgehogs and +$|l|$ for anti-
hedgehogs. _After accounting for the $2N$-fold degeneracy in flavor index, we
find the total number of topologically protected fermion zero-modes is equal
to $\sigma^{s}_{xy}\times|l|=2N|l|$._ This result is protected by Callias’s
index theorem. JackiwRebbi1 ; Callias1 ; Callias2

For the unit-strength, radial (anti-)hedgehog configurations, with
$\boldsymbol{\Omega}_{\mu}=\pm\;m(x)\;\hat{x}_{\mu}$, where $m$ is amplitude
of the order parameter, the zero-mode eigenfunctions are given by

$\phi^{\ast}_{+,L}=\phi_{-,R}^{T}=(0,1,-i,0)f(|x|),$ (10)

where $f(|x|)\propto\exp[-\int dx^{\prime}m(x^{\prime})]$. For a constant $m$,
we obtain $f(|x|)=\sqrt{\frac{|m|^{3}}{2\pi}}e^{-m|x|}$. GoswamiSi3  In this
case, one can also determine the full spectra of $\mathcal{D}$ by following
Refs. JackiwRebbi1, ; Callias1, . The existence of zero-modes for the dynamic
Weyl operator indicates the fermion determinant (or Pfaffian) vanishes in the
presence of isolated hedgehogs. Therefore, the coupling to Weyl fermions
suppresses the probability of any isolated hedgehog. In the vicinity of
quantum phase transitions, due to the divergent correlation length, we can
consider a dilute gas of hedgehogs. Therefore, the overlap between widely
separated zero-modes provides a clear idea about the nature of competing
orders, arising from the condensation of skyrmions.

Figure 1: The degeneracy of fermion zero-modes, localized on hedgehog
configurations of quantum spin Hall order parameter and the nature of paired
state. Here, $E_{1}$, $E_{2}$ and $E_{3}$ represent three different types of
flavor symmetry breaking chemical potentials.

_’ $t$ Hooft vertex and pairing_: Since a strength $l$ (anti-)hedgehog leads
to $2N|l|$ zero-modes in the (annihilation) creation channel, we anticipate a
(anti-)hedgehog creation operator will be coupled to $2N|l|$ number of fermion
(annihilation) creation operators. The calculation of such effective TV can be
performed by following Refs. 'T Hooft ; Shuryak ; GoswamiSi3  It is crucial to
average over the arbitrary orientations of hedgehogs for a disordered or para-
magnetic phase. The eigenfunction of arbitrarily oriented and radial hedgehogs
are related by $SU(2)$ rotations
$\psi(\hat{\boldsymbol{n}},\phi)=\mathcal{U}^{\dagger}\psi(\hat{x})$, where
$\mathcal{U}=\pm e^{i\phi/2\hat{\boldsymbol{n}}\cdot\boldsymbol{\sigma}}$,
such that
$\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}=\mathcal{U}^{\dagger}\hat{x}\cdot\boldsymbol{\sigma}\mathcal{U}$.
Since higher-$l$ hedgehogs will have lower probability, we will only consider
TV due to $l=\pm 1$ hedgehogs.

After performing the integral over $SU(2)$ group, the two-fermion, TV for MLG
with $N=1$ becomes

$Y=y\int
d^{3}kf^{2}(k)k^{2}\epsilon^{ij}[\psi_{i}^{T}(-k)\psi_{j}(k)+\bar{\psi}(k)\bar{\psi}^{T}_{j}(-k)],$
(11)

where $y$ is the fugacity of $l=\pm 1$ hedgehogs, $i,j\in\\{R,L\\}$, and
$\psi^{T}$ and $\bar{\psi}^{T}$ have $\gamma_{1}\otimes\sigma_{2}$ absorbed by
definition. The form factor
$f(k)=\frac{4\sqrt{2\pi}m^{5/2}}{(k^{2}+m^{2})^{2}}$ is the Fourier transform
of $f(r)=\sqrt{\frac{|m|^{3}}{2\pi}}e^{-|m|r}$. The TV describes _frequency-
momentum dependent charge $2e^{-}$ bilinear in the spin-singlet, $s$-wave
pairing channel_. Due to the exponentially localized behavior of zero-mode
eigenfunctions, $f(k)$ exhibits short-ranged behavior. Since the minimal
hedgehogs induce charge $2e^{-}$ pairing mass, the use of level-$1$ $SO(5)$
WZW theory for describing competing orders is justified.

