# Chern mosaic and ideal flat bands in equal-twist trilayer graphene

Daniele Guerci  Center for Computational Quantum Physics, Flatiron Institute,
New York, New York 10010, USA  Yuncheng Mao  Université Paris Cité, CNRS,
Laboratoire Matériaux et Phénomènes Quantiques, 75013 Paris, France
Christophe Mora  Université Paris Cité, CNRS, Laboratoire Matériaux et
Phénomènes Quantiques, 75013 Paris, France

###### Abstract

We study trilayer graphene arranged in a staircase stacking configuration with
equal consecutive twist angle. On top of the moiré cristalline pattern, a
supermoiré long-wavelength modulation emerges that we treat adiabatically. For
each valley, we find that the two central bands are topological with Chern
numbers $C=\pm 1$ forming a Chern mosaic at the supermoiré scale. The Chern
domains are centered around the high-symmetry stacking points ABA or BAB and
they are separated by gapless lines connecting the AAA points, where the
spectrum is fully connected. In the chiral limit and at a magic angle of
$\theta\sim 1.69^{\circ}$, we prove that the central bands are exactly flat
with ideal quantum curvature at ABA and BAB. Furthermore, we decompose them
analytically as a superposition of an intrinsic color-entangled state with
$\pm 2$ and a Landau level state with Chern number $\mp 1$. To connect with
experimental configurations, we also explore the non-chiral limit with finite
corrugation and find that the topological Chern mosaic pattern is indeed
robust and the central bands are still well separated from remote bands.

#### Introduction —

Stacking and twisting two layers of graphene realizes an extraordinary
platform [1] which in the magic angle region gives rise to flat bands [2, 3,
4, 5, 6, 7] hosting superconductivity [8, 9, 10, 11, 12], interaction-driven
insulating states [13, 14, 15, 16, 17, 18, 19, 20, 21, 22], anomalous Hall
effects [23, 24, 25, 26, 27, 28, 29, 30, 31, 32] and fractional Chern
insulators [33, 34, 35]. The intimate connection between the flat bands of
twisted bilayer graphene (TBG) and the properties of Landau levels [36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] played a key role for the
understanding of the interplay between correlation and topology in the
aforementioned correlated states. Following this guiding principle we
characterized the properties of the flat bands in equal-twist staircase
trilayer graphene (eTTG), see Fig. 1a, finding high-symmetry stacking ABA/BAB
configurations with total Chern number $\pm 1$ hosting an intrinsic color-
entangled state [51, 52, 53, 54] with Chern number $2$ and a Landau level like
state with Chern number $-1$.

Adding an additional graphene sheet to TBG rotated by a small relative twist
angle [twisted trilayer graphene (TTG)] gives rise to the superposition of two
moiré superlattices [55, 56]. With the exception of mirror-symmetric TTG [57,
58, 59, 60, 61, 62, 63, 64, 65, 66], the two moiré periodicities are
incommensurate [67, 68] leading to a quasicristalline structure that dominates
the electronic behavior at relevant energies [56]. The theoretical description
of twisted trilayer graphene runs into fundamental difficulties [67, 68] due
to the quasiperiodic nature of the low-energy Hamiltonian, disallowing all the
simplifications from Bloch’s theorem. Similar effects can also emerge in
encapsulated TBG when the hBN layers are nearly aligned with the moiré pattern
of TBG [69, 70].

The aim of this letter is to study the emergent effect of the superposition of
the two moiré patterns in eTTG, see Fig. 1a. The system is the simplest
example of a quasiperiodic moiré crystal [67] where the angle $\theta_{12}$,
between layer 1 (top) and 2 (middle), and $\theta_{23}$, between layer 2 and 3
(bottom), are equal $\theta_{12}=\theta_{23}\equiv\theta$. In the magic angle
region, where $\theta\approx 1^{\circ}$, the two incommensurate periodicity
can be decomposed in a fast modulation $\bm{q}_{j}$ on the moiré scale
$|\bm{q}_{j}|\propto\theta$ and a slow one $\delta\bm{q}_{j}$ with much larger
periodicity $|\delta\bm{q}_{j}|\propto\theta^{2}$ [68, 71]. Applying
semiclassical adiabatic approximation [72, 73, 74, 75, 76] we define a local
Hamiltonian $H_{\rm eTTG}(\bm{r})=H_{\rm eTTG}(\bm{r},{\bm{\phi}})$ where
${\bm{\phi}}$ depends on the slowly varying supermoiré scale [68]. In this
picture, we obtain a Chern number versus ${\bm{\phi}}$ real-space map Fig. 1b
that gives rise to a triangular lattice Chern mosaic of regions with $\pm 1$
Chern number. The domain walls separating the topological regions close the
gaps to the remote bands and form lines connecting the AAA centers.

Figure 1: a) Equal-twist angle trilayer graphene lattice in real space. b)
Real-space Chern mosaic in the moiré of moiré lattice scale. We distinguish
three high-symmetry regions of character AAA, ABA and BAB. The former shows a
fully connected spectrum while the latter is characterized by flat bands with
total Chern number $+1$ and $-1$, respectively. Domain walls regions marking
topological transition between Chern $\pm 1$ states show fully connected
spectrum.

There are three high-symmetry stacking configurations that are especially
significant and indicative of the Chern mosaic pattern: AAA, ABA and BAB. We
explore them analytically in the chiral limit to unveil the topological
features of the mosaic. We thereby derive analytical expressions for the
resulting ideal flat bands emerging at a magic angle. The AAA stacking,
considered in the preprint [77], is characterized by a vanishing Berry
curvature and a fully connected spectrum protected by $C_{2z}T$ [78]. At the
magic angle $\theta_{AAA}\approx 0.75^{\circ}$ a fourfold degenerate zero
energy flat band sector emerges, connected to a single Dirac cone. The
ABA(BAB) stacking, on the other hand, shows a flat band region detached from
the remote bands with total Chern number $C=1(-1)$. The origin of the finite
Chern number is readily traced out by the nature of the flat bands at the
magic angle $\theta_{ABA}\approx 1.69^{\circ}$ which, remarkably, is larger
than the one in mirror-symmetric TTG [79]. We prove that the flat band sector
decomposes into a Chern $+2(-2)$ color-entangled zero mode [54, 80, 81] and a
Chern $+1(-1)$ Landau level like state [36, 39, 38, 47, 82, 48, 50, 83]. The
resulting imbalance in Chern flux creates a Chern mosaic pattern in real
space, which could potentially be detected by measuring the local orbital
magnetization in real space [27, 28].

#### Chern Mosaic on the supermoiré scale —

When the twisting angle is small, non commensurability effects are
characterized by a length scale well separated from the moiré scale,
$|\delta\bm{q}_{j}|/|\bm{q}_{j}|\approx 0.02$ for $\theta=1^{\circ}$. As a
result the long wavelength modulation can be treated parametrically, leading
to the local Hamiltonian obtained in Ref. [68]:

$H_{\rm
eTTG}(\bm{r},{\bm{\phi}})=\begin{bmatrix}v_{F}\hat{\bm{k}}\cdot\bm{\sigma}&T(\bm{r},{\bm{\phi}})&0\\\
h.c.&v_{F}\hat{\bm{k}}\cdot\bm{\sigma}&T(\bm{r},-{\bm{\phi}})\\\
0&h.c.&v_{F}\hat{\bm{k}}\cdot\bm{\sigma}\end{bmatrix},$ (1)

where $v_{F}\approx 10^{6}$m/s is the graphene velocity, the phase
${\bm{\phi}}=(\phi_{1},\phi_{2},\phi_{3})$ defines the local stacking
configuration [68, 71]. Varying ${\bm{\phi}}$ maps out the supermoiré unit
cell in Fig. 1b, ${\bm{\sigma}}$ is the vector of Pauli matrices in the
sublattice space and $\hat{\bm{k}}=-i\nabla_{\bm{r}}$. The tunneling between
different layers is described by the moiré potential:

