††thanks: These authors contributed equally to the work.††thanks: These
authors contributed equally to the work.

# Pair-density-wave and chiral superconductivity in twisted bilayer
transition-metal-dichalcogenides

Yi-Ming Wu  Institute for Advanced Study, Tsinghua University, Beijing, China
Zhengzhi Wu  Institute for Advanced Study, Tsinghua University, Beijing, China
Hong Yao<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua
University, Beijing, China  State Key Laboratory of Low Dimensional Quantum
Physics, Tsinghua University, Beijing 100084, China

###### Abstract

We theoretically explore possible unconventional superconductivity induced by
weak repulsive interactions in twisted bilayer TMD (e.g. WSe2) in the presence
of an out-of-plane electric field. Using renormalization group (RG) analysis,
we show that topological chiral superconducting states with Chern number
$\mathcal{N}=1,2,4$ (namely $p+ip$, $d+id$, $g+ig$) appear over a large
parameter region with moiré filling factor around $n=1$. At some special
values of applied electric field and in the presence of a weak out-of-plane
Zeeman field, spin-polarized pair density wave (PDW) superconductivity can
emerge. This spin-polarized PDW state can be probed by experiments such as
spin-polarized STM measuring spin-resolved pairing gap and quasi-particle
interference. Moreover, the spin-polarized PDW could lead to spin-polarized
superconducting diode effect.

Introduction: The advent of engineering moiré structures from stacking 2D
materials starts an exciting path for studying intriguing electronic
properties in quantum materials. Through interlayer van der Waals coupling,
spatial moiré potential profoundly modifies electronic band structures, and in
certain circumstances results in low energy isolated narrow or even flat bands
(moiré bands) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], where interactions play
a vital role in low energy physics. From experimental aspect, various phases
have been found in magic-angle twisted bilayer graphene (TBG), including
correlated insulators [13, 14, 15, 16, 17], superconductors [13, 18, 19, 20],
strange metal[21, 22, 23], magnetic phases [24, 25, 26, 27] and quantum
anomalous Hall states [28, 29]. Soon after this, similar phases are also found
in other forms of moiré structures including twisted double bilayer and
trilayer graphene systems [30, 31, 32, 33, 34, 35, 36, 37, 38].

Figure 1: Various types of superconductivity from twisted bilayer WSe2. At
$\phi=\pi/3$ there is an emergent spin-polarized PDW with $\bm{Q}=\pm\bm{K}$
at small and large $n$. Centered around $n=1$ there exist different chiral SC
states which we identify as $p+ip$, $d+id$ and $g+ig$ based on their Chern
numbers. At large or small $n$, a six node or nodeless SC state dominates,
which we denote by $f$\- or $f^{\prime}$-wave according to their
representation [$A_{1}$ ($B_{1u}$) or $A_{2}$ ($B_{2u}$) of $C_{3v}$
($D_{6h}$) when $\phi>0$ ($\phi=0$)]. At $\phi\to 0(\pi/3)$, there are tiny
regimes of 12 node SC states ($i$\- and $i^{\prime}$-wave differ in their
representations). The white curve shows the locations of the VHSs.

Twisted bilayer transition metal dichalcogenides (TMD) were recently shown to
be another promising and sometimes advantageous platform for simulating
various correlated and topological states [39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49]. Monolayer TMD is a semiconductor with strong spin-orbit coupling [50,
51, 52, 53, 54, 55]. The spin splitting in its valence band is much larger
than that in the conduction band so that the topmost valence band can be used
as a model for effectively spinless or spin-polarized fermions on Fermi
surface [56, 57]. Moreover, the finite energy gap in monolayer TMD enables the
moiré band width to vary continuously with the twisted angle, implying a great
tunability of moiré flat bands in this system compared to TBG [58]. Very
recently, correlated insulators, antiferromagnetism, and stripe phase have
been observed in WSe2/WS2 heterostructure [59, 60, 61, 62, 63], where the
moiré lattice is formed by the mismatch of the lattice constants. In
MoTe2/WSe2 moiré heterostructures, metal-insulator transition was realized
[64]. For twisted bilayer WSe2 (tWSe2), many interesting phenomena including
metal-insulator transition, quantum critical behavior, and possible
superconductivity were also found [65, 66].

Figure 2: (a) Formation of the moiré Brillouin zone in twisted homo-bilayer
WSe2. In the upper left and right insets we show two schematic plots for both
the monolayer conduction and valence band structures near $\pm K_{0}$. (b) Two
fold degeneracy of the topmost moiré valence band, which can be lifted by an
out-of-plane electric field (c). (d) The electric field effectively induces a
stagger phase factor $\pm\phi$ on the bonds, leading to an accumulated flux
$\pm 3\phi$ on each triangular plaquette. (e) The physical bond phase $\phi$
as a function of $V_{z}$, from Ref.[67]. (f) Various types of Fermi surfaces
in the first Brillouin zone , obtained for different $n$ and $\phi$. Depending
on the parameters, there can be electron pockets, hole pockets, and disjoint
Fermi surfaces.

In this paper, we explore possible exotic superconductivity (SC) in tWSe2
induced by weak repulsive interactions, and investigate how they are affected
by the moiré filling factor $n$ as well as an out-of-plane electric field
parameterized by $\phi$. SC can arise from repulsive interaction if this
interaction can generate effective attractive components in one or more
pairing channels [68, 69]. For instance, it has been shown that $d+id$
topological SC can arise from repulsive interaction in doped graphene [70] and
in doped monolayer WSe2 [71] although their experimental realization is still
challenging so far partly due to the high requirement of Fermi level tuning to
van Hove singularity (VHS) in the proposed monolayer systems. Here we
investigate exotic SC in considering tWSe2 away from VHS (free from fine
tuning of Fermi level). Note that competing orders exactly at the VHS in the
twisted system were discussed in [72, 73].