For TBLG with $N=2$, the calculation of quartic TV due to four degenerate
zero-modes of minimal hedgehogs requires some algebraic manipulations. The
final result is given by

$\displaystyle\begin{split}Y=&y\int d^{3}k_{1}\int
d^{3}k_{2}f^{2}(k_{1})f^{2}(k_{2})k_{1}^{2}k_{2}^{2}\\\
&\epsilon^{ijkl}[\psi^{T}_{i}(-k_{1})\psi_{j}(k_{1})\psi^{T}_{k}(-k_{2})\psi_{l}(k_{2})+h.c.],\end{split}$
(12)

where $i,j\in\\{R_{1},R_{2},L_{1},L_{2}\\}$ and $\psi^{T}$ and
$\bar{\psi}^{T}$ have $\gamma_{1}\otimes\sigma_{2}$ absorbed by definition.
_The quartic, TV describes spin-singlet, composite, charge $4e^{-}$ paired
state that preserves $SU(4)$ symmetry in the combined valley and mini-valley
space (enforced by $\epsilon^{ijkl}$)_. Obviously such a phase cannot be
described by fermion bilinears. Similar conclusions can be drawn for the
quantum-disordered QVH and QMVH phases. _In summary, hedgehogs only break
$U(1)$ symmetry and induce pairing, without breaking flavor symmetry._

_Effects of flavor-symmetry-breaking_ : In order to realize charge $2e^{-}$
states for $N\geq 2$, the $SU(2N)$ symmetry must be broken through additional
mechanism. In contrast to the long-range tail of Coulomb interactions, the
generic short-range interactions do not respect flavor symmetry, inherited
from the valley/mini-valley degrees of freedom. Thus, depending on many non-
universal details of short-range interactions, different types of charge
$2e^{-}$ states can emerge out of charge $4e^{-}$ paired states.

The FSB by high-temperature orders can also be instrumental in selecting
charge $2e^{-}$ states. For example, we can consider flavor dependent chemical
potentials, which shift various Weyl points to different reference energies.
Such chemical potentials couple to the flavor-density operators and reduce the
strength of $\sigma^{s}_{xy}$ from being $2$ to $1$. For TBLG, in addition to
the conventional chemical potential ($E_{0}$) for the total number operator,
we can consider three types of spin-singlet, chemical potentials ($E_{j}$ with
$j=1,2,3$) in the valley and mini-valley space. When $E_{0}=\pm E_{j}$, the
degeneracy of zero-modes is reduced to $2$. As illustrated in Fig. 1,
depending on the explicit nature of FSB, charge $2e^{-}$ BCS (zero center of
mass momentum), pair-density-waves/PDW (large center of mass momentum or short
wavelength modulations), and Moiŕe pair-density-waves/MPDW (small center of
mass momentum or long wavelength modulations) states can be realized by
destroying QSH order. Very similar considerations for QVH and QMVH orders show
the possibilities of realizing spin-singlet and spin-triplet, charge $2e^{-}$
states in BCS and pair-density-wave channels (see Appendix A).

_Outlook_ : There are growing experimental and theoretical evidence in favor
of QFH order at various integer filling fractions. Cao1 ; Taniguchi ; Efetov1
; Efetov2 ; Saito ; Kerelsky ; Choi ; Jiang ; Xie ; Wong ; Zondiner ; LiuWang
; Cao2 ; ZhangPo ; KangVafek1 ; KangVafek3 ; Liao  Therefore, the condensation
of skyrmion textures of QFH insulators can provide valuable insights into the
nature of proximate superconducting states, which are not easily accessible
from weak or strong coupling analysis of microscopic models. The experiments
also suggest the presence of FSB chemical potentials, which can cause revival
of Weyl points, in the vicinity of Fermi level. Wong ; Zondiner  More concrete
evidence for the precise nature of QFH order and FSB chemical potentials are
required for making definitive predictions for pairing symmetry in TBLG. Our
work suggests exciting possibilities of realizing exotic charge $4e^{-}$
states around $\nu=0$ and pair-density-waves for all filling fractions through
topological pairing mechanism.

