# Higher vortexability: zero field realization of higher Landau levels

Manato Fujimoto  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Department of Applied Physics, The University of
Tokyo, Hongo, Tokyo, 113-8656, Japan  Daniel E. Parker  Department of Physics,
University of California at Berkeley, Berkeley, CA 94720, USA  Department of
Physics, University of California at San Diego, La Jolla, California 92093,
USA  Junkai Dong  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Eslam Khalaf  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Ashvin Vishwanath  Department
of Physics, Harvard University, Cambridge, Massachusetts 02138, USA  Patrick
Ledwith  Department of Physics, Harvard University, Cambridge, Massachusetts
02138, USA

###### Abstract

The rise of moiré materials has led to experimental realizations of integer
and FCIs in small or vanishing magnetic fields. At the same time, a set of
minimal conditions sufficient to guarantee an Abelian fractional state in a
flat band were identified, namely “ideal" or “vortexable" quantum geometry.
Such vortexable bands share essential features with the LLL, while excluding
the need for more fine-tuned aspects such as flat Berry curvature. A natural
and important generalization is to ask if such conditions can be extended to
capture the quantum geometry of higher Landau levels, particularly the first
(1LL), where non-Abelian states at $\nu=1/2,2/5$ are known to be competitive.
The possibility of realizing these states at zero magnetic field, and perhaps
even more exotic ones, could become a reality if we could identify the
essential structure of the 1LL in Chern bands. In this work, we introduce a
precise definition of 1LL quantum geometry, along with a figure of merit that
measures how closely a given band approaches the 1LL. Periodically strained
Bernal graphene is shown to realize such a 1LL structure even in zero magnetic
field.

Fractional Chern Insulators (FCIs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] realize the
fractionalized charges and anyonic statistics associated with fractional
quantum Hall physics in Chern bands. Such states do not require a magnetic
field and can potentially form at much higher temperatures [11, 12, 13, 14,
15, 16, 17]. They were observed in Harper-Hofstadter bands [11] at high
magnetic field, and in the partially filled Chern bands of twisted bilayer
graphene with the help of a small magnetic field [12]. Not only does the field
reduce the bandwidth [18], but it was also argued to improve the wavefunctions
of the Chern band, pushing them to an “ideal" or “vortexable" limit where the
band becomes a modulated version of the lowest LL (LLL) [19, 20, 21, 22, 23,
24, 25, 26, 27, 28]. More recently, Abelian FCIs at zero magnetic field,
dubbed fractional quantum anomalous Hall (FQAH) states, were observed in
twisted MoTe2 [13, 14, 15, 16], following theoretical works showcasing small
bandwidths and numerical works finding FCI ground states [29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43], and also in pentalayer graphene [17,
44, 45, 46, 47, 48, 49]. As these zero field devices improve, they will likely
run the gamut of Abelian topological orders found in the LLL.

Figure 1: Cartoon depicting the definition, (4), of a first vortexable band,
with band projector $P_{1}$.

To date, experimental realizations of non-Abelian topological order are the
almost-exclusive province of the first LL. Namely, the half-filled first LL in
GaAs [50, 51, 52] and Bernal graphene [53, 54, 55, 56, 57] realize a Pfaffian-
type state with non-Abelian particle statistics[58]. Also, the plateau at
$\nu=2+2/5$ in GaAs [59] may be a non-Abelian phase harboring Fibonacci anyons
[60], a scenario that has received numerical support [60, 61]. Beyond their
immense theoretical importance, such states have practical utility as a
platform for fault-tolerant quantum computation [62, 63]. Realizing non-
Abelian states in Chern bands, at energy scales unachievable with external
magnetic field, would be especially useful in this context. This raises the
question — are there Chern bands at zero magnetic field analogous to the first
LL?

In this work we answer this question by providing a precise definition of what
it means for a band to be similar to the first LL. Our definition makes use of
“vortexability" applied to multi-band systems. A Chern band has first LL
structure if we can find a partner band with a zeroth LL structure such that
the two bands taken together are vortexable. Vortex attachment applied to both
bands produces exact Jain-like [64] ground states in the limit of short-range
repulsive interactions, without the need for further projection when the bands
are degenerate. In general, the partner band is not required to be nearby in
energy. We argue that single-band definitions, such as those based on the
Fubini-Study metric used for the LLL case [65], cannot capture first LL
structure. Below we motivate and explicate our definition, then provide a
concrete example in zero-field superlattice system, periodically strained
Bernal graphene. Our construction is readily extendable to higher Chern bands,
by forming $(n+1)$ Chern band complexes to capture the geometry of the ‘$n$th’
LL.

_LLL-type Quantum Geometry Review—_ We begin by reviewing the point of view
that has recently emerged on “LLL band geometry," through ideal, or
vortexable, Chern bands [19, 23, 20, 21, 22, 66, 67, 24, 25, 28, 68, 69, 26,
27, 70]. These names refer to the following equivalent criteria. (i) The bands
are vortexable [24, 28]: $z\psi=Pz\psi$, where $P$ is the projector onto the
band(s) of interest, $\psi=P\psi$ is a wavefunction in those bands, and
$z=x+iy$ is the vortex function. (ii) The bands satisfy the trace condition
$\operatorname{tr}g({\bm{k}})=\Omega({\bm{k}})$ [71, 10], where
$g_{\mu\nu}({\bm{k}})$ is the Fubini-Study metric and $\Omega({\bm{k}})$ is
the Berry curvature. (iii) The cell-periodic states
$u_{{\bm{k}}a}({\bm{r}})=e^{-i{\bm{k}}\cdot{\bm{r}}}\psi_{{\bm{k}}a}({\bm{r}})$
can be written as holomorphic functions of $k=k_{x}+ik_{y}$ up to
orthonormality (here “$a$" is a band index) [72, 73, 22]. More general vortex
functions [28] with correspondingly generalized trace conditions (suitable for
other unit cell embeddings [74]) are omitted here for concision. For zero
dispersion vortexable bands, Laughlin-like trial states constructed by
attaching vortices (i.e. Jastrow factors) $\prod_{i<j}(z_{i}-z_{j})^{2m}$ to
integer filling states are exact many-body ground states in the limit of short
range repulsive interactions [75, 19, 28]. Thus, these generalizations of the
LLL provide a frustration-free route to realizing the Laughlin states. Note,
in contrast to naive intuition and early work on FCIs, homogeneous Berry
curvature is not a requirement.

Wavefunctions for $C=1$ ideal bands have the form [19, 23]

$\psi({\bm{r}})=f(z)\mathcal{N}_{0}({\bm{r}});\quad\mathcal{N}_{0}^{l}({\bm{r}})=e^{-K({\bm{r}})}\zeta^{l}({\bm{r}})$
(1)

where $f(z)$ is holomorphic, $\zeta^{l}({\bm{r}})$ is an orbital-space spinor
(with layer index $l$, e.g.), and $\phi({\bm{r}})=f(z)e^{-K({\bm{r}})}$ is a
zero mode of a Dirac particle in the inhomogeneous magnetic field
$B({\bm{r}})=\nabla^{2}\operatorname{Re}K({\bm{r}})$. Due to its normalization
$\sum_{l}\lvert\zeta^{l}({\bm{r}})\rvert^{2}=1$ the spinor part does not
influence the energy of the band under density-density interactions, and can
thus be largely ignored for our purposes. The class of $C=1$ vortexable bands
therefore has the same analytic structure as the LLL in an inhomogenous
magnetic field and metric that breaks continuous translation symmetry [19,
69]. Vortexability is thus a sufficient condition for a Chern band to be
“analogous to the LLL".

_First Landau Levels Are Diverse—_ To extend the success of ideal bands to the
first LL, one could imagine imposing band geometric criteria to mimic a 1LL
features. For example, the first LL of a $p^{2}/2m$ particle has
$\operatorname{tr}g=3\Omega$ [21]. However, this and other approaches fail to
capture the wide spectrum of “first LLs" in multilayer graphene. To wit, we
recall Bernal graphene at $B>0$:

$H_{\rm BG}=\left(\begin{array}[]{cc}0&D_{1}^{\dagger}\\\
D_{1}&0\end{array}\right),\quad
D_{1}=\left(\begin{array}[]{cc}D_{0}&\gamma_{1}\\\ 0&D_{0}\end{array}\right),$
(2)

written in the chiral basis
$(|A,1\rangle,|A,2\rangle,|B,1\rangle,|B,2\rangle)$. Here
$D_{0}=2v\left(-i\bar{\partial}_{z}-\bar{A}_{z}\right)=\sqrt{2}vl_{B}^{-1}\hat{a}$
is proportional to the annihilation operator and $\gamma_{1}$ is the
interlayer coupling. There are two bands of zero modes; $\ker D_{1}$ is
spanned by

$\Phi_{0}=\left(\begin{array}[]{c}\psi_{0}\\\
0\end{array}\right),\quad\Phi_{1}=\left(\begin{array}[]{c}v^{-1}\ell_{B}\gamma_{1}\hat{a}^{\dagger}\psi_{0}\\\
-\sqrt{2}\psi_{0}\end{array}\right),$ (3)

forming a zeroth and first LL respectively, where
$\psi_{0}({\bm{r}})=f(z)e^{-\lvert z\rvert^{2}/4\ell_{B}^{2}}$ in symmetric
gauge with $f(z)$ holomorphic.

Direct computation shows that, as $\gamma_{1}$ varies from $0$ to $\infty$,
the ratio $\operatorname{tr}g_{1}/\Omega_{1}$ of band $\Phi_{1}$ varies
continuously from $1$ to $3$. However, the wavefunctions have a similar
analytic structure throughout. Even if one insists the first LL corresponds
only to the $\gamma_{1}\to\infty$ limit, with
$\operatorname{tr}g_{1}=3\Omega_{1}$, the condition
$\operatorname{tr}g=3\Omega$ is also satisfied by the second LL of trilayer
graphene for a particular value of $v^{-1}\ell_{B}\gamma_{1}$. The single-band
Berry curvature and quantum metric therefore appear insufficient to specify a
first LL structure.

The fact that the first $n$ LLs are ideal, when taken together, was first
noticed in [21] by explicitly calculating $\operatorname{tr}g=\Omega$ for a
$p^{2}/2m$ particle. This property that adding vortices to the first LL gives
components in both first and zeroth LLs is the crux of our generalization.

_Definition of First LL Structure—_ We now precisely define the structure
bands must satisfy to be “first LL analogues". A Chern band, with projector
$P_{1}$, is first vortexable if the following two conditions are satisfied:

  1. 1.

($\exists$ Partner Band) There exists an orthogonal vortexable band with
projector $P_{0}$ (i.e. $P_{0}z\Phi_{0}~{}=~{}z\Phi_{0}$ for all
$P_{0}\Phi_{0}=\Phi_{0}$) such that

$\displaystyle(P_{1}+P_{0})z\Phi_{1}$ $\displaystyle=z\Phi_{1}$ (4)

for all states $P_{1}\Phi_{1}=\Phi_{1}$.

  2. 2.

(Indecomposability) there is no alternative basis of wavefunctions
$(\Phi_{0,{\bm{k}}}^{\prime},\Phi_{1,{\bm{k}}}^{\prime})$, where each band is
vortexable, that also spans the two band subspace of $P_{0}+P_{1}$.

The additional band, described by the projector $P_{0}$, functions as an
effective zeroth LL. Intuitively, acting with $z$ either keeps the
wavefunction within the ‘first’ band $P_{1}$ or lowers it to the ‘zeroth’ band
$P_{0}$ — but does not generate components in any other bands (see Fig. 1a).
The second condition is in principle necessary to avoid subtle situations
where the first condition can nominally be realized, but the system should be
thought of as a collection of two separate, individually vortexable, bands
instead.