$T(\bm{r},{\bm{\phi}})=\sum^{3}_{j=1}T_{j}e^{-i\bm{r}\cdot\bm{q}_{j}}e^{-i\phi_{j}},$
(2)

where $T_{j+1}=w_{\rm AA}\sigma^{0}+w_{\rm AB}[\sigma^{x}\cos 2\pi
j/3+\sigma^{y}\sin 2\pi j/3]$, $w_{\rm AB}=110$meV, using complex notation
$\bm{q}_{j+1}=ie^{2i\pi j/3}$ [78], $j=1,2,3$, in unit of $k_{\theta}=\theta
K_{D}$ with $K_{D}=4\pi/3a_{\rm G}$ and $a_{\rm G}\approx 2.46$Å. The moiré
lattice is characterized by the reciprocal lattice vectors
$\bm{b}_{1/2}=\bm{q}_{1}-\bm{q}_{2/3}$ and primitive vectors $\bm{a}_{1/2}$.
Bloch periodicity takes the form $H_{\rm
eTTG}(\bm{r}+\bm{a}_{1/2},{\bm{\phi}})=U_{\varphi/-\varphi}H_{\rm
eTTG}(\bm{r},{\bm{\phi}})U^{\dagger}_{\varphi/-\varphi}$ with $\varphi=2\pi/3$
and $U_{\varphi}=\text{diag}(\omega^{*},1,\omega)$ with $\omega=e^{2\pi i/3}$.
The spectrum is thus invariant upon shifting $\bm{k}$ by $\bm{a}_{1}$ and
$\bm{a}_{2}$ up to a layer dependent phase factor. The Hamiltonian (1) is also
invariant under the particle-hole transformation $PH_{\rm
eTTG}(\bm{r},{\bm{\phi}})P=-H_{\rm eTTG}({-\bm{r},\bm{\phi}})$ where

$P=\begin{bmatrix}0&0&1\\\ 0&-1&0\\\ 1&0&0\end{bmatrix}_{\rm
layer}\otimes\sigma^{0}\equiv\mathcal{M}\otimes\sigma^{0}.$ (3)

At low-energy Eq. 1 is characterized by three inequivalent Dirac cones at $K$,
$K^{\prime}$ and $\Gamma$ of the mini Brillouin zone (BZ) shown in Fig. 2a.
The central one at $\Gamma$ is protected by $P$ while $K$ and $K^{\prime}$ are
gapped for generic ${\bm{\phi}}$ [68].

We now obtain the spectrum of the Hamiltonian (1) and study the topological
properties of the nearly-flat bands around charge neutrality. Fig. 1b shows
the real-space mosaic pattern obtained by computing the Chern number
$\mathcal{C}({\bm{\phi}})$ for the two central bands at the magic angle
$\theta_{\rm ABA}=1.69^{\circ}$, for finite corrugation $w_{\rm AA}=0.8w_{\rm
AB}$. The mosaic exhibits a triangular periodic structure, which is generated
by the lattice vectors $\mathbf{a}_{1/2}^{\mathrm{MM}}=4\pi e^{\mp
i\pi/3}/3k{\theta}^{\mathrm{MM}}$, where
$k_{\theta}^{\mathrm{MM}}=\theta^{2}K_{D}$. The two central bands are
topological everywhere except along lines connecting the AAA centers, where
the spectrum is fully connected. Each topological region is centered around
$\mathbf{r}_{\mathrm{ABA}}=(\mathbf{a}_{1}^{\mathrm{MM}}-\mathbf{a}_{2}^{\mathrm{MM}})/3$
and $\mathbf{r}_{\mathrm{BAB}}=-\mathbf{r}_{\mathrm{ABA}}$, with opposite
Chern numbers of $\pm 1$. Therefore, we specifically focus on these two high-
symmetry points and consider the chiral limit, where the bands become exactly
flat, and an analytical solution can be obtained.

Figure 2: a) mini Brillouin zone (BZ). b) Renormalized velocity $v^{*}$ for
AAA (red) and ABA (blue) regions as a function of the dimensionless coupling
$\alpha=w_{AB}/v_{F}k_{\theta}$ and $w_{AA}=0$ (chiral limit). Dispersion
relation for ABA stacking panel c) and AAA stacking panel d) at the magic
angle $\theta_{\rm AAA}\approx 0.75^{\circ}$ and $\theta_{\rm ABA}\approx
1.69^{\circ}$, respectively.

#### ABA-stacking: color-entangled flat band—

The ABA region is described by the local Hamiltonian $\mathcal{H}_{\rm
ABA}=H_{\rm eTTG}({\bm{\phi}}_{\rm ABA})$ Eq. 1 with ${\bm{\phi}}_{\rm
ABA}=(0,\varphi,-\varphi)$. Here, the $C_{3z}$ symmetry is recovered while
$C_{2x}$ and $C_{2z}T$ are broken. The latter connects ${\rm ABA}$ to ${\rm
BAB}$, $C_{2z}T{\mathcal{H}_{\rm
ABA}}(\bm{r})(C_{2z}T)^{\dagger}=-{\mathcal{H}_{\rm BAB}}(-\bm{r})$,
explaining the opposite Chern number of the ABA and BAB regions. The
combination of $C_{2x}$ and $C_{2z}T$ is a symmetry for the model $C_{2y}T$
which together with $P$ and $C_{3z}$ protects three Dirac cones at $\Gamma$,
$K$ and $K^{\prime}$ [71]. We henceforth consider the chiral limit $w_{\rm
AA}=0$ where an inspiring mathematical structure emerges [36].
$\mathcal{H}_{\rm ABA}$ then anticommutes with the chiral operator
$\Lambda^{z}=\tau^{0}\otimes\sigma^{z}$ with $\tau^{0}$ the identity in the
layer basis. Denoting with $\psi_{l}$ and $\chi_{l}$ with $l=1,2,3$ the
wavefunction components polarized in the A and B sublattices, in the basis
$\Psi=\begin{pmatrix}\psi_{1}&\psi_{2}&\psi_{3}&\chi_{1}&\chi_{2}&\chi_{3}\end{pmatrix}^{T}$
the Hamiltonian $\mathcal{H}_{\rm ABA}$ reads:

$\frac{\mathcal{H}_{\rm
ABA}(\bm{r})}{v_{F}k_{\theta}}=\begin{pmatrix}0&\mathcal{D}_{1}(\bm{r})\\\
\mathcal{D}^{\dagger}_{1}(\bm{r})&0\end{pmatrix},$ (4)

we look for zero mode solutions:

$\mathcal{D}_{1}(\bm{r}){\bm{\chi}}_{\bm{k}}(\bm{r})=0,\quad\mathcal{D}^{\dagger}_{1}(\bm{r}){\bm{\psi}}_{\bm{k}}(\bm{r})=0,$
(5)

with boundary conditions:

$\begin{split}&\bm{\psi}_{\bm{k}}(\bm{r}+\bm{a}_{1/2})=e^{i\bm{k}\cdot\bm{a}_{1/2}}U_{\varphi}\bm{\psi}_{\bm{k}}(\bm{r}),\\\
&\bm{\chi}_{\bm{k}}(\bm{r}+\bm{a}_{1/2})=e^{i\bm{k}\cdot\bm{a}_{1/2}}U_{\varphi}\bm{\chi}_{\bm{k}}(\bm{r}).\end{split}$
(6)

In Eq. 4 the operator is:

$\mathcal{D}_{1}(\bm{r})=\begin{pmatrix}-i\sqrt{2}\partial&U_{\omega}(\bm{r})&0\\\
U_{0}(-\bm{r})&-i\sqrt{2}\partial&U_{0}(\bm{r})\\\
0&U_{\omega}(-\bm{r})&-i\sqrt{2}\partial\end{pmatrix},$ (7)

with $\partial=\partial_{x}-i\partial_{y}/(\sqrt{2}k_{\theta})$,
$U_{0}(\bm{r})=\alpha\sum^{3}_{j=1}e^{-i\bm{q}_{j}\cdot\bm{r}}$ and
$U_{\omega}(\bm{r})=\alpha\sum^{3}_{j=1}\omega^{j-1}e^{-i\bm{q}_{j}\cdot\bm{r}}$
with $\alpha=w_{\rm AB}/v_{F}k_{\theta}$. We also introduce
$\bar{\partial}=\partial_{x}+i\partial_{y}/(\sqrt{2}k_{\theta})$,
$z=k_{\theta}(x+iy)/\sqrt{2}$ and $\bar{z}=k_{\theta}(x-iy)/\sqrt{2}$. We
focus on the first magic angle $\theta_{ABA}\approx 1.69^{\circ}$ where the
renormalized velocity $v^{*}$ vanishes, see blue line in Fig. 2b, and
correspondingly the bands around charge neutrality becomes perfectly flat as
shown in Fig. 2c. Interestingly, the single particle gap that separates the
flat bands from remote ones is $E_{\rm gap}\approx 130$meV quite large if
compared with the typical value of the Coulomb interaction screened by
metallic gates [41]. The $C_{3z}$ symmetry yields $\chi_{\Gamma
1}(0)=\chi_{\Gamma 3}(0)=0$, while $\chi_{\Gamma 2}(0)$ is usually non-zero.
The magic angle $\theta_{\rm ABA}$ is exactly defined by
$\chi_{\Gamma,2}(0)=0$. As the spinor $\bm{\chi}_{\Gamma}(0)$ then fully
vanishes at $\theta_{\rm ABA}$, the B-polarized flat band has an analytical
expression

$\bm{\chi}_{\bm{k}}(\bm{r})=\bar{\eta}_{\bm{k}}(\bar{z})\bm{\chi}_{\Gamma}(\bm{r}),$
(8)

where the antiholomorphic $\bar{\eta}_{\bm{k}}(\bar{z})=\eta_{\bm{k}}^{*}(-z)$
is related to the meromorphic function