###### Acknowledgements.

This work was supported by the National Science Foundation MRSEC program
(DMR-1720139) at the Materials Research Center of Northwestern University.

##  Appendix A Pairing bilinears

In this appendix, we outline FSB fermion bilinears in particle-particle
channels. The relationship with appropriate $SO(5)$ and $SO(9)$ NLSMs of
competing orders will be identified. For the general case of $4N$-flavors of
Weyl fermions, all momentum-independent, charge $2e^{-}$ pairing terms can be
described as
$\phi_{l,1}\Psi^{\dagger}\alpha_{1}\hat{\Lambda}^{\prime}_{l}\Psi$ and
$\phi_{l,2}\Psi^{\dagger}\alpha_{2}\hat{\Lambda}^{\prime}_{l}\Psi$, where
$\Psi^{\dagger}=(\psi^{\dagger},\psi^{T})/\sqrt{2}$ is the $16N$-component
Nambu spinor, $\hat{\Lambda}^{\prime}_{l}$ are the $4N(8N-1)$ number of
$8N\times 8N$, imaginary, $SU(8N)$ Gell-Mann matrices, and the Pauli matrices
$\alpha_{j}$’s operate on the particle-hole index. Notice that the number of
pairing bilinears $4N(8N-1)$ exactly equals the number of generators for
$SO(8N)$ group. Out of these bilinears, only
$\Psi^{\dagger}\alpha_{j}\otimes\hat{M}\otimes\tau_{2}\Psi$, with symmetric
$4N\times 4N$ matrix $\hat{M}$ can anti-commute with the Hamiltonian of free
fermions and act as superconducting mass terms. There are $2N(4N+1)$
independent, charge $2e^{-}$, mass terms, which is equal to the number of
generators for $USp(4N)$ group. By absorbing the imaginary Pauli matrices
($\eta_{2}$, $\rho_{2}$ etc.) from all $SU(2)$ sectors on the lower component
of Nambu spinor, the pairing mass terms can be organized in a more convenient
singlet and triplet forms.

###  A.1 Spinful MLG

For MLG with $N=1$, we will absorb $\eta_{2}\otimes\sigma_{2}\otimes\tau_{2}$
into the lower component of Nambu spinor:
$\Psi^{T}_{1}=(\psi_{1},\tilde{\psi}^{*}_{1})^{T}/\sqrt{2}=(\psi_{1},\eta_{2}\otimes\sigma_{2}\otimes\tau_{2}\bar{\psi}^{T}_{1})^{T}/\sqrt{2}$,
and define the barred Nambu spinor with an additional $\alpha_{1}$ absorbed:
$\bar{\Psi}_{1}=\alpha_{1}(\bar{\psi}_{1},\tilde{\psi}^{T}_{1})/\sqrt{2}=(\psi^{T}_{1}\eta_{2}\otimes\sigma_{2}\otimes\tau_{2},\bar{\psi}_{1})/\sqrt{2}$.
In this basis, the QSH and QVH mass terms become

$\displaystyle\mathcal{O}_{QSH}$
$\displaystyle=\mathbf{\Omega}_{1}\cdot\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\eta_{0}\otimes\boldsymbol{\sigma}\otimes\tau_{0}\Psi,$
(13) $\displaystyle\mathcal{O}_{QVH}$
$\displaystyle=\mathbf{\Omega}_{2}\cdot\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\boldsymbol{\eta}\otimes\sigma_{0}\otimes\tau_{0}\Psi,$
(14)

while the Hamiltonian operator of free fermion acquires the form
$H_{f}=-i\sum_{j=1}^{2}\;\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\mathbb{1}_{4\times
4}\otimes\gamma_{j}\partial_{j}\Psi$. The ten possible pairing masses can be
rewritten as

