# Pseudospin Paramagnons and the Superconducting Dome
in Magic Angle Twisted Bilayer Graphene

Chunli Huang, Nemin Wei, Wei Qin, and Allan H. MacDonald  Department of
Physics, University of Texas at Austin, Austin TX 78712

###### Abstract

We present a theory of superconductivity in magic-angle twisted bilayer
graphene in which attraction is generated between electrons on the same
honeycomb sublattice when the system is close to a sublattice polarization
instability. The resulting Cooper pairs are spin-polarized valley-singlets.
Because the sublattice polarizability is mainly contributed by interband
fluctuations, superconductivity occurs over a wide range of moiré filling
fraction. It is suppressed by i) applying a sublattice polarizing field
(generated by an aligned BN substrate) or ii) changing moiré band filling to
favor valley polarization. The enhanced intrasublattice attraction close to
sublattice polarization instability is analogous to enhanced like-spin
attraction in liquid 3He near the melting curve and the enhanced valley-
singlet repulsion close to valley-polarization instabilities is analogous to
enhanced spin-singlet repulsion in metals that are close to a ferromagnetic
instability. We comment on the relationship between our pseudospin paramagnon
model and the rich phenomenology of superconductivity in twisted bilayer and
multilayer graphene.

Introduction:– Possible explanations for superconductivity in magic-angle
twisted bilayer (trilayer) graphene MATB(T)G cao2018unconventional ;
park2021tunable  are increasingly constrained by experimental data
yankowitz2019tuning ; lu2019superconductors ; saito2020independent ;
polshyn2019large ; cao2020strange ; de2021gate ; liu2021tuning ;
wong2020cascade ; stepanov2020competing ; zondiner2020cascade ;
saito2021isospin ; cao2021pauli ; choi2021interaction ; choi2021correlation ;
wong2020cascade . The normal state properties revealed by the weak-field Hall
effect and magnetoresistance oscillations saito2021isospin  are particularly
telling; the strongest superconductivity occurs within the moiré band filling
interval $\nu\in(-3,-2)$, within which the system has a hole-like Fermi
surfaces surrounding $|\nu|-2$ states per moiré period. The implied Fermi
surface reconstruction points to symmetry breaking that depopulates two of
four spin/valley flavors, an interpretation that is reinforced by Landau fans
that are only doubly degenerate for $\nu\in(-3,-2)$ in spite of the systems
four-fold spin/valley band degeneracy. The normal state symmetry breaking
provides a natural explanation for large in-plane critical magnetic fields
cao2021pauli  that are $\sim 2-3$ times larger than the Clogston-Chandrashekar
limit of spin-singlet superconductors. As summarized in Fig. 1,
superconductivity is suppressed as $\nu$ approaches $-3$ and as $\nu$
approaches $-2$, filling factors at which the system is known to tend toward
valley-ordered states, forming a dome in the system’s $(T,\nu)$ phase diagram
that is reminiscent of those observed in cuprate superconductors. For example,
the superconducting critical temperature $T_{c}$ of device 1 in Ref.
saito2020independent  is peaked at $\nu\sim-2.4$, and vanishes near
$\nu\rightarrow-2.7$ on the low-filling-factor side and near
$\nu\rightarrow-2.2$ on the high filling factor side. The suppression of
superconductivity on the low-filling factor side occurs in spite of an
increasing Fermi level density-of-states, and therefore argues against a
mechanism, like phonon-dressing, in which the pairing glue is external to the
electron system. Superconductivity actually seems to be suppressed when the
Fermi level is close to a the flat valence band van Hove singularity energy
stepanov2020competing ; park2021tunable . These experimental observations
point to an electronic pairing mechanism.

Figure 1: Exchange-enhanced sublattice pseudospin polarizatibility leads to an
attractive intrasublattice interaction that can support spin-triplet, valley-
singlet superconductors across a wide range of moiré band filling fraction
$\nu$. Superconductivity is suppressed when $\nu$ is tuned close to valley
polarization instabilities and when the density-of-states (DOS) is small.

In this Letter, we argue that the superconducting properties of MATBG are
consistent with a sublattice-pseudospin paramagnon pairing mechanism, and
interpret the shape of the superconducting dome in terms of sublattice and
valley paramagnons. Our main message is summarized in Fig. 1.

Figure 2:  $T_{c}$ v.s. $\Gamma^{AA}$ and $\Gamma^{AB}$.
$A^{a}=(\Gamma^{AA}-\Gamma^{AB})/2$ is attractive and strong in nearly
sublattice polarized (NSP) metals, whereas $A^{s}=(\Gamma^{AA}+\Gamma^{AB})/2$
is repulsive, but weak if screening is strong. Superconductivity occurs at low
temperature when $\Gamma^{AA}$ is attractive, even though $\Gamma^{AB}$ is
repulsive. The physically accessible parameter range above the dashed white
$A^{s}=0$ line includes superconductivity states. The inset plots the pairing
self-energy $\Delta_{\boldsymbol{k}}$ vs. $\boldsymbol{k}$ in the moiré
Brillouin-zone for a continuum model with $t_{AA}/t_{AB}=0.7$, at
$\theta=1.1^{\circ}$, at $\nu=-2.4$ SM . $\Delta_{\boldsymbol{k}}$ is largest
near the normal state Fermi surfaces centered on $\kappa$, $\kappa^{\prime}$.
When the system is close to a valley-Ising instability (near $-3$) or when it
is sublattice polarized (near $\nu=+3$) serlin2020intrinsic ;
sharpe2019emergent , $\nu=+1$ stepanov2020competing ), $\Gamma^{AA}$ will
increase and superconductivity will weaken. When $A^{s}$ is decreased by the
acoustic-phonon mediated interaction and/or screening of the long-range
Coulomb interactions, $T_{c}$ is enhanced. These trends are indicated by
arrows and discussed in the main text.

Pseudospin-paramagnons (PPM) and fermion pairing:- Our theory of MATBG
superconductivity is guided by experiments cao2021pauli ; saito2021isospin
and inspired by what is known about the relationship between fermion pairing
in liquid 3He and the strongly enhanced paramagnetic spin susceptibilities
that appear near the solidification curve levin1983phenomenological . The
enhanced susceptibility leads to a low-frequency neutral excitation spectrum
that is dominated by paramagon peaks, damped collective modes that generate a
strong attraction between like-spin nuclei, and leads to anisotropic spin-
triplet Anderson-Brinkman-Morel anderson1973anisotropic  superfluidity. Spin-
fluctuation mediated superfluidity in 3He is succinctly captured by the
paramagnon model, which uses a single exchange-interaction parameter $I$ to
describe the enhanced spin susceptibility berk1966effect . The paramagnon
model accounts quantitatively for the $T^{3}\ln T$ contribution to heat
capacity in 3He doniach1966low , and has been extended to understand the
interplay between magnetism and superconductivity in metals, including heavy-
fermion materials anderson1984heavy ; moriya2006developments ;
RevModPhys.84.1383 .

Unlike nuclei in 3He, which possess only a spin degree of freedom, flat-band
electrons in MATBG possess a
spin(s)$\times$valley($\tau$)$\times$sublattice($\sigma$) pseudospin octet
that generates eight distinct states for each momentum and an abundance of
potential pairing channels that has to be winnowed. Some progress can be
achieved by taking note of the time-reversal symmetry property
$\epsilon_{K}(\boldsymbol{k})=\epsilon_{K^{\prime}}(-\boldsymbol{k})$, which
strongly favors Cooper pairing between electrons in opposite valleys $K$ and
$K^{\prime}$
($\epsilon_{K}(\boldsymbol{k})\neq\epsilon_{K}(-\boldsymbol{k})$). If we
assume cao2021pauli ; saito2021isospin  that the normal state for $\nu<-2$ is
ferromagnetic (in spin), we can conclude that the Cooper pairs must be spin-
polarized valley-singlets. That still leaves the sublattice pseudospin, and
the need to overcome the strong sublattice independent Coulomb repulsion. A
route to superconductivity is provided by the properties of the flatband
spinors which imply, as illustrated in Fig. 2, that superconductivity occurs
when the intrasublattice interaction is attractive ($\Gamma^{AA}<0$) even if
the intersublattice interaction is strongly repulsive ($\Gamma^{AB}>0$) 111It
is also true that attractive intersublattice interactions $\Gamma^{AB}<0$
produce superconductivity when the intrasublattice interaction is repulsive.
See SM  for a discussion of this case. This property of magic-angle
superconductivity is analogous to the robustness of spin-triplet
superconductivity against repulsive opposite-spin interactions in systems with
weak spin-orbit coupling anderson2018theory , and can be traced to the $D_{6}$
point-group symmetry of the band Hamiltonian which decouples like-sublattice
and unlike-sublattice pairing in the linearized gap equation SM . We show
below that intrasublattice attraction is generated when the system is close to
a spontaneous sublattice polarization instability. This instability is common
in graphene multilayers because sublattice polarization simplifies the spinor
content of occupied states and increases exchange energies
macdonald2012pseudospin ; velasco2012transport . The same analysis that shows
that sublattice PPM bolster superconductivity, shows that valley PPM are
obstructive.

