# Time-Reversal Invariant Topological Moiré Flatband: A Platform for the
Fractional Quantum Spin Hall Effect

Yi-Ming Wu  Stanford Institute for Theoretical Physics, Stanford University,
Stanford, California 94305, USA  Daniel Shaffer  Department of Physics, Emory
University, 400 Dowman Drive, Atlanta, GA 30322, USA  Zhengzhi Wu  Institute
for Advanced Study, Tsinghua University, Beijing 100084, China  Luiz H. Santos
Department of Physics, Emory University, 400 Dowman Drive, Atlanta, GA 30322,
USA

###### Abstract

Motivated by recent experimental observation of the quantum spin Hall effect
in monolayer germanene, we study the topological phases of twisted bilayer
Kane-Mele model with time-reversal symmetry and spin $s_{z}$ conservation. For
large twist angles the helical edge states from the two layers are unstable
and the system is a trivial insulator. At small twist angles however, the
emergent moiré flatbands can be topologically nontrivial due to inversion
symmetry breaking from coupling to substrate. Each of these flatbands for each
spin projection admits a lowest-Landau-level description in the chiral limit
and at magic twist angle. This allows for the construction of a many-body
Laughlin state with time-reversal symmetry which can be stabilized by a short-
range pseudopotential, and therefore serves as an ideal platform for realizing
the so-far elusive fractional quantum spin Hall effect with emergent spin-1/2
U(1) symmetry.

Introduction.– Since the discovery of superconducting and correlated
insulating phases in magic angle twisted bilayer graphene (TBG)[1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12], the moiré engineering of 2D Van der Waals materials,
such as graphene and transition metal dichalcogenides (TMDs)[13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23], in a host of bilayer and multilayer
heterostructures[24, 25, 26, 27, 28, 29, 30, 31, 32], has attracted enormous
research attention. Moiré platforms are ideal hubs to forge the interplay
between strongly correlated effects and topological phases, giving rise to
rich phenomena including superconductivity[1, 6, 7, 8], strange metal
behavior[33, 34, 35], magnetic quantum anomalous Hall effect in TBG[36, 9, 37]
and more recently, long-sought after fractional Chern insulators (FCI)[38, 39,
40] first observed in twisted bilayer MoTe2[41, 42, 43].

The discovery of FCIs in TMD heterostructures highlights the importance of
moiré flat bands [44, 45, 46, 47, 48, 49, 50, 51] in the service of electron
fractionalization without a magnetic field. In particular, moiré flatbands in
the chiral limit [47] share properties akin to lowest Landau level (LLL)
wavefunctions, shedding light on the stability of time-reversal broken
topological states through local interactions[52, 53, 54, 55, 56]. Conversely,
a burning question arises: can moiré flat bands support time-reversal
symmetric (TRS) fractional topological order? While TRS $\mathbb{Z}_{2}$
flatbands have been theoretically proposed in twisted bilayer TMDs[57, 58, 59,
60, 61], and experimental signatures of the quantum spin Hall (QSH) effect
[62, 63, 64, 65] have been noted in twisted bilayer WSe2 [66], prospects for
fractional QSH effect in moiré systems remain terra incognita.

In this Letter, we propose a mechanism to realize TRS topological
$\mathbb{Z}_{2}$ moiré flatbands and establish them as potential platforms to
achieve fractional quantum spin Hall effect [64, 67, 68, 69] through electron
interactions in moiré heterostructures. Our departure is the paradigmatic
Kane-Mele (KM) model [62], recently realized in a monoelemental honeycomb
material–germanene [70]. Two layers of KM model are generally expected to form
a trivial insulator given the instability of the double pairs of helical edge
modes. [62] However, our analysis of the small-angle twisted bilayer KM
continuum model identifies topological phase transitions which can be tuned by
the twist angle, interlayer coupling strength and sublattice potential. The
resulting quasi-flat moiré bands are characterized by a TRS $\mathbb{Z}_{2}$
topological invariant signaling a new pair of helical edge states. Remarkably,
in the chiral limit where the $AA$ interlayer coupling vanishes, the
wavefunction for each flatband behave as a Kramer’s pair of time-reversal
invariant LLL wavefunctions containing both holomorphic and anti-holomorphic
coordinates related by time-reversal. As such, our work identifies key aspects
enabling TRS electron fractionalization in twisted moiré germanene. In
particular, we propose classes of many-body wavefunctions hosting fractional
QSH stabilized which ground states of short-range interactions.

This work thus puts forth a promising route to realize $\mathbb{Z}_{2}$
flatbands in germanene moiré heterostructure, uncovering a new platform to
explore fractional QSH. Our consideration may also apply to double twisted
bilayer TMDs systems where the moiré of moiré can develop at the lowest energy
scale, as well as cold atom platforms for which moiré engineering has also
been made possible recently[71].

Model.– The low energy description of monolayer KM model near the two valleys
$\pm\bm{K}$ rotated by a angle $\theta$ is given by [62]:

$h_{s\tau}(\theta,\bm{k})=\tau\hbar|\bm{k}|v_{F}\begin{pmatrix}0&e^{-i\tau(\theta_{k}-\theta)}\\\
e^{i\tau(\theta_{k}-\theta)}&0\\\
\end{pmatrix}+s\tau\lambda_{\text{so}}\sigma_{z}.$ (1)

Here we use $s=\pm 1$ to denote spin up and spin down, $\tau=\pm 1$ to denote
different valleys, and $\sigma$ the sublattice indices. $\lambda_{\text{so}}$
is the strength of spin-orbit coupling (SOC). The limit
$\lambda_{\text{so}}=0$ reduces to the massless Dirac Hamiltonian for a single
layer graphene. SOC opens a gap while preserving time-reversal ($\mathcal{T}$)
symmetry. Note this model has an emergent U(1) symmetry for the spin $s_{z}$
component. Following [44], the continuum model for twisted bilayer germanene
for spin $s$ and valley $\tau$ is described as top and bottom KM fermions with
a local interlayer coupling $T(\bm{r})$ that captures the moiré superlattice,