$\displaystyle\mathcal{O}_{s,j}$
$\displaystyle=\phi_{s,j}\bar{\Psi}\alpha_{1}\alpha_{j}\otimes\eta_{0}\otimes\sigma_{0}\otimes\tau_{0}\Psi,$
(15) $\displaystyle\mathcal{O}^{mn}_{t,j}$
$\displaystyle=\phi_{t,j}^{mn}\bar{\Psi}\alpha_{1}\alpha_{j}\otimes\eta_{m}\otimes\sigma_{n}\otimes\tau_{0}\Psi,$
(16)

with $m$ and $n$ being equal to $1,2,3$. Only the spin and valley singlet,
$s$-wave, pairing mass $\mathcal{O}_{s,j}$ can anti-commute with $H_{f}$,
$\mathcal{O}_{QSH}$ and $\mathcal{O}_{QVH}$. Hence, the condensation of
skyrmion textures of both QSH and QVH order parameters in the para-magnetic
phase can lead to a unique, charge $2e^{-}$, spin- and valley- singlet,
$s$-wave pairing mass. Consequently, we can construct two quintuples

$\displaystyle\mathbf{N}_{1}$ $\displaystyle=$
$\displaystyle(\Omega_{1,1},\Omega_{1,2},\Omega_{1,3},\phi_{s,1},\phi_{s,2}),$
(17) $\displaystyle\mathbf{N}_{2}$ $\displaystyle=$
$\displaystyle(\Omega_{2,1},\Omega_{2,2},\Omega_{2,3},\phi_{s,1},\phi_{s,2}),$
(18)

for describing competition between particle-hole and particle-particle
channels. Simple matrix algebra shows the resulting $SO(5)$ NLSMs support
level-$1$ WZW term. _This is in agreement with the TV being a charge $2e^{-}$
bilinear_ [see Eq. (11) ]. Since the spin and valley triplet pairing mass
terms $\mathcal{O}^{mn}_{t,j}$ do not anti-commute with $\mathcal{O}_{QSH}$ or
$\mathcal{O}_{QVH}$, they are not favored as emergent, pairing orders.

QFH | QSH | QVH | QMVH
---|---|---|---
PDW |  $\mathcal{O}_{vt,j}^{100}$, $\mathcal{O}_{vt,j}^{200}$ | N/A |  $\mathcal{O}_{vt,j}^{100}$, $\mathcal{O}_{vt,j}^{200}$
MPDW |  $\mathcal{O}_{mt,j}^{010}$, $\mathcal{O}_{mt,j}^{020}$ |  $\mathcal{O}_{mt,j}^{010}$, $\mathcal{O}_{mt,j}^{020}$ | N/A
BCS |  $\mathcal{O}_{vt,j}^{300}$, $\mathcal{O}_{mt,j}^{030}$ |  $\mathcal{O}_{mt,j}^{030}$, $\mathcal{O}_{st,j}^{00l}$ |  $\mathcal{O}_{vt,j}^{300}$, $\mathcal{O}_{st,j}^{00l}$
Table 1: Classification of pairing mass terms for each QFH order, with j=1,2
and l=1,2,3.

###  A.2 Spinful TBLG

For TBLG with $N=2$, we absorb
$\eta_{2}\otimes\rho_{2}\otimes\sigma_{2}\otimes\tau_{2}$ into the lower
component of Nambu spinor:
$\Psi^{T}_{2}=(\psi_{2},\tilde{\psi}^{*}_{2})^{T}/\sqrt{2}=(\psi_{2},\eta_{2}\otimes\rho_{2}\otimes\sigma_{2}\otimes\tau_{2}\bar{\psi}^{T}_{2})^{T}/\sqrt{2}$,
and define the barred Nambu spinor with an additional $\alpha_{1}$ absorbed:
$\bar{\Psi}_{2}=\alpha_{1}(\bar{\psi}_{2},\tilde{\psi}^{T}_{2})/\sqrt{2}=(\psi^{T}_{2}\eta_{2}\otimes\rho_{2}\otimes\sigma_{2}\otimes\tau_{2},\bar{\psi}_{2})/\sqrt{2}$.
In this basis, the QSH, QVH, QMVH mass terms become