Sublattice-Dependent Effective Interactions:-. The reduced pairing Hamiltonian
for spin $\uparrow$ electrons in opposite valleys is

$\displaystyle H_{\text{red}}=$
$\displaystyle\sum_{\boldsymbol{k}\boldsymbol{k}^{\prime},\sigma_{i}}\;\bigg{[}\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}(\sigma_{1}\sigma_{4};\sigma_{2}\sigma_{3})$
$\displaystyle\times
c_{\boldsymbol{K}+\boldsymbol{k}^{\prime}\sigma_{1}\uparrow}^{\dagger}c_{-\boldsymbol{K}-\boldsymbol{k}^{\prime}\sigma_{2}\uparrow}^{\dagger}c_{-\boldsymbol{K}-\boldsymbol{k}\sigma_{3}\uparrow}c_{\boldsymbol{K}+\boldsymbol{k}\sigma_{4}\uparrow}\bigg{]},$
(1)

where $\Gamma$ is the particle-particle channel irreducible 4-point vertex
function estimated below vignale1985effective ; bickers2004self ,
$\boldsymbol{k}$ labels a state in the moiré valence band, and $\sigma=(A,B)$
labels sublattice. As we explain below, the PPM contributions to $\Gamma$ are
dominantly diagonal in sublattice at each vertex. This property motivates a
model in which the dependence of $\Gamma$ on
$\boldsymbol{k},\boldsymbol{k}^{\prime}$ is neglected and $\mathcal{C}_{2}$
symmetry is recognized:

$\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}(\sigma_{1}\sigma_{4};\sigma_{2}\sigma_{3})\to\delta_{\sigma_{1},\sigma_{4}}\delta_{\sigma_{2},\sigma_{3}}\big{[}\Gamma^{AA}\delta_{\sigma_{1},\sigma_{2}}+\Gamma^{AB}\delta_{\sigma_{1},-\sigma_{2}}\big{]}.$
(2)

As illustrated in Fig. 2, superconductivity occurs in this model when the same
sublattice effective interaction $\Gamma^{AA}$ is attractive. The
superconductivity gap is approximately uniform on the valley-projected FS
(centered around $\kappa$ and $\kappa^{\prime}$), as shown in the inset.
$T_{c}$ is not suppressed by repulsive $\Gamma^{AB}$ interactions because the
susceptibility for pairs on opposite sublattices averages to zero on the Fermi
surface. Motivated by this observation, we now address the relationship of
$\Gamma^{AA}$ to PPMs.

Figure 3: a) The interaction potential $\Gamma$ relevant to superconductivity
has three contributions vignale1985effective ; bickers2004self : a particle-
particle vertex function $\Lambda_{\text{P}}$, a double-crossing diagram and
the Berk-Schrieffer (single-crossing) diagram. The crossing diagrams relate
$\Gamma$ to paramagnon interactions that diverage at particle-hole
instabilities. b) $u_{0}>0$ is a pseudospin-independent density-density
interaction and $u_{\perp}>0$ is an inter-valley scattering potential. c) The
particle-hole vertex function $\Lambda$ consists of the direct and exchange
scattering terms from both $u_{0}$ and $u_{\perp}$. d) Schematic Fermi surface
for $-3<\nu<-2$ where $c(v)$ are the spin and valley projected conduction
(valence) band.

Recognizing that the computation of $\Gamma$ from first-principles is a
formidable challenge even in relatively simple systems anderson2018theory ;
nava2012equation , we follow a phenomenological approach similar in spirit to
that employed successfully in 3He anderson1984heavy . In that case spin-
rotational invariance allows the spin-dependent effective interaction to be
parameterized by just two scattering amplitudes, one each for the spin-
symmetric and spin antisymmetric channel, (i.e.
$\Gamma=A^{s}+A^{a}\boldsymbol{s_{1}}\cdot\boldsymbol{s_{2}}$
levin1983phenomenological ). In MATBG, the Hamiltonian is not invariant under
rotations of the sublattice pseudospins. When the system is close to a
sublattice-polarization instability, however, effective interactions that are
diagonal in sublattice dominate and the analogous expressions are

$\Gamma^{AA}=A^{s}+A^{a}\;\;,\;\;\Gamma^{AB}=A^{s}-A^{a}.$ (3)

We estimate $A^{s}$ and $A^{a}$ by assuming a simple $\delta$-function
interaction model in order to sum the Feynman diagrams specified by Fig. 3.
The irreducible vertex function $\Lambda$ accounts for both direct and
exchange scattering (i.e. generalized-random-phase approximation) generated by
$u_{0}$ and $u_{\perp}$. $u_{0}$ is the fully symmetric Coulomb interaction
while the valley-exchange scattering term $u_{\perp}>0$ breaks independent
spin-rotation symmetry in opposite valleys and selects the spin-ferromagnet
over other possibilities as the flavor-polarized normal state. 222We use
crossing-symmetry bickers2004self  to relate the vertex functions in the
particle-particle channel ($\Lambda_{P}$) to the one in particle-hole channel
($\Lambda$).

Ignoring for the moment the Berk-Schrieffer transverse fluctuation diagram in
Fig. 3, we find that

$\displaystyle
A^{s}=\frac{1}{2}\left(T^{\tau_{0}\sigma_{0}}-T^{\tau_{z}\sigma_{0}}\right)\;,\;A^{a}=\frac{1}{2}\left(T^{\tau_{0}\sigma_{z}}-T^{\tau_{z}\sigma_{z}}\right),$
(4)

where

$\displaystyle T^{\tau_{0}\sigma_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{3u_{0}-u_{\perp}}{1+(3u_{0}-u_{\perp})\chi_{\boldsymbol{k}-\boldsymbol{k^{\prime}},\omega}^{K\sigma_{0},K\sigma_{0}}},$
(5) $\displaystyle T^{\tau_{z}\sigma_{0}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{u_{0}-u_{\perp}}{1-(u_{0}-u_{\perp})\chi_{\boldsymbol{k}-\boldsymbol{k^{\prime}},\omega}^{K\sigma_{0},K\sigma_{0}}},$
(6) $\displaystyle T^{\tau_{0}\sigma_{z}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{u_{0}+u_{\perp}}{1-(u_{0}+u_{\perp})\chi_{\boldsymbol{k}-\boldsymbol{k^{\prime}},\omega}^{K\sigma_{z},K\sigma_{z}}},$
(7) $\displaystyle T^{\tau_{z}\sigma_{z}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{u_{0}-u_{\perp}}{1-(u_{0}-u_{\perp})\chi_{\boldsymbol{k}-\boldsymbol{k^{\prime}},\omega}^{K\sigma_{z},K\sigma_{z}}},$
(8)

Here $\chi_{\boldsymbol{q},\omega=0}^{O,O^{\dagger}}$ is a bare susceptibility
for pseudospin polarization $O$. We discuss the valley-diagonal, sublattice
even ($\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{0},K\sigma_{0}}$) and
sublattice odd ($\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{z},K\sigma_{z}}$)
susceptibilities further below.

The random phase sums decouple contributions that are proportional to
different products of valley and sublattice Pauli matrices ($\tau_{0}$ or
$\tau_{z}$ and $\sigma_{0}$ or $\sigma_{z}$) of the interacting particle-hole
pairs and yield pseudospin-dependent interactions that are sums of the four
different geometric series. In each of the $T^{\tau_{a}\sigma_{b}}$’s a bare
interaction is modified by a dressing factor (i.e. denominator) identical to
the one that modifies the corresponding susceptibility. For the valley and
sublattice independent interaction $T^{\tau_{0}\sigma_{0}}$ the bare
interaction is dominated by the bare repulsive Coulomb interaction $u_{0}$ and
the modification is suppression due to static screening. For all other
interaction channels, the dominant bare contribution is an attractive exchange
contribution and the modification factor yields enhancement. At lowest order
$\Gamma^{AA}=u_{0}-u_{\perp}$ and $\Gamma^{AB}=u_{0}$ are both repulsive.
Superconductivity is possible when $T^{\tau_{0}\sigma_{z}}$, which contributes
attractively to $\Gamma^{AA}$, has much larger enhancement factors than
$T^{\tau_{z}\sigma_{0}}$ and $T^{\tau_{z}\sigma_{z}}$, both of which
contribute repulsively and diverge at valley polarization instabilities. We
conclude that attractive intra-sublattice effective interactions, and hence
superconductivity, is likely when the system is close to a valley-independent
sublattice polarization instability but far from a valley-polarization
instabilities.

We have so far neglected the Berk-Schrieffer (BS) transverse valley
fluctuations indicated in Fig. 3. The particle-hole channels in the BS
diagrams decouple into four channels $\tau_{+}\sigma_{0,x,y,z}$ in the long
wavelength limit. They change

$A^{s,a}\rightarrow
A^{s,a}\mp\frac{1}{2}\sum_{i=x,y}T^{\tau_{+}\sigma_{i}}-\frac{1}{2}\sum_{\delta=0,z}T^{\tau_{+}\sigma_{\delta}}.$
(9)

Because the susceptibilities in the $\delta=0,z$ channels excite particle-hole
pair from opposite Chern bands, their average transition matrix elements are
small bultinck2020ground ; lian2021iv . As shown in Ref. SM , they are in fact
smaller than the susceptibility in the pseudospin density fluctuation
channels. The other two channels in the BS diagrams ($i=x,y$) are off-diagonal
in sublattice and therefore do not affect the attractive interaction
$\Gamma^{AA}$. As a result, the net effect of transverse valley fluctuations
(which can be strong close to intervalley coherent order bultinck2020mechanism
; zhang2019nearly ; lee2019theory  ) is to increase $\Gamma^{AB}$ in the
vertical direction shown in Fig. 2 and do not modify the $T_{c}$. Hence,
transverse valley fluctuations can be neglected in our analysis. In making
these estimates of importance, we are not considering the possibility of a
nodal superconducting gap generated by repulsive interactions with strong
momentum dependence - something that cannot be ruled out.