While the definition above may appear computationally difficult to verify the
wavefunctions that satisfy it are tightly restricted. Our constructions will
all yield wavefunctions of the form

$\Phi_{1}({\bm{r}})=Q_{0}\left[\left(\overline{z}f(z)-2B_{0}^{-1}f^{\prime}(z)\right)\mathcal{N}_{0}({\bm{r}})+f(z)\mathcal{N}_{1}({\bm{r}})\right]$
(5)

which will be proven to be general in Ref. [76]. Here $Q_{0}=1-P_{0}$,
$\mathcal{N}_{0,1}$ are spinor-valued [cf. Eq. (1)], and $P_{0}$ is the
projector onto the “zeroth-LL" wavefunctions $f(z)\mathcal{N}_{0}({\bm{r}})$.
$B_{0}=\frac{2\pi}{A_{\mathrm{UC}}}=\frac{A_{\mathrm{BZ}}}{2\pi}$, and
$A_{\mathrm{UC}}$ and $A_{\mathrm{BZ}}$ are the areas of the unit cell and the
Brillouin Zone (BZ), respectively. In Appendix E, we prove that bands with
wavefunctions (5) are first vortexable. Bands that are suspected to be first
vortexable can be explicitly compared to an instance (5), as we will see
shortly. In Appendix A, we derive a zero-field concrete form of Eq.(5) in an
analytically solvable model motivated by magic angle twisted bilayer graphene.

First vortexability has immediate many-body implications. Consider a first
vortexable band that is flat and degenerate with its vortexable counterpart,
as in Bernal graphene and in a upcoming example. Let $\Psi_{\nu=2}$ be the
many-body state where both bands are full. Now consider the vortex attached
state $\Psi_{v}=\prod_{i<j}(z_{i}-z_{j})^{2s}\Psi_{\nu=p}$ which has filling
$p/(2ps+1)$, where $p=2$ (note, general $p$ follows analogously for $(p-1)$’th
vortexability). Thanks to vortexability of the combined bands, attaching this
vortex factor keeps the final state within the combined bands. The above
construction is similar to that of LLL Jain states [64], where projection into
the LLL is strictly speaking required at the end. Here however, LLL projection
is unnecessary; $\Psi_{v}$ is an exact zero mode of short-range interactions
[75, 19, 28].

We also introduce a refinement of our criterion, which is specifically geared
towards realizing states unique to the first LL. While we motivated our
definition through the inclusion of Bernal graphene for any value of the
interlayer tunneling, as a byproduct we have included bands that for all
practical purposes function as LLLs; Consider the decoupled $\gamma_{1}\to 0$
limit, which adiabatically connects Jain-like states with layer-singlet
Halperin-like states at $\gamma_{1}=0$. At half filling of the first-
vortexable band, however, $\gamma_{1}$ must be sufficiently large for a
Pfaffian to outcompete the composite Fermi liquid at half filling [77]. It
turns out that the $\gamma_{1}\to\infty$ limit, or more generally
$\mathcal{N}_{1}\to 0$, has some additional structure in non-Abelian band
geometry that leads to a natural quantification of how close a band is to the
“maximal" first LL. In Appendix F, we show that for $\mathcal{N}_{1}\to 0$,
the non-Abelian Berry curvature (in the space of the zeroth and first LLs)
becomes rank deficient, with only one nonzero entry $\hat{\Omega}_{11}$. We
therefore define a “maximality index"

$M({\bm{k}})=\frac{\lvert\lambda_{1}({\bm{k}})-\lambda_{2}({\bm{k}})\rvert}{\lambda_{1}({\bm{k}})+\lambda_{2}({\bm{k}})},$
(6)

where $\lambda_{1,2}$ are the eigenvalues of the $2\times 2$ non-Abelian Berry
curvature. As $\gamma_{1}$ varies from $0\to\infty$ in Bernal graphene,
$M({\bm{k}})=\gamma_{1}^{2}/(2(v\ell_{B}^{-1})^{2}+\gamma_{1}^{2})$ increases
from $0\to 1$ (see Appendix G). First vortexable bands with $M({\bm{k}})\to 1$
(i.e. $\mathcal{N}_{1}\to 0$) are effectively 1LLs with periodic modulation
$\lVert\mathcal{N}_{0}({\bm{r}})\rVert/e^{-B_{0}\lvert{\bm{r}}\rvert^{2}/4}$.
We expect them to host Pfaffian-type topological orders.

Figure 2: (a) Bernal graphene with vertical buckling and a strain field,
giving a periodic psuedomagnetic field $B$ (color map) with zero average. (b)
Average overlap deviation between numerical and analytical approximate zero
modes with $\eta=0.5$, $\langle S\rangle_{\rm BZ}$, versus $\alpha$ at
$\beta=0$. Inset: bandstructure for $(\alpha,\beta,\gamma_{1})=(0.5,0,0.5)$
with low-energy modes $E_{0,1}$ in purple. (c) Bandstructure at
$(\alpha,\beta,\gamma_{1})=(0.4,0.068,1)$. Displacement field $\beta>0$
isolates the two low-energy Chern bands (purple).

_Periodically Strained Bernal Graphene—_ We will now show how a first
vortexable band appears amongst the low energy bands of periodically strained
bilayer graphene [78]. We will take a similar approach to Ref. [79] which
studied periodically strained monolayer graphene with a $C_{2}$-breaking
substrate that induces $C_{2}$ breaking pseudomagnetic field

$\mathcal{B}(z,\bar{z})=\mathcal{B}_{0}\sum_{l=0}^{5}e^{i\bm{G}_{l}\cdot\bm{r}}=\mathcal{B}_{0}\sum_{l=0}^{5}e^{\frac{i}{2}\left(G_{l}\bar{z}+\bar{G}_{l}z\right)}$
(7)

where $\bm{G}_{l}=R_{2\pi
l/6}\bm{G}_{0},\bm{G}_{0}=\frac{4\pi}{\sqrt{3}L_{M}}(1,0)$ are smallest
reciprocal lattice vectors in terms of the period $L_{M}$; the addition of
higher harmonics is not expected to change our conclusions [79]. We note that
${\mathcal{B}}({\bm{r}})={\mathcal{B}}(-{\bm{r}})$, within a valley, implies
that the pseudomagnetic field is $C_{2}$-odd. Furthermore, ${\mathcal{B}}(r)$
averages to zero such that the bands we will obtain are Bloch bands, not LLs.
The periodic strain can be realized by placing monolayer graphene on a lattice
of nanorods [80], or by spontaneous buckling of a graphene sheet on $C_{2}$
breaking substrates such as NbSe2 [81]. We will also include a spatially-
varying scalar potential with the same periodicity proportional to
$V({\bm{r}})=\sum_{l}e^{i\bm{G}_{l}\cdot{\bm{r}}}$ and of strength $V_{0}$.
The scalar potential could come from patterned gates [82, 83, 84, 85, 86], or
an electric field that couples to the buckling height.

The (de-dimensionalized) Hamiltonian for AB-stacked graphene’s $K$ valley is
then

$\displaystyle H_{\rm SBG}$
$\displaystyle=E_{0}\left(\begin{array}[]{cc}h&\gamma_{1}\sigma_{+}\\\
\gamma_{1}\sigma_{-}&h\end{array}\right),$ (8) $\displaystyle h$
$\displaystyle=\bm{\sigma}\cdot[-i\bm{\nabla}+\alpha\bm{{\mathcal{A}}}]-\beta
V({\bm{r}}),$

where $E_{0}=\hbar v_{F}\lvert\bm{G}_{0}\rvert$,
$\sigma_{\pm}=\frac{1}{2}(\sigma_{+}\pm i\sigma_{-})$,
$\alpha=1/\ell_{B}^{2}\lvert G_{0}\rvert^{2}$, $\beta=V_{0}/E_{0}$,
${\mathcal{B}}_{0}=\hbar/e\ell_{B}^{2}$,
${\mathcal{A}}=\mathcal{A}_{x}+iA_{y}=\sum_{l}e^{2\pi
i/l}e^{i\bm{G}_{l}\cdot\bm{r}}$ We will neglect the interlayer potential for
simplicity, consistent with patterned dielectric gates sandwiching the
sample.. We take interlayer tunnelling $E_{0}\gamma_{1}\approx 370$ meV and,
following [81], $E_{0}\approx$300\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$.
The bandstructure, Fig. 2(b) inset, has a low energy subspace of four bands
(purple) with total bandwidth $O(e^{-6\alpha})$. In the chiral limit
$\beta=0$, $H_{\rm SBG}/E_{0}$ takes the form of Eq. (2) but now with
$D_{0}=-2i\overline{\partial}+\alpha{\mathcal{A}}({\bm{r}})$.

We explicitly construct trial wavefunctions on the $A$ sublattice for a zeroth
and first vortexable band that — up to exponentially small corrections
$O(e^{-6\alpha})$ — are zero modes of $D_{1}$. The two trial wavefunctions are
Chern bands whose layer spinors take the form

$\Phi_{0,\bm{k}}(\bm{r})=\left(\begin{array}[]{c}\psi_{0,\bm{k}}(\bm{r})\\\
0\end{array}\right),\Phi_{1,\bm{k}}(\bm{r})=\left(\begin{array}[]{c}\psi_{1,\bm{k}}(\bm{r})\\\
2i\gamma_{1}^{-1}\psi_{0,\bm{k}}(\bm{r})\end{array}\right),$ (9)

with precise expressions given in Appendix C. The construction of (9) follows
the same reasoning as Ref. [79], making use of $[D_{0},z]=0$ and
$\psi_{\Gamma}({\bm{r}}=0)=e^{-6\alpha}\approx 0$, where $V({\bm{r}}=0)=6$. In
Appendix C we also construct two topologically trivial near zero modes of
$D_{1}^{\dagger}$ for the $B$-sublattice. We now use these trial wavefunctions
to identify a first vortexable band of $H_{\mathrm{SBG}}$ and isolate it from
the rest of the spectrum.

Figure 3:  (a),(c): SCHF bandstructures for $\nu=-1$ and $\nu=-2$,
respectively, with parameters corresponding to Fig. 2(c), and
$(d,\epsilon_{r})=($250\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,15)$. The
dashed lines represent the Fermi energy. (b) The band gaps $V_{g}$ for
$\nu=-1$ and $\nu=-2$ are plotted with green and orange lines, respectively.
(d) Average overlap deviation between numerical and analytical eigenstates
with $\eta=0.5$ at $\nu=-1$ as a function of $\alpha$. Here $\langle
T_{0}\rangle_{\rm BZ}$ (blue), and $\langle T_{1}\rangle_{\rm BZ}$ (red)
correspond to the zeroth and first vortexable bands, whereas $\langle
S\rangle_{\rm BZ}$ is tow-band complex.

We now numerically verify that the low-energy subspace of $H_{\mathrm{SBG}}$
on the $A$-sublattice is spanned by the trial wavefunctions Eq.(9) up to
$O(e^{-6\alpha})$ corrections. As the energy eigenstates are mixed between the
sublattices (at $\beta=0$), they cannot correspond to the trial wavefunctions
directly. Instead, we use the sublattice basis
$\Psi^{A,B}_{n,{\bm{k}}}=\frac{1}{\sqrt{2}}(\Psi_{E_{n},{\bm{k}}}\pm\sigma_{z}\Psi_{E_{n},{\bm{k}}})$,
where $n=0,1$ labels the two eigenstates of smallest $E_{n}>0$, and
$\sigma_{z}\Psi_{E_{n}}\propto\Psi_{-E_{n}}$. The overlap deviation between
the span of the trial wavefunctions
$P_{\Phi{\bm{k}}}=\sum_{n}|\Phi_{n,{\bm{k}}}\rangle\langle\Phi_{n,{\bm{k}}}|$
and the $A$-subspace
$P_{\Psi{\bm{k}}}=\sum_{n}|\Psi^{A}_{n,{\bm{k}}}\rangle\langle\Psi^{A}_{n,{\bm{k}}}|$
is quantified by
$S_{\bm{k}}[\Phi_{{\bm{k}}},\Psi^{A}_{{\bm{k}}}]=1-\mathrm{Tr}[P_{\Phi{\bm{k}}}P_{\Psi{\bm{k}}}]/2$.
The trial wavefunctions span the $A$-subspace if and only if the Brillouin
zone average vanishes, $\braket{S}_{\mathrm{BZ}}\approx 0$. Fig. 2(b) shows
$\langle S\rangle_{\mathrm{BZ}}$ decreases as $e^{-6\alpha}$, giving overlaps
$>0.99$ whenever $\alpha>0.2$. We therefore conclude that the low-energy
$A$-sublattice subspace is spanned by “almost" vortexable and first vortexable
bands, satisfying the definition up to small $O(e^{-6\alpha})$ corrections.