$\displaystyle\mathcal{O}_{QSH}$
$\displaystyle=\mathbf{\Omega}_{1}\cdot\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\eta_{0}\otimes\rho_{0}\otimes\tau_{0}\otimes\boldsymbol{\sigma}\Psi,$
(19) $\displaystyle\mathcal{O}_{QVH}$
$\displaystyle=\mathbf{\Omega}_{2}\cdot\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\boldsymbol{\eta}\otimes\rho_{0}\otimes\tau_{0}\otimes\sigma_{0}\Psi,$
(20) $\displaystyle\mathcal{O}_{QMVH}$
$\displaystyle=\mathbf{\Omega}_{3}\cdot\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\eta_{0}\otimes\boldsymbol{\rho}\otimes\tau_{0}\otimes\sigma_{0}\Psi,$
(21)

while the Hamiltonian operator of free fermion acquires the form
$H_{f}=-i\sum_{j=1}^{2}\;\bar{\Psi}\alpha_{1}\alpha_{3}\otimes\mathbb{1}_{8\times
8}\otimes\gamma_{j}\partial_{j}\Psi$. The pairing mass terms can be grouped
into four categories:

$\displaystyle\mathcal{O}_{st,j}^{00l}$
$\displaystyle=\phi_{st,j}^{00l}\bar{\Psi}\alpha_{1}\alpha_{j}\otimes\eta_{0}\otimes\rho_{0}\otimes\sigma_{l}\otimes\tau_{0}\Psi,$
(22) $\displaystyle\mathcal{O}_{vt,j}^{l00}$
$\displaystyle=\phi_{vt,j}^{l00}\bar{\Psi}\alpha_{1}\alpha_{j}\otimes\eta_{l}\otimes\rho_{0}\otimes\sigma_{0}\otimes\tau_{0}\Psi,$
(23) $\displaystyle\mathcal{O}_{mt,j}^{0l0}$
$\displaystyle=\phi_{mt,j}^{0l0}\bar{\Psi}\alpha_{1}\alpha_{j}\otimes\eta_{0}\otimes\rho_{l}\otimes\sigma_{0}\otimes\tau_{0}\Psi,$
(24) $\displaystyle\mathcal{O}_{at,j}^{lmn}$
$\displaystyle=\phi_{at,j}^{lmn}\bar{\Psi}\alpha_{1}\alpha_{j}\otimes\eta_{l}\otimes\rho_{m}\otimes\sigma_{n}\otimes\tau_{0}\Psi,$
(25)

where $l$, $m$, and $n$ can take values $1,2,3$. They respectively describe
pairing bilinears, which are spin-triplet, valley-triplet, mini-valley-
triplet, and triplets in all flavor channels. The physical significance of
such pairing mass terms are described in Table 1.

The QSH mass for $N=2$ model anti-commutes with charge $2e^{-}$ bilinears
$\mathcal{O}_{vt}^{l00}$ and $\mathcal{O}_{mt}^{0l0}$. There are six possible
ways to form _quintuples_

$\displaystyle\mathbf{N}^{l}_{1}$ $\displaystyle=$
$\displaystyle(\Omega_{1,1},\Omega_{1,2},\Omega_{1,3},\phi_{vt,1}^{l00},\phi_{vt,2}^{l00}),$
(26) $\displaystyle\mathbf{N}^{l}_{2}$ $\displaystyle=$
$\displaystyle(\Omega_{1,1},\Omega_{1,2},\Omega_{1,3},\phi_{mt,1}^{0l0},\phi_{mt,2}^{0l0}),$
(27)

with $l=1,2,3$, and _all six types of $SO(5)$ NLSMs will support level-$2$ WZW
term_. Due to the presence of multiple pairing mass terms, the skyrmion
condensation cannot select any specific charge $2e^{-}$ pairing channel,
without any additional mechanism of breaking $SU(4)$ flavor symmetry.