$H_{s\tau}=\begin{pmatrix}h_{s\tau}(\frac{\theta}{2},\bm{\nabla})+\delta\sigma_{z}&T_{\tau}(\bm{r})\\\
T_{\tau}^{\dagger}(\bm{r})&h_{s\tau}(-\frac{\theta}{2},\bm{\nabla})+\delta\sigma_{z}\end{pmatrix},$
(2)

where $h_{s\tau}(\theta/2,\bm{\nabla})=-i\tau\hbar
v_{F}R[\bm{\sigma}\cdot\bm{\nabla}]R^{-1}+s\tau\lambda_{\text{so}}\sigma_{z}$
(with $R=e^{-i\frac{\theta}{4}\sigma_{z}}$) is the real space representation
of Eq.(1). We have introduced an inversion-symmetry breaking term $\delta$ in
addition to the SOC term, which can arise due to coupling to the
substrate111Here we approximately assume that $\delta$ is the same for both
layers.. The Hamiltonian acts on a spinor
$\psi_{s\tau}=(\psi_{A1},\psi_{B1},\psi_{A2},\psi_{B2})^{T}$, where $A,B$ and
$1,2$ are sublattice and layer indices respectively. As in TBG the interlayer
coupling can be approximated by
$T_{+}(\bm{r})=\sum_{n=1}^{3}T_{n}e^{-i\bm{q}_{n}\cdot\bm{r}}$ where
$\bm{q}_{1}=k_{\theta}(0,-1)$, $\bm{q}_{2}=k_{\theta}(\sqrt{3}/2,1/2)$ and
$\bm{q}_{2}=k_{\theta}(-\sqrt{3}/2,1/2)$ with
$k_{\theta}=2k_{D}\sin(\theta/2)$ being the moiré Brillouin zone length scale.
Note that for the other valley $T_{-}(\bm{r})=T_{+}^{*}(\bm{r})$. The three
coefficients $T_{n}$ are
$T_{n+1}=w_{AA}I+w_{AB}(\sigma_{x}\cos\frac{2n\pi}{3}+\sigma_{y}\sin\frac{2n\pi}{3})$,
and $w_{AA}$ and $w_{AB}$ are the interlayer tunneling strength in AA- and AB-
stacked areas respectively.

Figure 1: (a)Moiré pattern with system size given by $L_{1}$ and $L_{2}$. (b)
Moiré Brillouin zone at $\pm K$ valleys. The definition of $\pm\kappa$ is
chosen in such a way that $\kappa$ at $+K$ is related to $-\kappa$ at $-K$
valley by time-reversal. (c)Moiré band structure for $+K$ valley at the first
magic angle $\theta=1.09^{\circ}$, obtained by choosing $w_{AB}=80$meV and
$\delta=0$. (d) Phase diagram with $\delta\neq 0$. For a fixed $w_{AA}$ and
$w_{AB}$, tuning the twist angle can result in a topological phase transition
from a $\mathbb{Z}_{2}$ band to a trivial band.

$\mathbb{Z}_{2}$ flatbands.– The moiré pattern and Brillouin zone of the
twisted bilayer germanene is shown in Fig.1(a) and (b) with low energy theory
given by Eq.(2). The corresponding moiré band structure is shown in Fig.1 (c).
Here we take $v_{F}=5.6\times 10^{5}m/s$, about $70\%$ of that in graphene
[73], and $\lambda_{\text{SO}}=24meV$ [70]. Assuming the interlayer coupling
is also around $70\%$ of that in TBG, we approximately take $w_{AB}=80meV$ and
set $r=w_{AA}/w_{AB}$ as a variable. The two bands marked in red are the moiré
flatbands, separated by a gap due to $\lambda_{\text{so}}\neq 0$. The first
magic angle $\theta\approx 1.09^{\circ}$ is defined as the one for which the
moiré bandwidth is minimized. Furthermore, for a given $s\tau$ configuration,
the flatbands have nonzero Chern numbers given by
$\mathcal{C}=\pm\operatorname{sgn}(s)$, where ‘$+$’ is for the upper bands and
‘$-$’ is for the lower bands. This gives rise to nontrival topology with TRS
which is charaterized by the $\mathbb{Z}_{2}$ topological invariant [74, 75]

$\nu_{\pm}=\frac{\mathcal{C}_{\uparrow,\pm}-\mathcal{C}_{\downarrow,\mp}}{2}\text{mod}2$
(3)

Thus the upper moiré band has $\nu_{\pm}=1$ while the lower moiré band has
$\nu_{\pm}=-1$. The topological phase can be tuned by $\theta$, $w_{AB}$ and
$w_{AA}$, as shown in Fig.1(d). We also confirmed that at even larger $\theta$
(not shown) the system becomes trivial as well, consistent with the
$\mathbb{Z}_{2}$ classification of two layer KM model.

Chiral limit.– The Hamiltonian in Eq.(2) has an emergent chiral symmetry when
$w_{AA}\to 0$ even in the presence of a finite $\lambda_{\text{so}}$. To see
this, one can perform a unitary transformation for each layer
$Uh_{s\tau}(\pm\frac{\theta}{2},\bm{r})U^{-1}$, such that the new $h_{s\tau}$
becomes off-diagonal. Next one can interchange the second and the third
components of the spinor which $H_{s\tau}(\bm{r})$ acts on, putting
$H_{s\tau}(\bm{r})$ into an explicitly chiral form.

At the magic angle and in the chiral limit there are two exactly flat bands
for each spin and valley [see Fig.1(c)] of which the energy is determined
solely by $\lambda_{\text{so}}$ and $\delta$. To see this, we can first rotate
the basis to
$\tilde{\chi}_{s\tau}=\text{diag}(e^{i\theta\sigma_{z}/4},e^{-i\theta\sigma_{z}/4})\psi_{s\tau}\equiv(\chi_{A1},\chi_{B1},\chi_{A2},\chi_{B2})^{T}$,
and then reshuffle the second and the third component, to obtain a new basis
$\chi_{s\tau}=(\chi_{A1},\chi_{A2},\chi_{B1},\chi_{B2})^{T}\equiv(\chi_{A},\chi_{B})^{T}$.
The resulting Hamiltonian in this basis and the corresponding eigenvalue
equation become