Our main results from RG analysis of the extended triangular lattice Hubbard
model [Eq. (1) below] are summarized in the phase diagram shown in Fig. 1. In
a large portion of the parameter space with filling factor around $n=1$ there
exist chiral SC states with Chern numbers $\mathcal{N}=1,2,4$, corresponding
to $p+ip$, $d+id$ and $g+ig$ states, respectively. For small and large $n$,
the ground state is either $f$\- or $f^{\prime}$\- wave SC state,
corresponding to $A_{1}$ or $A_{2}$ representation of the underlying $C_{3v}$
group (or $D_{6h}$ group when $\phi=0$). The $i$\- and $i^{\prime}$-wave
pairing states with 12 nodes belonging to different representations are also
obtained [74]. Intriguingly, at $\phi=\pi/3$ and for small or large $n$, the
ground state features pair-density-wave (PDW) superconductivity [75, 76, 77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92] with equal-spin
pairing, which is degenerate with the opposite-spin pairing at zero momentum
when no magnetic field is applied. By applying a weak out-of-plane Zeeman
field to the tWSe2, this degeneracy is lifted and the equal-spin pairing (or
spin-polarized) PDW is the unique ground state. This spin-polarized PDW state
can be probed by experiments including spin-polarized STM measuring spin-
resolved pairing gap and electric transport measurement of superconducting
diode effect (SDE).

Model: The monolayer WSe2 is a triangular lattice semiconductor with broken
inversion symmetry [50]. The valence band top is located at $\pm\bm{K}_{0}$,
as shown in Fig. 2(a). Due to strong spin-orbit coupling, single particle
states near $\bm{K}_{0}$ and $-\bm{K}_{0}$ have opposite spin polarization.
When two layers of WSe2 are AA stacked together and twisted by a small angle,
a moiré pattern and a moiré Brillouin zone develops, as is shown in Fig. 2(a)
[93, 94, 39, 40, 95]. The spin up states in the top (down) layer near the
$\bm{K}_{0}$ points are mapped to the states near $\bm{K}$ ($-\bm{K}$) in the
moiré Brillouin zone , while the spin down in the top (down) layer near the
$-\bm{K}_{0}$ points are mapped to the states near $-\bm{K}$ ($\bm{K}$). The
hybridization due to the interlayer coupling leads to a narrow moiré band, as
shown in Fig. 2(b).

Because the twisted bilayer system respects both inversion and time-reversal
symmetries, it gives rise to double degeneracy for each band; however, the
double degeneracy can be lifted by a finite out-of-plane electric field [see
Fig. 2(c)] through the layer potential difference $V_{z}$ which explicitly
breaks inversion symmetry. A tight binding model for the moiré band can be
obtained by constructing a set of Wannier states and fitting with DFT
calculations[40, 65, 67]. The hopping parameter $t_{ij}$ of electrons with
spin polarization $\sigma$ between moiré lattice sites in the presence of a
finite out-of-plane electric field picks up a nontrivial phase as
$t^{\sigma}_{ij}=|t_{ij}|e^{i\phi^{\sigma}_{ij}}$, where
$\phi^{-\sigma}_{ij}=-\phi^{\sigma}_{ij}$ required from TRS. The amplitude
$|t_{ij}|$ decays exponentially with distance between $i$ and $j$, which
allows for a nearest-neighbor hopping approximation; namely $|t_{ij}|=t$ on NN
bonds and zero otherwise. The spin-dependent phase on NN bonds is $\pm\phi$,
giving rise to an accumulated flux $\pm 3\phi$ on each triangular plaquette,
as shown in Fig. 2(d). The magnitudes of $\phi$ depends on the strength of
electric field (or $V_{z}$) in a monotonic way [67], and was shown in Fig.
2(e) for clarity.

Moreover, it is found the onsite Coulomb repulsion is much larger than the
interaction between two adjacent sites [67, 65], which validates the following
triangular-lattice Hubbard model description:

$H=\sum_{\bm{k},\sigma=\pm}\epsilon_{\bm{k}}^{\sigma}c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},$
(1)

where the single-particle dispersion is given by
$\epsilon_{\bm{k}}^{\sigma}=-2t\sum_{\bm{a}_{m}}\cos(\bm{k}\cdot\bm{a}_{m}+\sigma\phi)$.
Hereafter we set $\bm{a}_{1}=(1,0)$, $\bm{a}_{2}=(-1/2,\sqrt{3}/2)$,
$\bm{a}_{3}=(-1/2,-\sqrt{3}/2)$ in unit of moiré lattice constant. For generic
$\phi\neq 0$, the model respects time-reversal symmetry, spin-rotational
symmetry along the z axis, and lattice point-group symmetry $C_{3v}$ (for
$\phi=0$ the point-group symmetry is $D_{6h}$). Moreover, the model has the
following symmetries regarding $\phi$. First, the model is invariant by
interchanging spin polarizations and changing $\phi$ to $-\phi$. In addition,
changing the total flux on each triangular plaquette by $2\pi$ should leave
the phase diagram invariant, and this corresponds to shifting $\phi$ by $\pm
2\pi/3$. Finally, particle-hole transformation leads to $n\to 2-n$ and
$\phi\to\pi-\phi$. Consequently, the ground state phase diagram of the model
is symmetric under $\phi\to\phi\pm\pi/3$ and $n\to 2-n$.