In the analysis of spin-fluctuation mediated Cooper pairing in liquid 3He and
other Fermi-systems with weak spin-orbit coupling, there is often no need to
distinguish the longitudinal spin-fluctuation double-crossing diagram from the
transverse spin-fluctuation Berk-Schrieffer diagram because the pairing
interaction is invariant under SU$(2)$ spin rotation. This distinction is
important in our present analysis because the pseudospin ($\tau\otimes\sigma$)
anisotropic energy arose from the band Hamiltonian is very large and as a
result, the transverse and longitudinal pseudospin fluctuations can contribute
independently to the pairing interaction.

The Superconducting Dome:–. We are now in a position to explain how the
distinct filling factor dependencies of the different
$T^{\tau_{a}\sigma_{b}}$’s forms the superconducting dome. An enhanced
sublattice polarization susceptibility, which strengthens an attractive
interaction, is present when the corresponding Stoner criterion
$(u_{0}+u_{\perp})\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{z},K\sigma_{z}}=1$,
is nearly satisfied. Repulsive interactions are strengthened when one of the
valley polarization susceptibilities, with Stoner criteria
$(u_{0}-u_{\perp})\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{0},K\sigma_{0}}=1$
or
$(u_{0}-u_{\perp})\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{z},K\sigma_{z}}=1$,
are close to being satisfied. Because $u_{0}+u_{\perp}$ is larger than
$u_{0}-u_{\perp}$ the attractive $\tau_{0}\sigma_{z}$ channel interaction is
always stronger than the repulsive $\tau_{z}\sigma_{z}$ channel. Physically,
the $\tau_{0}\sigma_{z}$ susceptibility is enhanced and the
$\tau_{z}\sigma_{z}$ susceptibility suppressed by $u_{\perp}$ because its
exchange energy is maximized when opposite valleys have identical sublattice
polarization. The competition between normal and superconducting states is
therefore between the sublattice polarization Stoner criteria, which involves
the $\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{z},K\sigma_{z}}$ polarization,
and the valley polarization Stoner criterion, which involves
$\chi_{\boldsymbol{q},\omega=0}^{K\sigma_{0},K\sigma_{0}}$. Because of
$\mathcal{C}_{2}T$ symmetry (where $T$ is a spinless time-reversal symmetry)
these two polarizations have very distinct dependencies on band filling. In
particular $\chi_{\boldsymbol{q}\rightarrow
0,\omega=0}^{K\sigma_{z},K\sigma_{z}}$ does not have an intraband
contributions because the expectation value of $\sigma_{z}$ is zero in all
moiré band states. Its value is therefore not related to peaks in the band
density-of-states and, as illustrated in Fig. 3d, should instead be larger
closer to $\nu=-2$ where the majority spins are nearly half-filled and the
interband transition phase space is therefore largest. On the other hand,
$\chi_{\boldsymbol{q}\rightarrow 0,\omega=0}^{K\sigma_{0},K\sigma_{0}}$ is
proportional to the Fermi level density of states and increases rapidly as
$|\nu|-2$ increases and the Fermi level approaches the Van Hove peak in the
valence moiré band. We attribute the suppression of superconductivity on the
large $|\nu|-2$ side of the superconducting dome to the increasingly strong
repulsive interaction contribution to $A^{s}$ from $T^{\tau_{z}\sigma_{0}}$,
which is proportional to the valley polarization susceptibility enhancement
factor and suppresses superconductivity, as indicated in Fig. 2. On the other
hand, we attribute the suppression of superconductivity on the small $|\nu|-2$
side of the dome to the decreasing Fermi-level density-of-states.

We emphasize that our picture of superconductivity in graphene moiré
superlattice requires that the normal state is nearly sublattice polarized
metal over a wide range of filling fraction. Enhanced sublattice polarization,
and sometimes sublattice polarization instabilities, are in fact common in all
graphene multilayer electron gas systems macdonald2012pseudospin . For
example, neutral Bernal bilayer graphene has $\sigma_{z}s_{z}$ order, i.e.
spin-dependent sublattice-polarization in the antiferromagnetic state
velasco2012transport . In twisted bilayers the gaps between flat and remote
bands open an opportunity for flavor polarization and hence for sublattice
polarization not only near neutrality but also near integer $\nu$. In moiré
flat bands itinerant exchange energies favor maximum spin-polarization before
orbital-polarization, allowing the majority spin projected conduction and
valence bands to mix and produce finite sublattice polarization over a wide
range of filling fraction.

Summary and Discussions:– Paramagnon-mediated pairing physics in MATBG is
enriched by the relevance of spin, valley, and sublattice pseudospins. Because
the valley-projected bands are not time-reversal invariant, pairing is likely
to occur between time-reversal partner states in opposite valleys and yield
valley-singlets. We have shown that valley-singlet superconductivity occurs in
MATBG when the effective interaction between electrons on the same sublattice
is attractive, and that this condition is satisfied when the system is close
to a sublattice-polarization instability, but far from a valley-polarization
instability. There is no need for the interaction between electrons on
opposite sublattice, or the total interaction summed over sublattices, to be
attractive. Enhanced intrasublattice attraction close to sublattice
polarization instabilities is analogous to the enhanced like-spin attraction
in 3He anderson1973anisotropic  near the melting curve, and enhanced valley-
singlet repulsion close to valley-polarization instabilities is analogous to
enhanced spin-singlet repulsion berk1966effect  in metals that are close to a
ferromagnetic instability. Together these two effects explain the prominent
superconducting domes seen in most MATBG samples inside the filling factor
interval $\nu\in(2,3)$ 333When the filling factor $\nu$ approaches $\nu=-3$,
the enhanced valley polarizability not only generates a valley-singlet
repulsion but also an attraction between electrons in the same valley. When
this attraction leads to condensation of Cooper pairs from the same valley,
the resulting superconducting order parameter cannot be uniform-s-wave because
of Fermi statistics. (We assume the normal state is always spin-polarized).
This means the order parameter might undergo a transition from valley-singlet
to valley-triplet as $\nu\rightarrow-3$ and might explain the $U-V$ transition
recently observed in local-tunneling spectroscopy kim2021spectroscopic. In
this scenario, valley-triplet superconductivity close to $\nu=-3$ is
eventually suppressed by the enhanced pair-breaking (trigonal-warping)
potential from the band-Hamiltonian..

We now comment on other aspects of the phenomenlogy of MATBG superconductivity
seen through the pseudospin paramagnon lens:

i) Influence of Boron-Nitride (hBN) Alignment:– Aligned hBN induces a finite
sublattice polarization in a graphene sheet, and weakens sublattice pseudospin
fluctuations. Since all other contributions to the intrasublattice interaction
are repulsive, this theory predicts that superconductivity is suppressed by
hBN alignment, in agreement with current experimental findings.

ii) Coulomb screening:– When MATBG devices are surrounded by nearby material
that is conducting saito2020independent ; stepanov2020untying ; liu2021tuning
or has a large dielectric constant arora2020superconductivity , insulating
states at integer filling factors become less prominent in the phase diagram
and superconductivity is found over a broader range of filling factors.
Insulating states are understood in terms of exchange-splitting between bands
associated with different flavor states that is larger than the flatband
bandwidth. Since the exchange splittings are dominated by the Coulomb
interaction $u_{0}$, they are expected to be reduced by enhanced environmental
screening. The reduction in insulating state gaps seen experimentally is
therefore expected. On the other hand, the net attractive interaction
contributed by $T^{\tau_{0}\sigma_{z}}-T^{\tau_{z}\sigma_{z}}$ scales with the
intervalley-exchange interaction $u_{\perp}$, which should not be influenced
by environmental screening. In addition the enhancement factors in
$T^{\tau_{0}\sigma_{z}}$ and $T^{\tau_{z}\sigma_{z}}$, which together
contribute attractively to $\Gamma^{AA}$, both involve the
$\chi_{\boldsymbol{q},\omega}^{K\sigma_{z},K\sigma_{z}}$ susceptibility which
does not have a $\boldsymbol{q}=0$ Fermi surface contribution.
$\chi_{\boldsymbol{q},\omega}^{K\sigma_{z},K\sigma_{z}}$ is instead dominated
by interband fluctuations that are less sensitive to remote screening. On the
other hand, $T^{\tau_{0}\sigma_{0}}$, the repulsive screened Coulomb
interaction, is certainly weakened by enhanced dielectric screening. The net
result of enhanced remote screening could therefore be to enhance
superconductivity, as sometimes observed experimentally. Although both
superconductivity and insulating behavior require interactions, remote
screening influences only $u_{0}$, and can therefore have opposite influences
on the two states.