$\eta_{\bm{k}}(z)=e^{ik_{1}z/a_{1}}\frac{\vartheta_{1}[z/a_{1}-k/b_{2},\omega]}{\vartheta_{1}(z/a_{1},\omega)},$
(9)

with the notation $k_{1}=\bm{k}\cdot\bm{a}_{1}$ and $\vartheta_{1}(z,\omega)$
is the Jacobi theta-function [71], which vanishes at $z=0$ and results in a
Bloch periodicity Eq. (6). The Bloch wavefunction associated with Eq. (8) is
$k$-antiholomorphic

$\begin{split}{\bm{u}}_{\bar{k}}(\bm{r})=e^{-i\bm{r}\cdot{\bm{b}}_{2}\bar{k}/b^{*}_{2}}\frac{\vartheta_{1}[{\bar{z}}/a_{1}^{*}+{\bar{k}}/b_{2}^{*},-\omega^{*}]}{\vartheta_{1}[{\bar{z}}/a_{1}^{*},-\omega^{*}]}{\bm{\chi}}_{\Gamma}(\bm{r})\end{split}$
(10)

corresponding to an ideal flat band [39, 47]. The Chern number of the band can
be readily read off from the $k$-space boundary conditions

${\bm{u}}_{\bar{k}+\bar{b}_{j}}(\bm{k})=e^{-i{\bm{b}}_{j}\cdot\bm{r}}e^{i\phi_{{\bar{k},\bar{b}_{j}}}}{\bm{u}}_{\bar{k}}(\bm{r}),$
(11)

where $\phi_{{\bar{k},\bar{b}_{1}}}=-2\pi\bar{k}/b^{*}_{2}+\pi-\pi
b^{*}_{1}/b^{*}_{2}$ and $\phi_{{\bar{k},\bar{b}_{1}}}=\pi$ which implies
$C_{B}=-1$ where the Chern number has been computing employing [47, 54]:

$C=\frac{\phi_{\bm{k}_{0}+\bm{b}_{2},\bm{b}_{1}}+\phi_{\bm{k}_{0},\bm{b}_{2}}-\phi_{\bm{k}_{0},\bm{b}_{1}}-\phi_{\bm{k}_{0}+\bm{b}_{1},\bm{b}_{2}}}{2\pi}.$
(12)

We turn to the exact solution for the A-polarized wavefunction ${\bm{\psi}}$.
$C_{3z}$ yields again $\psi_{\Gamma 1/3}(0)=0$ at $\Gamma$ and
$\psi_{K/K^{\prime}2}(0)=0$ at $K$ ($K^{\prime}$), however $\psi_{\Gamma
2}(0)$ does not vanish at the magic angle. Nevertheless, we numerically find
that $\psi_{K,1}(0)=-\psi_{K,3}(0)$, right at the magic angle $\theta_{\rm
ABA}$ which, combined with particle-hole symmetry $P$, can be used to prove
that the two spinors

$\psi_{K}(0)=\psi_{K^{\prime}}(0)$ (13)

at $\theta_{\rm ABA}$. This remarkable identity allows us to exhibit an exact
analytical expression for the A-polarized flat band (up to $\bm{k}$-dependent
prefactor)

${\bm{\psi}}_{\bm{k}}(\bm{r})=a_{k}\eta_{\bm{k}-\bm{q}_{1}}(z){\bm{\psi}}_{K}(\bm{r})+a_{-k}\eta_{\bm{k}+\bm{q}_{1}}(z){\bm{\psi}}_{K^{\prime}}(\bm{r}),$
(14)

satisfying the Bloch periodicity Eq. (6), with the holomorphic function
defined in Eq. (9) and $a_{k}=\vartheta_{1}[(k+q_{1})/b_{2},\omega]$. For more
details on the symmetry conditions leading to Eq. (14) we refer to SM [71]. In
Eq. (14), we set the $K$ and $K^{\prime}$ points at $\pm\bm{q}_{1}$
respectively. Thanks to the following property of the theta-function
$\vartheta_{1}(-z,\omega)=-\vartheta_{1}(z,\omega)$, it is readily checked
that the poles of $\eta_{\bm{k}\pm\bm{q}_{1}}(z)$ at $z=0$ cancel each other
in Eq. (14), as a result of Eq. (13), and the wavefunction is finite
everywhere. We note that, the corresponding unnormalized Bloch function
${\bm{u}}_{k}(\bm{r})=\bm{\psi}_{\bm{k}}(\bm{r})e^{-i\bm{k}\cdot\bm{r}}$ is
$k$-holomorphic and thus constitutes an ideal flat band [39, 47]. In addition,
momentum space boundary conditions 11 give $\phi_{k,b_{1}}=4\pi k/b_{2}+2\pi
b_{1}/b_{2}$ and $\phi_{k,b_{2}}=0$ leading to a Chern number $C_{A}=2$ which
gives the total Chern number $C=C_{A}+C_{B}=+1$ characteristic of the
triangular regions centered at the ABA sites of the real-space pattern in Fig.
1b. Remarkably, the Chern $2$ band of Eq. (14) describes a color-entangled
wavefunction [54] and, upon translation of a lattice vector
$\bm{r}_{0}\to\bm{r}_{0}+\bm{a}_{1}$, the $k$-space zeros of
$|\psi_{\bm{k}2}(\bm{r})|$ get swapped, see Fig. 3.

Figure 3: Each figure shows $|\psi_{\bm{k}2}(\bm{r})|$ as a function of
$\bm{k}$ for a fixed $\bm{r}$. From left top to right bottom the position
$\bm{r}$ evolves from $\bm{r}_{0}=(0.1,0.3)$ (unit of $1/k_{\theta}$) to
$\bm{r}_{0}+\bm{a}_{1}$. There are in total $C_{A}=2$ zeros in the Brillouin
zone at fixed position $\bm{r}$, and their pattern is lattice translational
invariant but indices are be exchanged going from $\bm{r}_{0}$ to
$\bm{r}_{0}+\bm{a}_{1}$. The effect resembles Thouless charge pumping but in
reciprocal space. The solid gray line denotes the diamond shaped BZ.

The emergence of the Chern bands $+2$ and $-1$ can be intuitively understood
as a direct consequence of the original three Dirac cones of each layer,
similar to twisted monolayer-bilayer graphene [84, 85].

#### AAA-stacking and domain wall lines–

The local Hamiltonian describing the AAA points is obtained by setting
${\bm{\phi}}=0$ in Eq. (1). It satisfies all symmetries [78]: $C_{2z}T$,
$C_{2x}$, $C_{3z}$ and particle-hole symmetry $P$, protecting the Dirac cones
at $K$, $K^{\prime}$ and $\Gamma$. $C_{2z}T$ furthermore enforces [78] a fully
connected spectrum as an odd number of Dirac cones cannot form isolated
minibands [86, 87, 68].

Figure 4: a) Trace of the non-Abelian Berry curvature for the A sublattice
$\Lambda=+1$ (due to $C_{2z}T$ we have
$\text{tr}\Omega_{B}=-\text{tr}\Omega_{A}$). We check numerically that the
trace condition is satisfied. b) Spectrum obtained moving away from the AAA
stacking along to the $\mathbf{a}^{\rm MM}_{1}-\mathbf{a}^{\rm MM}_{2}$
direction. A gap opens in the spectrum, we highlight in red the bands giving
rise to a total Chern number $+1$. c) Bands obtained along the domain wall
regions $\mathbf{a}^{\rm MM}_{1}+\mathbf{a}^{\rm MM}_{2}$ direction. The
results are obtained at the magic angle $\theta_{AAA}\approx 0.75^{\circ}$ and
for $w_{\rm AA}=0.$

In the chiral limit ($w_{\rm AA}=0$) the Hamiltonian $\mathcal{H}_{\rm AAA}$
in the basis
$\Psi=\begin{pmatrix}\psi_{1}&\psi_{2}&\psi_{3}&\chi_{1}&\chi_{2}&\chi_{3}\end{pmatrix}^{T}$
takes the form:

$\mathcal{H}_{\rm AAA}(\bm{r})=\begin{pmatrix}0&\mathcal{D}_{2}(\bm{r})\\\
\mathcal{D}^{\dagger}_{2}(\bm{r})&0\end{pmatrix},$ (15)

where the operators reads:

$\mathcal{D}_{2}(\bm{r})=\begin{pmatrix}-i\sqrt{2}\partial&U_{\omega^{*}}(\bm{r})&0\\\
U_{\omega^{*}}(-\bm{r})&-i\sqrt{2}\partial&U_{\omega^{*}}(\bm{r})\\\
0&U_{\omega^{*}}(-\bm{r})&-i\sqrt{2}\partial\end{pmatrix},$ (16)

and $U_{\omega^{*}}(\bm{r})=U^{*}_{\omega}(-\bm{r})$. At the magic angle
taking place at $\theta_{\rm AAA}\approx 0.75^{\circ}$, see red line in Fig.
2b, the spectrum shown in Fig. 2d is composed by a fourfold degenerate zero
mode subspace and a renormalized Dirac cone located at $\Gamma$.
Interestingly, the wavefunction of the zero modes can be exactly expressed in
terms of meromorphic functions as shown in Ref. [77]. We here focus on the
topological properties of the fourfold degenerate flat band sector. Away from
the $\Gamma$ point the flat bands are isolated and the degeneracy can be
partially resolved by $\Lambda_{z}$ which gives two dimensional subspaces with
opposite sublattice polarization $\Lambda_{z}=\pm 1$. For a given sublattice
the topological properties are characterized by the non-Abelian quantum
geometric tensor
$Q^{ab}_{nm}(\bm{k})=\innerproduct{D_{a}u_{n\bm{k}}}{D_{b}u_{m\bm{k}}}$ with
$D_{a}$ the covariant derivative [88]. The non-Abelian trace condition [35]
reads as $\Tr\text{tr}\,g=\text{tr}\,\Omega$ with $\Tr$ trace over space
directions and tr over the 2D subspace. Fig. 4a shows the Berry curvature
$\text{tr}\,\Omega_{A}$, we check numerically that the trace condition is
satisfied everywhere at the exclusion of the $\Gamma$ point where the Berry
curvature is ill-defined. $C_{2z}T$ imposes that the two sublattice sectors
yield opposite Berry curvature $\text{tr}\Omega_{A}=-\text{tr}\Omega_{B}$. The
points AAA are however singular. The fourfold degeneracy is lifted by any
small but finite ${\bm{\phi}}$ and the flat band sector with Chern number $\pm
1$ is recovered, see Fig. 4b. The Chern mosaic of Fig. 1b is thus largely
governed by the topology of the ABA and BAB points.

Finally, the different topological regions extended around ABA and BAB meet
along lines where the gaps to the remote bands vanish. These lines form a
triangular lattice originating from the AAA lattice sites as shown in Fig. 1b.
One can prove that, along these lines, the $C_{2x}$ symmetry, combined with
$C_{3z}$ yields a fully connected band structure for Eq. (1), as seen in Fig.
4c. The proof, which follows Ref. [78], is given in the SM [71]. However,
breaking these symmetries can move these domain walls but not suppress them
since the distinct topological domain must be separated by gap-closing
contours.

#### Stability away from the chiral limit —

Our predictions formally derived in the chiral limit are stable and persists
for finite value of $w_{\rm AA}$. We see that at finite $w_{\rm AA}$ the low-
energy bands at ABA regions acquire a finite dispersion as shown in Fig. 5b.
However, the two flat bands highlighted in red in Fig. 5b are characterized by
a total Chern number $1$ as shown by the winding $2\pi$ of the center of mass
of the Wilson loop $\mathcal{W}(k_{2})$ [89], gray dots in Fig. 5c while red
and blue dots show the evolution of the two eigenvalues. The Chern number
mosaic pattern over the entire supermoiré space was previously presented in
Fig. 1b with $w_{\rm AA}=0.8w_{\rm AB}$.

Figure 5: Results away from chiral limit ($w_{\rm AA}/w_{\rm AB}=0.7$) for
twist angle $\theta=1.69^{\circ}$ corresponding to the magic angle for ABA
stacking in the chiral limit. a) Fully connected spectrum for AAA stacking. b)
Spectrum for ABA stacking, the flat bands separated from remote ones are
highlighted in red. c) Wilson loop $\mathcal{W}(k_{2})$ spectrum for the flat
bands as a function of $k_{2}=\bm{k}\cdot\bm{a}_{2}/2\pi$. Red and blue dots
shows the eigenvalues for the two flat bands, grey dots, instead, show the
evolution of the center of mass characterized by a 2$\pi$ winding
corresponding to a total Chern number $1$.

#### Conclusions —

In summary, we showed that in equal-twist angle trilayer graphene separation
between length scale leads to an interesting concept of supermoiré lattice
where the local registry corresponds to twisting around AAA or ABA stacking
but has a local-to-local variation over long range. The local adiabatic
Hamiltonian which depends parametrically on the supermoiré lattice coordinate
shows topologically distinct regions where the low-energy flat bands have
finite and opposite Chern number. Remarkably, the non-vanishing Chern number
at ABA regions originates from a zero modes composed by a Chern $+2$ color-
entangled wavefunction and a Chern $-1$ Landau level like state. The
intermediate domain wall regions originating from the AAA stacking
configurations are characterized by a fully connected spectrum which is
gapless at all energies. We conjecture that similar properties can be realized
also for unequal twist angle configurations once the decoupling between slow
and fast length scales is performed. The large energy gap between flat and
remote bands compared to the typical Coulomb energy scale makes ABA stacking
eTTG an ideal playground for studying Fractional Chern insulators in higher
Chern number bands.

#### Acknowledgments —

We are grateful to Jie Wang, Andrei Bernevig, Jed Pixley, Nicolas Regnault and
Raquel Queiroz for insightful discussions. D.G. also acknowledge discussions
held at the 2023 Quantum Geometry Working Group meeting that took place at the
Flatiron institute where he was introduced to some of the concepts presented
here. We acknowledge support by the French National Research Agency (project
TWISTGRAPH, ANR-21-CE47-0018). The Flatiron Institute is a division of the
Simons Foundation.

Note added: after the completion of the draft we become aware of the work by
Trithep Devakul et al. [90] which overlapped with part of our results.

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Supplementary material for: “ ”

Daniele Guerci1, Yuncheng Mao2, and Christophe Mora2

1 Center for Computational Quantum Physics, Flatiron Institute, 162 5th
Avenue, NY 10010, USA
2 Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques,
75013 Paris, France

These supplementary materials contain the details of analytic calculations as
well as additional numerical details supporting the results presented in the
main text. It is structured as follows. In Sec. A we introduce the local
Hamiltonian providing the effective lattice model describing adiabatically the
slowly varying supermoiré length scale. In Sec. B we set the notation used in
the main text. In Sec. C and D we provide the symmetries for the ABA and AAA
stacking regions and in Sec. C.1 the symmetry constraint on the $C=2$ zero
mode solution. Moreover, in Sec. D.1 we prove that the spectrum is fully
connected along the high-symmetry lines connected AAA sites. Sec. E introduce
complex notation useful for finding the zero mode solutions in the chiral
limit. In Sec. F we provide some basic properties of the Jacobi functions.
Finally, in Sec. G we introduce the Wronskian which plays a crucial role for
finding the analytical expression of the Chern $2$ flat band.

##  Appendix A Length scales decoupling and local Hamiltonian

In this section we recall some results derived in Ref. [68] for generic
relative twist $\theta_{12}$ and $\theta_{23}$. Specializing the discussion to
the case of equal twist angle $\theta_{1}/\theta_{2}\simeq 1$ we define the
moiré modulation:

$\bm{q}_{1}=\bar{\theta}\,{\bf u}_{z}\wedge{\bf K}$ (1)

with ${\bf u}_{z}$ axis perpendicular to the TBG plane, ${\bf K}$ graphene
Dirac point of length $|{\bf K}|=K_{D}=4\pi/3a_{\rm G}$,
$\bar{\theta}=(\theta_{1}+\theta_{2})/2$. The supermoiré wave vector is

$\delta{\bm{q}}_{1}=\bar{\theta}^{2}\,{\bf u}_{z}\wedge{\bf u}_{z}\wedge{\bf
K}+\delta\theta\,{\bf u}_{z}\wedge{\bf K},$ (2)

where $\delta\theta=\theta_{2}-\theta_{1}$ is the angle deviation. Their
principal axis are rotated by $90^{\circ}$ with respect to each other in the
equal-angle case $\delta\theta=0$ and parallel in the opposite limiting case
of $\bar{\theta}^{2}\ll\delta\theta$ [67]. It is worth stressing that the
supermoiré pattern develops even when the twist angles are exactly equal
$\theta_{1}=\theta_{2}$. The single-particle band spectrum is described within
a continuum model where the Dirac cones in each layer are coupled by the
transverse tunneling of electrons. In the ${\bf K}$ valley, it reads:

$H_{K}=\begin{pmatrix}\hat{\bm{k}}\cdot\bm{\sigma}&\alpha\sum_{j=1}^{3}T_{j}\,e^{-i\bm{q}_{j}\cdot\bm{r}}e^{-i\delta\bm{q}_{j}\cdot\bm{r}/2}&0\\\
h.c.&\hat{\bm{k}}\cdot\bm{\sigma}&\alpha\sum_{j=1}^{3}T_{j}\,e^{-i\bm{q}_{j}\cdot\bm{r}}e^{+i\delta\bm{q}_{j}\cdot\bm{r}/2}\\\
0&h.c.&\hat{\bm{k}}\cdot\bm{\sigma}\end{pmatrix}.$ (3)