$\begin{pmatrix}s\tau\lambda_{\text{so}}+\delta&D_{\tau}^{*}(-\bm{r})\\\
D_{\tau}(\bm{r})&-s\tau\lambda_{\text{so}}-\delta\end{pmatrix}\begin{pmatrix}\chi_{A}\\\
\chi_{B}\end{pmatrix}=\varepsilon\begin{pmatrix}\chi_{A}\\\
\chi_{B}\end{pmatrix},$ (4)

where

$D_{+}(\bm{r})=\begin{pmatrix}-2iv_{F}\partial_{\bar{z}}&U_{+}(\bm{r})\\\
U_{+}(-\bm{r})&-2iv_{F}\partial_{\bar{z}}\end{pmatrix},~{}D_{-}(\bm{r})=D_{+}^{*}(\bm{r}),$
(5)

$U_{+}(\bm{r})=w_{AB}(e^{-i\bm{q}_{1}\cdot\bm{r}}+e^{i\phi-i\bm{q}_{2}\cdot\bm{r}}+e^{-i\phi-i\bm{q}_{3}\cdot\bm{r}})$
and
$\partial_{\bar{z}}\equiv\frac{\partial}{\partial\bar{z}}=(\partial_{x}+i\partial_{y})/2$.
Eq.(4) has an apparent solution which is in a fully sublattice-polarized form
with either $\chi_{A}$ or $\chi_{B}$ remains zero for all $\bm{r}$. If we
assume $\chi_{B}=0$, and the eigenvalue problem is solved by
$\varepsilon=s\tau\lambda_{\text{so}}+\delta$ and
$D_{\tau}(\bm{r})\chi_{A}=0$. The latter condition is in fact identical to the
zero energy flatband equation for chiral TBG[46]. Note the other set of
solution with the opposite sublattice polarization is found by assuming
$\chi_{A}=0$, which has the energy
$\varepsilon=-s\tau\lambda_{\text{so}}-\delta$, and therefore the the low
energy gap is $\Delta=2\lambda_{\text{so}}-2\delta$.

Figure 2: (a) Sublattice polarization of the upper moiré flatband for
$s\tau=\uparrow+$ and for two different chiral ratio $r$, at the first magic
angle. The polarization is maximized in the chiral limit. (b) Illustration of
the four exact flatbands in the chiral limit. (c) Schematics of the phase
diagram when the Fermi level is at the dashed line shown in (b). (d)Plots of
the wavefunction amplitude at the moiré $\kappa$ point in log-scale, which
show the zeros are located at $\pm(\bm{a}_{1}-\bm{a}_{2})/3$ depending on
spin-valley index.

The sublattice-polarization is not just a fine tuning effect of the chiral
limit and a similar effect has been discussed in lattice models [76, 77, 78].
In Fig.2 (a), we plot the sublattice weight for the topmost flat band at
different chiral ratios $r$. The sublattice weight for the $n$-th band is
defined as $u^{(n)}_{A/B}=\sum_{I}|u^{(A/B)}_{I,n}|^{2}$ where
$u^{(A/B)}_{I,n}$ is the eigenstate for the $n$-th band projected to $A/B$
sublattice and $I$ is the shorthand of the indices other than sublattice. We
clearly see that even for $r=0.5$ there already exists a substantial
sublattice polarization for the whole band, which is more prominent near the
$\gamma$ point. In the chiral limit $r=0$, the polarization is maximized and
in the upper band all the fermions with $s\tau=\uparrow+$ come from
$A$-sublattice only, which is consistent with our analysis above. In Fig.2 (b)
we show the Chern numbers for the four flat bands. With sublattice
polarization, each of these four flat bands hosts only one Kramers’ pair of
fermions and is therefore topological with $\nu_{A,B}=\pm 1$ as indicated in
the plot. If the Fermi level is located in the middle of the lower two bands,
then tuning $\mu$ results in a topological phase transition as indicated in
Fig.2 (c), where in the chiral limit the trivial regime shrinks to a point at
$\delta=0$.

Similar to TBG [46], the wave function for these chiral flat bands can be
obtained. For simplicity we consider the topmost band. For $\tau=+1$, the
equation $D_{+}(\bm{r})\chi_{A}(\bm{r})=0$ has a $C_{3}$-rotation symmetry
protected solution at the moiré Brillouin corner $\pm\kappa$, which we denote
as $\chi_{\uparrow+}^{\pm\kappa}$ (the wavefunction at $\kappa$ and $-\kappa$
are related by $C_{3}$ rotation so below we use $\chi_{\uparrow+}^{\kappa}$
only). Because $D_{+}(\bm{r})$ contains only the anti-holomorphic derivative
$\partial_{\bar{z}}$, the general solution can be written as
$\chi_{\uparrow+,\bm{k}}(\bm{r})=f_{\bm{k}}(z)\chi_{\uparrow+}^{\kappa}(\bm{r})$
where $f_{\bm{k}}(z)$ is some holomorphic function. The function
$f_{\bm{k}}(z)$ can only be meromorphic in order to preserve the moiré lattice
translation symmetry, which must have simple poles. But remarkably each
component of $\chi_{\uparrow+}^{\kappa}(\bm{r})$ has zeros at some particular
$\bm{r}_{0}$ right at the magic twist angle. Therefore, one can locate the
poles of $f_{\bm{k}}(z)$ at these $\bm{r}_{0}$ to make
$\chi_{\uparrow+,\bm{k}}(\bm{r})$ a bounded function. In Fig.2 (c) we show the
plot of the norm of the two-component wavefunction
$\chi_{s\tau}^{\kappa}(\bm{r})$ for the two upper flat bands on a logarithmic
scale. The zeros are located at either $AB$\- or $BA$-stacking points,
depending on the spin-valley index. Note that for a fixed $s\tau$, the
wavefunction at $\kappa$ and $-\kappa$ have zeros at opposite positions since
they are related by $C_{3}$ rotation. For A-sublattice polarized flat bands,
$\bm{r}_{0}=(\bm{a}_{1}-\bm{a}_{2})/3$ when measured from the AA-stacking
center.