iii) Superconductivity near $\nu=0$:– Although superconductivity is most
robustly observed for $\nu\in(2,3)$, the hole-like and electron-like Fermi
surfaces on opposite side of $\nu=0$ also sometimes host superconductivity as
shown in Ref. lu2019superconductors . Since our proposed theory of
superconductivity between $-3<\nu<-2$ only involves states in a spin-projected
Hilbert space (c.f. Fig. 3d), it can be extended to $-1<\nu<+1$ by assuming an
identical pairing mechanism occurs in in both spin projected Hilbert spaces.
This state would be similar to the equal spin pairing state in 3He. The
stronger superconductivity for $-3<\nu<-2$, could be attributed to band-
renormalizations that flatten bands away from half-filling.
choi2021interaction ; PhysRevB.100.205113 ; guinea2018electrostatic ;
xie2020weak .

iv) Particle-hole asymmetry of superconductivity:– Experiment shows that the
tendency toward valley polarization, which opposes superconductivity, is
stronger for electron-doped than for hole-doped MATBG. This tendency is due in
part to conduction bands that are less dispersive than the valence bands, and
have a different shape choi2021interaction ; PhysRevB.100.205113 ;
guinea2018electrostatic ; xie2020weak . It is quite likely that the normal
state on the electron-doped side is a sublattice ferromagnet, as is supported
by the observation of the anomalous Hall effect over a wide range of filling
factors near $\nu=3$ in devices with aligned hBN serlin2020intrinsic ;
sharpe2019emergent  and in $\nu=1$ stepanov2020competing  without aligned hBN.
In addition, the valley PPMs that suppress superconductivity are certain to be
stronger on the electron-doped side.

v) Phonon mediated pairing choi2018strong ; wu2018theory ; qin2021critical ;
qin2021critical ; abanin2007quantized ; codecido2019correlated ;
ojajarvi2018competition :– Optical (acoustic) phonon mediated interactions are
outside (inside) the interaction parameter space spanned by $\Gamma^{AA}$ and
$\Gamma^{AB}$. Explicit gap equation solutions wu2018theory ; qin2021critical
show that optical phonons can induce superconductivity if the repulsive
Coulomb interaction is somehow strongly suppressed. When the acoustic phonon
propagator is not screened by electrons, it leads to a $\nu$-independent
attraction $\Gamma^{AA}=A^{s}\sim-0.5$meV qin2021critical  that is too small
to explain the superconductivity dome. As shown in Fig. 2, since acoustic
phonons reduce $A_{s}$ Ceae2107874118 , they can nevertheless play a role in
enhancing $T_{c}$.

The pseudospin parmagnon is able to account for many aspects of the rich
phenomenology of MATBG, including the mismatch between conditions that favor
superconductivity and the anomalous Hall effect, the influence of enhanced
remote screening, and particle-hole asymmetry. Our proposal can be tested by
systematically studying how encapsulating hBN alignment and remote screening
influence superconductivity. It is interesting to contrast superconductivity
in MATBG with the recently discovered zhou2021superconductivity
superconductivity in ABC trilayer graphene, which also has spin, valley, and
sublattice pseudospins and has the advantage of simpler underlying electronic
structure. In both cases, pairing is likely to occur between electrons in
opposite valleys because the valley-projected bands are not time-reversal
invariant. In ABC trilayers superconductivity appears to be enhanced by
proximity to a transition that occurs between unbroken-symmetry and partially
flavor polarized states. The difference between the two cases may lie in the
underlying physics that controls sublattice polarization instabilities.

Acknowledgement:– We acknowledge informative conversations with Youngjoon
Choi, Andrea Young and Haoxin Zhou. This work was supported by the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, under Award
DE-SC0022106.

## References

  * (1) Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, “Unconventional superconductivity in magic-angle graphene superlattices,” Nature, vol. 556, no. 7699, pp. 43–50, 2018.
  * (2) J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, “Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene,” Nature, vol. 590, no. 7845, pp. 249–255, 2021.
  * (3) M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, “Tuning superconductivity in twisted bilayer graphene,” Science, vol. 363, no. 6431, pp. 1059–1064, 2019.
  * (4) X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al., “Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene,” Nature, vol. 574, no. 7780, pp. 653–657, 2019.
  * (5) Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young, “Independent superconductors and correlated insulators in twisted bilayer graphene,” Nature Physics, vol. 16, no. 9, pp. 926–930, 2020.
  * (6) H. Polshyn, M. Yankowitz, S. Chen, Y. Zhang, K. Watanabe, T. Taniguchi, C. R. Dean, and A. F. Young, “Large linear-in-temperature resistivity in twisted bilayer graphene,” Nature Physics, vol. 15, no. 10, pp. 1011–1016, 2019\.
  * (7) Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, “Strange metal in magic-angle graphene with near planckian dissipation,” Physical review letters, vol. 124, no. 7, p. 076801, 2020.
  * (8) F. K. de Vries, E. Portolés, G. Zheng, T. Taniguchi, K. Watanabe, T. Ihn, K. Ensslin, and P. Rickhaus, “Gate-defined josephson junctions in magic-angle twisted bilayer graphene,” Nature Nanotechnology, pp. 1–4, 2021.
  * (9) X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. Li, “Tuning electron correlation in magic-angle twisted bilayer graphene using coulomb screening,” Science, vol. 371, no. 6535, pp. 1261–1265, 2021.
  * (10) 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, vol. 582, no. 7811, pp. 198–202, 2020.
  * (11) P. Stepanov, M. Xie, T. Taniguchi, K. Watanabe, X. Lu, A. H. MacDonald, B. A. Bernevig, and D. K. Efetov, “Competing zero-field chern insulators in superconducting twisted bilayer graphene,” arXiv preprint arXiv:2012.15126, 2020.
  * (12) U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern, et al., “Cascade of phase transitions and dirac revivals in magic-angle graphene,” Nature, vol. 582, no. 7811, pp. 203–208, 2020.
  * (13) Y. Saito, F. Yang, J. Ge, X. Liu, T. Taniguchi, K. Watanabe, J. Li, E. Berg, and A. F. Young, “Isospin pomeranchuk effect in twisted bilayer graphene,” Nature, vol. 592, no. 7853, pp. 220–224, 2021.
  * (14) Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, “Pauli-limit violation and re-entrant superconductivity in moiré graphene,” Nature, vol. 595, no. 7868, pp. 526–531, 2021.
  * (15) Y. Choi, H. Kim, C. Lewandowski, Y. Peng, A. Thomson, R. Polski, Y. Zhang, K. Watanabe, T. Taniguchi, J. Alicea, et al., “Interaction-driven band flattening and correlated phases in twisted bilayer graphene,” arXiv preprint arXiv:2102.02209, 2021.
  * (16) Y. Choi, H. Kim, Y. Peng, A. Thomson, C. Lewandowski, R. Polski, Y. Zhang, H. S. Arora, K. Watanabe, T. Taniguchi, et al., “Correlation-driven topological phases in magic-angle twisted bilayer graphene,” Nature, vol. 589, no. 7843, pp. 536–541, 2021.
  * (17) Supplementary-materials
  * (18) M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, “Intrinsic quantized anomalous hall effect in a moiré heterostructure,” Science, vol. 367, no. 6480, pp. 900–903, 2020.
  * (19) A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. Kastner, and D. Goldhaber-Gordon, “Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene,” Science, vol. 365, no. 6453, pp. 605–608, 2019.
  * (20) K. Levin and O. T. Valls, “Phenomenological theories of liquid 3he,” Physics Reports, vol. 98, no. 1, pp. 1–56, 1983.
  * (21) P. W. Anderson and W. Brinkman, “Anisotropic superfluidity in he 3: A possible interpretation of its stability as a spin-fluctuation effect,” Physical Review Letters, vol. 30, no. 22, p. 1108, 1973.
  * (22) N. Berk and J. Schrieffer, “Effect of ferromagnetic spin correlations on superconductivity,” Physical Review Letters, vol. 17, no. 8, p. 433, 1966\.
  * (23) S. Doniach and S. Engelsberg, “Low-temperature properties of nearly ferromagnetic fermi liquids,” Physical Review Letters, vol. 17, no. 14, p. 750, 1966.
  * (24) P. W. Anderson, “Heavy-electron superconductors, spin fluctuations, and triplet pairing,” Phys. Rev. B, vol. 30, pp. 1549–1550, Aug 1984.
  * (25) T. Moriya, “Developments of the theory of spin fluctuations and spin fluctuation-induced superconductivity,” Proceedings of the Japan Academy, Series B, vol. 82, no. 1, pp. 1–16, 2006.
  * (26) D. J. Scalapino, “A common thread: The pairing interaction for unconventional superconductors,” Rev. Mod. Phys., vol. 84, pp. 1383–1417, Oct 2012.
  * (27) It is also true that attractive intersublattice interactions $\Gamma^{AB}<0$ produce superconductivity when the intrasublattice interaction is repulsive. See SM  for a discussion of this case.
  * (28) P. Anderson and W. Brinkman, “Theory of anisotropic superfluidity in he3,” in Basic Notions of Condensed Matter Physics, pp. 287–388, CRC Press, 2018\.
  * (29) A. H. MacDonald, J. Jung, and F. Zhang, “Pseudospin order in monolayer, bilayer and double-layer graphene,” Physica Scripta, vol. 2012, no. T146, p. 014012, 2012.
  * (30) J. Velasco, L. Jing, W. Bao, Y. Lee, P. Kratz, V. Aji, M. Bockrath, C. Lau, C. Varma, R. Stillwell, et al., “Transport spectroscopy of symmetry-broken insulating states in bilayer graphene,” Nature nanotechnology, vol. 7, no. 3, pp. 156–160, 2012.
  * (31) G. Vignale and K. Singwi, “Effective two-body interaction in coulomb fermi liquids,” Physical Review B, vol. 32, no. 4, p. 2156, 1985.
  * (32) N. Bickers, “Self-consistent many-body theory for condensed matter systems,” in Theoretical Methods for Strongly Correlated Electrons, pp. 237–296, Springer, 2004.
  * (33) M. Nava, A. Motta, D. Galli, E. Vitali, and S. Moroni, “Equation of state of two-dimensional 3 he at zero temperature,” Physical Review B, vol. 85, no. 18, p. 184401, 2012.
  * (34) We use crossing-symmetry bickers2004self  to relate the vertex functions in the particle-particle channel ($\Lambda_{P}$) to the one in particle-hole channel ($\Lambda$).
  * (35) N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath, and M. P. Zaletel, “Ground state and hidden symmetry of magic-angle graphene at even integer filling,” Physical Review X, vol. 10, no. 3, p. 031034, 2020.
  * (36) B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yazdani, and B. A. Bernevig, “Twisted bilayer graphene. iv. exact insulator ground states and phase diagram,” Phys. Rev. B, vol. 103, p. 205414, May 2021.
  * (37) N. Bultinck, S. Chatterjee, and M. P. Zaletel, “Mechanism for anomalous hall ferromagnetism in twisted bilayer graphene,” Physical Review Letters, vol. 124, no. 16, p. 166601, 2020.
  * (38) Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, “Nearly flat chern bands in moiré superlattices,” Physical Review B, vol. 99, no. 7, p. 075127, 2019.
  * (39) J. Y. Lee, E. Khalaf, S. Liu, X. Liu, Z. Hao, P. Kim, and A. Vishwanath, “Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene,” Nature communications, vol. 10, no. 1, pp. 1–10, 2019.
  * (40) When the filling factor $\nu$ approaches $\nu=-3$, the enhanced valley polarizability not only generates a valley-singlet repulsion but also an attraction between electrons in the same valley. When this attraction leads to condensation of Cooper pairs from the same valley, the resulting superconducting order parameter cannot be uniform-s-wave because of Fermi statistics. (We assume the normal state is always spin-polarized). This means the order parameter might undergo a transition from valley-singlet to valley-triplet as $\nu\rightarrow-3$ and might explain the $U-V$ transition recently observed in local-tunneling spectroscopy kim2021spectroscopic. In this scenario, valley-triplet superconductivity close to $\nu=-3$ is eventually suppressed by the enhanced pair-breaking (trigonal-warping) potential from the band-Hamiltonian.
  * (41) 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,” Nature, vol. 583, no. 7816, pp. 375–378, 2020.
  * (42) H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi, H. Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu, et al., “Superconductivity in metallic twisted bilayer graphene stabilized by wse 2,” Nature, vol. 583, no. 7816, pp. 379–384, 2020.
  * (43) T. Cea, N. R. Walet, and F. Guinea, “Electronic band structure and pinning of fermi energy to van hove singularities in twisted bilayer graphene: A self-consistent approach,” Phys. Rev. B, vol. 100, p. 205113, Nov 2019\.
  * (44) F. Guinea and N. R. Walet, “Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers,” Proceedings of the National Academy of Sciences, vol. 115, no. 52, pp. 13174–13179, 2018.
  * (45) M. Xie and A. H. MacDonald, “Weak-field hall resistivity and spin/valley flavor symmetry breaking in matbg,” arXiv preprint arXiv:2010.07928, 2020\.
  * (46) Y. W. Choi and H. J. Choi, “Strong electron-phonon coupling, electron-hole asymmetry, and nonadiabaticity in magic-angle twisted bilayer graphene,” Physical Review B, vol. 98, no. 24, p. 241412, 2018.
  * (47) F. Wu, A. MacDonald, and I. Martin, “Theory of phonon-mediated superconductivity in twisted bilayer graphene,” Physical review letters, vol. 121, no. 25, p. 257001, 2018.
  * (48) W. Qin, B. Zou, and A. H. MacDonald, “Critical magnetic fields and electron-pairing in magic-angle twisted bilayer graphene,” arXiv preprint arXiv:2102.10504, 2021.
  * (49) D. Abanin and L. Levitov, “Quantized transport in graphene pn junctions in a magnetic field,” Science, vol. 317, no. 5838, pp. 641–643, 2007.
  * (50) E. Codecido, Q. Wang, R. Koester, S. Che, H. Tian, R. Lv, S. Tran, K. Watanabe, T. Taniguchi, F. Zhang, et al., “Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle,” Science advances, vol. 5, no. 9, p. eaaw9770, 2019.
  * (51) R. Ojajärvi, T. Hyart, M. A. Silaev, and T. T. Heikkilä, “Competition of electron-phonon mediated superconductivity and stoner magnetism on a flat band,” Physical Review B, vol. 98, no. 5, p. 054515, 2018.
  * (52) T. Cea and F. Guinea, “Coulomb interaction, phonons, and superconductivity in twisted bilayer graphene,” Proceedings of the National Academy of Sciences, vol. 118, no. 32, 2021.
  * (53) H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. Young, “Superconductivity in rhombohedral trilayer graphene,” arXiv preprint arXiv:2106.07640, 2021.
  * (54) W. Qin and A. H. MacDonald, “In-plane critical magnetic fields in magic-angle twisted trilayer graphene,” 2021.
  * (55) D. Scalapino, “Superconductivity and spin fluctuations,” Journal of Low Temperature Physics, vol. 117, no. 3, pp. 179–188, 1999.