Renormalization group analysis: To analysis possible pairing instability of
the system in the regime of weak Hubbard repulsion, we employ the Raghu-
Kivelson-Scalapino RG analysis [69] to identify the leading pairing channel
for the ground state. Here we outline the key ingredients of the method and
leave the calculation details in the Supplemental Material [74]. For weak
repulsive interaction $U$, there is no attractive channel in the leading order
of $U$; one needs to start with the second order of $U$ in the interaction
vertex part, and sum over an infinite ladder series to obtain the total
interaction vertex in the pairing channel. This procedure results in an
renormalization group equation $\frac{dg}{dl}=-g\star g$, where
$l=\rho\log(W/\Lambda)$ with $\rho$ being density of states and $W$ the band
width. The energy scale $\Lambda^{*}$ when the RG procedure breaks down is the
same as the transition temperature $T_{c}$ and the zero-$T$ gap amplitude
$\Delta$[69]. Here $g$ is the interaction vertex, which can be written as [69,
96]

$g_{\bm{k},\bm{p}}=\sqrt{\frac{\bar{v}_{F}}{|v_{F}(\bm{k})|}}\Gamma(\bm{k},\bm{p})\sqrt{\frac{\bar{v}_{F}}{|v_{F}(\bm{p})|}},$
(2)

where $v_{F}({\bm{k}})$ is the Fermi velocity at $\bm{k}$ on the FS and
$\bar{v}_{F}\equiv\int_{FS}dk_{\parallel}/[\int_{FS}dk_{\parallel}v_{F}^{-1}(\bm{k})]$
is the average Fermi velocity. The interaction vertex for the opposite-spin
pairing channel, to the order of $U^{2}$, is just the static particle-hole
bubble $\Gamma(\bm{k},\bm{p})=U^{2}\chi(\bm{k}+\bm{p})$, where $\chi(\bm{q})$
is the spin susceptibility at momentum $\bm{q}$. Note that for $\phi=0$ and
$\phi=\pi/3$ we also need to consider the same-spin pairing channel for which
$\Gamma(\bm{k},\bm{p})=-U^{2}\chi(\bm{k}-\bm{p})$, where the minus sign comes
from an additional fermion loop. We are at liberty to choose both $\bm{k}$ and
$\bm{p}$ on, e.g. the spin up FS, and the Fermi velocities then must be
calculated from the same choice of FS.

To solve the RG equation with $g$ given in Eq. (2), one needs to first
diagonalize the real symmetric matrix $g_{\bm{k},\bm{p}}$, whose eigenvalues
$\lambda_{n}$ are decoupled and respect the same equation
$d\lambda_{n}/dl=-\lambda_{n}^{2}$. If the eigenvalue $\lambda_{n}$ is
negative, the corresponding channels will flow to strong coupling regime in
low energy limit, implying tendency towards SC. The most negative eigenvalue
$\lambda_{n}$ corresponds to the SC state with the largest pairing strength
and transition temperature. In general, different eigenvalues correspond to
different pairing symmetry categorized by the irreducible representations of
the underlying symmetry group. This is because $\Gamma(\bm{k},\bm{p})$ is the
kernel of the linearized gap equation, and the eigenvectors of $g$ is nothing
but the linearized gap function $\Delta(\bm{k})$ on the FS.

Figure 3: (a) and (b). Emergence of finite $\bm{Q}$ pairing at $\phi=\pi/3$.
(c) Fermi surface in the presence of an out-of-plane Zeeman field at
$\phi=\pi/3$. (d) Pairing strength of $\bm{Q}=\pm\bm{K}$ PDW order and
$\bm{Q}=0$ SC state at $\phi=\pi/3$.(e) Experimental setup for measuring the
spin polarized PDW using SDE.

Using this RG method [69], we analyze pairing instabilities in the tWSe2
system, and identify the most dominant SC channels in the ground states as a
function of $\phi$ and $n$, which we show in Fig. 1. We mainly focus on two
particularly interesting cases: pair-density-wave and chiral
superconductivity, respectively.

Pair-density-wave: For the particular case $\phi=\pi/3$ (more generally
$\pm\pi/3$ modulo $\pi$), both spin-up and spin-down Fermi surfaces possess an
intriguing property: Fermi surface is nested in the particle-particle channel;
namely FSs for each spin are symmetric with respect to the finite momentum
points at $\pm\bm{K}/2$ [shown as blue and red points in Fig. 3(a) and (b)],
rending to possible pairing whose center of mass momentum is finite with
$\bm{Q}=\pm\bm{K}$, i.e. a PDW superconductivity. Indeed, for $0<n<0.5$,
$0.54<n<0.68$ and $1.29<n<2$, our RG analysis shows that the ground state
features same-spin pairing at finite momentum $\pm\bm{K}$ and opposite-spin
pairing at zero momentum, which are degenerate in energy.

The emergence of PDW order of pairing between electrons with the same spin
polarization at $\phi=\pi/3$ is actually no surprise because this particular
$\phi=\pi/3$ is related to $\phi=0$ by symmetry transformation. A particle-
hole transformation plus a local gauge transformation corresponds to
$\phi\to\phi\pm\pi/3$ and $n\to 2-n$. It is the local transformation that maps
a triplet pairing with total $S^{z}=\pm 1$ at $\phi=0$ and $2-n$ to a PDW
order at $\pi/3$ and $n$. To see how this happens, consider the model with
$\phi=0$ featuring a zero-momentum triplet pairing ground state, for which the
triplet order with $S^{z}=+1$ have the form of
$\Delta_{\uparrow\uparrow}(\bm{R},\bm{r})c_{i\uparrow}c_{j\uparrow}$
[$\bm{R}=(\bm{r}_{i}+\bm{r}_{j})/2$ the position of the center of mass, and
$\bm{r}=\bm{r}_{i}-\bm{r}_{j}$ the relative position]. In momentum space, this
fermion bilinear term is
$\Delta_{\uparrow\uparrow}(\bm{Q},\bm{k})c_{\uparrow}(\bm{k}-\bm{Q}/2)c_{\uparrow}(-\bm{k}-\bm{Q}/2)$.
For $\phi=0$, triplet pairing has zero momentum $\bm{Q}=0$ and its real space
order parameter is independent of $\bm{R}$. After the particle-hole
transformation ($c_{j\sigma}\to c^{\dagger}_{j\sigma}$) and a local gauge
transformation ($c_{j\sigma}\to e^{i\sigma\eta_{j}}c_{j\sigma}$ with
$\eta_{i}=\bm{K}\cdot\bm{r}_{i}/2$), the model with $\phi=0$ and $2-n$ is
mapped to the one with $\phi=\pi/3$ and $n$, for which the corresponding order
parameter acquires a spatial phase modulation,namely,