It is possible to combine all competing mass terms into a _nonuple_

$\mathbf{L}=(\Omega_{1},\Omega_{2},\Omega_{3},\phi_{vt,1}^{100},\phi_{vt,1}^{200},\phi_{vt,1}^{300},\phi_{mt,2}^{010},\phi_{mt,2}^{020},\phi_{mt,2}^{030}),$
(28)

and write down $SO(9)$ NLSM for $\hat{\mathbf{L}}$. The $SO(9)$ NLSM treats
all mutually anti-commuting mass orders on an equal footing. The resulting
coset space $SO(9)/SO(8)$ is closely tied to the octonion gauge theories and
details of such exotic aspects will be provided elsewhere. Within the $SO(9)$
model, the pairing mass terms are embedded as a _sextuple_. The appearance of
$SO(6)$ is a natural consequence of $SU(4)$ being the double cover of $SO(6)$.
Similar results can be obtained for the combinations
$(\mathcal{O}_{QVH},\mathcal{O}_{st,j}^{00l},\mathcal{O}_{mt,j}^{0l0})$ and
$(\mathcal{O}_{QMVH},\mathcal{O}_{vt,j}^{l00},\mathcal{O}_{st,j}^{00l})$. This
type of _triality_ is a consequence of the underlying $SU(8)$ flavor symmetry.