The total wavefunction can be more conveniently expressed when the spatial
origin is shifted to $\bm{r}_{0}$, so that

$\chi_{\uparrow+,\bm{k}}(\bm{r})=F_{\uparrow+}(\bm{r})N(\bm{k})\left[e^{i\frac{k^{*}z}{2}}\sigma\left(z+i\frac{S}{2\pi}k\right)\right]$
(6)

where

$F_{\uparrow+}(\bm{r})=\frac{\chi_{\uparrow+}^{\kappa}(\bm{r}+\bm{r}_{0})e^{-\frac{\pi
a_{1}^{*}z^{2}}{2Sa_{1}}}}{\vartheta_{1}(\frac{z}{a_{1}},\frac{a_{2}}{a_{1}})},~{}N(\bm{k})=\frac{\pi\vartheta^{\prime}(0,\frac{a_{2}}{a_{1}})}{a_{1}}e^{\frac{Sa_{1}^{*}k^{2}}{8\pi
a_{1}}}.$ (7)

Here the plain forms of $z$, $k$, etc. are the complex version of the
corresponding vectors, for instance $k\equiv k_{x}+ik_{y}$, and
$S=|\bm{a}_{1}\times\bm{a}_{2}|$ is the area of the moiré unit cell.
$\theta_{1}(u,\eta)$ is one of the Jacobi theta functions
$\vartheta_{1}(u,\eta)$[79] which has the double-periodic properties
$\vartheta_{1}(u+n,\eta)=(-1)^{n}\vartheta_{1}(u,\eta)$ and
$\vartheta_{1}(u+n\eta,\eta)=(-1)^{n}e^{-i\pi(2nu+n^{2}\eta)}\vartheta_{1}(u,\eta)$
for $n\in\mathbb{Z}$, and vanishes at $u=n+m\eta$ for $n,m\in\mathbb{Z}$ such
that $1/\vartheta_{1}(u,\eta)$ has simple poles at these positions. As a
result, $F(\bm{r})$ becomes regular. The universal part, $[...]$ in the above
equation, contains a modified Weierstrass sigma function [80], which has zeros
on the moiré lattice sites, and is related to $\vartheta_{1}$ via
$\sigma(z)=\frac{a_{1}}{\pi}\exp(\frac{\pi
a_{1}^{*}z^{2}}{2Sa_{1}}){\vartheta_{1}(\frac{z}{a_{1}},\eta)}/{\vartheta_{1}^{\prime}(0,\eta)}$.
Translation properties of $\sigma(z)$ can thus be readily deduced from that of
$\vartheta_{1}$222See the supplementary material. The time-reversal
counterpart $\chi_{\downarrow-,\bm{k}}(\bm{r})$ is constructed in the same
manner. We first notice that the zeros of $\chi_{\uparrow+}^{\kappa}$ and
$\chi_{\downarrow-}^{-\kappa}$ coincide in space [see Fig.2 (c)], and they can
be made complex conjugate to each other. Furthermore, the operator
$D_{-}(\bm{r})$ contains only holomorphic derivatives, indicating the
wavefunction for $\tau=-1$ valley contains only anti-holomorphic functions.
Clearly this construction is equivalent to taking the complex conjugate of
$\chi_{\uparrow+,\bm{k}}$, and we thus have

$\chi_{\downarrow-,\bm{k}}(\bm{r})=F_{\downarrow-}(\bm{r})N^{*}(\bm{k})\left[e^{-i\frac{kz^{*}}{2}}\sigma\left(z^{*}-i\frac{S}{2\pi}k^{*}\right)\right]$
(8)

with $F_{\downarrow-}(\bm{r})=F_{\uparrow+}^{*}(\bm{r})$. Using the quasi-
periodic properties of the theta function, it is straightforward to check that
both of these wavefunctions indeed satisfy Bloch’s theorem. One caveat here is
that $\chi_{s\tau,\bm{k}}$ is a two-component wavefunction, with each
component from distinct layers shifted by $\bm{q}_{1}$ from each other.
Bloch’s theorem for $\chi_{s\tau}$ thus implies
$\chi_{s\tau,\bm{k}}(\bm{r}+\bm{a})=e^{i\bm{k}\cdot\bm{a}}\text{diag}(1,e^{i\bm{q}_{1}\cdot\bm{a}})\chi_{s\tau,\bm{k}}(\bm{r})$.

Eq.(6) and (8) are quite similar to the wavefunction of the LLL of the 2D
electron gas on a torus subject to an out-of-plane magnetic field, written in
the symmetric gauge [82]. The difference is that for the LLL
$F(\bm{r})=e^{-|z|^{2}/4}$ and $N(\bm{k})=e^{-|k|^{2}/4}$ (setting magnetic
length $l_{B}=1$ for simplicity), and $\sigma(z)$ is defined for arbitrary
lattice as long as the unit-cell area is $S=2\pi$ (one flux quantum for each
unit cell). In fact, for any 2D ideal flat band with Chern number
$\mathcal{C}=1$, its wavefunction can be written in the form of Eq.(6) with
some properly chosen $F(\bm{r})$ and $N(\bm{k})$ [83, 84]. The flat band being
ideal means that the cell-periodic part, defined as
$u_{s\tau,\bm{k}}(\bm{r})=e^{i\tau\bm{k}\cdot\bm{r}}\chi_{s\tau,\bm{k}}(\bm{r})$
is holomorphic in $k$ for spin up and anti-holomorphic for spin down. This
follows directly from the identity $\bm{k}\cdot\bm{r}=(k^{*}z+z^{*}k)/{2}$. As
a key consequence, the quantum geometric tensor
$\eta_{\alpha\beta}(\bm{k}):=\braket{D_{\alpha}u_{\bm{k}}}{D_{\beta}u_{\bm{k}}}$
has vanishing determinant at every $\bm{k}$; here
$D_{\alpha}:=\partial_{k_{\alpha}}-iA_{\alpha}$,
$A_{\alpha}=i\braket{u_{\bm{k}}}{\partial_{k_{\alpha}}u_{\bm{k}}}$ being the
Berry connection. This in turn implies that the Fubini-Study metric
$g_{ab}(\bm{k})$, the real part of $\eta(\bm{k}),$ is related to the Berry
curvature $\bm{\Omega}(\bm{k})=\nabla_{\bm{k}}\times\bm{A}$ via
$g_{\alpha\beta}(\bm{k})=\frac{1}{2}|\Omega(\bm{k})|\delta_{\alpha\beta}$. It
is known that these properties make the flat band an ideal system to mimic the
Girvin-MacDonald-Platzman (GMP) algebra [85] for the LLL in a strong magnetic
field:
$[\rho_{\bm{q}_{1}},\rho_{\bm{q}_{2}}]=i\Omega\bm{q}_{1}\times\bm{q}_{2}\rho_{\bm{q}_{1}+\bm{q}_{2}}$
with $\rho_{\bm{q}}$ being the density operator projected to the flat band, if
we identify the average Berry curvature $\Omega=\frac{1}{S_{BZ}}\int
d\bm{k}\Omega(\bm{k})$ as the square of the magnetic length $l_{B}^{2}$ [86,
87, 88].It is this similarity that makes it possible to obtain quantum (spin)
Hall effect in the full or partially filled moiré flat bands.