$\Delta_{\uparrow\uparrow}(\bm{r})\to\Delta_{\uparrow\uparrow}(\bm{r})e^{i(\eta_{i}+\eta_{j})}=\Delta_{\uparrow\uparrow}(\bm{r})e^{i\bm{Q}\cdot\bm{R}}$
(3)

where $\bm{Q}=\bm{K}=(\frac{4\pi}{3},0)$. For pairing between two spin down
fermions, the argument is completely parallel, but we will have
$\bm{Q}=-\bm{K}$ instead. One can further see that for the opposite-spin
pairing ($S^{z}=0$) of the model with $\phi=0$ is mapped to usual zero-
momentum paring of the model with $\phi=\pi/3$. This is because spin up and
spin down sectors transform in opposite direction,
$\Delta_{\uparrow\downarrow}(\bm{r})\to\Delta_{\uparrow\downarrow}(\bm{r})e^{i(\eta_{i}-\eta_{j})}\sim\Delta_{\uparrow\downarrow}(\bm{r})$,
which does not lead to non-trivial $\bm{R}$ dependence. Therefore, a opposite-
spin pairing state ($S^{z}=0$) at $\phi=0$ is still a $\bm{Q}=0$ SC state at
$\phi=\pi/3$, while a triplet pairing state with $S^{z}=\pm 1$ at $\phi=0$
transforms into a $S_{z}=\pm 1$ PDW order at $\phi=\pi/3$.

Since at $\phi=0$ the same-spin triplet pairing with $S^{z}=\pm 1$ is
degenerate with the opposite-spin triplet pairing with $S^{z}=0$ due to the
full spin $SU(2)$ rotational symmetry, the system at $\phi=\pi/3$ also
maintains this degeneracy between the same-spin PDW order and the opposite-
spin zero-momentum pairing. The degeneracy between finite-$\bm{Q}$ PDW and
zero-$\bm{Q}$ SC state can be lifted by applying a weak out-of-plane magnetic
field, which generates a Zeeman coupling to the spin of electrons. Moreover,
for the twisted angle $\theta\gtrsim 3^{\circ}$, the orbital effect of the
applied magnetic field can be neglected [72], leaving the Zeeman coupling the
only dominant effect. In Fig. 3(c) we draw as an example the FS configurations
with a finite Zeeman coupling, which differentiates the sizes of the spin up
and spin down FSs. As a result, the $\bm{Q}=0$ pairing between $\bm{k}$ and
$-\bm{k}$ is suppressed, and the PDW becomes the unique ground state.
Moreover, the degeneracy between spin-up and spin-down PDW is also lifted by
the Zeeman field: fermions with the larger FS tend to have a stronger pairing
strength and hence a higher transition temperature. In Fig. 3(d) we plot the
dimensionless pairing strength $V_{eff}$ of the spin-polarized PDW order as
well as the $\bm{Q}=0$ SC state in a weak magnetic field. $V_{eff}$ is defined
such that the transition temperature scales as
$\exp[-1/(V_{eff}U^{2}/t^{2})]$. It shows the competition between the spin-
polarized PDW and another $\bm{Q}=0$ SC state, since they belong to different
irreducible representations. From the result we see $V_{eff}$ at small $n$ is
much larger than that at large $n$. The PDW order has been suggested as
intrapocket pairing in monolayer system [71, 90, 91, 92]. Here in our case,
the PDW order for large $n$ can also be interpreted as intrapocket pairing,
while the at small $n$, it is the interpocket pairing between electrons with
the same spin polarizations that leads to PDW. The pairing strength of
interpocket PDW for small $n$ is much larger than the large $n$ case, and is
thus more promising to realize.

Figure 4: SC phase variation along the FS parametrized by the angle
$\vartheta$. Distinct topological phases are characterized by the number of
$2\pi$ slips when moving around the FS in a full circle.

Chiral superconductivity: In a large part of the parameter space around $n=1$,
we find that the smallest eigenvalue always comes in pairs, corresponding to
the $E_{2}$ representation. In the most general case, the pairing structure of
the SC ground state is a linear combination of two corresponding eigenstates,
i.e. $\Delta(\bm{k})=\Delta_{1}v_{1}(\bm{k})+\Delta_{2}v_{2}(\bm{k})$, where
both $\Delta_{1}$ and $\Delta_{2}$ are complex numbers.