## References

  * (1) R. Rajaraman, Solitons and Instantons (North Holland, 1987).
  * (2) J. Goldstone, and F. Wilczek, _Fractional Quantum Numbers on Solitons_ , Phys. Rev. Lett. 47, 986 (1981).
  * (3) T. Jaroszewicz, _Induced fermion current in the $\sigma$ model in (2 + 1) dimensions_, Phys. Lett. B 146, 337 (1984).
  * (4) T. Jaroszewicz, _Induced topological terms, spin and statistics in (2 + 1) dimensions_ , Phys. Lett. B 159, 299 (1985).
  * (5) Y. H. Chen, and F. Wilczek, _Induced Quantum Numbers In Some 2+1 Dimensional Models_ , Int. J. Mod. Phys. B 3, 117 (1989).
  * (6) A. G. Abanov and P. B. Wiegmann, _Theta-terms in nonlinear sigma-models_ , Nucl. Phys. B 570, 685 (2000).
  * (7) A. Tanaka and X. Hu, _Many-Body Spin Berry Phases Emerging from the $\pi$-Flux State: Competition between Antiferromagnetism and the Valence-Bond-Solid State_, Phys. Rev. Lett. 95, 036402 (2005).
  * (8) T. Senthil and M. P. A. Fisher, _Competing orders, nonlinear sigma models, and topological terms in quantum magnets_ , Phys. Rev. B 74, 064405 (2006).
  * (9) T. Grover, and T. Senthil, _Topological Spin Hall States, Charged Skyrmions, and Superconductivity in Two Dimensions_ , Phys. Rev. Lett. 100, 156804 (2008).
  * (10) H. Yao and D. H. Lee, _Topological insulators and topological nonlinear $\sigma$ models_, Phys. Rev. B 82, 245117 (2010).
  * (11) L. Fu, S. Sachdev, C. Xu, _Geometric phases and competing orders in two dimensions_ , Phys. Rev. B 83, 165123 (2011).
  * (12) I. F. Herbut, C. K. Lu, B. Roy, _Conserved charges of order-parameter textures in Dirac systems_ , Phys. Rev. B 86, 075101 (2012).
  * (13) C. K. Lu, and I. F. Herbut, _Zero Modes and Charged Skyrmions in Graphene Bilayer_ , Phys. Rev. Lett. 108, 266402 (2012).
  * (14) E. G. Moon, _Skyrmions with quadratic band touching fermions: A way to achieve charge 4e superconductivity_ , Phys. Rev. B 85, 245123 (2012).
  * (15) C. H. Hsu, and S. Chakravarty, _Charge-2e skyrmion condensate in a hidden-order state_ , Phys. Rev. B 87, 085114 (2013).
  * (16) P. Goswami and Q. Si, _Topological defects of Néel order and Kondo singlet formation for Kondo-Heisenberg model on a honeycomb lattice_ , Phys. Rev. B 89, 045124 (2014).
  * (17) P. Goswami, and Q. Si, _Dynamic zero modes of Dirac fermions and competing singlet phases of antiferromagnetic order_ , Phys. Rev. B 95, 224438 (2017).
  * (18) C. C. Liu, P. Goswami, and Q. Si, _Skyrmion defects and competing singlet orders in a half-filled antiferromagnetic Kondo-Heisenberg model on the honeycomb lattice_ , Phys. Rev. B 96, 125101 (2017).
  * (19) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P. A. Fisher, _Deconfined Quantum Critical Points_ , Science 303, 1490 (2004).
  * (20) A. Nahum, P. Serna, J. T. Chalker, M. Ortuño, and A. M. Somoza, _Emergent SO(5) Symmetry at the Néel to Valence-Bond-Solid Transition_ , Phys. Rev. Lett. 115, 267203 (2015).
  * (21) C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, _Deconfined quantum critical points: Symmetries and dualities_ , Phys. Rev. X 7, 031051 (2017).
  * (22) Z. Wang, M. P. Zaletel, R. S. K. Mong, and F. F. Assaad, _Phases of the (2+1) dimensional SO(5) non-linear sigma model with topological term_ , Phys. Rev. Lett. 126, 045701 (2021).
  * (23) Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, _Unconventional superconductivity in magic-angle graphene superlattices_ , Nature 556, 43 (2018).
  * (24) T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, _Tuning superconductivity in twisted bilayer graphene_ , Science, 24, eeaav1910 (2019).
  * (25) X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das,C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold,A. H. MacDonald, and D. K. Efetov, _Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene_ , Nature 574, 653 (2019).
  * (26) P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K.Efetov, _Untying the insulating and superconducting orders in magic-angle graphene_ , 583, 375 (2020).
  * (27) Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young, _Decoupling superconductivity and correlated insulators in twisted bilayer graphene_ , Nat. Phys. 16, 926 (2020).
  * (28) A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean, A. Rubio, and A. N. Pasupathy, _Maximized electron interactions at the magic angle in twisted bilayer graphene_ , Nature 572, 95 (2019).
  * (29) Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora, R. Polski, Y. Zhang, H. Ren, J. Alicea, G. Refael, F. von Oppen, K. Watanabe, T. Taniguchi, and S. Nadj-Perge, _Electronic correlations in twisted bilayer graphene near the magic angle_ , Nat. Phys. 15, 1174 (2019).
  * (30) Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule, J. Mao, and E. Y. Andrei, _Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene_ , Nature 573, 91 (2019).
  * (31) Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, _Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene_ , Nature 572, 101 (2019).
  * (32) D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, _Cascade of electronic transitions in magic-angle twisted bilayer graphene_ , Nature 582, 198 (2020).