Many-body wavefunctions.\- Since $\Omega$ plays the same role as $l_{B}^{2}$
in the GMP algebra, we can identify each moiré unit cell as the magnetic unit
cell which hosts a single magnetic flux. We can then define the system as a
parallelogram with the widths $\bm{L}_{1}=N_{1}\bm{a}_{1}$ and
$\bm{L}_{2}=N_{2}\bm{a}_{2}$, such that there are $N_{1}N_{2}$ fluxes in
total, as shown in Fig.1(a). The general twisted periodic boundary condition
for each particle on the many-body wavefunction implies
$\Psi(\\{z_{i}\\}|z_{j}=L_{1,2})=e^{i\phi_{1,2}}\Psi(\\{z_{i}\\}|z_{j}=0)$,
where $\phi_{1,2}$ are not necessarily zero. Following the logic of Ref.[82,
52], we have for spin-up fermions

$\displaystyle\Psi_{\uparrow,m}(\\{z_{j}\\})=$ $\displaystyle
e^{iKZ}\prod_{j=1}^{N_{e}}F_{\uparrow+}(\bm{r}_{j})\prod_{\nu=1}^{m}\sigma_{L}(Z-iZ_{\nu})\times$
(9) $\displaystyle\prod_{i<j}\left[\sigma_{L}(z_{i}-z_{j})\right]^{m}$

where we have assumed there are $N_{e}$ spin-up fermions so that the filling
fraction is $1/m=N_{e}/N_{s}$. Note that we need to keep $m$ an odd integer in
order to maintain fermionic properties. Here $Z=\sum_{j}z_{j}$ and the values
of $K$ and $Z_{0}:=\sum_{\nu}Z_{\nu}$ are chosen to satisfy

$e^{iKL_{1,2}}=(-1)^{N_{s}+N_{1,2}}e^{i\frac{\pi
L_{1,2}^{*}Z_{0}}{N_{s}S}+i\phi_{1,2}}.$ (10)

The sigma function with subscript ‘L’ is defined similarly to the previous
discussion, but with the unit cell enlarged to the whole sample spanned by
$L_{1}$ and $L_{2}$ instead of $a_{1}$ and $a_{2}$. Since the sigma function
vanishes as $\sigma_{L}(z)\sim z$ when $z\to 0$, this wavefunction scales as
$(z_{i}-z_{j})^{m}$ whenever there are two particles approaching each other,
so it can be stablized by some pseudopotential similar to Haldane’s. The
difference from Haldane’s pseudopotential is that the ideal flat band
pseudopotential not only depends on the relative angular momentum between two
particles, but also on their center-of-mass (COM), so that the general form of
the interactions can be written as
$V(\bm{r}_{1},\bm{r}_{2})=\sum_{M,m}v_{M,m}\hat{P}_{M,m}$ where
$\hat{P}_{M,m}$ is the projector [83]. This can be traced back to the fact
that the LLL wavefunction obeys the magnetic translation group [89] while the
ideal flatband wavefunction in our case does not. One simple realization that
stablize the wavefunction in Eq.(9) is to consider sufficiently short range
(Hubbard-like) interactions, for which the COM gets frozen, and the
pseudopotential can be modeled by
$V(\bm{r})=\sum_{m^{\prime}=0}^{m^{\prime}<m}v_{m^{\prime}}(\nabla^{2})^{m^{\prime}}\sum_{i}\delta(\bm{r}-\bm{R}_{i})$
where $\bm{R}_{i}$ denote all lattice sites and all $m^{\prime}>0$ should be
odd. For the spin-down fermions, the construction is exactly the same,

$\displaystyle\Psi_{\downarrow,m^{\prime}}(\\{\bar{w}_{j}\\})=$ $\displaystyle
e^{-i\bar{Q}\bar{W}}\prod_{j=1}^{N_{e}^{\prime}}F_{\downarrow-}(\bm{r}_{j})\prod_{\nu=1}^{m^{\prime}}\sigma_{L}(\bar{W}-i\bar{W}_{\nu})\times$
(11)
$\displaystyle\prod_{i<j}\left[\sigma_{L}(\bar{w}_{i}-\bar{w}_{j})\right]^{m^{\prime}}$

with $\bar{W}=\sum_{j}\bar{w}_{j}$, and $\bar{Q}$ and
$\bar{W}_{0}:=\sum_{\nu}\bar{W}_{\nu}$ satisfying conditions similar to
Eq.(10).

Upon combining the two spin components (9) and (11), we identify

$\Psi_{\text{FQSH}}(\\{z_{j},w_{j}\\})=\Psi_{\uparrow,m}(\\{z_{j}\\})\Psi_{\downarrow,m}(\\{\bar{w}_{j}\\})$
(12)

as a candidate wavefunction describing a FQSH state with spin filling fraction
$\nu_{spin}=\frac{1}{m}$, which hosts conserved spin current and fractional
spin excitations [64, 67, 68, 69]. Remarkably, the ideal flat band condition
[52, 83] ensures that (12) is the ground state of a local time-reversal
symmetric Haldane pseudo-potential interaction, which establishes a
microscopic mechanism for time-reversal invariant fractional topological order
in moiré flat band systems. The many-body wavefunction (12) can be generalized
by multiplication by terms $\sim\prod_{r<s}(z_{r}-\bar{w}_{s})^{n}$, which
represent correlations between opposite spins[64, 69]. Achieving such FQSH
states would require different local interactions, a question that merits
further investigation.