The true ground state should minimize the Ginzburg-Landau free energy, and
this helps to determine the two complex amplitudes $\Delta_{1}$ and
$\Delta_{2}$. To this end, we start with the action in pairing channel from
(1), and then perform Hubbard-Stratonavich transformation by introducing the
gap function $\Delta(\bm{k})$ as an auxiliary field. Integrating out fermions
and expanding $\Delta$ to the quartic order, we obtain,

$\displaystyle\mathcal{F}[\Delta_{1},\Delta_{2}]$
$\displaystyle=\alpha(T-T_{c})(|\Delta_{1}|^{2}+|\Delta_{2}|^{2})$ (4)
$\displaystyle+\beta_{1}(|\Delta_{1}|^{2}+|\Delta_{2}|^{2})^{2}+\beta_{2}|\Delta_{1}^{2}+\Delta_{2}^{2}|^{2}$

where $\alpha(T-T_{c})=-\rho\ln[W/T]-\lambda_{min}^{-1}$ with
$\lambda_{min}<0$ being the most negative degenerate eigenvalue of the
$g$-matrix [Eq. ((2))], and $\beta_{1}=(1/3)K\text{Tr}(v_{1}^{4})$ and
$\beta_{2}=(1/2)K\text{Tr}(v_{1}^{2}v_{2}^{2})$ with $K>0$ [74]. Since $v_{1}$
and $v_{2}$ are eigenvectors of a real symmetric matrix, we always have
$\beta_{1}>0$ and $\beta_{2}>0$. This fact leads to nontrivial consequences.
First $\beta_{1}$ being positive guarantees the existence of a SC ground
state. Furthermore, in order to minimize the free energy with $\beta_{2}>0$,
the term $|\Delta_{1}^{2}+\Delta_{2}^{2}|^{2}$ must vanish. It vanishes only
when $|\Delta_{1}|=|\Delta_{2}|$ and $\arg(\Delta_{1}/\Delta_{2})=\pi/2$,
corresponding to a chiral SC state: the phase of pairing
$\varphi=\arg[\Delta(\bm{k})]$ winds up multiple times of $2\pi$ when $\bm{k}$
goes around the whole FS while the amplitude $|\Delta(\bm{k})|$ remains
nonzero on the FS (nodeless), and it spontaneously breaks the time reversal
symmetry.

The nodeless chiral superconductor is topological, which supports chiral
Majorana modes on edges of the system. The topological invariant is
characterized by the Chern number defined as:

$\mathcal{N}=\frac{1}{4\pi}\int_{BZ}d\bm{k}[\hat{\bm{h}}\cdot(\partial_{k_{x}}\hat{\bm{h}}\times\partial_{k_{y}}\hat{\bm{h}})],$
(5)

where
$\hat{\bm{h}}=(\Re[\Delta(\bm{k})],\Im[\Delta(\bm{k})],\xi_{\bm{k}})/E_{\bm{k}}$
and $E_{\bm{k}}=\sqrt{\xi_{\bm{k}}^{2}+\Delta^{2}(\bm{k})}$. $2\pi\mathcal{N}$
is the Berry flux of the two-level Hamiltonian
$h_{k}=\hat{h}(\bm{k})\cdot\bm{\sigma}$, or the monopole charge located at the
torus center, which is just the line integral of a gauge transformation on the
FS, according to the Stokes theorem. In other words, Eq. ((5)) is the same as
the winding numbers defined by the number of times that the SC phase $\varphi$
slips $2\pi$ when $\bm{k}$ sweeps around the FS [97]. The number of chiral
Majorana modes is $2\mathcal{N}$, as the chiral central charge equals to the
Chern number [98] and the central charge of a single Majorana mode is
$\frac{1}{2}$. In Fig. 4 we show some examples of the chiral SC states
obtained in our model. We plot the SC phase $\varphi$ as a function of the FS
parameter $\vartheta$. Since the FS here forms a closed loop (either centered
at $\bm{\Gamma}$ point or $\pm\bm{K}$ point), we can parametrize the points on
FS by the angle $\vartheta$ formed by $\bm{k}_{F}$ (or $\bm{k}_{F}\pm\bm{K}$)
and $\hat{x}$.

Discussions and concluding remarks: In this paper, we have shown that an out-
of-plane electric field and magnetic field induced spin-polarized PDW order
can arise in the twisted bilayer TMD system, which is the unique ground state.
This PDW state supports nonzero spin-polarized SC diode current due to the
absence of inversion symmetry[See 3(e) for clarity], similar to the
superconducting diode effect (SDE) of the Flude-Fellel state [99], and can
thus be directly probed using the SDE experiment [100, 101, 102]. We also find
various topological chiral SC states in this system with the Chern number
$|\mathcal{N}|=1,2,4$. Here the number of chiral Majorana edge modes is equal
to 2$|\mathcal{N}|$. The model we studied is experimentally accessible and is
promising to be realized in experimental setups. Various other exciting
physics, such as the charge-4e superconductor or possible quantum critical
behavior between different phases tuned by the out-of-plane electric field, is
left for future study.

Acknowledgement.—We would like to sincerely thank Linhao Li, Zi-Xiang Li,
Fengcheng Wu, and Ya-Hui Zhang for useful discussions. This work is supported
in part by the NSFC under Grant No. 11825404 (YMW, ZZW and HY), the MOSTC
Grant No. 2018YFA0305604 (HY), the CAS Strategic Priority Research Program
under Grant No. XDB28000000 (HY). YMW is also supported in part by Shuimu
Fellow Foundation at Tsinghua.

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Supplemental Materials

##  .1 A. Renormalization group analysis

The specific shape of the FSs for both spin up and spin down fermions depend
on the value of $\phi$. For the regular case free of van Hove singularities,
there is no nesting effect along the FS. As a result, we can consider only SC
instabilities. Since we are interested in the possible pairing symmetry, we
need to take all momenta on the FS into account.