Supplementary Material: Pseudospin Paramagnons and the Superconducting Dome in
Magic Angle Twisted Bilayer Graphene

##  I Non-interacting susceptibilitites in various pseudsopsin channels

In the main text, we explain the superconducting dome in magic angle twisted
bilayer graphene (MATBG) by considering a sublattice-dependent interaction
potential that depends sensitively on the moiré band filling factor $\nu$. We
focus on $-2<\nu<-3$. We show that MATBG is at the boarder of sublattice
polarization over a wide range of $\nu$ and this generates an attractive
intra-sublattice interaction between spin-polarized charge carriers in
opposite valley and leads to superconductivity. This superconducting channel
is suppressed as $\nu\rightarrow-3$ because the valley polarizibility
increases and as $\nu\rightarrow-2$ because the density of states at the Fermi
level diminish.

In this section, we use a simple bandstructure model to calculate the non-
interacting susceptibilities in various pseudospin channels to support our
explanation to the superconducting dome. The bandstructure is obtained from
the Bistritzer-MacDonald continuum model at twist angle $\theta=1.15^{\circ}$
parametrized by interlayer tunneling parameters $t_{AB}=110$meV and
$t_{AA}=77$meV for AB and AA bilayer stacking, respectively. We also include a
nonlocal momentum-dependent correction to the interlayer tunneling,
$dt_{AA}/dk=dt_{AB}/dk=-0.1$ qin2021inplane . Because there is no consensus on
the Fermiology in MATBG (e.g. the location of the Fermi-surface after reset
transition at $\nu=-2$), we will only make conclusions that are not sensitive
to the details of the bandstructure.

As discussed in the main text, the normal state is spin-polarized so we
consider only particle-hole excitations in the majority-spin projected Hilbert
space. The pseudospin susceptibility of a particle-hole channel $\mathcal{O}$
is defined as follows:

$\displaystyle\chi_{\boldsymbol{q},\omega}^{\mathcal{O},\mathcal{O}^{\dagger}}=-i\int_{0}^{\infty}dt\
e^{i\omega
t}\left\langle\frac{1}{A}\left[\sum_{\boldsymbol{p},\alpha\beta}c_{\boldsymbol{p},\alpha}^{\dagger}\mathcal{O}_{\alpha\beta}c_{\boldsymbol{p}+\boldsymbol{q},\beta}\
,\sum_{\boldsymbol{p}^{\prime},\alpha^{\prime}\beta^{\prime}}c_{\boldsymbol{p}^{\prime}+\boldsymbol{q},\alpha^{\prime}}^{\dagger}\mathcal{O}_{\alpha^{\prime}\beta^{\prime}}^{\dagger}c_{\boldsymbol{p}^{\prime},\beta^{\prime}}\right]\right\rangle$
(S1) $\displaystyle\qquad\ \
=\frac{1}{A}\sum_{\boldsymbol{k}\in\text{B.Z.},\tau
m,\tau^{\prime}n}\frac{n_{F}(\epsilon_{\tau
m\boldsymbol{k}+\boldsymbol{q}})-n_{F}(\epsilon_{\tau^{\prime}n\boldsymbol{k}})}{\omega+\epsilon_{\tau^{\prime}n\boldsymbol{k}}-\epsilon_{\tau
m\boldsymbol{k}+\boldsymbol{q}}+i\delta}|\mathcal{M}_{\tau^{\prime}n\boldsymbol{k},\tau
m\boldsymbol{k}+\boldsymbol{q}}|^{2}$ (S2)
$\displaystyle\mathcal{M}_{\tau^{\prime}n\boldsymbol{k},\tau
m\boldsymbol{k}+\boldsymbol{q}}=\sum_{\boldsymbol{G},\alpha\beta}\psi_{n\alpha}^{*}(\boldsymbol{k}+\boldsymbol{G})\mathcal{O}_{\alpha\beta}\psi_{m\beta}(\boldsymbol{k}+\boldsymbol{q}+\boldsymbol{G}),\qquad\alpha=\tau^{\prime}\sigma^{\prime}l^{\prime}(\beta=\tau\sigma
l)$ (S3)