Discussion.\- We have shown that the would-be topologically trivial two layers
of KM model can become non-trivial at small twist angles, and can give rise to
ideal flat bands with TRS at magic twist angle and in the chiral limit,
serving as an ideal platform for realizing FQSH effect. There are, however,
two obstacles that can potentially destroy the FQSH state. The first one is
the competition with other symmetry breaking phases, in particular
ferromagnetism. The true ground state depends on the details of the
interactions, so given that moiré systems have much higher tunability of
interactions compared to other platforms, we expect that the FQSH state
considered here is indeed in a physically accessible regime. The second
obstacle is that a finite $w_{AA}$ spoils the ideal flat band condition and
renders the Berry curvature more inhomogeneous in $\bm{k}$-space. Comparing
energies of different competing states in this case, e.g. using exact
diagonalization and DMRG, is an interesting question which we leave for a
future study.

Acknowledgements.\- We thank Fengcheng Wu, Trithep Devakul, Ben Feldman, Sri
Raghu and Steven Kivelson for useful discussions. This research was supported
by the Gordon and Betty Moore Foundation’s EPiQS Initiative through GBMF8686
(Y.-M. W), and by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, under Award DE-SC0023327 (D. S. and L. H. S.).

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ONLINE SUPPLEMENTARY MATERIAL FOR
Time-reversal invariant topological moiré flatband and fractional quantum spin
Hall effect

Yi-Ming Wu,1, Daniel Shaffer2, Zhengzhi Wu3 and Luiz H. Santos1

1 Stanford Institute for Theoretical Physics, Stanford University, Stanford,
California 94305, USA

2 Department of Physics, Emory University, 400 Dowman Drive, Atlanta, GA
30322, USA

3 Institute for Advanced Study, Tsinghua University, Beijing 100084, China

In this Supplemental Material we show i) the construction of LLL wave function
on a torus using both theta function and sigma function, ii) the construction
of many body Laughlin state on a torus and iii) similar construction for the
moiré flatband.

##  I A.LLL wavefunction on a torus and the construction of Laughlin
wavefunction.

The LLL wavefunction on a torus was first studied in Ref.[82], where the theta
function was used to account for the double-periodicity nature of the
wavefunction. There the theta functions is periodic in terms of the boundaries
$L_{1}$ and $L_{2}$, which necessarily involves a product of many theta
functions. Alternatively, one can define the problem on a lattice, and the
unit cell is chosen arbitrarily but must enclose an area of $2\pi l_{B}^{2}$
where a unit flux passes. The introducing of lattice makes it easy to compare
with real lattice system with a flat Chern band. Below we discuss both of
these two pictures separately.

###  I.1 LLL without lattice

####  I.1.1 single-particle wavefunction

Here we closely follow the logic of Ref.[82]. The single particle LLL
wavefunction in Landau gauge $\bm{A}(\bm{r})=(-yB,0,0)$ can be written as

$\psi(x,y)=e^{-{y^{2}}/{2}}f(z).$ (13)

Here we have set $l_{B}^{2}=1$, and $f(z)$ is a holomorphic function defined
on the whole plane. Due to the presence of the exponentially decaying
prefactor $e^{-{y^{2}}/{2}}$, $f(z)$ can be unbounded in the $y$-direction, so
the total wavefunction $\psi(x,y)$ can still be normalizable. The magnetic
translation operator $t(\bm{L})$ acting on a wavefunction is (assuming
$\bm{B}$ is in the $z$-direction)

$t(\bm{L})\psi(\bm{r})\equiv\psi(\bm{r}+\bm{L})=e^{i\bm{L}\cdot[-i\bm{\nabla}-e\bm{A}(\bm{r})]+ie\bm{B}\cdot(\bm{r}\times\bm{L})}\psi(\bm{r}).$
(14)

Suppose we defined the system size as a parallelogram with width
$|\bm{L}_{1}|$ and $|\bm{L}_{2}|$ and and angle $\varphi$ between them. Then
the total number of flux piercing this sample is given by

$N_{s}=\frac{A}{2\pi
l_{B}^{2}}=\frac{|\bm{L}_{1}||\bm{L}_{2}|\sin\varphi}{2\pi}.$ (15)

The twisted boundary condition on the wave function implies,

$\psi(\bm{L}_{1,2})=\psi(0)e^{i\phi_{1,2}}.$ (16)

Let’s further assume $\bm{L}_{1}$ is in $x$-direction without loss of
generality. Then we can immediately see

$\boxed{f(L_{1})=f(0)e^{i\phi_{1}}}.$ (17)

The boundary condition in $\bm{L}_{2}$ direction also applies to the prefactor
$e^{-{y^{2}}/{2}}$, so we will have

$e^{-(|\bm{L}_{2}|\sin\varphi)^{2}/2}f(L_{2})=f(0)e^{i\phi_{2}}.$ (18)

Using the definition of $N_{s}$, we can also write the above expression as

$\boxed{e^{-\pi
N_{s}\frac{|\bm{L}_{2}|}{|\bm{L}_{1}|}\sin\varphi}f(L_{2})=f(0)e^{i\phi_{2}}}.$
(19)

The solution of the above two boxed equations are given by the Jacobi theta
function,

$f(z)=e^{ikz}\prod_{\nu=1}^{N_{s}}\vartheta_{1}\left(\frac{z-z_{\nu}}{L_{1}},\tilde{\eta}\right)$
(20)

where $\tilde{\eta}=L_{2}/L_{1}$ and the theta function $\vartheta_{1}$ is
defined as

$\vartheta_{1}(u,\tilde{\eta}):=-i\sum_{l\in\mathbb{Z}}(-1)^{l}q^{(l+1/2)^{2}}e^{i\pi(2l+1)u},~{}q:=e^{i\pi\tilde{\eta}},$
(21)

which has the following properties,

$\displaystyle\vartheta_{1}(u+n,\tilde{\eta})$
$\displaystyle=(-1)^{n}\vartheta_{1}(u,\tilde{\eta}),$ (22)
$\displaystyle\vartheta_{1}(u+n\tilde{\eta},\tilde{\eta})$
$\displaystyle=(-1)^{n}e^{-i\pi(2nu+n^{2}\tilde{\eta})}\vartheta_{1}(u,\tilde{\eta}),$
$\displaystyle\vartheta_{1}(-u,\tilde{\eta})$
$\displaystyle=-\vartheta_{1}(u,\tilde{\eta}),$
$\displaystyle\vartheta_{1}(u,\tilde{\eta})$
$\displaystyle=0~{}~{}~{}\text{for}~{}~{}u=n+m\tilde{\eta},~{}m,n\in\mathbb{Z}$
$\displaystyle\vartheta_{1}(u,\tilde{\eta})$ $\displaystyle\sim
u~{}~{}~{}\text{for}~{}~{}u\to 0.$