The possible SC from a repulsive Hubbard $U$ is a higher order effect. This
should be in contrast to the attractive $U$ case, where an arbitrarily small
attractive $U$ could lead to Cooper instability. To see this, it is convenient
to consider the one-loop renormalization group equation for the interaction
vertex in Cooper channel:

$\frac{dg}{dl}=-g\star g,$ (S1)

which is also shown pictorially in Fig. S1(a). Here if we dopt finite
temperature RG scheme, $l=\rho\log{W/T}$ with $\rho$ being density of states
and $W$ the band width, is the Cooper logarithm from the particle-particle
bubble and plays the role of the running parameter. In zero temperature RG
scheme, the running parameter is $l=\rho\log W/\Lambda$ where $\Lambda$ is the
running energy scale which changes as one continuously integrates out fast
fermion modes. The transition temperature $T_{c}$ and the zero-$T$ gap
amplitudes are the same with the energy scale $\Lambda^{*}$ when the RG
procedure breaks down. Here $g$ is the interaction vertex, which can be bare
or dressed. The notation $\star$ is a shorthand of the integral convolution
between the two involved vertices, but for the simplest case when $g$ is the
bare interaction $U$, it is equivalent to scalar multiplication.

Let’s first consider the case when $g=\Gamma=U$. Solving the differential
equation with the initial condition $\Gamma(0)=U$, one obtains

$\Gamma(l)=\frac{1}{l+1/U}=\frac{U}{1+lU}.$ (S2)

Notice that exactly the same expression can also be obtained by summing up the
infinite series of the ladder diagrams shown in Fig. S1(b). This has
demonstrated the equivalence between the one-loop RG analysis and the
diagrammatic calculation. Now it becomes manifest that in the low energy
limit, the interaction either flows to zero, or enter strong coupling regime,
depending on whether $U>0$ or $U<0$. Indeed, for any arbitrarily small
attractive $U$, the point when $1+lU=0$ is where RG fails to work, and this is
the Cooper instability which marks the onset of superconductivity. The
relevant temperature can then be found readily from this condition, which is
$T_{c}\sim\exp[-1/(\rho|U|)]$. For a positive $U$, however, there is no such
an instability at the bare interaction level.

Figure S1: Diagrammatic interpretation of the renormalization group analysis.

SC could arise for $U>0$ if we consider, instead of the bare $U$, the dressed
static interaction vertex shown in Fig. S1(c), and use it as the building
block in the ladder series depicted in Fig. S1(d). Again, this ladder series
is an equivalent way to interpret the one-loop RG equation in (S1). The
dressed interaction vertices in Fig. S1(c) are those which do not contain
Cooper logarithm, and we only show the diagrams up to the second order in the
Hubbard $U$. For weak interacting system which we consider in this work, these
two are the leading order corrections to the bare $U$. The full interaction
vertex in the pairing channel, contains contributions from both $\Gamma_{1}$
and $\Gamma_{2}$, as is shown in Fig. S1(e). It is easy to see the the
combination of $\Gamma_{1}$ with another Cooper logarithm gives a regular
contribution, and as a result, the leading possible instability now is solely
determined by $\Gamma_{2}$, which is a function of the external momenta.
Unlike the case with bare interaction, the resulting pairing symmetry is
usually no longer $s$-wave. For practical purpose, one should also distinguish
different contributions in $\Gamma_{2}$ to pairing states with different spin
configurations. In the Hubbard model, the bare interaction $U$ connects
fermions with opposite spin projections. Consequently, the first diagram in
Fig. S1(c) contributes to $S_{z}=0$ pairing state, while the second one
contributes to $|S_{z}|=1$ state.

For $\phi\neq 0$, fermions with different spins have different energy
dispersions. A closer look into the particle-particle bubble from this model
yields the conclusion that the Cooper logarithm shows up only in $S_{z}=0$
channel. In $|S_{z}|=1$ channel, the particle-particle bubble remains finite
even in the limit $T\to 0$. There is one exception when $\phi$ can be related
to $\phi=0$ via symmetry transformation. In this case the competition between
a finite $\bm{Q}$ PDW in $|S_{z}|=1$ channel and a $\bm{Q}=0$ SC state in
$S_{z}=0$ channel becomes possible. Thus we will be focusing only on the
$S_{z}=0$ channel when $\phi$ is neither zero nor in other high symmetry
point. In the special case of $\phi=0,\pi/3$, pairing with same spin becomes
possible, and accordingly the second diagram in Fig. S1(c) needs to be
considered together with the first one, leading to a competition between
singlet and triplet pairing.

Based on the above discussion the resulting interaction vertex to be input
into (S1) can be written as [69, 96]

$g_{k,p}=\sqrt{\frac{\bar{v}_{F}}{|v_{F}(k)|}}\Gamma(k,p)\sqrt{\frac{\bar{v}_{F}}{|v_{F}(p)|}},$
(S3)

We are at liberty to choose both $k$ and $p$ on, e.g. the spin up FS, and the
Fermi velocities then must be calculated from this particular choice of FS.
One is also free to flip the sign of one or both of these external momenta,
but it changes nothing since the FS is fixed as we fix the spin projection,
and $k$ and $-k$ are just two different notations for the same FS. The
interaction vertex for $\phi\neq 0,\pi/3$, to the order of $U^{2}$, is just
the static particle-hole bubble:

$\Gamma(\bm{k},\bm{p})=U^{2}\chi_{S_{z}=0}(\bm{k}+\bm{p}).$ (S4)

For $\phi=0$ and $\phi=\pi/3$ we need to replace
$\chi_{S_{z}=0}(\bm{k}+\bm{p})$ in the above expression with
$-\chi_{|S_{z}|=1}(\bm{k}-\bm{p})$, where the minus sign comes from the
fermion loop in Fig. S1(c). For our model, the particle-hole bubbles in
different channels are:

$\displaystyle\chi_{S_{z}=0}(\bm{p}+\bm{k})$
$\displaystyle=\int_{BZ}\frac{d^{2}q}{8\pi^{2}}\frac{\tanh[\beta\xi_{\bm{p}+\bm{k}+\bm{q}}^{+}/2]-\tanh[\beta\xi_{\bm{q}}^{-}/2]}{\xi_{\bm{p}+\bm{k}+\bm{q}}^{+}-\xi_{\bm{q}}^{-}},$
(S5) $\displaystyle\chi_{S_{z}=1}(\bm{p}-\bm{k})$
$\displaystyle=\int_{BZ}\frac{d^{2}q}{8\pi^{2}}\frac{\tanh[\beta\xi_{\bm{p}-\bm{k}+\bm{q}}^{+}/2]-\tanh[\beta\xi_{\bm{q}}^{+}/2]}{\xi_{\bm{p}-\bm{k}+\bm{q}}^{+}-\xi_{\bm{q}}^{+}}.$

Here $\beta$ is the inverse temperature which can be taken as infinitely large
in zero temperature limit, $\xi=\epsilon-\mu$ is from the energy dispersion
discussed in our main text.

The additional factor $\sqrt{\bar{v}_{F}/|v_{F}(k)|}$ in (S3) comes from the
need of the evaluation of

$\int\frac{d^{2}k}{(2\pi)^{2}}\Gamma(\bm{k},\bm{p})\left[\frac{1}{\beta}\sum_{n}G(k,\omega_{n})G(-k,-\omega_{n})\right]$
(S6)

when summing up the series in FigS1.(d). The particle-particle and particle-
hole bubbles are convoluted, but in low energy limit can be separated by
considering a thin shell around the FS where integration over momentum can be
carried out along two directions: along and perpendicular to the FS. In the
perpendicular direction $\chi(k+p)$ is treated as a constant, and the
integration leads to Cooper logarithm. To see this it’s convenient to re-
express the integration as follows,

$\int dk_{\bot}=\frac{(2\pi)^{2}}{L_{F}}\frac{\bar{v}_{F}}{|v_{F}(k)|}\int\rho
d\varepsilon,$ (S7)

where $L_{F}$ is the length of the FS (the FS area in 2D). The average Fermi
velocity on FS is obtained via

$\frac{1}{\bar{v}_{F}}=\int\frac{dk_{\|}}{L_{F}}\frac{1}{|v_{F}(k)|}.$ (S8)

The remaining integration along FS thus incorporates the factor
$\bar{v}_{F}/|v_{F}(k)|$. Distribute it symmetrically to the two adjacent
$\Gamma_{S_{z}}$ connected by a particle-particle bubble, we immediately
arrive at (S3). The $\star$ operator in (S1) can also be easily deduced from
the remaining integration on FS. Written explicitly we have

$(g\star
g)_{k,p}=\int\frac{dk^{\prime}_{\|}}{L_{F}}g_{k,k^{\prime}}g_{k^{\prime},p}.$
(S9)

For a set of discrete momentum points, this is nothing but a matrix product,
accompanied by a constant factor $\delta k^{\prime}_{\|}/L_{F}$.

To solve the RG equation (S1) where $g$ is in fact a matrix with infinite
dimension in the continuum limit, it is easy to first diagonalize it since $g$
is a real symmetric matrix. The eigenvalues $\lambda_{n}$ from diagonalization
are all decoupled in the RG equation, and each of them is constrained by the
same equation $d\lambda_{n}/dl=-\lambda_{n}^{2}$. The solution for each
$\lambda(l)$ obeys exactly the same form as (S2). From this we can easily
determine whether or not the system will develop a SC instability. If one or a
few eigenvalues are negative this channel will flow to strong coupling regime
in low energy limit, then indeed there will be SC phase transition in low
energy limit. The smallest negative eigenvalues corresponds to the SC state
with the largest transition temperature. In general, different eigenvalues
corresponds to different pairing symmetry. To appreciate this, consider the
linearized gap equation from the dressed interaction:

$\Delta(p)=-\int\frac{d^{2}k}{(2\pi)^{2}}\Gamma(p,k)\left[\frac{1}{\beta}\sum_{n}G(k,\omega_{n})G(-k,-\omega_{n})\right]\Delta(k).$
(S10)

Using the same integration tactic discussed above, we can easily see that the
eigenvectors $g_{p,k}$ are the gap functions up to some nonzero factor
$\tilde{\Delta}(p)=\sqrt{\bar{v}_{F}/|v_{F}(p)|}\Delta(p)$. Therefore, the
pairing symmetry is determined by the structure of the eigenfunction from
diagonalization of $g$.

As an example, we plot in Fig. S2 the pairing strength $V_{eff}$ in different
channels at $\phi=0$ when the symmetry is $D_{6h}$. The dimensionless pairing
strength is defined through the eigenvalue $V_{eff}=\rho|\lambda|$ such that
the SC transition temperature scales as $\exp[-1/(V_{eff}U^{2}/t^{2})]$. In
the same figure, we also show some typical examples of the gap function when
$\phi\neq 0$ and $n$ is small and large. Under this circumstance, the ground
state is either a six node or nodeless SC state, which dictates the $A_{1}$
and $A_{2}$ representation of the underlying $C_{3v}$ group. Note that $A_{1}$
representation is symmetric while $A_{2}$ representation is anti-symmetric
under $\sigma_{v}$ operation. For a fixed $n$ (say, $n=1.7$), as $\phi$
increases, the gap function undergoes a transition from $A_{1}$ representation
to $A_{2}$ representation. The transition is not sharp as there is a finite
range around $\phi=\pi/6$ where the pairing strength of the two
representations are very comparable.