Here $\alpha,\beta$ labels sublattice $\sigma$, valley $\tau$ and layer $l$
degree of freedoms. In Eq. (S2), the basis is transformed from the plane-wave
basis to the single-particle band basis. $|\tau
n\boldsymbol{k}\rangle=\sum_{\boldsymbol{G}\sigma
l}\psi_{n\tau\sigma\l}(\boldsymbol{k}+\boldsymbol{G})|\tau\sigma
l,\boldsymbol{k+\boldsymbol{G}}\rangle$ is a state in band $n$ with energy
$\epsilon_{n\boldsymbol{k}}$, where $\boldsymbol{k}$ is restricted to the
first Brillouin zone (chosen to be centered around the Moiré $m$ point) and
the Bloch wave functions in two valleys are related by $\psi_{n+\sigma
l}(\boldsymbol{k}+\boldsymbol{G})=\psi_{n-\sigma
l}^{*}(-\boldsymbol{k}-\boldsymbol{G})$. $n_{F}$ is the Fermi-Dirac
distribution and we set temperature $T=0.44$K. We find that most
susceptibilities except for$\chi^{K\sigma_{0},K\sigma_{0}}$ and
$\chi^{+\sigma_{y},-\sigma_{y}}$ receive a large contribution from the
interband transitions and gradually increases with the ultraviolet energy
cutoff macdonald2012pseudospin . We truncate to 508 bands per valley
(corresponding to energy cutoff $\sim 2.5eV$) and the results are summarized
in Fig.S1.

Figure S1:  The static uniform susceptibilities of twisted bilayer graphene in
a) pseudospin density channels b) transverse valley channels. $\tau,\sigma,l$
represent valley, sublattice and layer degrees of freedom, respectively.

Fig. S1a) illustrates the competition between sublattice even
($\chi_{\boldsymbol{q}=0,\omega=0}^{K\sigma_{0},K\sigma_{0}}$) and odd
($\chi_{\boldsymbol{q}=0,\omega=0}^{K\sigma_{z},K\sigma_{z}}$)
susceptibilities as a function $\nu$. The former (at zero temperature) is
equal to the Fermi surface density of states in a single valley $N_{F}$ and
therefore, it depends sensitively on $\nu$. It vanishes when the Fermi level
approaches the Dirac points (which is at $\nu=-2$ in our bandstructure model)
and increases rapidly as the hole densities increases. In contrast,
$\chi_{\boldsymbol{q}=0,\omega=0}^{K\sigma_{z},K\sigma_{z}}$ shows very weak
$\nu$ dependence because the $q=0$ susceptibility does not have any intra-band
contribution when the band has a $\mathcal{C}_{2}T$ symmetry where
$\mathcal{C}_{2}$ and $T$ are inversion and spinless time-reversal symmetry.
As a result, it must be larger than $N_{F}$ as $\nu\rightarrow-2$. This shows
that the sublattice polarizability enhanced by a factor of
$[1-(u_{0}+u_{\perp})\chi_{\boldsymbol{q}=0,\omega=0}^{K\sigma_{z},K\sigma_{z}}]^{-1}$
is typically stronger than the enhanced valley polarizability
$N_{F}/[1-(u_{0}-u_{\perp})N_{F}]$ except for filling factor close to
$\nu=-3$. Recall $u_{\perp}>0$.

Fig. S1b) shows the susceptibility associated with neutral excitations where
the particle and hole are drawn from opposite valley. There are four layer-
diagonal, sublattice dependent susceptibility denoted by
$\chi^{{\tau_{+}}\sigma_{i},\tau_{-}\sigma_{i}},i=x,y,z,0$. The mixing between
different channels are not considered, a justified approximation in the limit
$\boldsymbol{q}=\omega=0$ due to the $D_{6}$ and spinless time reversal
symmetry of twisted bilayer graphene. As an example, we demonstrate that the
sublattice even and odd intervalley susceptibilities decouple: The $C_{3}$
symmetry acts as exp$(-i\frac{2\pi}{3}\tau_{z}\sigma_{z})$ in the valley-
sublattice pseudospin space. It commutes with $\tau_{\pm}\sigma_{x,y}$ but
rotate $\tau_{\pm}\sigma_{0,z}$ by a phase factor exp$(\pm i\frac{2\pi}{3})$.
Therefore,
$\chi_{\boldsymbol{q},\omega}^{\tau_{+}\sigma_{0,z},\tau_{-}\sigma_{x,y}}=$
exp$(\pm
i\frac{2\pi}{3})\chi_{R_{2\pi/3}\boldsymbol{q},\omega}^{\tau_{+}\sigma_{0,z},\tau_{-}\sigma_{x,y}}$
so it must vanish at $\boldsymbol{q}=0$. Since intervalley sublattice even
channels do not mix with sublattice odd ones and Fig. S1b) shows that the
former have small susceptibilities, they can be neglected in the effective
interaction in the main text. When the particle and hole drawn from opposite
valleys are also from opposite layers, the susceptibility is significantly
larger than those that are layer-diagonal as shown in Fig. S1b). The origin of
the enhanced susceptibility in this channel is due to a (small-twist-angle)
emergent $C_{2}P=\tau_{x}\sigma_{x}l_{y}$ symmetry in twisted bilayer graphene
discovered in Ref. lian2021iv . Because of $C_{2}P$, $\langle
K,-n,\boldsymbol{k}|\tau_{+}\sigma_{x}l_{y}|K^{\prime},n,\boldsymbol{k}\rangle\approx
1$ where $P$ is a unitary particle-hole symmetry lian2021iv  and $\pm n$ bands
are $C_{2}P$ partners. The large matrix elements between the $C_{2}P$ partners
result in a large $\chi^{+\sigma_{x}l_{y},-\sigma_{x}l_{y}}$ according to Eq.
S3. Note that this susceptibility dominates other pseudospin particle-hole
channels when the Fermi level approached the Dirac points and might lead to an
intervalley exciton condensation. The resulting valley XY-like order parameter
$\tau^{+}\sigma_{x}l_{y}$ preserves a spinless time reversal symmetry
$\mathcal{T}=\tau_{y}\mathcal{K}$($\mathcal{T}^{2}=-1$), and hence is
consistent with the previously proposed Kramers-intervalley coherent order at
$\nu=-2$ bultinck2020ground ; lian2021iv . Note the tendency towards valley-XY
order generates a strong valley-singlet repulsion via the Berk-Schrieffer
diagram, as discussed in the main text. However, because this repulsion is
also off-diagonal in sublattice (like $\Gamma^{AB}$) and the $C_{2}$ and
$C_{3}$ symmetry decouple interaction according to their sublattice parity in
the linearized gap equation (as shown in the next section), superconductivity
prevails as long as intra-sublattice is attractive ($\Gamma^{AA}<0$). In the
main text and in what follows, we neglect valley-singlet repulsion generated
by all the valley-XY fluctuations since the channels that modify intra-
sublattice attraction is very small. In this case, suppression of valley-
singlet superconductivity close to $\nu=-2$ will only require a mundane
explanation – a reduced density of states.

We would also like to point out that in the definition of the susceptibility
in Eq. (S2) the particle-hole momentum $\boldsymbol{q}$ is, rigorously
speaking, conserved only up to moiré reciprocal vectors $\boldsymbol{G}$ and
therefore the bare susceptibility $\chi_{\boldsymbol{q},\omega}$ used in the
generalized RPA formula in the main text should be understood as a matrix in
the momentum space,
$\hat{\chi}_{\boldsymbol{q},\omega}(\boldsymbol{G},\boldsymbol{G}^{\prime})$.
However, for the purpose of comparing the relative strength of
susceptibilities in various channels, it is adequate to focus on the small
$\boldsymbol{q}$ and $\boldsymbol{G}=\boldsymbol{G}^{\prime}=0$.

##  II Linearized Gap equation

In this section, we study the linearized gap equation within the reduced
interaction model described by Eqs. (1) and (2) in the main text. The real-
space interaction model is given by the Hamiltonian

$\displaystyle H_{\text{red}}$ $\displaystyle=\int
d^{2}\boldsymbol{r}\sum_{l,l^{\prime}\tau\sigma\sigma^{\prime}}\Gamma^{\sigma\sigma^{\prime}}\delta_{l,l^{\prime}}\delta_{\tau,-\tau^{\prime}}\,\psi^{\dagger}_{l\tau\sigma}(\boldsymbol{r})\psi^{\dagger}_{l^{\prime}\tau^{\prime}\sigma^{\prime}}(\boldsymbol{r})\psi_{l^{\prime}\tau^{\prime}\sigma^{\prime}}(\boldsymbol{r})\psi_{l\tau\sigma}(\boldsymbol{r}),$
(S4)

where $l,l^{\prime}$ label layer, $\tau,\tau^{\prime}$ label valley and
$\sigma,\sigma^{\prime}$ denote sublattice. The interaction matrix
$\Gamma^{\sigma\sigma^{\prime}}=\Gamma^{AA}\delta_{\sigma,\sigma^{\prime}}+\Gamma^{AB}\delta_{\sigma,-\sigma^{\prime}}$.
This leads to the following linearized gap equation