Given $|\delta\bm{q}_{j}|\ll|\bm{q}_{j}|$ the slow variables are the phases:

$\phi_{j}(\bm{r})=\delta\bm{q}_{j}\cdot\bm{r}/2$ (4)

which vary over the supermoiré lengthscale. On shorter scales, we can
approximate them to be constant $\phi_{j}(\bm{r})\simeq\phi_{j}$ introducing
the local Hamiltonian:

$H_{\rm
eTTG}(\bm{r},{\bm{\phi}})=\begin{pmatrix}v_{F}\hat{\bm{k}}\cdot\bm{\sigma}&T(\bm{r},{\bm{\phi}})&0\\\
h.c.&v_{F}\hat{\bm{k}}\cdot\bm{\sigma}&T(\bm{r},-{\bm{\phi}})\\\
0&h.c.&v_{F}\hat{\bm{k}}\cdot\bm{\sigma}\end{pmatrix},$ (5)

given in the manuscript. In fact we emphasize that $\phi_{j}$ are
approximately constant on the moiré scale but still depend on the position
${\bf R}$ evolving on the supermoiré superlattice and leading to the Chern
mosaic discussed in the manuscript.

##  Appendix B Conventions

We defined $\bm{q}_{j}=ie^{2(j-1)\pi/3}$ where momenta are measured in unit of
$k_{\theta}=\theta K_{D}$ with $K_{D}=4\pi/3a_{\rm G}$ and we employed the
complex number notation. In unit of $k_{\theta}$ the reciprocal lattice
vectors are given by:

$\bm{b}_{1}=\bm{q}_{1}-\bm{q}_{2}=\sqrt{3}e^{i\pi/3},\quad\bm{b}_{2}=\bm{q}_{1}-\bm{q}_{3}=\sqrt{3}e^{2i\pi/3}.$
(6)

In unit of $1/k_{\theta}$ the lattice vectors are given by:

$\bm{a}_{1}=4\pi e^{i\pi/6}/3,\quad\bm{a}_{2}=4\pi e^{5i\pi/6}/3.$ (7)

We define $K=\bm{q}_{1}$ and $K^{\prime}=-\bm{q}_{1}$.

$\begin{array}[]{c||c|c|c|c|c|c}r&a_{1}&a_{2}&a_{1}+a_{2}&r_{0}&r_{1}&r_{2}\\\
\hline\cr\hline\cr e^{iK\cdot r}&\omega&\omega&\omega&1&\omega&\omega^{*}\\\
\hline\cr e^{iK^{\prime}\cdot
r}&\omega^{*}&\omega^{*}&\omega^{*}&1&\omega^{*}&\omega\end{array}\,\,,\quad\omega=e^{2i\pi/3},$
(8)

where $r_{j}=C^{j}_{3z}r_{0}$ with $j=0,1,2$ and $r_{0}=(a_{1}-a_{2})/3$.
Finally, the original Hamiltonian is written in the basis:

$\Phi=\begin{pmatrix}\psi_{1}&\chi_{1}&\psi_{2}&\chi_{2}&\psi_{3}&\chi_{3}\end{pmatrix}^{T},$
(9)

where $1,2,3$ refer to top, middle and bottom layers, respectively. The two
components $\psi$ and $\chi$ refer to A and B sublattice, respectively.

##  Appendix C Symmetries and protected Dirac cones for ABA stacking

In the following we discuss the symmetry properties of eTTG with respect to
the ABA stacking configuration. The Hamiltonian representative of the ABA
region is obtained setting the values of the phases ${\bm{\phi}}_{\rm
ABA}=(\phi_{1},\phi_{2},\phi_{3})=(0,\varphi,-\varphi)$ with $\varphi=2\pi/3$,
notice that one of the phases say $\phi_{1}$ can be gauged away by global
phase shift $\phi_{j}\to\phi_{j}+\phi_{0}$. The Hamiltonian in the basis 9
reads:

$\mathcal{H}_{\rm ABA}\equiv H_{\rm eTTG}({\bm{\phi}}_{\rm
ABA})=\begin{pmatrix}\hat{\bm{k}}\cdot\bm{\sigma}&T_{\omega^{*}}(\bm{r})&0\\\
h.c.&\hat{\bm{k}}\cdot\bm{\sigma}&T_{\omega}(\bm{r})\\\
0&h.c.&\hat{\bm{k}}\cdot\bm{\sigma}\end{pmatrix}.$ (10)

where the tunneling matrices are:

$T_{\omega^{*}}(\bm{r})\equiv
T(\bm{r},{\bm{\phi}})=\alpha\sum^{3}_{j=1}\left(\omega^{*}\right)^{j-1}T_{j}e^{-i\bar{\bm{q}}_{j}\cdot\bm{r}},\quad
T_{\omega}(\bm{r})\equiv
T(\bm{r},-{\bm{\phi}})=\alpha\sum^{3}_{j=1}\omega^{j-1}T_{j}e^{-i\bar{\bm{q}}_{j}\cdot\bm{r}}.$
(11)

The $C_{3z}$ symmetry takes the form:

$C_{3z}=\begin{pmatrix}\omega^{*}e^{i\varphi\sigma_{z}}&0&0\\\
0&e^{i\varphi\sigma_{z}}&0\\\
0&0&\omega^{*}e^{i\varphi\sigma_{z}}\end{pmatrix}=\begin{pmatrix}\omega^{*}&0&0\\\
0&1&0\\\ 0&0&\omega^{*}\end{pmatrix}\otimes
e^{i\varphi\sigma_{z}}=\Omega\otimes e^{i\varphi\sigma_{z}},$ (12)

and

$\mathcal{H}_{\rm ABA}(C_{3z}\bm{r})=C_{3z}\mathcal{H}_{\rm
ABA}(\bm{r})C^{\dagger}_{3z}.$ (13)

We observe that $C_{2x}$ is defined as:

$C_{2x}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\otimes\sigma_{x}=\mathcal{M}_{x}\otimes\sigma_{x},$ (14)

is broken, since under $C_{2x}$ we map the ABA to the BAB stacking. This can
be readily realized observing that:

$C_{2x}{\mathcal{H}_{\rm
ABA}}(\bm{r})C^{\dagger}_{2x}=1\otimes(C_{2x}\bm{k})\cdot\bm{\sigma}+\begin{pmatrix}0&T_{\omega}(C_{2x}\bm{r})&0\\\
h.c.&0&T_{\omega^{*}}(C_{2x}\bm{r})\\\ 0&h.c.&0\end{pmatrix}={\mathcal{H}_{\rm
BAB}}(C_{2x}\bm{r}).$ (15)

Similarly, we find:

$C_{2z}T{\mathcal{H}_{\rm
ABA}}(\bm{r})(C_{2z}T)^{\dagger}=-1\otimes\bm{k}\cdot\bm{\sigma}+\begin{pmatrix}0&T_{\omega}(-\bm{r})&0\\\
h.c.&0&T_{\omega^{*}}(-\bm{r})\\\ 0&h.c.&0\end{pmatrix}={\mathcal{H}_{\rm
BAB}}(-\bm{r}).$ (16)

We now notice that the combination $C_{2x}C_{2z}T=C_{2y}T$

$C_{2y}T=\mathcal{M}_{x}\otimes\sigma_{0}\mathcal{K}$ (17)

is a symmetry of the model:

$C_{2y}T{\mathcal{H}_{\rm
ABA}}(\bm{r})(C_{2y}T)^{\dagger}=1\otimes\left(C_{2y}\bm{k}\right)\cdot\bm{\sigma}+\begin{pmatrix}0&T_{\omega^{*}}(C_{2y}\bm{r})&0\\\
h.c.&0&T_{\omega}(C_{2y}\bm{r})\\\ 0&h.c.&0\end{pmatrix}={\mathcal{H}_{\rm
ABA}}(C_{2y}\bm{r}).$ (18)

We summarize the symmetries in Table 1.