Figure S2: Left: Pairing strength in different channels at $\phi=0$. Right:
Nodal and nodeless gap functions belonging to $A_{1}$ and $A_{2}$
representations at small and large $n$ with $\phi\neq 0$.

##  .2 B. Ginzburg-Landau free energy

The true ground state should minimize the Ginzburg-Landau free energy, and
this criterion is used to determine the two complex amplitudes $\Delta_{1}$
and $\Delta_{2}$. To this end, we start with the action in pairing channel,
then perform Hubbard-Stratonavich transformation by introducing the gap
function $\Delta(\bm{k})$ as an auxiliary field. The resulting Lagrangian is

$\displaystyle\mathcal{L}=$
$\displaystyle-\sum_{\bm{p},\bm{k}}\bar{\Delta}(\bm{p})g_{\bm{p},\bm{k}}^{-1}\Delta(\bm{k})$
(S11)
$\displaystyle-\text{Tr}_{\bot}\left[\Psi^{\dagger}(\bm{k})\begin{pmatrix}G_{+}^{-1}&\Delta_{d}(\bm{k})\\\
\bar{\Delta}_{d}(\bm{k})&G_{-}^{-1}\end{pmatrix}\Psi(\bm{k})\right].$

Here $\Delta(\bm{k})$ should be understood as a column vector in momentum
space with $\bm{k}\in\text{FS}$. In the second line
$\Psi(\bm{k})=[c_{\bm{k}_{1}},c_{\bm{k}_{2}},...,c^{\dagger}_{-\bm{k}_{1}},c^{\dagger}_{-\bm{k}_{2}},...]^{T}$
is a generalized Nambu-Gorkov basis with all $\bm{k}_{i}\in\text{FS}$.
Therefore the off-diagonal term
$\Delta_{d}(\bm{k})=\Delta_{1}d_{1}+\Delta_{2}d_{2}$ is a diagonal matrix with
$d_{1}=\text{diag}[v_{1}(\bm{k})]$ and $d_{2}=\text{diag}[v_{2}(\bm{k})]$,
where $v_{1}$ and $v_{2}$ are the two degenerate eigenstates defined in the
main text. The diagonal terms in (S11) are the particle and hole Green’s
functions defined by

$G_{\pm}=\frac{1}{i\omega_{m}\mp\xi_{\pm\bm{k}}^{\pm}}.$ (S12)

Unlike in $\Psi$ and $\Delta_{d}$, the momentum $\bm{k}$ in $G_{\pm}$ needn’t
to be restricted on FS. But all momenta have to be integrated out. We again
use the same strategy applied to (S6) already discussed above. The spirit is
to approximately separate the integration in two orthogonal directions. Here
we already adopted this in (S11), noting that while the matrix product in the
second line of (S11) is in fact a convenient way of expressing the integration
along the FS, the partial trace $\text{Tr}_{\bot}$ takes care of the
integration perpendicular to the FS as well as dimensional factors.

If we substitute $\Delta(\bm{k})$ from the combination of the two degenerate
eigenvectors of $g$ into the Lagrangian, then the first term can be simplified
using the orthogonality between these two eigenvectors. To obtain the
effective Ginzburt-Landau free energy, we need to further integrate out
fermion degrees of freedom $\Psi$, and expand the action to the order of
$O(\Delta^{4})$. In the intermediate stage one encounters a quadratic term
$\text{Tr}_{\|}(G_{+}G_{-}\Delta_{d}\Delta_{d}^{*})$ as well as a quartic term
$\text{Tr}_{\|}(G_{+}G_{-}G_{+}G_{-}\Delta_{d}\Delta_{d}^{*}\Delta_{d}\Delta_{d}^{*})$
where $\text{Tr}_{\|}$ denotes the integration along the FS. Given that
$G_{\pm}$ does not depend on $\bm{k}\in\text{FS}$, the Green’s functions can
be pulled out of the $\text{Tr}_{\|}$ operation. Thus the partial trace
operation to the FS acts only on the gap function and it can be further
simplified using the fact that
$\text{Tr}_{\|}(d_{1}^{2})=\text{Tr}_{\|}(d_{2}^{2})=1$,
$\text{Tr}_{\|}(d_{1}^{4})=\text{Tr}_{\|}(d_{2}^{4})=3\text{Tr}_{\|}(d_{1}^{2}d_{2}^{2})$
while other cross terms such as $\text{Tr}_{\|}(d_{1}d_{2})$ and
$\text{Tr}_{\|}(d_{1}^{3}d_{2})$ all vanish. All these considerations lead us
to the following free energy:

$\displaystyle\mathcal{F}[\Delta_{1},\Delta_{2}]$
$\displaystyle=\alpha(T-T_{c})(|\Delta_{1}|^{2}+|\Delta_{2}|^{2})$ (S13)
$\displaystyle+\beta_{1}(|\Delta_{1}|^{2}+|\Delta_{2}|^{2})^{2}+\beta_{2}|\Delta_{1}^{2}+\Delta_{2}^{2}|^{2}$

where $\alpha(T-T_{c})=\text{Tr}_{\bot}(G_{+}G_{-})-\lambda_{min}^{-1}$ with
$\lambda_{min}<0$ being the smallest degenerate eigenvalue of the $g$-matrix,
and $\beta_{1}=(1/3)K\text{Tr}_{\|}(d_{1}^{4})$ and
$\beta_{2}=(1/2)K\text{Tr}_{\|}(d_{1}^{2}d_{2}^{2})$ with
$K=\text{Tr}_{\bot}(G_{+}G_{-}G_{+}G_{-})$. Note that
$\text{Tr}_{\bot}(G_{+}G_{-})=-\rho\ln[W/T]$ is the Cooper logarithm from
which we can determine $T_{c}$, and the quartic term $K$ is always positive.