Finally, we show how interactions can spectrally isolate the almost first-
vortexable band. Chiral symmetry is lifted by displacement fields $\beta>0$,
breaking the Dirac cone and separating out a complex of two bands
$\Psi^{A}_{(0,1)}$ with $C_{\rm total}=2$. The complex’s total bandwidth is
minimized at $\beta=0.068$ [Fig 2(c)], but the two bands within it are not
isolated without adding interactions. We now add gate-screened Coulomb
interactions in the standard way, and study the resulting self-consistent
Hartree-Fock (SCHF) bandstructures at integer fillings (details in Appendix
H). We find interactions split the complex into two isolated bands
$\Psi_{0}^{\prime}$ and $\Psi_{1}^{\prime}$ that approximately are vortexable
and first-vortexable, respectively. The overlap deviation $\braket{S}$, shown
in Fig. 3(d), decreases as $e^{-5.6\alpha}$. This is directly analogous to
Bernal graphene in field, where interactions split the degenerate zeroth and
first LLs since the more-concentrated LLL wavefunctions have less exchange
energy [87, 88, 89].

Fig. 3 shows SCHF bandstructures at fillings $\nu=-1$ and $-2$, each of which
shows two isolated $C=1$ bands. At $\nu=-2$, the SCHF bandgap closes near
$\alpha\approx 0.45$, enabling a topological transition $C=(1,1)\to(1,0)$ by
mixing with the third band. Conversely, the top two bands hybridize to give
isolated bands with $C=(2,0)$ below $\alpha\approx 0.3$. Hence the two $C=1$
bands are isolated over a range of $\alpha$ for both $\nu=-1$ & $-2$.

To confirm vortexability, we compare the numerical wavefunctions to the
analytical ansätze $\Phi_{(0,1)}$ from Eq. (9). Fig. 3(d) shows $\langle
T_{n}\rangle$, the average of
$1-\left|\braket{\Psi_{n{\bm{k}}}^{A}}{\Phi_{n{\bm{k}}}}\right|$ over the BZ
and the two band overlap deviation $\langle S\rangle$ defined above. In the
regime $0.3<\alpha<0.45$ with $C=(1,1)$, the overlap deviation decreases as
$O(e^{-5.6\alpha})$, giving deviations $<1\%$ for $\alpha>0.38$. The numerical
wavefunctions therefore match the analytic ones to extremely good precision,
and the latter are almost vortexable and first vortexable. We compute the
maximality index, Eq. (6), to be $\langle M\rangle_{\rm BZ}\approx 0.79$ at
$\alpha=0.4$ and dimensionless tunneling $\gamma_{1}=1$. This is close to the
optimal value of $1$.

In this work we have proposed “first vortexability" — a precise definition of
when a Chern band has the essential character of the first LL. First
vortexable bands are defined by their quantum geometric data, and each
requires a partner band analogous to the zeroth LL. Our examples, both
analytic and physically reasonable, suggest such bands are experimentally
realizable. Future work will investigate the many-body physics of partially-
filled first vortexable bands, particularly at half filling where non-Abelian
states are likely.

###### Acknowledgements.

We thank Jie Wang, Bruno Mera, Tomohiro Soejima, and Mike Zaletel for
enlightening discussions. D.E.P. is supported by the Simons Collaboration on
UltraQuantum Matter, which is a grant from the Simons Foundation. D.E.P. is
supported by startup funds from the University of California, San Diego. This
work was supported by JSPS KAKENHI grant no. JP23KJ0339 and the Center for the
Advancement of Topological Semimetals (CATS), an Energy Frontier Research
Center at the Ames National Laboratory. Work at the Ames National Laboratory
is supported by the U.S. Department of Energy (DOE), Basic Energy Sciences
(BES) and is operated for the U.S. DOE by Iowa State University under Contract
No. DE-AC02-07CH11358. A.V. is supported by a Simons Investigator grant.

##  Appendix A Analytic Example at Zero Field

To demonstrate first vortexable bands without magnetic fields, we consider an
analytically solvable (albeit artificial) example inspired by chiral twisted
bilayer graphene. Let

$H_{1}=\left(\begin{array}[]{cc}0&D^{\dagger}_{1}\\\
D_{1}&0\end{array}\right),\quad
D_{1}=\left(\begin{array}[]{cc}D_{0}&\gamma_{1}I_{2\times 2}\\\
0&D_{0}\end{array}\right).$ (10)

In particular, we assume that $D_{0}$ is the chiral Hamiltonian of twisted
bilayer graphene (TBG), given by

$D_{0}=\left(\begin{array}[]{cc}-2i\bar{\partial}&\alpha
U_{\omega^{*}}(\bm{r})\\\ \alpha
U_{\omega^{*}}(-\bm{r})&-2i\bar{\partial}\end{array}\right),\quad
U_{\omega}(\bm{r})=\alpha\sum_{j=0}^{2}\omega^{j}e^{-i\bm{q}_{j}\cdot\bm{r}}$
(11)

with $\omega=e^{2\pi i/3}$ and $q_{j,x}+iq_{j,y}=-ie^{2\pi j/3}$. While (10)
may appear strange due to the upper triangular hoppings connecting non-
nearest-“layers", after a suitable unitary transformation it arises exactly as
a $k_{z}=0$ band of three dimensional twisted AB-BA double bilayer graphene
(see Sec B). At each magic angle (such as $\alpha\approx 0.586$), $D_{0}$ has
a zero mode [73, 19, 90]

$\psi_{0,{\bm{k}}}(r)=\frac{\sigma\left(z+iB_{0}^{-1}k\right)}{\sigma(z)}e^{\frac{i}{2}\bar{k}z}\psi_{\Gamma}({\bm{r}}),$
(12)

where $\sigma(z)$ is the Weierstrass $\sigma$-function [91, 90]. The
Weierstrass sigma function $\sigma(z)$ satisfies $\sigma(-z)=-\sigma(z)$ and
$\sigma(z+R)=\eta_{\bm{r}}e^{\frac{B}{2}\bar{R}(z+R/2)}\sigma(z)$, where
$\eta_{\bm{R}}=+1$ if $\bm{R}/2$ is a lattice vector and $-1$ otherwise.

We can now construct $\Phi_{0,\bm{k}}=\big{[}\psi_{0,\bm{k}},0\big{]}^{T}$ and
$\Phi_{1,\bm{k}}=\big{[}\psi_{1,\bm{k}},2i\gamma_{1}^{-1}\psi_{0,\bm{k}}\big{]}^{T}$,
where

$\psi_{1,{\bm{k}}}(\bm{r})=\left[\bar{z}-\frac{2B_{0}^{-1}\partial_{z}\sigma\left(z+iB_{0}^{-1}k\right)}{\sigma\left(z+iB_{0}^{-1}k\right)}-\alpha({\bm{k}})\right]\psi_{0,{\bm{k}}}(\bm{r}).$
(13)

with $\alpha({\bm{k}})$ such that
$\braket{\psi_{0,{\bm{k}}}}{\psi_{1,{\bm{k}}}}=0$. Then the identity
$D_{0}\psi_{1,{\bm{k}}}=-\psi_{0,{\bm{k}}}$ immediately implies
$D_{1}\Phi_{1,\bm{k}}=0$, giving two bands of zero modes analogous to the LLL
and 1LL. We note $\psi_{1,{\bm{k}}}$ obeys the Bloch condition because the
shift in $z\rightarrow z+R$ in the first term is cancelled out by

$\partial_{z}\sigma(z+R)=-e^{\frac{B_{0}}{2}\bar{R}(z+R/2)}\left[\partial_{z}\sigma(z)-\frac{B_{0}}{2}\bar{R}\sigma(z)\right].$
(14)

We numerically confirm that $\Phi_{0}$ and $\Phi_{1}$ are the zero-modes of
$H_{1}$ [Eq. (10)] at the magic angle of TBG. There $H_{1}$ has four low-
energy bands isolated from the remote bands, whose bandwidth vanishes at the
same magic angle as TBG, $\alpha=0.586$ [Fig. 5(b)]. Due to the chiral
symmetry of the Hamiltonian, the four eigenstates, denoted by $\Psi_{\pm
E_{n}}$ with the energy $E_{n}>0$ $(n=0,1)$, can be decomposed into sublattice
polarized states such as
$\Psi_{n\bm{k}}^{A,B}=\frac{1}{\sqrt{2}}\left(\Psi_{E_{n},{\bm{k}}}\pm\sigma_{z}\Psi_{E_{n},{\bm{k}}}\right)$.
In Fig. 5(b), we plot the bandwidth and the average of
$1-\left|\braket{\Psi_{n{\bm{k}}}^{A}}{\Phi_{n{\bm{k}}}}\right|$ over the BZ,
denoted by $\langle T_{n}\rangle_{\rm BZ}$. At the magic angle
$\Psi_{n\bm{k}}^{A}=\Phi_{n{\bm{k}}}$ exactly, as expected.

##  Appendix B Three dimensional twisted AB-BA double bilayer

Figure 4: Schematic picture of three-dimensional alternatively ABBA twisted
Bernal graphene.

The Hamiltonian [Eq.(10)] corresponds to the model where the layers are
twisted by $(0,\theta,\theta,2\theta)$ and hosts a hopping between different
sublattices of the next nearest layers. Using the unitary transformation such
as changing the order of layer basis
$(|\sigma,1\rangle,|\sigma,2\rangle,|\sigma,3\rangle,|\sigma,4\rangle)$ into
$(|\sigma,1\rangle,|\sigma,3\rangle,|\sigma,4\rangle,|\sigma,2\rangle)$, where
$\sigma=\pm$ denotes $A/B$, we obtain the following Hamiltonian

$\mathcal{H}_{\rm{3DATDBG}}=\mathcal{H}_{\rm{AB-
BA}}+\left(\begin{array}[]{cccc}0&0&0&T^{\dagger}(\bm{r})\\\ 0&0&0&0\\\
0&0&0&0\\\ T(\bm{r})&0&0&0\end{array}\right).$ (15)

where the first term is the Hamiltonian of twisted AB-BA double bilayer (TDBG)
[92]

$\mathcal{H}_{\rm{AB-
BA}}=\left(\begin{array}[]{cccc}-i\bm{\sigma}\cdot\bm{\nabla}&\Gamma_{1}^{\dagger}&0&0\\\
\Gamma_{1}&-i\bm{\sigma}\cdot\bm{\nabla}&T^{\dagger}(\bm{r})&0\\\
0&T(\bm{r})&-i\bm{\sigma}\cdot\bm{\nabla}&\Gamma_{1}\\\
0&0&\Gamma_{1}^{\dagger}&-i\bm{\sigma}\cdot\bm{\nabla}\end{array}\right)$ (16)

with $\Gamma_{1}=(\sigma_{x}-i\sigma_{y})\gamma_{1}/2$.