$\Delta^{l}_{\sigma\sigma^{\prime}}(\boldsymbol{G})=\Gamma^{\sigma\sigma^{\prime}}\sum_{l^{\prime}\sigma_{1}\sigma_{1}^{\prime}\boldsymbol{G}^{\prime}}\Pi_{l\sigma\sigma^{\prime},l^{\prime}\sigma_{1}\sigma_{1}^{\prime}}(\boldsymbol{G},\boldsymbol{G}^{\prime})\Delta^{l^{\prime}}_{\sigma_{1}\sigma_{1}^{\prime}}(\boldsymbol{G}^{\prime}),$
(S5)

where $\boldsymbol{G}$ and $\boldsymbol{G}^{\prime}$ are moiré reciprocal
lattice vectors. The layer and sublattice-dependent pair susceptibility

$\Pi_{l\sigma\sigma^{\prime},l^{\prime}\sigma_{1}\sigma_{1}^{\prime}}(\boldsymbol{G},\boldsymbol{G}^{\prime})=\sum_{mm^{\prime}\boldsymbol{k}^{\prime}}\frac{n_{F}(\epsilon_{m\boldsymbol{k}^{\prime}})-n_{F}(-\epsilon_{m^{\prime}\boldsymbol{k}^{\prime}})}{\epsilon_{m\boldsymbol{k}^{\prime}}+\epsilon_{m^{\prime}\boldsymbol{k}^{\prime}}}\langle
m^{\prime}\boldsymbol{k}^{\prime}|e^{i\boldsymbol{G}\cdot\boldsymbol{r}}|m\boldsymbol{k}^{\prime}\rangle_{l\sigma^{\prime}\sigma}\langle
m^{\prime}\boldsymbol{k}^{\prime}|e^{i\boldsymbol{G}^{\prime}\cdot\boldsymbol{r}}|m\boldsymbol{k}^{\prime}\rangle_{l^{\prime}\sigma_{1}^{\prime}\sigma_{1}}^{*},$
(S6)

where $m$ and $m^{\prime}$ are band indices. For the above contact-interaction
model, the interaction matrix is independent of moiré momentum
$\boldsymbol{k}$, suggesting the superconducting pairing potential exhibits
moiré periodicity in the real space. The critical temperature is obtained by
equating the largest eigenvalue of matrix $-\Gamma\cdot\Pi$ to 1. We
numerically diagonalized this matrix as a function of the input
$(\Gamma^{AA},\Gamma^{AB})$ and this leads to the result shown in Fig. 1 of
the main text. The results we obtained are not qualititively modified by the
factor $\delta_{l,l^{\prime}}$ we used to simplify Eq. (S4).

For MATBG, the largest contribution to the pair susceptibility come from the
flat conduction and valence bands and the Bloch wavefunctions from the these
bands are localized at small moiré $G$-vectors (i.e. they vary slowly in the
moiré unit cell). Therefore, although the eigenvector of the Eq. 5 has a very
large dimension, most of the properties of the superconducting order parameter
can be understood in the subspace of
$\boldsymbol{G}=\boldsymbol{G}^{\prime}=0$. In this subspace, the pairing
amplitude of a spin-polarized valley-singlet superconductor can be
conveniently described by the following $2\times 2$ matrix,

$F_{\sigma\sigma^{\prime}}(\boldsymbol{k})=\langle
c_{K\sigma\uparrow}(\boldsymbol{k})c_{K^{\prime}\sigma^{\prime}\uparrow}(-\boldsymbol{k})\rangle,$
(S7)

where we assume all electrons are from the same layer. The linearized gap
equation takes a much simpler form,

$\Delta_{\sigma\sigma^{\prime}}(\boldsymbol{k})=\sum_{\boldsymbol{k}^{\prime}\sigma_{1}\sigma_{1}^{\prime}}\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}^{\sigma\sigma^{\prime}}\Pi_{\sigma\sigma^{\prime},\sigma_{1}\sigma_{1}^{\prime}}(\boldsymbol{k}^{\prime})\Delta_{\sigma_{1}\sigma_{1}^{\prime}}(\boldsymbol{k}^{\prime}).$
(S8)

We reinserted the momentum $\boldsymbol{k},\boldsymbol{k}^{\prime}$ dependence
above for the clarity of analysis below. This sublattice dependent gap
equation for MATBG in the $\boldsymbol{G}=\boldsymbol{G}^{\prime}=0$ subspace
is reminiscent to the spin-dependent gap equation in 3He. However, unlike 3He,
the sublattice pseudospin conservation is broken in MATBG by the non-
interaction single-particle Hamiltonian. Because the order parameter matrix
must be anti-symmetric under transposition, i.e.
$\Delta(\boldsymbol{p})=-\Delta^{\text{T}}(-\boldsymbol{p})$, where
$\boldsymbol{p}=\boldsymbol{K}+\boldsymbol{k}$, and the parity operator
$\mathcal{C}_{2}$ flips sublattice polarization $\sigma\rightarrow-\sigma$ and
momentum $\boldsymbol{p}\rightarrow-\boldsymbol{p}$, we have
$\mathcal{C}_{2}\Delta_{\sigma,\sigma^{\prime}}(\boldsymbol{p})\rightarrow\Delta_{-\sigma,-\sigma^{\prime}}(-\boldsymbol{p})=-\Delta_{-\sigma^{\prime},-\sigma}(\boldsymbol{p})$.
Here we note that, unlike spin in 3He, the pseudospin is flipped by the parity
operator and this means the even-parity is a singlet while the odd-parity is a
triplet,

$\displaystyle\Delta(\boldsymbol{p})$
$\displaystyle=g_{0}(\boldsymbol{p})\sigma^{z}+(\boldsymbol{d}(\boldsymbol{p})\cdot\boldsymbol{\sigma})\sigma^{z}=\begin{pmatrix}g_{0}+d^{z}&d^{+}\\\
d^{-}&-g_{0}+d^{z}\end{pmatrix},$ (S9)

where $g_{0}(-\boldsymbol{p})=g_{0}(\boldsymbol{p})$ and
$\boldsymbol{d}(-\boldsymbol{p})=-\boldsymbol{d}(\boldsymbol{p})$. The normal
state $\mathcal{C}_{2}$ symmetry then decouples the (sublattice-pseudospin)
singlet-triplet channel of the gap equation:

$g_{0}(\boldsymbol{k})=\sum_{\boldsymbol{k}^{\prime}}\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}^{AA}\Pi^{zz}(\boldsymbol{k}^{\prime})g_{0}(\boldsymbol{k}^{\prime}),$
(S10) $\begin{pmatrix}d^{z}\\\ d^{+}\\\
d^{-}\end{pmatrix}_{\boldsymbol{k}}=\sum_{\boldsymbol{k}^{\prime}}\hat{\Gamma}_{\boldsymbol{k}\boldsymbol{k}^{\prime}}\begin{pmatrix}\Pi^{00}&\Pi^{0-}&\Pi^{0-}\\\
\Pi^{0-}&\Pi^{++}&\Pi^{+-}\\\
\Pi^{0-}&\Pi^{+-}&\Pi^{++}\end{pmatrix}_{\boldsymbol{k}^{\prime}}\begin{pmatrix}d^{z}\\\
d^{+}\\\ d^{-}\end{pmatrix}_{\boldsymbol{k}^{\prime}}$ (S11)

where
$\hat{\Gamma}_{\boldsymbol{k}\boldsymbol{k}^{\prime}}=\text{diag}\\{\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}^{AA},\
2\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}^{AB},\
2\Gamma_{\boldsymbol{k}\boldsymbol{k}^{\prime}}^{AB}\\}$, and

$\Pi^{ij}(\boldsymbol{k})=\frac{1}{2}\sum_{\sigma\sigma^{\prime},\sigma_{1}\sigma_{1}^{\prime}}(\sigma^{i})_{\sigma\sigma^{\prime}}(\sigma^{j})_{\sigma_{1}\sigma_{1}^{\prime}}\Pi_{\sigma\sigma^{\prime},\sigma_{1}\sigma_{1}^{\prime}}(\boldsymbol{k}),\
\ i,j=0,z,\pm.$ (S12)

Since superconductivity is an instability of the Fermi surface, it is unlikely
to happen in the even-parity channel (i.e. $g_{0}=0$) due to the following
reasons. First, the intra-band contribution to even-parity pairing
susceptibility $\Pi^{zz}$ vanishes since the sublattice polarization is not
established in the normal state, indicating $\langle
m\boldsymbol{k}|\sigma^{z}|m\boldsymbol{k}\rangle=0$ in Eq. (S6). Secondly,
the inter-band contribution to $\Pi^{zz}$ is negligible because the critical
temperature $T_{c}\ll$ band separation. When $g_{0}=0$, $\boldsymbol{d}$ alone
cannot give rise to finite sublattice polarization, so
$\langle\sigma_{z}\rangle=0$ in the superconducting state and it remains
nearly sublattice polarized for $T<T_{c}$.