We now need to choose proper $k$ and $z_{\nu}$ in order to satisfy the boxed
boundary condition in Eq.(17) and (19). Using the properties of
$\vartheta_{1}$ listed above, and defining $z_{0}=\sum_{\nu}^{N_{s}}z_{\nu}$,
it is easy to see that $k$ and $z_{0}$ satisfy the following equations,

$\displaystyle e^{ikL_{1}}$ $\displaystyle=(-1)^{N_{s}}e^{i\phi_{1}},$ (23)
$\displaystyle e^{ikL_{2}}$ $\displaystyle=(-1)^{N_{s}}e^{i\left(\phi_{2}-2\pi
z_{0}/L_{1}+\pi N_{s}\frac{|\bm{L}_{2}|}{|\bm{L}_{1}|}\cos\varphi\right)}.$

Note the second equation is slightly different from that in Ref.[82], and by
setting $\varphi=\pi/2$, i.e. when the $\bm{L}_{1}$ and $\bm{L}_{2}$ are
perpendicular to each other, the equation is reduced to the one in Ref.[82].
From Eq.(23) it is easy to see that, if $(k,z_{0})$ is the solution, so is
$(k+n_{1}2\pi/L_{1},z_{0}-n_{1}L_{2}+n_{2}L_{1})$ with
$n_{1},n_{2}\in\mathbb{Z}$. The number of the linearly independent solutions
is equal to the number of zeros of $f(z)$, which is $N_{s}$.

####  I.1.2 Laughlin wavefunction

The construction of the many-body Laughlin wavefunction proceeds as follows.
Suppose there are $N_{e}$ electrons, so the filling factor is given by
$N_{e}/N_{s}=1/m$. We will assume $m\geq 3$ is some odd integer. The
wavefunction is given by the ansatz,

$\Psi(\\{z_{i}\\})=F(Z)\prod_{i<j}g(z_{i}-z_{j})$ (24)

where $F(Z)$ is a function which depends on the center-of-mass coordinate $Z$
and $g(z_{i}-z_{j})$ is the Jastrow factor which only depends on the relative
coordinate. Recall the usual Laughlin wavefunction is

$\Psi_{\text{LW}}(\\{z_{i}\\})=e^{-\sum_{i}^{N_{e}}|z_{i}|^{2}/4}\prod_{i<j}(z_{i}-z_{j})^{m},$
(25)

hence we need

$g(z_{i}-z_{j})\sim(z_{i}-z_{j})^{m}~{}~{}\text{for}~{}~{}z_{i}\to z_{j}.$
(26)

One possibility is

$g(z)=\left[\vartheta_{1}\left(\frac{z}{L_{1}},\tilde{\eta}\right)\right]^{m},$
(27)

which leads to the following boundary condition for a generic single particle,
say $z_{1}$

$\displaystyle g(L_{1}-z_{j})$ $\displaystyle=(-1)^{m}g(-z_{j})$ (28)
$\displaystyle g(L_{2}-z_{j})$
$\displaystyle=(-1)^{m}e^{-im\pi\left(-2\frac{z_{j}}{L_{1}}+\frac{L_{2}}{L_{1}}\right)}g(-z_{j}).$

Therefore, for the product of $N_{e}-1$ particles, we have

$\displaystyle\prod_{j}g(L_{1}-z_{j})$
$\displaystyle=(-1)^{N_{s}-m}\prod_{j}g(-z_{j})$ (29)
$\displaystyle\prod_{j}g(L_{2}-z_{j})$ $\displaystyle=(-1)^{N_{s}-m}e^{i2\pi
m\frac{Z}{L_{1}}-i(N_{s}-m)\pi\frac{L_{2}}{L_{1}}}\prod_{j}g(-z_{j})$

If we require that the total wavefunction satisfies

$\Psi_{\text{LW}}(\\{z_{j}|z_{i}=L_{1,2}\\})=\Psi_{\text{LW}}(\\{z_{j}|z_{i}=0\\})e^{i\phi_{1,2}}$
(30)

then the center-of-mass factor $F(Z)$ must satisfy

$\displaystyle F(Z+L_{1})$ $\displaystyle=F(Z)(-1)^{N_{s}-m}e^{i\phi_{1}},$
(31) $\displaystyle F(Z+L_{2})$ $\displaystyle=F(Z)(-1)^{N_{s}-m}e^{-i2\pi
m\frac{Z}{L_{1}}+i(N_{s}-m)\pi\frac{L_{2}}{L_{1}}}e^{i\phi_{2}}.$

The general solution for $F(Z)$ can be expressed as

$F(Z)=e^{iKZ}\prod_{\nu=1}^{m}\vartheta_{1}\left(\frac{Z-Z_{\nu}}{L_{1}},\tilde{\eta}\right)$
(32)

Likewise, we need to properly choose $K$ and $Z_{0}:=\sum_{\nu}Z_{\nu}$ to
solve Eq.(31). This puts constraints on $K$ and $Z_{0}$, namely

$\displaystyle e^{iKL_{1}}$ $\displaystyle=(-1)^{N_{s}}e^{i\phi_{1}},$ (33)
$\displaystyle e^{iKL_{2}}$
$\displaystyle=(-1)^{N_{s}}e^{-i2\pi\frac{Z_{0}}{L_{1}}}e^{iN_{s}\pi\tilde{\eta}}e^{i\phi_{2}}.$

###  I.2 Another choice: LLL wave function with lattice

####  I.2.1 Symmetric gauge

Here it is useful to switch to symmetric gauge where $\bm{A}=(-yB/2,xB/2,0)$,
the magnetic translation operator acting on $\psi(\bm{r})$ has a simple form
(again assuming $eB=l_{B}^{-2}=1$)

$t(\bm{L})=e^{i\bm{L}\cdot(-i\bm{\nabla})+\frac{i}{2}(\bm{r}\times\bm{L})\cdot\hat{z}}.$
(34)