$T(\bm{r})=\sum_{n=1}^{3}T_{n}e^{-i\bm{q}_{n}\cdot\bm{r}}$ (17)

where $T_{n+1}=\alpha\left(\sigma_{x}\cos n\phi+\sigma_{y}\sin n\phi\right)$
is the interlayer hopping in the chiral limit. The second term is the
interlayer coupling between layer 1 and 4, implying imposing the periodic
boundary condition to the twisted AB-BA double bilayer in the stacking
direction. This model can be realized in three-dimensional alternatively
twisted AB-BA double bilayer graphene shown in Fig. 4. In an infinite number
of stackings of TDBG, the wavenumber $k_{z}$ along the stacking direction is a
good quantum number to label the energy state. At $k_{z}=0$, the Hamiltonian
is equivalent to Eq.(15).

##  Appendix C Trial wavefunction of zeroth and first vortexable state of
strained Bernal graphene

Let us construct the trial wavefunction of zeroth and first vortexable state
of strained Bernal graphene. At $\beta=0$ the chiral symmetry of the model
allows us to write

$\displaystyle H_{\rm SBG}$
$\displaystyle=E_{0}\begin{pmatrix}0&D_{1}^{\dagger}\\\
D_{1}&0\end{pmatrix},\,\,\,D_{1}=\begin{pmatrix}D_{0}&\gamma_{1}\\\
0&D_{0}\end{pmatrix}$ (18) $\displaystyle D_{0}$
$\displaystyle=-2i\overline{\partial}+\alpha{\mathcal{A}}.$

We will obtain the low energy subspace through sublattice polarized near-zero
modes of $D_{1}$ and $D_{1}^{\dagger}$. We will begin with $D_{1}$. At the
$\Gamma$ point, there is an exact zero mode of $D_{0}$,
$\psi_{0,\Gamma}({\bm{r}})=e^{-\alpha V({\bm{r}})}$, and an associated zero
mode of $D_{1}$ (and thus $H$) by placing $\psi_{0,\Gamma}$ in the top layer
component. We now create two near zero modes, for large $\alpha$, under the
approximation $\psi_{0,\Gamma}({\bm{r}}=0)=e^{-6\alpha}\approx 0$; where here
and below “$\approx$" specifically means “up to terms of
$O\left(e^{-6\alpha}\right)"$.

We begin by constructing an approximate zero mode of $D_{0}$, following Ref.
[79]. We will use the modified state $\psi_{\eta}=f_{\eta}({\bm{r}})e^{-\alpha
V({\bm{r}})}$, where $f_{\eta}({\bm{r}})=0$ near ${\bm{r}}=0$ but is $1$
otherwise (e.g. $f_{\eta}({\bm{r}})=\tanh\eta\lvert{\bm{r}}\rvert$ for
$\eta\gg 1$). Then, $D_{0}\psi_{\eta}\approx 0$. Furthermore, since
$\psi_{\eta}({\bm{r}}=0)=0$ by construction, we have

$D_{0}\mathcal{N}({\bm{r}})=0,\qquad\mathcal{N}({\bm{r}})=\frac{\psi_{\eta}({\bm{r}})}{\sigma(z)}$
(19)

where $\sigma(z)$ is the Weierstrass $\sigma$-function [91, 90, 79] that we
used in Sec. A; see the discussion below (12). Since
$\lvert\sigma(z)\rvert^{-1}$ decreases exponentially as $\lvert
z\rvert\to\infty$, like the usual Gaussian factor of the LLL, we can now
multiply by holomorphic functions, that necessarily blow up at infinity, and
thereby construct a vortexable band $\psi_{0}=f(z)\mathcal{N}({\bm{r}})$.

These wavefunctions satisfy $D_{0}\psi_{0}\approx 0$ by construction. To
construct Bloch states we write

$\displaystyle\psi_{0,{\bm{k}}}({\bm{r}})$
$\displaystyle=f_{\bm{k}}(z)\mathcal{N}_{0}({\bm{r}})$ (20) $\displaystyle
f_{\bm{k}}(z)$
$\displaystyle=e^{\frac{i}{2}\overline{k}z}\sigma(z+iB_{0}^{-1}k).$

The construction (20) directly yields an approximate zero mode of $D_{1}$, by
placing $\psi_{0,{\bm{k}}}({\bm{r}})$ in the top component, analogously to the
zeroth LL of $B>0$ Bernal graphene and the analytical zero field model of the
previous section. To obtain the second approximate zero mode, we write

$\psi_{1,{\bm{k}}}({\bm{r}})=Q_{\psi_{0}}(\overline{z}f_{\bm{k}}(z)-2B_{0}^{-1}\partial_{z}f_{\bm{k}}(z))\mathcal{N}({\bm{r}})$
(21)

, $Q_{\psi_{0}}$ projects out the component proportional to
$\psi_{0,{\bm{k}}}$. Since $D_{0}\psi_{0,{\bm{k}}}\approx 0$ and
$D_{0}\psi_{1,{\bm{k}}}\approx-\psi_{0,{\bm{k}}}$, the states

$\Phi_{0,\bm{k}}(\bm{r})=\left(\begin{array}[]{c}\psi_{0,\bm{k}}(\bm{r})\\\
0\end{array}\right),\Phi_{1,\bm{k}}(\bm{r})=\left(\begin{array}[]{c}\psi_{1,\bm{k}}(\bm{r})\\\
2i\gamma_{1}^{-1}\psi_{0,\bm{k}}(\bm{r})\end{array}\right),$ (22)

satisfy $D_{1}\Phi_{(0,1),\bm{k}}(\bm{r})\approx 0$, giving two almost flat
bands of zero modes analogous to the LLL and 1LL. We may also write them in
the general form we quoted for zeroth and first vortexable bands,

$\displaystyle\Phi_{0,\bm{k}}(\bm{r})$
$\displaystyle=f_{{\bm{k}}}(z)\mathcal{N}_{0}({\bm{r}})$ (23)
$\displaystyle\Phi_{0,\bm{k}}(\bm{r})$
$\displaystyle=Q_{\psi_{0}}\left[(\overline{z}f_{\bm{k}}(z)-2B_{0}^{-1}\partial_{z}f_{\bm{k}}(z))\mathcal{N}_{0}({\bm{r}})+f_{\bm{k}}(z)\mathcal{N}_{1}({\bm{r}})\right],$

where

$\mathcal{N}_{0}({\bm{r}})=\begin{pmatrix}1\\\
0\end{pmatrix},\quad\mathcal{N}_{1}({\bm{r}})=\begin{pmatrix}0\\\
2i\gamma_{1}^{-1}\end{pmatrix}.$ (24)

The two $B$-sublattice zero modes of $D_{1}^{\dagger}$ are topologically
trivial. The exact zero mode of $D_{0}$, $\chi_{0,\Gamma}=e^{+\alpha V}$, is
strongly localized at a single site ${\bm{r}}={\bm{R}}$ in each unit cell
centered at the lattice vector ${\bm{R}}$. Let us define
$\chi_{0,{\bm{R}}}({\bm{r}})$ to be equal to $\chi_{\Gamma}({\bm{r}})$ if
${\bm{r}}$ is in the unit cell centered at ${\bm{R}}$, and zero otherwise. It
is exponentially localized around ${\bm{r}}={\bm{R}}$, and satisfies
$D_{0}=\chi_{0,{\bm{R}}}({\bm{r}})\approx 0$ as well; it should be thought of
as a Wannier state of the strained monolayer. Now we construct

$\mathcal{X}_{0,{\bm{R}}}({\bm{r}})=\begin{pmatrix}\chi_{0,{\bm{R}}}({\bm{r}})\\\
0\end{pmatrix},\quad\mathcal{X}_{1,{\bm{R}}}({\bm{r}})=\begin{pmatrix}z\chi_{0,{\bm{R}}}({\bm{r}})\\\
2i\gamma^{-1}\chi_{0,{\bm{R}}}({\bm{r}})\end{pmatrix},$ (25)

as exponentially localized Wannier states at each site which satisfy
$D_{1}\mathcal{X}_{0,1{\bm{R}}}({\bm{r}})\approx 0$.

##  Appendix D Multi-band Vortexable Band Geometry

In this section we review ideal, or vortexable, band geometry for multiple
bands and study the non-Abelian quantum geometric tensor in this context for
later use. The non-Abelian quantum geometric tensor (QGT) is given, in an
orthonormal basis, as

$\displaystyle\hat{\eta}^{\mu\nu}_{ab}$
$\displaystyle=\bra{\partial_{k_{\nu}}u_{{\bm{k}}b}}Q({\bm{k}})\ket{\partial_{k_{\mu}}u_{{\bm{k}}a}},$
(26) $\displaystyle Q({\bm{k}})$
$\displaystyle=1-\sum_{a}\ket{u_{{\bm{k}}a}}\bra{u_{{\bm{k}}a}}$

where $a,b$ are band-indices.

We will denote its trace by removing its hat, or with a capital
$\operatorname{Tr}$, i.e.

$\eta^{\mu\nu}=\operatorname{Tr}\hat{\eta}^{\mu\nu}=\sum_{a}\eta^{\mu\nu}_{aa}.$
(27)

Note that our gauge-covariance based definition of $\hat{\eta}$ ensures that
$\eta^{\mu\nu}$ is gauge invariant, even in non-orthonormal bases. The non-
Abelian Berry curvature and metric are defined with the symmetric and
antisymmetric components in the spatial indices

$\begin{aligned}
\hat{g}^{\mu\nu}({\bm{k}})&=\frac{1}{2}(\hat{\eta}^{\mu\nu}+\hat{\eta}^{\nu\mu})\\\
\hat{\Omega}_{ab}\varepsilon^{\mu\nu}&=i(\hat{\eta}^{\mu\nu}-\hat{\eta}^{\nu\mu})\end{aligned}.$
(28)

We now claim the following conditions are equivalent [28, 22, 21, 23, 24]

  1. (V1)

The bands are vortexable: $z\psi=Pz\psi$, where $\psi=P\psi$ is any
wavefunction in the band and $P$ is the band projector.

  2. (V2)

The inequality $\operatorname{tr}g\geq\Omega$, or
$\operatorname{tr}\operatorname{Tr}\hat{g}\geq\operatorname{Tr}\hat{\Omega}$,
is saturated, where $z+x+iy$ is the vortex function.

  3. (V3)

There is a gauge, in which the wavefunctions are generically non-orthonormal,
such that $\ket{u_{{\bm{k}}a}}=\ket{u_{ka}}$ are holomorphic functions of
$k=k_{x}+ik_{y}$.