ABA | Original | Chiral | $k_{+}$ | $k_{-}$ | $r_{+}$ | $r_{-}$
---|---|---|---|---|---|---
$C_{3z}$ | $\Omega\otimes e^{i\varphi\sigma_{z}}$ | $e^{i\varphi\sigma_{z}}\otimes\Omega$ | $\omega$ | $\omega^{*}$ | $\omega$ | $\omega^{*}$
$P$ | $\mathcal{M}\otimes\sigma_{0}$ | $\sigma_{0}\otimes\mathcal{M}$ | $-$ | $-$ | $-$ | $-$
$C_{2x}C_{2z}T$ | $\mathcal{M}\otimes\mathcal{K}$ | $\mathcal{K}\otimes\mathcal{M}$ | $-k_{-}$ | $-k_{+}$ | $-r_{-}$ | $-r_{+}$
$\Lambda_{z}$ | $\tau_{0}\otimes\sigma_{z}$ | $\sigma_{z}\otimes\tau_{0}$ | $+$ | $+$ | $+$ | $+$
$P\otimes\Lambda_{z}$ | $\mathcal{M}\otimes\sigma_{z}$ | $\sigma_{z}\otimes\mathcal{M}$ | $-$ | $-$ | $-$ | $-$
AAA | Original | Chiral | $k_{+}$ | $k_{-}$ | $r_{+}$ | $r_{-}$
$C_{2x}$ | $\mathcal{M}_{x}\otimes\sigma_{x}$ | $\sigma_{x}\otimes\mathcal{M}_{x}$ | $k_{-}$ | $k_{+}$ | $r_{-}$ | $r_{+}$
$C_{3z}$ | $\mathbf{1}\otimes e^{i\varphi\sigma_{z}}$ | $e^{i\varphi\sigma_{z}}\otimes\mathbf{1}$ | $\omega$ | $\omega^{*}$ | $\omega$ | $\omega^{*}$
$C_{2z}T$ | $\mathbf{1}\otimes\sigma_{x}\mathcal{K}$ | $\sigma_{x}\otimes\mathbf{1}\mathcal{K}$ | $+$ | $+$ | $-$ | $-$
$P$ | $\mathcal{M}\otimes\sigma_{0}$ | $\sigma_{0}\otimes\mathcal{M}$ | $-$ | $-$ | $-$ | $-$
$\Lambda_{z}$ | $\tau_{0}\otimes\sigma_{z}$ | $\sigma_{z}\otimes\tau_{0}$ | $+$ | $+$ | $+$ | $+$
$P\otimes\Lambda_{z}$ | $\mathcal{M}\otimes\sigma_{z}$ | $\sigma_{z}\otimes\mathcal{M}$ | $-$ | $-$ | $-$ | $-$
Table 1: Symmetries of the local Hamiltonian for the two different high-
symmetry stacking configurations for the K-valley. Original and chiral shows
the matrix expression of the symmetry in the two different basis. Finally,
last columns show how the symmetry acts on $k_{\pm}=k_{x}\pm ik_{y}$ and
$r_{\pm}=x\pm iy$. We further notice that $\mathcal{K}$ is the complex
conjugation operator and the ABA stacking breaks $C_{2z}T$ and $C_{2x}$
symmetries. The combination $C_{2x}C_{2z}T$ is still a symmetry of the model.

To the aim of showing the protection of three Dirac cones at $\Gamma$, $K$ and
$K^{\prime}$ we further discuss the spatial symmetries in the ABA
configuration and the symmetry group they generate. As detailed above, the
model exhibits two spatial symmetries in that case, $C_{3z}$ and $C_{2y}T$.
They form the magnetic space group $P32^{\prime}1$ ($\\#150.27$ in the BNS
notation). The irreducible representations at the high-symmetry momenta
$\Gamma$ and $K$ ($K^{\prime}$) verify the $C_{3}$ point group character table
and are all one-dimensional. The absence of two-dimensional representations
noticeably prevents the stabilization of Dirac cones in the spectrum.

It can be understood more directly by considering an eigenstate of $C_{3z}$,
$C_{3z}|\psi_{+}\rangle=\omega|\psi_{+}\rangle$. From the relation

$(C_{2y}T)C_{3z}(C_{2y}T)=C_{3z}^{-1},$ (19)

we obtain $C_{3z}C_{2y}T|\psi_{+}\rangle=\omega\,C_{2y}T|\psi_{+}\rangle$.
$C_{2y}T$ thus does not circulate between the eigenstates of $C_{3z}$ and
cannot protect a twofold degeneracy.

The three zero-energy Dirac cones arising in the band spectrum of ABA trilayer
graphene are therefore only stable in the presence of the additional particle-
hole symmetry $P$. Since $P$ and $C_{3z}$ commute, applying $P$ on a given
state does not change its $C_{3z}$ eigenvalue. Therefore, if the spectrum, at
$\Gamma$ or $K/K^{\prime}$, hosts two states with $C_{3z}$ eigenvalues
$\omega$, $\omega^{*}$ near charge neutrality (and no other states), particle-
hole symmetry $P$ automatically pins these two states at zero energy. This is
proven by contradiction: if we assume that the two states sit at opposite non-
vanishing energies, then $P$ permutes them. This is however impossible since
$P$ cannot change the $C_{3z}$ eigenvalue which completes the proof. In fact,
$P$ restricted to these two states must be the identity as it commutes with
$C_{3z}$. It further shows that breaking $C_{3z}$ does not lift the Dirac
crossings as the trivial (identity) representation of $P=\mathbb{I}_{2}$
cannot deform continuously to the traceless $\sigma_{x}$ matrix permuting
states with opposite non-zero energies.

In summary, the Dirac cone at $\Gamma$ is explained using $C_{3z}$ and $P$.
$K$ and $K^{\prime}$ are however not stable under $P$ \- which permutes $K$
and $K^{\prime}$ \- and the stability of their Dirac cones must come from a
different operator. Since the symmetry $C_{2y}T$ also permutes $K$ and
$K^{\prime}$, the combination $P^{\prime}=PC_{2y}T$ leaves them invariant and
acts as a (anti-unitary) particle-hole operator. Using that

$P^{\prime}C_{3z}P^{\prime}=C_{3z}^{-1},$ (20)

one verifies that $P^{\prime}$ cannot permute the $C_{3z}$ eigenvalue between
$\omega$ and $\omega^{\prime}$ which implies that an isolated pair of states
at $K$ or $K^{\prime}$ with $C_{3z}$ eigenvalues $(\omega,\omega^{*})$ is
degenerate and pinned at zero energy. In this subspace,
$P^{\prime}=\mathcal{K}\,\mathbb{I}_{2}$ and the Dirac cone survives a
breaking of $C_{3z}$.

###  C.1 Symmetry constraints on the Chern $2$ flat band

Here we provide the symmetry constraint on the $C=2$ zero mode that leads to
the analytical expression 14 given in the main text.

To start with we observe that the $C_{3z}$ symmetry implies $\psi_{\Gamma
1/3}(\bm{r})=\omega\psi_{\Gamma 1/3}(C_{3z}\bm{r})$ and $\psi_{\Gamma
2}(\bm{r})=\psi_{\Gamma 2}(C_{3z}\bm{r})$ resulting in the condition
$\psi_{\Gamma 1/3}(0)=0$. However, differently from the B-sublattice component
we find $\psi_{\Gamma 2}(0)\neq 0$ at the magic angle. To proceed further we
now observe that ${\cal M}\mathcal{D}^{\dagger}_{1}(-\bm{r}){\cal
M}=-\mathcal{D}^{\dagger}_{1}(\bm{r})$ with $\mathcal{M}$ defined in Eq. 3 of
the manuscript. At the $\Gamma$ point, it leads to $\psi_{\Gamma
2}(\bm{r})=\psi_{\Gamma 2}(-\bm{r})$, and $\psi_{\Gamma
1}(\bm{r})=\psi_{\Gamma 3}(-\bm{r})$ which connects the top and bottom
components at inversion-symmetric positions. Looking at $K$ and
$K^{\prime}=-K$ points the $C_{3z}$ symmetry leads to the identities
$\psi_{\pm K1/3}(\bm{r})=\psi_{\pm K1/3}(C_{3z}\bm{r})$ and $\psi_{\pm
K2}(\bm{r})=\omega^{*}\psi_{\pm K2}(C_{3z}\bm{r})$ consistent with
$\psi_{K}=(1,0,0)$ and $\psi_{K^{\prime}}=(0,0,-1)$ at $w_{\rm AB}=0$. A
direct consequence is that $\psi_{\pm K2}(0)=0$. In contrast to $C_{3z}$, the
particle-hole symmetry $\mathcal{M}$ couples $K$ and $K^{\prime}$ we find
$\psi_{K1/3}(\bm{r})=-\psi_{-K1/3}(-\bm{r})$ and
$\psi_{K2}(\bm{r})=\psi_{-K2}(-\bm{r})$ resulting in $\psi_{\pm
K1}(0)=-\psi_{\mp K3}(0)$. Moreover and remarkably, we numerically find that,
right at the magic angle $\theta_{\rm ABA}$

$\psi_{K1}(0)=-\psi_{K3}(0),$ (21)

which results in the fact that the two spinors $\psi_{K}(\bm{r}=0)$ and
$\psi_{K^{\prime}}(\bm{r}=0)$ become in fact identical. We now generalize the
argument in Ref. [38, 50] defining the Wronskian for the case of three layers.
To this aim we consider three zero mode solutions ${\bm{\psi}}_{\bm{k}_{j}}$
with momenta ${\bm{k}}_{1,2,3}$ and we introduce the triple product