  * (33) U. Zondiner, A. Rozen, D. R.-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern, E. Berg, P. J.-Herrero, and S. Ilani, _Cascade of phase transitions and Dirac revivals in magic angle graphene_ , Nature 582, 203 (2020).
  * (34) X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. I. A. Li, _Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening_ , Science 371, 1261 (2021).
  * (35) Y. Cao, D. R.-Legrain, J. M. Park, F. N. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. J.-Herrero, _Nematicity and competing orders in superconducting magic-angle graphene_ , Science 372, 264 (2021).
  * (36) F. Wu, A. H. MacDonald, and I. Martin, _Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene_ , Phys. Rev. Lett. 121, 257001 (2018).
  * (37) H. Isobe, N. F. Q. Yuan, and L. Fu, _Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene_ , Phys. Rev. X 8, 041041 (2018).
  * (38) B. Lian, Z. Wang, and B. A. Bernevig, _Twisted Bilayer Graphene: A Phonon Driven Superconductor_ , Phys. Rev. Lett. 122, 257002 (2019).
  * (39) Y.-Z. You and A. Vishwanath, _Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene_ , npj Quantum Materials 4, 16 (2019).
  * (40) L. Classen, A. V. Chubukov, C. Honerkamp, and M. M. Scherer, _Competing orders at higher-order Van Hove points_ , Phys. Rev. B 102, 125141 (2020).
  * (41) D. V. Chichinadze, L. Classen, and A. V. Chubukov, _Valley magnetism, nematicity, and density wave orders in twisted bilayer graphene_ , Phys. Rev. B 102, 125120 (2020).
  * (42) V. Kozii, M. P. Zaletel, and N. Bultinck, _Superconductivity in a doped valley coherent insulator in magic angle graphene: Goldstone-mediated pairing and Kohn-Luttinger mechanism_ , arXiv:2005.12961 (2020).
  * (43) Y. Wang, J. Kang, and R. M. Fernandes, _Topological and nematic superconductivity mediated by ferro-SU(4) fluctuations in twisted bilayer graphene_ , Phys. Rev. B 103, 024506 (2021).
  * (44) E. Khalaf, S. Chatterjee, N. Bultinck, M. P. Zaletel, and A. Vishwanath, Science Advances 7, eabf5299 (2021).
  * (45) M. Christos, S. Sachdev, and M. S. Scheurer, Proceedings of the National Academy of Sciences 117, 29543 (2020).
  * (46) S. Chatterjee, M. Ippoliti, and M. P. Zaletel, Skyrmion superconductivity: DMRG evidence for a topological route to superconductivity, arXiv:2010.01144 (2020).
  * (47) E. Khalaf, P. Ledwith, and A. Vishwanath, _Symmetry constraints on superconductivity in twisted bilayer graphene: Fractional vortices, 4e condensates or non-unitary pairing_ , arXiv:2012.05915 (2020).
  * (48) F. D. M. Haldane, _O(3) Nonlinear $\sigma$ Model and the Topological Distinction between Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions_, Phys. Rev. Lett. 61, 1029 (1988).
  * (49) G. Murthy and S. Sachdev, _Action of hedgehog instantons in the disordered phase of the (2 + 1)-dimensional $CP^{N-1}$ model_, Nucl. Phys. B 344, 557 (1990).
  * (50) N. Read and S. Sachdev, _Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets_ , Phys. Rev. B 42, 4568 (1990).
  * (51) T. Schaefer and E. Shuryak, _Instantons in QCD_ , Rev. Mod. Phys. 70, 328 (1998).
  * (52) D. Diakonov, _Instantons at work_ , Prog. Part. Nucl. Phys. 51, 173 (2003).
  * (53) G. ’t Hooft, _Computation of the quantum effects due to a four-dimensional pseudoparticle_ , Phys. Rev. D 14, 3432 (1976).
  * (54) R. Bistritzer and A. MacDonald, Proceedings of the National Academy of Sciences 108, 12233 (2011).
  * (55) R. D. Gail, M. Goerbig, F. Guinea, G. Montambaux, and A. Neto, Phys. Rev. B 84, 045436 (2011).
  * (56) J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. B 86, 155449 (2012).
  * (57) H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Phys. Rev. X 8, 031089 (2018).
  * (58) A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
  * (59) A. A. Belavin, and A. M. Polyakov, Metastable states of two-dimensional isotropic magnets, JETP Lett., 22, 245 (1975).
  * (60) J. Arafune, P. G. O. Freund, and C. J. Goebel, _Topology of Higgs fields_ , J. Math. Phys. 16, 433 (1975).
  * (61) R. Jackiw and C. Rebbi, _Solitons with fermion number 1/2_ , Phys. Rev. D 13, 3398 (1976).
  * (62) C. J. Callias, _Spectra of fermions in monopole fields—Exactly soluble models_ , Phys. Rev. D 16, 3068 (1977).
  * (63) C. J. Callias, _Axial anomalies and index theorems on open spaces_ , Commun. Math. Phys. 62, 213 (1978).
  * (64) Y. H. Zhang, H. C. Po, T. Senthil, _Landau level degeneracy in twisted bilayer graphene: Role of symmetry breaking_ , Phys. Rev. B 100, 125104 (2019).
  * (65) J. Kang, and O. Vafek, _Strong coupling phases of partially filled twisted bilayer graphenenarrow bands_ , Phys. Rev. Lett. 122, 246401 (2019).
  * (66) J. Kang, O. Vafek, _Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene_ , Phys. Rev. B 102, 035161 (2020).
  * (67) Yuan Da Liao, Jian Kang, Clara N. Breiø, Xiao Yan Xu, Han-Qing Wu, Brian M. Andersen, Rafael M. Fernandes, Zi Yang Meng, _Correlation-induced insulating topological phases at charge neutrality in twisted bilayer graphene_ , Phys. Rev. X 11, 011014 (2021).