  4. (V4)

The quantum metric factorizes as
$\hat{\eta}^{\mu\nu}_{ab}=\frac{1}{2}\hat{\Omega}_{ab}(\delta_{\mu\nu}-i\varepsilon_{\mu\nu})$,
where
$\hat{\Omega}_{ab}=\bra{\partial_{k}u_{{\bm{k}}b}}Q({\bm{k}})\ket{\partial_{k}u_{{\bm{k}}a}}$
is the non-Abelian Berry curvature and
$2\partial_{k}=\partial_{k_{x}}+i\partial_{k_{y}}$ is the derivative with
respect to $k=k_{x}+ik_{y}$

Ref. [28] showed the equivalence between (V1) and (V2), and reviewed the
equivalence between (V2) and (V3) through an explicit construction of the
required gauge transformation, which was previously shown to exist by Ref.
[22]. It remains to show (V4) is equivalent to (V1-3). The implication
$\rm{(V4)}\implies\rm{(V2)}$ follows immediately by computing
$\operatorname{tr}g$ and $\Omega$ from the traced symmetric and antisymmetric
parts of
$\hat{\eta}^{\mu\nu}_{ab}=\frac{1}{2}\hat{\Omega}_{ab}(\delta_{\mu\nu}-i\varepsilon_{\mu\nu})$
respectively. We conclude by showing $\rm{(V3)}\implies\rm{(V4)}$; to see
this, we use $(V3)$ to write
$\ket{u_{{\bm{k}}}}=S_{\bm{k}}\ket{\tilde{u}_{k}}$, where $S_{\bm{k}}$ is a
matrix that is not necessarily unitary and $\ket{\tilde{u}_{k}}$ is
holomorphic. Then we have

$\displaystyle Q({\bm{k}})\partial_{k_{x}}\ket{u_{{\bm{k}}a}}$
$\displaystyle=Q({\bm{k}})(\partial_{k}+\overline{\partial}_{k})S_{\bm{k}}\ket{\tilde{u}_{k}}$
(29) $\displaystyle=S_{\bm{k}}Q({\bm{k}})\partial_{k}\ket{\tilde{u}_{ka}}$
$\displaystyle=Q({\bm{k}})\partial_{k}\ket{u_{{\bm{k}}a}}$

where we used that the term coming from differentiating $S({\bm{k}})$ is
annihilated by $Q({\bm{k}})$. Similarly we can write
$Q({\bm{k}})\partial_{k_{y}}\ket{u_{{\bm{k}}}}=iQ({\bm{k}})\partial_{k}\ket{u_{{\bm{k}}}}$.
Substituting into (27) then yields (V4).

##  Appendix E First vortexability of trial wavefunctions

In this section we show explicitly that the trial wavefunctions that we have
used throughout the main text are first vortexable. For readability, we recall
the definition now. A band is first vortexable if

  1. (P1)

The band is not vortexable by itself

  2. (P2)

There exists an orthogonal vortexable band with projector $P_{0}$ (i.e.
$P_{0}z\psi_{0}~{}=~{}z\psi_{0}$ for all $P_{0}\psi_{0}=\psi_{0}$) such that

$(P_{1}+P_{0})z\psi_{1}=z\psi_{1}$ (30)

for all states $P\psi_{1}=\psi_{1}$

  3. (P3)

There is no alternative basis of wavefunctions
$(\phi_{0,{\bm{k}}},\phi_{1,{\bm{k}}})$, where each band is vortexable, that
also spans the two band subspace $P_{0}+P_{1}$

We begin with a useful Lemma used repeatedly below.

######  Lemma 1.

A function of crystal momentum that satisfies
$f({\bm{k}}+{\bm{G}})=f({\bm{k}})-i\lambda\overline{G}$ under translation by
reciprocal lattice vectors, for some $\lambda\neq 0$, is not holomorphic in
$k_{x}+ik_{y}$.

###### Proof.

Consider a contour integral of $f$ around the parallelogram boundary of the
BZ, with side lengths $G_{1}$ and $G_{2}$, that vanishes for holomorphic
functions by Cauchy’s theorem. On the other hand, the shift-periodicity of $f$
relates the integrals on opposite sides of the parallelogram such that

$\oint_{\partial{\mathrm{BZ}}}f({\bm{k}})dk=-i\lambda(\overline{G_{1}}G_{2}-\overline{G_{2}}G_{1})=\lambda
A_{\mathrm{BZ}}\neq 0.$ (31)

We conclude that $f$ is not holomorphic ∎

We begin by outlining how we will show that the trial wavefunctions
$(\Phi_{0,{\bm{k}}},\Phi_{1,{\bm{k}}})$ used in the main text are vortexable
and first vortexable, respectively. The first step is to show that the various
$\Phi_{1,{\bm{k}}}({\bm{r}})$ we have constructed are not vortexable
themselves, property (P2) but are when combined with
$\Phi_{0,{\bm{k}}}({\bm{r}})$, (P1). It is possible to show this in real
space, by acting with $z$ on $\Phi_{1,{\bm{k}}}({\bm{r}})$ and expressing the
result as a sum of $\Phi_{1,{\bm{k}}}({\bm{r}})$ and
$\Phi_{0,{\bm{k}}}({\bm{r}})$. However, (P3) demands that is no “decomposed"
basis, in which the space spanned by $\Phi_{(0,1){\bm{k}}}({\bm{r}})$ is split
into two bands that are vortexable on their own. To show that such a basis
doesn’t exist, it is convenient to work in a holomorphic gauge in momentum
space as there are powerful no-go results on holomorphic functions. Thus,
throughout this section we will use the equivalence of vortexability with the
existence of a holomorphic gauge (V3). We note that the wavefunctions in the
holomorphic gauge are generically neither normalized nor orthogonal, and gauge
transformations are invertible but generically non-unitary.

We note that all trial wavefunctions that we have used (c.f. Secs. A and C)
have periodic wavefunctions
$u_{\bm{k}}({\bm{r}})=e^{-i{\bm{k}}\cdot{\bm{r}}}\Phi_{\bm{k}}({\bm{r}})$ of
the form

$\displaystyle u_{0,k}({\bm{r}})$
$\displaystyle=f_{k}(z)\mathcal{N}_{0}({\bm{r}})$ (32) $\displaystyle
u_{1,{\bm{k}}}({\bm{r}})$
$\displaystyle=\left[\left(\overline{z}-2B_{0}^{-1}\partial_{z}-\alpha({\bm{k}})\right)f_{k}({\bm{r}})\right]\mathcal{N}_{0}({\bm{r}})$
$\displaystyle+f_{k}({\bm{r}})\mathcal{N}_{1}({\bm{r}})$

where

$f_{k}({\bm{r}})=e^{-\frac{i}{2}k\overline{z}}\sigma(z+iB_{0}^{-1}k).$ (33)

We have used that multiplication by $e^{i{\bm{k}}\cdot{\bm{r}}}$ converts
$e^{\frac{i}{2}\overline{k}z}$ to $e^{-\frac{i}{2}k\overline{z}}$. The
function $\alpha({\bm{k}})$ was chosen so that the states are orthogonal, and
the functions $\mathcal{N}_{0,1}^{l}({\bm{r}})$ are functions of the layer and
real-space but not ${\bm{k}}$. We note that we can write
$\left(\overline{z}-2B_{0}^{-1}\partial_{z}\right)f_{k}({\bm{r}})=2i\partial_{k}f_{k}({\bm{r}})$.
This implies that we can form the $k$-holomorphic states

$\displaystyle
v_{0,k}=u_{0,k},\,\,v_{1,k}({\bm{r}})=2i\partial_{k}u_{0,k}({\bm{r}})+f_{k}({\bm{r}})\mathcal{N}_{1}({\bm{r}})$
(34)

as an alternative holomorphic basis through dropping $\alpha({\bm{k}})$.

It is important to emphasize that the wavefunctions $v_{1,k}^{l}({\bm{r}})$ do
not describe a vortexable band despite the fact that they are holomorphic in
$k$. Indeed, they do not define a single Bloch band because $v_{1,k+G}$ is not
proportional to $v_{1,k}$. Using
$\sigma(z+R)=\eta_{\bm{R}}e^{\frac{B}{2}\bar{R}(z+R/2)}\sigma(z)$, we have
that

$v_{0,k+G}({\bm{r}})=e^{-i{\bm{G}}\cdot{\bm{r}}}\xi_{\bm{G}}(k)v_{0,k}({\bm{r}})$

for
$\xi_{G}(k)=\eta_{\bm{G}}e^{\frac{1}{2}B^{-1}\overline{G}(k+\frac{G}{2})}$. In
contrast,

$\displaystyle v_{1,k+G}({\bm{r}})$
$\displaystyle=e^{-i{\bm{G}}\cdot{\bm{r}}}\left(2i\partial_{k}(\xi_{{\bm{G}}}(k)v_{0,k}+\xi_{{\bm{G}}}(k)f_{k}({\bm{r}})\mathcal{N}_{1}\right)$
(35)
$\displaystyle=e^{-i{\bm{G}}\cdot{\bm{r}}}\xi_{{\bm{G}}}(k)(v_{1,k}+iB_{0}^{-1}\overline{G}v_{0,k})$

where we used
$2i\xi_{\bm{G}}^{-1}\partial_{k}\xi_{\bm{G}}=iB_{0}^{-1}\overline{G}$.

The existence of the above holomorphic gauge shows that the entire two band
subspace is vortexable, (P2). However, the nonperiodicity allows us to show
that $u_{1,{\bm{k}}}=v_{1,k}({\bm{r}})-\alpha({\bm{k}})v_{0,k}({\bm{r}})$ is
not vortexable on its own. Indeed, $v_{1,k}$ is vortexable if and only if
$\alpha({\bm{k}})$ is holomorphic. However, using boundary conditions for
$v_{nk}$, we find $\alpha({\bm{k}})=\lVert
v_{0,k}\rVert^{-2}\braket{v_{0,k}}{v_{1,k}}$ satisfies
$\alpha({\bm{k}}+{\bm{G}})=\alpha({\bm{k}})+iB_{0}^{-1}\overline{G}$. Thus
Lemma 1 implies $\alpha$ is not holomorphic, and (P1) is satisfied.

We must now prove (P3). This amounts to showing that

$\begin{pmatrix}w_{0,k}\\\
w_{1,k}\end{pmatrix}=S_{k}^{-1}\begin{pmatrix}v_{0,k}\\\ v_{1,k}\end{pmatrix}$
(36)

cannot have diagonal boundary conditions,

$\begin{pmatrix}w_{0,k+G}\\\
w_{1,k+G}\end{pmatrix}=e^{-i{\bm{G}}\cdot{\bm{r}}}\begin{pmatrix}\tilde{\xi}_{\bm{G}}^{(0)}(k)&0\\\
0&\tilde{\xi}_{\bm{G}}^{(1)}(k)\end{pmatrix}\begin{pmatrix}w_{0,k}\\\
w_{1,k}\end{pmatrix},$ (37)

for any holomorphic invertible matrix $S_{k}$.

Our strategy will be to assume diagonal boundary conditions (37) and
holomorphic $S_{k}$ and seek a contradiction. The boundary conditions of
$v_{mk}$ that we computed above, phrased in matrix notation, are

$\begin{pmatrix}v_{0,k+G}\vspace{0.5em}\\\
v_{1,k+G}\end{pmatrix}=e^{-i{\bm{G}}\cdot{\bm{r}}}\xi_{G}(k)\begin{pmatrix}1&0\\\
iB_{0}^{-1}\overline{G}&1\end{pmatrix}\begin{pmatrix}v_{0,k}\vspace{0.5em}\\\
v_{1,k}\end{pmatrix}.$ (38)

The boundary conditions for $w_{k}$ are then related to the gauge
transformation $S_{k}$ as

$S_{k+G}\begin{pmatrix}\tilde{\xi}_{\bm{G}}^{(0)}(k)&0\\\
0&\tilde{\xi}_{\bm{G}}^{(1)}(k)\end{pmatrix}=\xi_{\bm{G}}(k)\begin{pmatrix}1&0\\\
i\overline{G}&1\end{pmatrix}S_{k}.$ (39)

We first focus on the top left entry of (39), which reads

$S_{00}(k+G)\tilde{\xi}_{{\bm{G}}}^{(0)}(k)=S_{00}(k)\xi_{\bm{G}}(k),$ (40)

where $S_{00}$ is the top left entry of $S$. Without loss of generality assume
that the Chern number $C_{0}$ of the band described by $w^{0,k}$ is zero or
one (the Chern numbers must sum to two, and the Chern numbers of vortexable
bands are positive). If $C_{0}=0$, then the band admits a tight-binding
description with one delta-function-localized Wannier state per unit cell [28]
and thus a gauge with $w_{0,k}=1$, and $\tilde{\xi}^{(0)}_{{\bm{G}}}(k)=1$.
Since $S_{k}$ is invertible, we can demand that $h(k)=\partial_{k}\log
S_{00}(k)$ is holomorphic. However, by (40), with
$\tilde{\xi}^{(0)}_{{\bm{G}}}(k)=1$, we have
$h(k+G)=h(k)+\frac{1}{2}\overline{G}$, so that by Lemma 1 $h(k)$ is not
holomorphic. We therefore rule out the $C_{0}=0$ case, and focus on $C_{0}=1$.