$W(\bm{r})={\bm{\psi}}_{\bm{k}_{1}}(\bm{r})\cdot\left[{\bm{\psi}}_{\bm{k}_{2}}(\bm{r})\times{\bm{\psi}}_{\bm{k}_{3}}(\bm{r})\right],$
(22)

which satisfies $\bar{\partial}W(\bm{r})=0$, see Sec. G, proving the
holomorphy of $W(\bm{r})$. Since $W(\bm{r})$ also satisfies Bloch periodicity
(being the product of three Bloch-periodic functions) and cannot have any pole
in the complex plane, by Lioville’s theorem it must be constant in space.
Taking $\bm{k}_{2}=K$ and $\bm{k}_{3}=K^{\prime}$ and an arbitrary momentum
$\bm{k}_{1}$, the quantity $W(\bm{r}=0)=0$ since ${\bm{\psi}}_{K}(0)$ and
${\bm{\psi}}_{K^{\prime}}(0)$ are collinear as discussed above. Therefore the
triple product $W(\bm{r})$ vanishes in this case and the three spinors
${\bm{\psi}}_{\bm{k}_{1}}(\bm{r})$, ${\bm{\psi}}_{K}(\bm{r})$ and
${\bm{\psi}}_{K^{\prime}}(\bm{r})$ are coplanar for all $\bm{r}$. Since this
is true for arbitrary $\bm{k}_{1}$, it implies that all zero-energy states
belong to the same plane generated by ${\bm{\psi}}_{K}(\bm{r})$ and
${\bm{\psi}}_{K^{\prime}}(\bm{r})$ and the triple product $W(\bm{r})$ is
identically vanishing irrespective of the values of $\bm{r}$ and
$\bm{k}_{1,2,3}$. Thus, an analytical expression also emerges from the
previous analysis and the wavefunction for the A-polarized flat band takes the
form (up to $\bm{k}$-dependent prefactor) in Eq. 14 of the manuscript.

##  Appendix D Symmetries for AAA stacking

In the following we discuss the symmetry properties of eTTG with respect to
the AAA stacking configuration.

The AAA stacking configuration ${\bm{\phi}}_{\rm AAA}=(0,0)$ recently
discussed in the preprint [77] is described by the Hamiltonian [78]:

$\mathcal{H}_{\rm AAA}\equiv H_{\rm eTTG}({\bm{\phi}}_{\rm
AAA})=\begin{pmatrix}\hat{\bm{k}}\cdot\bm{\sigma}&T_{0}(\bm{r})&0\\\
h.c.&\hat{\bm{k}}\cdot\bm{\sigma}&T_{0}(\bm{r})\\\
0&h.c.&\hat{\bm{k}}\cdot\bm{\sigma}\end{pmatrix}.$ (23)

In this case the tunneling matrix simply reads:

$T_{0}(\bm{r})=\alpha\sum^{3}_{j=1}T_{j}e^{-i\bar{\bm{q}}_{j}\cdot\bm{r}},$
(24)

under $C_{3z}$ it transforms as:

$e^{-i\varphi\sigma_{z}}T_{0}(C_{3z}\bm{r})e^{i\varphi\sigma_{z}}=T_{0}(\bm{r}).$
(25)

We readily understand that the three layers in this case are characterized by
the same $C_{3z}$ eigenvalue. Thus, the $C_{3z}$ operator reads:

$C_{3z}=\begin{pmatrix}e^{i\varphi\sigma_{z}}&0&0\\\
0&e^{i\varphi\sigma_{z}}&0\\\
0&0&e^{i\varphi\sigma_{z}}\end{pmatrix}=\mathbf{1}\otimes
e^{i\varphi\sigma_{z}},$ (26)

and

$\mathcal{H}_{\rm AAA}(R_{\varphi}\bm{r})=C_{3z}\mathcal{H}_{\rm
AAA}(\bm{r})C^{\dagger}_{3z}.$ (27)

For the sake of completeness we list the remaining symmetries:

$C_{2x}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{pmatrix}\otimes\sigma_{x}=\mathcal{M}_{x}\otimes\sigma_{x},$ (28)

so that

$\mathcal{H}_{\rm AAA}(C_{2x}\bm{r})=C_{2x}\mathcal{H}_{\rm
AAA}(\bm{r})C^{\dagger}_{2x}.$ (29)

In the previous expression $\mathcal{M}_{x}$ corresponds to the operator which
flips bottom and top layer leaving the mid one invariant. Table 1 summarizes
the symmetries of the model. We emphasize that the symmetries $P$, $C_{3z}$
and $C_{2z}T$ protect the Dirac cones at $\Gamma$, $K$ and $K^{\prime}$ of the
moiré Brillouin zone.

###  D.1 Fully connected spectrum at the domain wall regions

In this section we prove by symmetry arguments that the spectrum is fully
connected at the domain wall regions. To this aim we shift the position of the
middle layer along the $x$ axis with respect to the AAA stacking. By
construction, this choice, with $\phi_{2}=\phi_{3}$ where we used a gauge
transformation to set $\phi_{1}=0$, does not break the $C_{2x}$ symmetry but
breaks $C_{3z}$ and $C_{2z}T$. The particle-hole symmetry $P$ 3 is also not
broken such that there is particle-hole symmetry at all exceptional points.
The corresponding moire spectrum is displayed in the right panel of Fig. 4c in
the manuscript for $\phi_{2}=\phi_{3}=-2\pi/5$ where a gap opening at $K$
($K^{\prime}$) is observed resulting from the breaking of $C_{3z}$ and
$C_{2z}T$. In contrast, $P$ still applies which stabilizes the Dirac point at
$\Gamma$.

| $E$ |  2$C_{3}$ |  3$C_{2}$ |  |  $E$ |  3$C_{2}$
---|---|---|---|---|---|---
$\quad\Gamma_{1}\quad$ |  1 |  1 |  1 |  $M_{1}$ |  1 |  1
$\quad\Gamma_{2}\quad$ |  1 |  1 |  -1 |  $M_{2}$ |  1 |  -1
$\quad\Gamma_{3}\quad$ |  2 |  -1 |  0 |  |  |
Table 2: Character table of $C_{3v}$ at $\Gamma$ and $M$. $E$, $C_{3}$ and
$C_{2}$ represent the conjugation classes of the identity, $C_{3z}$ and
$C_{2x}$, respectively.

Furthermore, we also see that the bands are all connected. The proof of this
full connectivity goes along the same lines as in Ref. [78]. Suppose we take
an isolated set of $N$ bands - $P$ allows us to choose this set symmetrically
around zero energy. At the $\Gamma$ point, there are two one-dimensional
representations of $C_{2x}$, $\Gamma_{1}$ and $\Gamma_{2}$, with the
characters $+1$ and $-1$ see Tab. 2. $P$ implies a third two-dimensional
representation (the Dirac point) with character $0$, analytical continued from
$C_{2x}=\sigma_{x}$ as obtained for $\phi_{1}=\phi_{2}=0$. At the $M$ point
and on the $\Gamma$-$M$ line, again two representations $M_{1}=+1$, $M_{2}=-1$
see Tab. 2. Writting that $m_{\Gamma_{1}}+m_{\Gamma_{2}}+2m_{\Gamma_{3}}=N$
(where $m_{\Gamma}$ is the multiplicity of the representation $\Gamma$),
$m_{M_{1}}+m_{M_{2}}=N$, and that the full character is preserved along the
$\Gamma$-$M$ line, namely

$m_{\Gamma_{1}}-m_{\Gamma_{2}}=m_{M_{1}}-m_{M_{2}},$ (30)

we find

$m_{M_{2}}=m_{\Gamma_{2}}+m_{\Gamma_{3}}.$ (31)

This last expression can be checked modulo two on the zero-energy axis - as
particle-hole partners within the $N$ bands share the same representation and
thus count as a multiple of two, i.e. do not change the parity. The proof by
contradiction is completed using that, for $w_{\rm AA}=w_{\rm AB}=0$, there is
a Dirac point at $\Gamma$ and not at $M$ such that $m_{\Gamma_{3}}=1$ but
$m_{M_{2}}=m_{\Gamma_{2}}=0$, in contradiction with Eq. (31).