In the odd-parity triplet subspace, when $\Gamma^{AA}$ and $\Gamma^{AB}$ are
momentum independent and $\Gamma^{AB}\gg-\Gamma^{AA}>0$, the gap equation
admits a simple self-consistent solution with $d^{\pm}(\boldsymbol{k})=0$ and
$d^{z}(\boldsymbol{k})=d^{z}$ (i.e. independent of $k$). This paramaterization
simplifies the $3\times 3$ matrix equation (Eq. (S11)) to the following
equation:

$\Gamma^{AA}\sum_{\boldsymbol{k}}\Pi^{00}(\boldsymbol{k},T_{c})=-1\;\;,\;\;\Gamma^{AA}<0.$
(S13)

This simplification occurs because
$\sum_{\boldsymbol{k}}\Pi^{0-}(\boldsymbol{k})=0$ which follows from
$\mathcal{C}_{3}$ rotation symmetry:
$\Pi^{0-}(\mathcal{C}_{3}\boldsymbol{k})=e^{-4\pi
i/3}\Pi^{0-}(\boldsymbol{k})$. Clearly, $T_{c}$ obtained from this simplified
gap equation is independent of $\Gamma^{AB}$. This means when the two members
of the Cooper pairs are from the same sublattice, they can avoid the strong
inter-sublattice repulsion $\Gamma^{AB}$ if the pairing amplitude is uniform
on the Fermi surface.

Figure S2:  Color plot of critical temperature $T_{c}$ as function of
$\Gamma_{AA}$ and $\Gamma_{AB}\leq 0$, where $A_{m}$ denotes the area of moiré
supercell. The dashed curve highlights the phase boundary between the intra-
and inter-sublattice pairing states. These numerical results are obtained by
using a continuum model for MATBG with $t_{AA}/t_{AB}=0.7$ and filling factor
$\nu=-2.4$.

While we do not have compelling arguments to think that MATBG will be located
in the parameter space where $\Gamma^{AB}$ is attractive, it nevertheless lead
to superconductivity, as shown in Fig. S2. Let us discuss the structure of the
superconducting order parameter in this parameter space without dwelling with
the microscopic origin of an attractive $\Gamma^{AB}$. In the parameter space
where $\Gamma^{AA}\gg-\Gamma^{AB}>0$, we can write down two degenerate self-
consistent solutions
$(d^{z}(\boldsymbol{k}),d^{+}(\boldsymbol{k}),d^{-}(\boldsymbol{k}))=(0,d^{+},0)$
and
$(d^{z}(\boldsymbol{k}),d^{+}(\boldsymbol{k}),d^{-}(\boldsymbol{k}))=(0,0,d^{-})$
to reduce the $3\times 3$ matrix equation to a single gap equation which is
independent of $\Gamma^{AA}$:

$\Gamma^{AB}\sum_{\boldsymbol{k}}\Pi^{++}(\boldsymbol{k},T_{c})=\Gamma^{AB}\sum_{\boldsymbol{k}}\Pi^{--}(\boldsymbol{k},T_{c})=-1\;\;,\;\;\Gamma^{AB}<0\;\;.$
(S14)

This is again due to the property
$\sum_{\boldsymbol{k}}\Pi^{+-}(\boldsymbol{k})=0$ which follows from
$\mathcal{C}_{3}$: $\Pi^{+-}(\mathcal{C}_{3}\boldsymbol{k})=e^{4\pi
i/3}\Pi^{+-}(\boldsymbol{k})$. The phase boundary marked in Fig. 2 denotes the
change of superconducting order parameter from intrasublattice-like
$\boldsymbol{d}(\boldsymbol{k})=d^{z}$ to intersublattice-like
$\boldsymbol{d}(\boldsymbol{k})=d^{+}\;\text{or}\;d^{-}$. In this simple
analysis, we can always reduce the $3\times 3$ linearized gap equation for the
$\mathcal{C}_{2}$ triplet to a scalar equation which only involves either
$\Gamma^{AA}$ or $\Gamma^{AB}$. Hence, intra-sublattice Cooper pairs do not
‘see’ $\Gamma^{AB}$ and the inter-sublattice Cooper pairs do not ‘see’
$\Gamma^{AA}$. When we account for non-local correction from larger moiré
reciprocal vector, such perfect decoupling of $\Gamma^{AA}$ and $\Gamma^{AB}$
is no longer exact and $T_{c}$ can indeed be suppressed by the repulsion. For
reasons that are unclear to us, such suppression occurs in a very asymmetrical
way. The suppression of intra-sublattice pairing ($\Gamma^{AA}<0$) via
repulsive $\Gamma^{AB}$ only becomes significant when the repulsion exceeds
the attraction by a factor of $10$, $\Gamma^{AB}\sim-10\Gamma^{AA}>0$. On the
other hand, the suppression of inter-sublattice pairing ($\Gamma_{AB}<0$) from
repulsive $\Gamma^{AA}$ is very effective – when the repulsion $\Gamma^{AA}$
is roughly the same as the attraction $\Gamma^{AB}$, the $T_{c}$ is already
negligibly small.

##  III Valley-triplet, spin-singlet Pairing and Berk-Schrieffer diagrams

In the maintext we assume the two members of a Cooper pair are from opposite
valleys because they are related by a (spinless) time-reversal symmetry. When
they are drawn from the same valley, the band-energy is usually pair breaking
but it is nevertheless interesting to carry out the paramagnon analysis in
this valley-triplet channel. When the superconducting order parameter is
uniform, valley-triplet has to be a spin-singlet so the four external
particle-lines of the particle-particle irreducible interaction $\Gamma$ shown
in Fig. 3a of the main text are given by
$1,2,3,4=K\uparrow,K\downarrow,K\downarrow,K\uparrow$. Since pairing occurs in
the valley-projected Hilbert space, the diagrammatics is similar to the
original Berk-Schrieffer analysis berk1966effect  with an extra sublattice
degree of freedom. In particular, the pseudospin-density fluctuation
contribution to $A^{a}$ is

$\displaystyle A^{a}=$
$\displaystyle\frac{1}{2}\left(T^{\sigma_{z}s_{0}}-T^{\sigma_{z}s_{z}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{-u_{0}}{1-u_{0}\chi^{\sigma_{z}s_{0},\sigma_{z}s_{0}}}+\frac{u_{0}}{1-u_{0}\chi^{\sigma_{z}s_{z},\sigma_{z}s_{z}}}\right)=0.$
(S15)

The last equation is due to
$\chi^{\sigma_{z}s_{z},\sigma_{z}s_{z}}=\chi^{\sigma_{z}s_{0},\sigma_{z}s_{0}}$
and it happens because the valley-projected space is fully spin-polarized. As
a result, sublattice fluctuations do not generate attractive interaction in
the spin-singlet valley-triplet channel.

Let us now elaborate on the discuss of the Berk-Schrieffer (BS) transverse
fluctuation contributions indicated in Fig. 3 of the main text. The particle-
hole channels in the BS diagrams decouple into four channels
$\tau_{+}\sigma_{0,x,y,z}$ in the long wavelength limit and they change

$A^{s,a}\rightarrow
A^{s,a}\mp\frac{1}{2}\sum_{i=x,y}T^{\tau_{+}\sigma_{i}}-\frac{1}{2}\sum_{\delta=0,z}T^{\tau_{+}\sigma_{\delta}},$
(S16)

where the intervalley intrasublattice interaction
$T^{\tau_{+}\sigma_{\delta}}$ is suppressed by small matrix-elements at the
vertex because they mix states from opposite Chern band while the inter-valley
component is given by the following:

$\displaystyle T^{\tau_{+}\sigma_{i}}$
$\displaystyle=-\frac{u_{0}^{2}\chi_{\boldsymbol{k}+\boldsymbol{k}^{\prime},\omega}^{\tau_{+}\sigma_{i},\tau_{-}\sigma_{i}}}{1-u_{0}\chi_{\boldsymbol{k}+\boldsymbol{k}^{\prime},\omega}^{\tau_{+}\sigma_{i},\tau_{-}\sigma_{i}}}\;,\;i=x,y.$
(S17)

Here the susceptibility $\chi$ is shown in Fig. 1b. While this leads to a
repulsive $\Gamma^{AB}$, it does not modify $\Gamma^{AA}$ hence
superconductivity prevails as shown in the maintext (Fig. 1). If the
interaction is mainly enhanced at zero paramagnon momentum $q=0$, then valley-
Ising paramagnon ($T^{\tau^{z}\sigma_{0}}$) with
$\boldsymbol{q}=\boldsymbol{k}-\boldsymbol{k}^{\prime}$ scatters Cooper pairs
locally around the Fermi surface while the valley-XY paramagnon
($T^{\tau_{+}\sigma_{x,y}}$) with
$\boldsymbol{q}=\boldsymbol{k}+\boldsymbol{k}^{\prime}$ scatters Cooper pairs
across the Fermi surface, the latter is similar to the short-range repulsive
interaction generated by antiferromagnetic paramagnon in cuprates
scalapino1999superconductivity . We summarize important information about
different pseudospin paramagnons in Table. 1.

Table 1: A summary of important properties for the charge fluctuation and 7 pseudospin paramagnons. $T$ | $\tau_{0}\sigma_{0}$ | $\tau_{0}\sigma_{z}$ | $\tau_{z}\sigma_{z}$ | $\tau_{z}\sigma_{0}$ | $\tau_{+}\sigma_{x,y}$ | $\tau_{+}\sigma_{0,z}$
---|---|---|---|---|---|---
Types of fluctuation | Charge and Pseudospin Density | Berk-Schrieffer
Dominant scattering vertex | Direct | Exchange
Exchange enchancement | – | $-4<\nu<4$ | $|\nu|-2\rightarrow 1$ | $|\nu|-2\rightarrow 0$ | –
Types of Order | – | Sublattice-Polarized | Orbital-Moment | Valley-Ising | Valley-XY (IVC) | –