Vortexable bands with $C_{0}=1$ have been classified [23, 26], and have
boundary conditions
$\tilde{\xi}_{\bm{G}}^{(0)}(k)=\eta_{\bm{G}}e^{\frac{1}{2}B^{-1}\overline{G}(k-k_{0}+\frac{G}{2})}=\xi_{\bm{G}}(k-k_{0})$,
for some offset $k_{0}$. Then, Lemma 1 applied to $\log S_{00}(k)$ this time
implies $k_{0}=0$, so that $S_{00}(k+G)=S_{00}(k)$ is periodic. Periodic
holomorphic functions are constant, so we can set $S_{00}(k)=1$ without loss
of generality. The lower-left component of (39) then yields
$S_{10}(k+G)=S_{10}(k)+iB_{0}^{-1}\overline{G}$, which again leads to a
contradiction by Lemma 1. We have thus proven (P3), so that we can conclude
that the trial wavefunctions used in the main text are first vortexable.

##  Appendix F QGT of “maximal" first vortexability

In this section, we compute the quantum geometric tensor of the combined space
of a “maximal" first vortexable band and its zeroth counterpart. By maximal,
we specifically mean that there is no zeroth LL portion in the wavefunction
other than what is required for the orthogonal basis. For example, in the
language of the holomorphic periodic states of the previous section,

$\displaystyle v_{kl}^{(0)}$
$\displaystyle=f_{k}({\bm{r}})\mathcal{N}_{0}({\bm{r}}),$ (41) $\displaystyle
v_{kl}^{(1)}$
$\displaystyle=2i\partial_{k}f_{k}({\bm{r}})\mathcal{N}_{0}({\bm{r}})+f_{k}({\bm{r}})\mathcal{N}_{1}({\bm{r}}),$

we demand $\mathcal{N}_{1}({\bm{r}})\to 0$. This corresponds to the limit
$\gamma_{1}\ell_{B}/v\to\infty$ in the Bernal graphene based Hamiltonians in
the main text. In an orthonormal basis we then obtain

$\displaystyle u_{0,{\bm{k}}}$
$\displaystyle=\frac{1}{N_{0,{\bm{k}}}}f_{k}({\bm{r}})\mathcal{N}_{0}({\bm{r}}),$
(42) $\displaystyle u_{1,{\bm{k}}}$
$\displaystyle=\frac{1}{N_{1,{\bm{k}}}}Q_{0}({\bm{k}})\partial_{k}u_{0,{\bm{k}}}$

where $Q_{0}({\bm{k}})=1-\ket{u_{0,k}}\bra{u_{0,k}}$ orthogonalizes $u_{1,k}$
relative to $u_{0,k}$ and the factors $N_{m,{\bm{k}}}$ are such that the
wavefunctions are normalized.

We recall property (V4), which states that vortexability of the bands is
equivalent to
$\hat{\eta}^{\mu\nu}_{ab}=\frac{1}{2}\hat{\Omega}_{ab}(\delta_{\mu\nu}-i\varepsilon_{\mu\nu})$,
where the Berry curvature is

$\hat{\Omega}_{ab}({\bm{k}})=\bra{\partial_{k}u_{{\bm{k}}b}}Q({\bm{k}})\ket{\partial_{k}u_{{\bm{k}}a}}$
(43)

We now show that $\hat{\Omega}_{00}=\hat{\Omega}_{01}=\hat{\Omega}_{10}=0$, in
the maximal $\mathcal{N}^{(1)}\to 0$ limit, so that the only nonzero entry is
$\hat{\Omega}_{11}$, as claimed in the main text.

From (43), it sufficies to show
$Q({\bm{k}})\ket{\partial_{k}u_{0,{\bm{k}}}}=0$. This in turn follows from
writing

$\displaystyle\ket{\partial_{k}u_{0,{\bm{k}}}}$
$\displaystyle=P_{0}\ket{\partial_{k}u_{0,{\bm{k}}}}+Q_{0}\ket{\partial_{k}u_{0,{\bm{k}}}},$
(44)
$\displaystyle=P_{0}\ket{\partial_{k}u_{0,{\bm{k}}}}+N_{1,{\bm{k}}}\ket{u_{1,{\bm{k}}}},$

where we used (42). The first term is proportional to $u_{0,{\bm{k}}}$ by
construction while the second term is proportional to $u_{1,{\bm{k}}}$, so
that $Q({\bm{k}})\ket{\partial_{k}u_{0,{\bm{k}}}}=0$ as claimed.

##  Appendix G $\gamma_{1}$ dependence for $M(\mathbf{k})$

Figure 5: (a)$\langle M\rangle_{\rm BZ}$ for the analytical example [Eq.(10)]
(green points) and Bernal graphene with magnetic field [Eq.(53)] (black line)
as a function of $\gamma_{1}$, and $\gamma_{1}\ell_{B}/\sqrt{2}v$,
respectively. (b) The gray line represents the bandwidth of the set of four
low-energy bands of Eq.(10) as a function of $\alpha$ for $\gamma_{1}=1$. The
bandwidth becomes zero at the magic angle of TBG, $\alpha=0.586$. The blue and
red lines are the averaged overlaps $\langle T_{n}\rangle_{\rm BZ}$ for $n=0$
and $n=1$, respectively.

In this section, we compute the “maximality index" which measures how close
one is to the “maximal" first LL:

$M({\bm{k}})=\frac{\lvert\lambda_{1}({\bm{k}})-\lambda_{2}({\bm{k}})\rvert}{\lambda_{1}({\bm{k}})+\lambda_{2}({\bm{k}})},$
(45)

where $\lambda_{1,2}$ are the eigenvalues of the $2\times 2$ non-Abelian Berry
curvature. The previous subsection implies $M({\bm{k}})=1$ when
$\mathcal{N}_{1}\to 0$. Here we will compute $M$ for Bernal graphene with
constant $B>0$ analytically. We will also discuss the $M({\bm{k}})$ of the
chiral TBG example, which ends up closely related.

Bernal graphene in an external magnetic field has a continuous magnetic
translation symmetry; the magnetic translation operators are
$T(\bm{l})=e^{-i\xi_{\bm{l}}(\bm{r})}t(\bm{l})$ where
$t(\bm{l})=e^{i\bm{l}\cdot\bm{\nabla}}$ is the ordinary, $B=0$ translation
operator. The phase satisfies
$\bm{A}(\bm{r}+\bm{l})=\bm{A}(\bm{r})+\nabla\xi_{\bm{l}}(\bm{r})$, where
$\bm{A}$ is the vector potential. The magnetic translation operators satisfy

$T(\bm{l}_{1})T(\bm{l}_{2})=T(\bm{l}_{2})T(\bm{l}_{1})e^{iB\hat{z}\cdot\bm{l}_{1}\times\bm{l}_{2}}$
(46)

A commuting subset, generated by $T_{{\bm{R}}_{1}}$, $T_{{\bm{R}}_{2}}$ with
$B\hat{z}\cdot{\bm{R}}_{1}\times{\bm{R}}_{2}=2\pi$, can then be used to define
Bloch states $\Phi_{{\bm{k}}}({\bm{r}})$. The algebra (46) implies that
magnetic translations at other $\ell\neq{\bm{R}}$ act as “ladder operators" on
the Bloch states,
$T_{\bm{l}}\Phi_{{\bm{k}}}\propto\Phi_{{\bm{k}}-B\hat{z}\times\bm{l}}$ (see
e.g. Appendix B of Ref. [26] for a pedagogical discussion). Thus, all Bloch
momenta are symmetry related. We will shortly use this fact to specialize to
calculating the Berry curvature at ${\bm{k}}=0$.

The periodic states
$\mathcal{U}_{m,{\bm{k}}}=e^{-i{\bm{k}}\cdot{\bm{r}}}\Phi_{m,{\bm{k}}}({\bm{r}})$
of Bernal graphene have the form

$\displaystyle\mathcal{U}_{0,{\bm{k}}}$
$\displaystyle=\left(\begin{array}[]{c}u_{0,{\bm{k}}}\\\ 0\end{array}\right),$
(47) $\displaystyle\mathcal{U}_{1,{\bm{k}}}$
$\displaystyle=\frac{1}{\sqrt{\gamma_{1}^{2}+2(v\ell_{B}^{-1})^{2}}}\left(\begin{array}[]{c}\gamma_{1}u_{1,{\bm{k}}}\\\
\sqrt{2}v\ell_{B}^{-1}u_{0,{\bm{k}}}\end{array}\right),$

where $u_{0,{\bm{k}}}$ is the periodic state of the usual LLL and
$u_{1,{\bm{k}}}$ is that of the usual first LL, obtained from the lowest
through the raising operator. Both states are normalized. We will shortly use

$u_{1,{\bm{k}}}=\sqrt{B}Q_{0}({\bm{k}})\partial_{k}u_{0,{\bm{k}}}$ (48)

which can be computed directly or indirectly. The indirect route begins by
writing $u_{1,{\bm{k}}}=\lambda Q_{0}({\bm{k}})\partial_{k}u_{0k}$. One then
solves for $\lambda=\sqrt{B}$ through
$1=\braket{u_{1,{\bm{k}}}}{u_{1,{\bm{k}}}}=\lvert\lambda\rvert^{2}\bra{\partial_{k}u_{0,{\bm{k}}}}Q({\bm{k}})\ket{\partial_{k}u_{0,{\bm{k}}}}=\lvert\lambda\rvert^{2}\Omega_{\text{LLL}}$,
where we have identified $\Omega_{\text{LLL}}=1/B$ as the Berry curvature of
the LLL. We will also use

$\displaystyle\langle u_{1,{\bm{k}}}|\partial_{k}u_{0,{\bm{k}}}\rangle$
$\displaystyle=\sqrt{B}\langle\partial_{k}u_{0,{\bm{k}}}|Q_{0}({\bm{k}})|\partial_{k}u_{0,{\bm{k}}}\rangle,$
(49) $\displaystyle=\sqrt{B}\Omega_{\text{LLL}}=1/\sqrt{B}.$

We are now ready to compute the non-Abelian Berry curvature

$\hat{\Omega}_{mn}({\bm{k}})=\left\langle\partial_{k}\mathcal{U}_{n,{\bm{k}}}|Q({\bm{k}})|\partial_{k}\mathcal{U}_{m,{\bm{k}}}\right\rangle.$
(50)

Let us fix ${\bm{k}}=0$ without loss of generality, so that we can more easily
apply rotation symmetry (recall that all momenta are related by magnetic
translation symmetry, such that $\hat{\Omega}$ is ${\bm{k}}$-independent). We
note that $\mathcal{U}_{1,{\bm{k}}=0}$ has a different angular momentum than
$\mathcal{U}_{0,{\bm{k}}=0}$. This implies that the off-diagonal elements of
$\hat{\Omega}$ must vanish.