##  Appendix E The kinetic term in complex notation and chiral limit tunneling
matrices

The kinetic term reads:

$\hat{\bm{k}}\cdot\bm{\sigma}=\begin{pmatrix}0&k_{x}-ik_{y}\\\
k_{x}+ik_{y}&0\end{pmatrix},\quad k_{a}=-i\partial_{a}/k_{\theta},$ (32)

where $k_{\theta}=\theta K_{D}$. We use the complex notation:

$z=k_{\theta}\frac{x+iy}{\sqrt{2}},\quad\bar{z}=k_{\theta}\frac{x-iy}{\sqrt{2}},$
(33)

so that $x=(z+\bar{z})/[\sqrt{2}k_{\theta}]$ and
$y=-i(z-\bar{z})/[\sqrt{2}k_{\theta}]$. Finally, we have:

$\partial_{x}-i\partial_{y}=\sqrt{2}k_{\theta}\partial,\quad\partial_{x}+i\partial_{y}=\sqrt{2}k_{\theta}\bar{\partial}.$
(34)

As a result we have

$\hat{\bm{k}}\cdot\bm{\sigma}=\begin{pmatrix}0&-i\sqrt{2}\partial\\\
-i\sqrt{2}\bar{\partial}&0\end{pmatrix}.$ (35)

In the chiral limit we have:

$\begin{split}T_{1}=\alpha\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\,T_{2}=\alpha\begin{pmatrix}0&\omega^{*}\\\
\omega&0\end{pmatrix},\,T_{3}=\alpha\begin{pmatrix}0&\omega\\\
\omega^{*}&0\end{pmatrix},\end{split}$ (36)

where $\alpha=w_{\rm AB}/v_{F}k_{\theta}$. As a result of the different
$C_{3z}$ properties at the AAA and ABA-stacking we have three different scalar
potentials:

$\begin{split}U_{0}(\bm{r})=\alpha\sum^{3}_{j=1}e^{-i\bm{q}_{j}\cdot\bm{r}},\,U_{\omega}(\bm{r})=\alpha\sum^{3}_{j=1}\omega^{j-1}e^{-i\bm{q}_{j}\cdot\bm{r}},\,U_{\omega^{*}}(\bm{r})=\alpha\sum^{3}_{j=1}\left(\omega^{*}\right)^{j-1}e^{-i\bm{q}_{j}\cdot\bm{r}}.\end{split}$
(37)

We have:

$\begin{split}T_{0}=\begin{pmatrix}0&U_{\omega^{*}}\\\
U_{\omega}&0\end{pmatrix},\,T_{\omega}=\begin{pmatrix}0&U_{0}\\\
U_{\omega^{*}}&0\end{pmatrix},\,T_{\omega^{*}}=\begin{pmatrix}0&U_{\omega}\\\
U_{0}&0\end{pmatrix}.\end{split}$ (38)

Moreover, we have the following relations:

$\begin{split}U^{*}_{0}(\bm{r})=U_{0}(-\bm{r}),\,U^{*}_{\omega}(\bm{r})=U_{\omega^{*}}(-\bm{r}),\,U^{*}_{\omega^{*}}(\bm{r})=U_{\omega}(-\bm{r}).\end{split}$
(39)

##  Appendix F Basic properties of Jacobi theta function

In the maintext we have utilized the Jacobi theta-function:

$\vartheta_{1}(z,\omega)=\sum_{n\in\mathbb{Z}}e^{i\pi\omega(n+1/2)^{2}}e^{2i\pi(z-1/2)(n+1/2)}.$
(40)

Here we recall some basic properties useful to determine the boundary
conductions of the ideal flat bands. We first notice that

$\begin{split}&\vartheta_{1}[z+\omega,\omega]=-e^{-i\pi\omega}e^{-2\pi
iz}\vartheta_{1}[z,\omega],\\\
&\vartheta_{1}[z-\omega,\omega]=-e^{-i\pi\omega}e^{2\pi
iz}\vartheta_{1}[z,\omega].\end{split}$ (41)

In addition we also need:

$\vartheta_{1}[z\pm 1,\omega]=-\vartheta_{1}[z,\omega].$ (42)

Other useful relations are:

$\begin{split}&\vartheta_{1}[\bar{z}-\omega^{*},-\omega^{*}]=-e^{i\pi\omega^{*}}e^{-2\pi
i\bar{z}}\vartheta_{1}[\bar{z},-\omega^{*}],\\\
&\vartheta_{1}[\bar{z}+\omega^{*},-\omega^{*}]=-e^{i\pi\omega^{*}}e^{2\pi
i\bar{z}}\vartheta_{1}[\bar{z},-\omega^{*}].\end{split}$ (43)

Finally, we also have

$\vartheta_{1}[\bar{z}\pm 1,-\omega^{*}]=-\vartheta_{1}[\bar{z},-\omega^{*}].$
(44)

##  Appendix G Wronskian for the ABA-case: counting the number of independent
components

In this Section we introduce the Wronskian [38, 50] which determines the
vector space dimension of the zero mode solution in $\bm{k}$-space. Given
three solutions of the zero mode equations

$\mathcal{D}_{1}{\bm{\chi}}_{\bm{k}}=0,\quad\mathcal{D}^{\dagger}_{1}{\bm{\psi}}_{\bm{k}}=0,$
(45)

$\bm{\psi}_{a,b,c}$ and $\bm{\chi}_{a,b,c}$ with $a,b,c$ denoting different
$\bm{k}$ points we define the triple product:

$W=\bm{\psi}_{a}\cdot\left(\bm{\psi}_{b}\times\bm{\psi}_{c}\right),\quad\overline{W}=\bm{\chi}_{a}\cdot\left(\bm{\chi}_{b}\times\bm{\chi}_{c}\right).$
(46)

Generically $W=W(z,\bar{z})$ and $\overline{W}=\overline{W}(z,\bar{z})$, in
the following we show that the two quantities are holomorphic and
antiholomorphic functions, respectively. To this aim we first observe that the
zero mode equation can be written as:

$\left(\bar{\partial}+\overline{A}\right)\bm{\psi}_{\bm{k}}=0,\quad\left(\partial+A\right)\bm{\chi}_{\bm{k}}=0,$
(47)

where we have introduced the gauge fields:

$\overline{A}=\frac{i}{\sqrt{2}}\begin{pmatrix}0&U_{0}(\bm{r})&0\\\
U_{\omega^{*}}(-\bm{r})&0&U_{\omega^{*}}(\bm{r})\\\
0&U_{0}(-\bm{r})&0\end{pmatrix},\quad
A=\frac{i}{\sqrt{2}}\begin{pmatrix}0&U_{\omega}(\bm{r})&0\\\
U_{0}(-\bm{r})&0&U_{0}(\bm{r})\\\ 0&U_{\omega}(-\bm{r})&0\end{pmatrix}.$ (48)

Without losing generality we now focus on $\overline{W}$, in this case:

$\partial\overline{W}=\sum_{ljk}\varepsilon_{ljk}\partial\left[\chi_{a,l}\chi_{b,j}\chi_{c,k}\right]=-\sum_{ljks}\varepsilon_{ljk}A_{ks}\left[\chi_{b,l}\chi_{c,j}\chi_{a,s}-\chi_{a,l}\chi_{c,j}\chi_{b,s}+\chi_{a,l}\chi_{b,j}\chi_{c,s}\right].$
(49)

Now we observe that $A$ is off-diagonal. Thus, we have:

$\begin{split}\partial\overline{W}=&-\sum^{k\neq
s}_{ks}A_{ks}\sum_{j}\left[\varepsilon_{sjk}\left(\chi_{b,s}\chi_{c,j}\chi_{a,s}-\chi_{a,s}\chi_{c,j}\chi_{b,s}+\chi_{a,s}\chi_{b,j}\chi_{c,s}\right)+\right.\\\
&\left.\varepsilon_{jsk}\left(\chi_{b,j}\chi_{c,s}\chi_{a,s}-\chi_{a,j}\chi_{c,s}\chi_{b,s}+\chi_{a,j}\chi_{b,s}\chi_{c,s}\right)\right]\\\
=&-\sum^{k\neq
s}_{ksj}A_{ks}\varepsilon_{sjk}\left[\chi_{b,s}\chi_{c,j}\chi_{a,s}-\chi_{a,s}\chi_{c,j}\chi_{b,s}+\chi_{a,s}\chi_{b,j}\chi_{c,s}-\chi_{b,j}\chi_{c,s}\chi_{a,s}+\chi_{a,j}\chi_{c,s}\chi_{b,s}-\chi_{a,j}\chi_{b,s}\chi_{c,s}\right]=0.\end{split}$
(50)

As a result $\overline{W}=\overline{W}(\bar{z})$ is antiholomorphic function.
Similarly, it is possible to prove $\bar{\partial}W=0$ so that $W=W(z)$. Since
$W$ is regular it must be a constant. The only constant consistent with the
boundary conditions is zero. The latter condition implies that the zero mode
solution leaves in a 2D plane. For the B-sublattice component the vanishing of
${\bm{\chi}}_{\Gamma}(0)=0$ at the magic angle implies that solutions are
collinear $\bm{\chi}_{b}\times\bm{\chi}_{c}=0$ and described by a Landau level
like state with $C=1$.