The diagonal terms follow from

$\displaystyle\hat{\Omega}_{00}$
$\displaystyle=\langle\partial_{k}\mathcal{U}_{0,{\bm{k}}}|(1-P_{0}-P_{1})|\partial_{k}\mathcal{U}_{0,{\bm{k}}}\rangle$
(51)
$\displaystyle=\langle\partial_{k}\mathcal{U}_{0,{\bm{k}}}|Q_{0}|\partial_{k}\mathcal{U}_{0,{\bm{k}}}\rangle-\langle\partial_{k}\mathcal{U}_{0,{\bm{k}}}|P_{1}|\partial_{k}\mathcal{U}_{0,{\bm{k}}}\rangle$
$\displaystyle=\Omega_{\text{LLL}}-\frac{\gamma_{1}^{2}}{\gamma_{1}^{2}+2(v\ell_{B}^{-1})^{2}}\lvert\langle
u_{1,k}|\partial_{k}u_{0,k}\rangle\rvert^{2}$
$\displaystyle=B^{-1}\frac{2(v\ell_{B}^{-1})^{2}}{\gamma_{1}^{2}+2(v\ell_{B}^{-1})^{2}}.$

Because the trace of the non-Abelian Berry curvature must be $2/B$, to be
consistent with $C=2$ for the two-band system, we must have

$\hat{\Omega}_{11}=B^{-1}\frac{2(v\ell_{B}^{-1})^{2}+2\gamma_{1}^{2}}{\gamma_{1}^{2}+2(v\ell_{B}^{-1})^{2}}.$
(52)

We therefore obtain

$M=\frac{\gamma_{1}^{2}}{\gamma_{1}^{2}+2(v\ell_{B}^{-1})^{2}},$ (53)

as claimed in the main text. At $\gamma_{1}\rightarrow 0$ which corresponds to
the decoupling of Bernal graphene into two identical monolayer graphene, the
ingredients of non-Abelian Berry curvature are
$\hat{\Omega}_{01}=\hat{\Omega}_{10}=0$ and
$\hat{\Omega}_{00}=\hat{\Omega}_{11}=1/B$ so the maximal index is $M=0$. On
the other hand, at $\gamma_{1}\rightarrow\infty$,
$\hat{\Omega}_{00}=\hat{\Omega}_{01}=\hat{\Omega}_{10}=0$ and
$\hat{\Omega}_{11}=2/B$, leading to $M=1$.

We numerically calculate the $\langle M(\bm{k})\rangle_{\rm BZ}$ for the
analytical example [Eq.(10)] at magic angle $\alpha=0.586$ in Fig.5. The index
depends on $\gamma_{1}$ almost as same as that of Bernal graphene $B>0$.

##  Appendix H Self-consistent Hatree-Fock calculation

We consider realistic gate-screened Coulomb interactions:

$\hat{H}_{\rm
tot}=\hat{H}_{SBG}+\frac{1}{2A}\sum_{\bm{q}}V_{\bm{q}}:\hat{\rho}_{\bm{q}}\hat{\rho}_{-\bm{q}}:,\quad
V_{q}=\frac{2\pi\tanh(qd)}{\epsilon_{r}\epsilon_{0}q}$ (54)

with density operator $\rho_{\bm{q}}$ at wavevector $\bm{q}$, normal ordering
relative to filling $\nu=0$, sample area $A$, gate distance
$d=$250\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, and relative permittivity
$\epsilon_{r}=15$. To obtain the two isolated Chern bands, we employ self-
consistent Hartree-Fock (SCHF) calculations on $24\times 24$ unit cells. We
assume spin and valley polarization [79], and project to well-isolated
subspace of the three highest valence bands.

## References

  * Parameswaran  _et al._ [2013a] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum hall physics in topological flat bands, Comptes Rendus Physique 14, 816 (2013a).
  * Bergholtz and Liu [2013] E. J. Bergholtz and Z. Liu, Topological flat band models and fractional chern insulators, International Journal of Modern Physics B 27, 1330017 (2013).
  * Neupert  _et al._ [2015] T. Neupert, C. Chamon, T. Iadecola, L. H. Santos, and C. Mudry, Fractional (chern and topological) insulators, Physica Scripta T164, 014005 (2015).
  * Liu and Bergholtz [2023] Z. Liu and E. J. Bergholtz, Recent developments in fractional chern insulators, in  _Reference Module in Materials Science and Materials Engineering_ (Elsevier, 2023).
  * Neupert  _et al._ [2011] T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  * Sheng  _et al._ [2011] D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum hall effect in the absence of landau levels, Nature communications 2, 389 (2011).
  * Regnault and Bernevig [2011] N. Regnault and B. A. Bernevig, Fractional chern insulator, Physical Review X 1, 021014 (2011).
  * Tang  _et al._ [2011] E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum hall states, Physical review letters 106, 236802 (2011).
  * Sun  _et al._ [2011] K. Sun, Z. Gu, H. Katsura, and S. D. Sarma, Nearly flatbands with nontrivial topology, Physical review letters 106, 236803 (2011).
  * Roy [2014a] R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014a).
  * Spanton  _et al._ [2018] E. M. Spanton, A. A. Zibrov, H. Zhou, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Observation of fractional chern insulators in a van der waals heterostructure, Science 360, 62 (2018).
  * Xie  _et al._ [2021] Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester, _et al._ , Fractional chern insulators in magic-angle twisted bilayer graphene, Nature 600, 439 (2021).
  * Cai  _et al._ [2023] J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, _et al._ , Signatures of fractional quantum anomalous hall states in twisted MoTe2, Nature 622, 63 (2023).
  * Zeng  _et al._ [2023] Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional chern insulator in moiré MoTe2, Nature 622, 69 (2023).
  * Park  _et al._ [2023] H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, _et al._ , Observation of fractionally quantized anomalous hall effect, Nature 622, 74 (2023).
  * Xu  _et al._ [2023] F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, _et al._ , Observation of integer and fractional quantum anomalous hall effects in twisted bilayer MoTe2, Physical Review X 13, 031037 (2023).
  * Lu  _et al._ [2023] Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous hall effect in a graphene moire superlattice, arXiv preprint arXiv:2309.17436 10.48550/arXiv.2309.17436 (2023).
  * Parker  _et al._ [2021] D. Parker, P. Ledwith, E. Khalaf, T. Soejima, J. Hauschild, Y. Xie, A. Pierce, M. P. Zaletel, A. Yacoby, and A. Vishwanath, Field-tuned and zero-field fractional chern insulators in magic angle graphene, arXiv preprint arXiv:2112.13837 10.48550/arXiv.2112.13837 (2021).
  * Ledwith  _et al._ [2020] P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional chern insulator states in twisted bilayer graphene: An analytical approach, Physical Review Research 2, 023237 (2020).
  * Ledwith  _et al._ [2021] P. J. Ledwith, E. Khalaf, and A. Vishwanath, Strong coupling theory of magic-angle graphene: A pedagogical introduction, Annals of Physics 435, 168646 (2021).
  * Ozawa and Mera [2021] T. Ozawa and B. Mera, Relations between topology and the quantum metric for chern insulators, Physical Review B 104, 045103 (2021).
  * Mera and Ozawa [2021a] B. Mera and T. Ozawa, Kähler geometry and chern insulators: Relations between topology and the quantum metric, Physical Review B 104, 045104 (2021a).
  * Wang  _et al._ [2021a] J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact landau level description of geometry and interaction in a flatband, Physical review letters 127, 246403 (2021a).
  * Ledwith  _et al._ [2022] P. J. Ledwith, A. Vishwanath, and E. Khalaf, Family of ideal chern flatbands with arbitrary chern number in chiral twisted graphene multilayers, Physical Review Letters 128, 176404 (2022).
  * Wang and Liu [2022] J. Wang and Z. Liu, Hierarchy of ideal flatbands in chiral twisted multilayer graphene models, Physical Review Letters 128, 176403 (2022).
  * Dong  _et al._ [2023a] J. Dong, P. J. Ledwith, E. Khalaf, J. Y. Lee, and A. Vishwanath, Many-body ground states from decomposition of ideal higher chern bands: Applications to chirally twisted graphene multilayers, Phys. Rev. Res. 5, 023166 (2023a).
  * Wang  _et al._ [2023] J. Wang, S. Klevtsov, and Z. Liu, Origin of model fractional chern insulators in all topological ideal flatbands: Explicit color-entangled wave function and exact density algebra, Physical Review Research 5, 023167 (2023).
  * Ledwith  _et al._ [2023] P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vortexability: A unifying criterion for ideal fractional chern insulators, Physical Review B 108, 205144 (2023).
  * Wu  _et al._ [2019] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. MacDonald, Topological insulators in twisted transition metal dichalcogenide homobilayers, Physical review letters 122, 086402 (2019).
  * Li  _et al._ [2021] H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Spontaneous fractional chern insulators in transition metal dichalcogenide moiré superlattices, Physical Review Research 3, L032070 (2021).
  * Crépel and Fu [2023] V. Crépel and L. Fu, Anomalous hall metal and fractional chern insulator in twisted transition metal dichalcogenides, Physical Review B 107, L201109 (2023).
  * Dong  _et al._ [2023b] J. Dong, J. Wang, P. J. Ledwith, A. Vishwanath, and D. E. Parker, Composite fermi liquid at zero magnetic field in twisted MoTe2, Phys. Rev. Lett. 131, 136502 (2023b).
  * Goldman  _et al._ [2023] H. Goldman, A. P. Reddy, N. Paul, and L. Fu, Zero-field composite fermi liquid in twisted semiconductor bilayers, Phys. Rev. Lett. 131, 136501 (2023).
  * Hu  _et al._ [2023] X. Hu, D. Xiao, and Y. Ran, Hyperdeterminants and composite fermion states in fractional chern insulators, arXiv preprint arXiv:2312.00636 10.48550/arXiv.2312.00636 (2023).
  * Song  _et al._ [2023] X.-Y. Song, C.-M. Jian, L. Fu, and C. Xu, Intertwined fractional quantum anomalous hall states and charge density waves, arXiv preprint arXiv:2310.11632 10.48550/arXiv.2310.11632 (2023).
  * Mao  _et al._ [2023] N. Mao, C. Xu, J. Li, T. Bao, P. Liu, Y. Xu, C. Felser, L. Fu, and Y. Zhang, Lattice relaxation, electronic structure and continuum model for twisted bilayer MoTe2, arXiv preprint arXiv:2311.07533 10.48550/arXiv.2311.07533 (2023).
  * Yu  _et al._ [2023] J. Yu, J. Herzog-Arbeitman, M. Wang, O. Vafek, B. A. Bernevig, and N. Regnault, Fractional chern insulators vs. non-magnetic states in twisted bilayer MoTe2, arXiv preprint arXiv:2309.14429 10.48550/arXiv.2309.14429 (2023).
  * Morales-Durán  _et al._ [2023] N. Morales-Durán, N. Wei, and A. H. MacDonald, Magic angles and fractional chern insulators in twisted homobilayer tmds, arXiv preprint arXiv:2308.03143 10.48550/arXiv.2308.03143 (2023).
  * Jia  _et al._ [2023] Y. Jia, J. Yu, J. Liu, J. Herzog-Arbeitman, Z. Qi, N. Regnault, H. Weng, B. A. Bernevig, and Q. Wu, Moiré fractional chern insulators i: First-principles calculations and continuum models of twisted bilayer MoTe2, arXiv preprint arXiv:2311.04958 10.48550/arXiv.2311.04958 (2023).
  * Xu  _et al._ [2024] C. Xu, J. Li, Y. Xu, Z. Bi, and Y. Zhang, Maximally localized wannier functions, interaction models, and fractional quantum anomalous hall effect in twisted bilayer MoTe2, Proceedings of the National Academy of Sciences 121, e2316749121 (2024).
  * Crépel  _et al._ [2023] V. Crépel, N. Regnault, and R. Queiroz, The chiral limits of moiré semiconductors: origin of flat bands and topology in twisted transition metal dichalcogenides homobilayers (2023), arXiv:2305.10477 [cond-mat.mes-hall] .
  * Wang  _et al._ [2024] C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, Fractional chern insulator in twisted bilayer MoTe2, Physical Review Letters 132, 036501 (2024).
  * Li  _et al._ [2024] H. Li, Y. Su, Y. B. Kim, H.-Y. Kee, K. Sun, and S.-Z. Lin, Contrasting twisted bilayer graphene and transition metal dichalcogenides for fractional chern insulators: an emergent gauge picture, arXiv preprint arXiv:2402.02251 10.48550/arXiv.2402.02251 (2024).
  * Dong  _et al._ [2023c] Z. Dong, A. S. Patri, and T. Senthil, Theory of fractional quantum anomalous hall phases in pentalayer rhombohedral graphene moiré structures, arXiv preprint arXiv:2311.03445 10.48550/arXiv.2311.03445 (2023c).
  * Zhou  _et al._ [2023] B. Zhou, H. Yang, and Y.-H. Zhang, Fractional quantum anomalous hall effects in rhombohedral multilayer graphene in the moiréless limit and in coulomb imprinted superlattice, arXiv preprint arXiv:2311.04217 10.48550/arXiv.2311.04217 (2023).
  * Dong  _et al._ [2023d] J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous hall crystals in rhombohedral multilayer graphene I: Interaction-driven chern bands and fractional quantum hall states at zero magnetic field, arXiv preprint arXiv:2311.05568 10.48550/arXiv.2311.05568 (2023d).
  * Herzog-Arbeitman  _et al._ [2023] J. Herzog-Arbeitman, Y. Wang, J. Liu, P. M. Tam, Z. Qi, Y. Jia, D. K. Efetov, O. Vafek, N. Regnault, H. Weng, _et al._ , Moiré fractional chern insulators II: First-principles calculations and continuum models of rhombohedral graphene superlattices, arXiv preprint arXiv:2311.12920 10.48550/arXiv.2311.12920 (2023).
  * Kwan  _et al._ [2023] Y. H. Kwan, J. Yu, J. Herzog-Arbeitman, D. K. Efetov, N. Regnault, and B. A. Bernevig, Moiré fractional chern insulators III: Hartree-fock phase diagram, magic angle regime for chern insulator states, the role of the Moiré potential and goldstone gaps in rhombohedral graphene superlattices, arXiv preprint arXiv:2312.11617 10.48550/arXiv.2312.11617 (2023).
  * Guo  _et al._ [2023] Z. Guo, X. Lu, B. Xie, and J. Liu, Theory of fractional chern insulator states in pentalayer graphene moiré superlattice, arXiv preprint arXiv:2311.14368 10.48550/arXiv.2311.14368 (2023).
  * Willett  _et al._ [1987] R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. English, Observation of an even-denominator quantum number in the fractional quantum hall effect, Physical review letters 59, 1776 (1987).
  * Pan  _et al._ [2011] W. Pan, N. Masuhara, N. Sullivan, K. Baldwin, K. West, L. Pfeiffer, and D. Tsui, Impact of disorder on the 5/2 fractional quantum hall state, Physical review letters 106, 206806 (2011).
  * Eisenstein  _et al._ [2002] J. Eisenstein, K. Cooper, L. Pfeiffer, and K. West, Insulating and fractional quantum hall states in the first excited landau level, Physical Review Letters 88, 076801 (2002).
  * Ki  _et al._ [2014] D.-K. Ki, V. I. Fal’ko, D. A. Abanin, and A. F. Morpurgo, Observation of even denominator fractional quantum hall effect in suspended bilayer graphene, Nano letters 14, 2135 (2014).
  * Kim  _et al._ [2019] Y. Kim, A. C. Balram, T. Taniguchi, K. Watanabe, J. K. Jain, and J. H. Smet, Even denominator fractional quantum hall states in higher landau levels of graphene, Nature Physics 15, 154 (2019).
  * Zibrov  _et al._ [2017] A. A. Zibrov, C. Kometter, H. Zhou, E. Spanton, T. Taniguchi, K. Watanabe, M. Zaletel, and A. Young, Tunable interacting composite fermion phases in a half-filled bilayer-graphene landau level, Nature 549, 360 (2017).
  * Li  _et al._ [2017] J. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, and C. Dean, Even-denominator fractional quantum hall states in bilayer graphene, Science 358, 648 (2017).
  * Huang  _et al._ [2022] K. Huang, H. Fu, D. R. Hickey, N. Alem, X. Lin, K. Watanabe, T. Taniguchi, and J. Zhu, Valley isospin controlled fractional quantum hall states in bilayer graphene, Physical Review X 12, 031019 (2022).
  * Moore and Read [1991] G. Moore and N. Read, Nonabelions in the fractional quantum hall effect, Nuclear Physics B 360, 362 (1991).
  * Kumar  _et al._ [2010] A. Kumar, G. Csáthy, M. Manfra, L. Pfeiffer, and K. West, Nonconventional odd-denominator fractional quantum hall states in the second landau level, Physical review letters 105, 246808 (2010).
  * Read and Rezayi [1999] N. Read and E. Rezayi, Beyond paired quantum hall states: Parafermions and incompressible states in the first excited landau level, Physical Review B 59, 8084 (1999).
  * Mong  _et al._ [2017] R. S. Mong, M. P. Zaletel, F. Pollmann, and Z. Papić, Fibonacci anyons and charge density order in the 12/5 and 13/5 quantum hall plateaus, Physical Review B 95, 115136 (2017).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Non-abelian anyons and topological quantum computation, Reviews of Modern Physics 80, 1083 (2008).
  * Mong  _et al._ [2014] R. S. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, P. Fendley, C. Nayak, Y. Oreg, A. Stern, E. Berg, K. Shtengel, _et al._ , Universal topological quantum computation from a superconductor-abelian quantum hall heterostructure, Physical Review X 4, 011036 (2014).
  * Jain [1989] J. K. Jain, Composite-fermion approach for the fractional quantum hall effect, Phys. Rev. Lett. 63, 199 (1989).
  * Roy [2014b] R. Roy, Band geometry of fractional topological insulators, Physical Review B 90, 165139 (2014b).
  * Mera and Ozawa [2021b] B. Mera and T. Ozawa, Engineering geometrically flat chern bands with fubini-study kähler structure, Physical Review B 104, 115160 (2021b).
  * Varjas  _et al._ [2022] D. Varjas, A. Abouelkomsan, K. Yang, and E. J. Bergholtz, Topological lattice models with constant Berry curvature, SciPost Phys. 12, 118 (2022).
  * Mera and Ozawa [2023] B. Mera and T. Ozawa, Uniqueness of landau levels and their analogs with higher chern numbers, arXiv preprint arXiv:2304.00866 10.48550/arXiv.2304.00866 (2023).
  * Estienne  _et al._ [2023] B. Estienne, N. Regnault, and V. Crépel, Ideal chern bands as landau levels in curved space, Phys. Rev. Res. 5, L032048 (2023).
  * Liu  _et al._ [2024] H. Liu, K. Yang, A. Abouelkomsan, Z. Liu, and E. J. Bergholtz, Broken symmetry in ideal chern bands, arXiv preprint arXiv:2402.04303 10.48550/arXiv.2402.04303 (2024).
  * Parameswaran  _et al._ [2013b] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum hall physics in topological flat bands, Comptes Rendus. Physique 14, 816–839 (2013b).
  * Claassen  _et al._ [2015] M. Claassen, C. H. Lee, R. Thomale, X.-L. Qi, and T. P. Devereaux, Position-momentum duality and fractional quantum hall effect in chern insulators, Physical review letters 114, 236802 (2015).
  * Tarnopolsky  _et al._ [2019] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett. 122, 106405 (2019).
  * Simon and Rudner [2020] S. H. Simon and M. S. Rudner, Contrasting lattice geometry dependent versus independent quantities: Ramifications for berry curvature, energy gaps, and dynamics, Phys. Rev. B 102, 165148 (2020).
  * Trugman and Kivelson [1985] S. A. Trugman and S. Kivelson, Exact results for the fractional quantum hall effect with general interactions, Phys. Rev. B 31, 5280 (1985).
  * [76] P. Ledwith, A. Vishwanath, and D. Parker, in Preparation.
  * Zhu  _et al._ [2020] Z. Zhu, D. N. Sheng, and I. Sodemann, Widely tunable quantum phase transition from moore-read to composite fermi liquid in bilayer graphene, Phys. Rev. Lett. 124, 097604 (2020).
  * Wan  _et al._ [2023] X. Wan, S. Sarkar, K. Sun, and S.-Z. Lin, Nearly flat chern band in periodically strained monolayer and bilayer graphene, Phys. Rev. B 108, 125129 (2023).
  * Gao  _et al._ [2023] Q. Gao, J. Dong, P. Ledwith, D. Parker, and E. Khalaf, Untwisting moiré physics: Almost ideal bands and fractional chern insulators in periodically strained monolayer graphene, Physical Review Letters 131, 096401 (2023).
  * Jiang  _et al._ [2017] Y. Jiang, J. Mao, J. Duan, X. Lai, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Visualizing strain-induced pseudomagnetic fields in graphene through an hbn magnifying glass, Nano letters 17, 2839 (2017).
  * Mao  _et al._ [2020] J. Mao, S. P. Milovanović, M. Anđelković, X. Lai, Y. Cao, K. Watanabe, T. Taniguchi, L. Covaci, F. M. Peeters, A. K. Geim, Y. Jiang, and E. Y. Andrei, Evidence of flat bands and correlated states in buckled graphene superlattices, Nature 584, 215 (2020).
  * Forsythe  _et al._ [2018] C. Forsythe, X. Zhou, K. Watanabe, T. Taniguchi, A. Pasupathy, P. Moon, M. Koshino, P. Kim, and C. R. Dean, Band structure engineering of 2d materials using patterned dielectric superlattices, Nature nanotechnology 13, 566 (2018).
  * Shi  _et al._ [2019] L.-k. Shi, J. Ma, and J. C. Song, Gate-tunable flat bands in van der waals patterned dielectric superlattices, 2D Materials 7, 015028 (2019).
  * Cano  _et al._ [2021] J. Cano, S. Fang, J. Pixley, and J. H. Wilson, Moiré superlattice on the surface of a topological insulator, Physical Review B 103, 155157 (2021).
  * Guerci  _et al._ [2022] D. Guerci, J. Wang, J. Pixley, and J. Cano, Designer meron lattice on the surface of a topological insulator, Physical Review B 106, 245417 (2022).
  * Ghorashi and Cano [2023] S. A. A. Ghorashi and J. Cano, Multilayer graphene with a superlattice potential, Physical Review B 107, 195423 (2023).
  * Barlas  _et al._ [2008] Y. Barlas, R. Côté, K. Nomura, and A. H. MacDonald, Intra-landau-level cyclotron resonance in bilayer graphene, Phys. Rev. Lett. 101, 097601 (2008).
  * Barlas  _et al._ [2012] Y. Barlas, K. Yang, and A. H. MacDonald, Quantum hall effects in graphene-based two-dimensional electron systems, Nanotechnology 23, 052001 (2012).
  * Kousa  _et al._ [2024] B. M. Kousa, N. Wei, and A. H. MacDonald, Orbital competition in bilayer graphene’s fractional quantum hall effect, arXiv preprint arXiv:2402.10440 10.48550/arXiv.2402.10440 (2024).
  * Wang  _et al._ [2021b] J. Wang, Y. Zheng, A. J. Millis, and J. Cano, Chiral approximation to twisted bilayer graphene: Exact intravalley inversion symmetry, nodal structure, and implications for higher magic angles, Phys. Rev. Res. 3, 023155 (2021b).
  * Haldane [2018] F. Haldane, A modular-invariant modified weierstrass sigma-function as a building block for lowest-landau-level wavefunctions on the torus, Journal of Mathematical Physics 59 (2018).
  * Koshino [2019] M. Koshino, Band structure and topological properties of twisted double bilayer graphene, Physical Review B 99, 235406 (2019).