The wavefunction that simultaneously diagonalize the Hamiltonian and this
translation operator can be given by the modified Weierstrass sigma
function[80]

$\sigma(z)=a_{1}e^{\frac{\pi
a_{1}^{*}z^{2}}{2Aa_{1}}}\frac{\vartheta_{1}\left(\frac{z}{a_{1}},\eta\right)}{\vartheta_{1}^{\prime}(0,\eta)}$
(35)

where $\vartheta_{1}$ is the theta function defined above, and
$\vartheta_{1}^{\prime}$ denotes its derivative. $A$ is the area of the unit
cell, and in the lattice spanned by $a_{1}$ and $a_{2}$ we have $A=2\pi$. Upon
translated by a lattice $l=ma_{1}+na_{2}$ with $m,n\in\mathbb{Z}$, it changes
as

$\sigma(z+l)=\xi(l)e^{\frac{\pi l^{*}}{A}(z+l/2)}\sigma(z)$ (36)

where $\xi(l)=1$ if $l/2$ is also on the lattice and $\xi(l)=-1$ otherwise. In
particular, if $l=a_{j}$ with $j=1,2$, we have

$\sigma(z+a_{j})=-e^{\frac{a_{j}^{*}}{2}(z+a_{j}/2)}\sigma(z).$ (37)

Now if we write the wavefunction as

$\boxed{\psi_{S}(x,y)=e^{-|z|^{2}/4}f_{k}(z)}$ (38)

then the holomorphic function $f_{k}(z)$ is given by

$\boxed{f_{k}(z)=\sigma(z+ik)e^{-|k|^{2}/4+ik^{*}z/2}}.$ (39)

An important property of writing the LLL wave function using this modified
sigma function is that it does not dependent on the specific choice of the
lattice, i.e. it’s modular-invariant. One can design an artificial lattice
spanned by the lattice vector $\bm{a}_{1}$ and $\bm{a}_{2}$ with the unit cell
area being $2\pi$ (it is $2\pi l_{B}^{2}$ when $l_{B}$ is not set to unity).
Then the system under consideration can be described using two integers
$N_{1}$ and $N_{2}$, such that $L_{1}=N_{1}a_{1}$ and $L_{1}=N_{2}a_{2}$, and
the total number of fluxes passing through the system is given by
$N_{s}=N_{1}N_{2}$. Under translation $\bm{l}=m\bm{a}_{1}+n\bm{a}_{2}$, this
wavefunction transforms as

$\displaystyle\psi_{S}(\bm{r}+\bm{l})=e^{-|z+l|^{2}/4}f_{k}(z+l)$ (40)
$\displaystyle=e^{-\frac{|z|^{2}+|k|^{2}}{4}-\frac{|l|^{2}}{4}-\frac{z^{*}l+l^{*}z}{4}+i\frac{k^{*}z+k^{*}l}{2}}\sigma(z+l+ik)$
$\displaystyle=\xi(l)e^{i\frac{k^{*}l+l^{*}k}{2}+\frac{l^{*}z-z^{*}l}{4}}\psi_{S}(\bm{r})$
$\displaystyle=\xi(l)e^{i\bm{k}\cdot\bm{l}+\frac{i}{2}(\bm{r}\times\bm{l})\cdot\hat{z}}\psi_{S}(\bm{r}).$

Note this wavefunction transforms in a similar but not exactly the same way as
Bloch wavefunction transforms under spatial translation. Using this
properties, one can explicitly show that

$t(\bm{l}_{1})t(\bm{l}_{2})\psi_{S}(\bm{r})=t(\bm{l}_{2})t(\bm{l}_{1})\psi_{S}(\bm{r})e^{i(\bm{l}_{1}\times\bm{l}_{2})\cdot\hat{z}}$
(41)

as it should be. The periodic boundary condition in Eq.(16) then implies
$\bm{k}$ must obey

$\displaystyle\bm{k}\cdot\bm{L}_{1}$ $\displaystyle=2\pi
n_{1}+N_{1}\pi+\phi_{1},$ (42) $\displaystyle\bm{k}\cdot\bm{L}_{2}$
$\displaystyle=2\pi n_{2}+N_{2}\pi+\phi_{2},$

with $n_{1},n_{2}\in\mathbb{Z}$.

Using the relation between sigma function and the theta function, it is
possible to express the wavefunction only in terms of theta function. In
addition, without loss of generality, we can always choose $a_{1}$ to be real.
It is then easy to see

$\boxed{\psi_{S}(x,y)=e^{i\frac{\bm{k}\cdot\bm{a}_{1}}{a_{1}}z}\frac{a_{1}e^{-\frac{|z|^{2}-z^{2}}{4}-\frac{|k|^{2}+k^{2}}{4}}}{\vartheta_{1}^{\prime}(0,\eta)}{\vartheta_{1}\left(\frac{z+ik}{a_{1}},\omega\right)}}.$
(43)

Note this expression is different from Eq.(20) in a sense that it contains
only one holomorphic theta function. But the number of zeros enclosed by the
sample boundary remains the same. By making use of Eq.(LABEL:eq:zk), it is
easy to realize that the function

$u_{\bm{k}}:=\psi_{S}(\bm{r})e^{-i\bm{k}\cdot\bm{r}}\equiv
N_{\bm{k}}\tilde{u}_{k}(\bm{r})$ (44)

is a normalized $k$-holomorphic function $\tilde{u}_{k}(\bm{r})$ times a
$\bm{k}$-dependent complex normalization factor $N_{\bm{k}}$ (such that
$||u_{\bm{k}}||^{2}=|N_{\bm{k}}|^{2}$). This observation is useful, since the
quantum geometric tensor $\eta(\bm{k})$, defined as

$\eta_{ab}(\bm{k})=\frac{\braket{\partial_{k_{a}}u_{\bm{k}}}{\partial_{k_{b}}u_{\bm{k}}}}{|N_{\bm{k}}|^{2}}-\frac{\braket{\partial_{k_{a}}u_{\bm{k}}}{u_{\bm{k}}}\braket{u_{\bm{k}}}{\partial_{k_{b}}u_{\bm{k}}}}{|N_{\bm{k}}|^{4}}$
(45)

is actually independent of $N_{\bm{k}}$. A simple manipulation shows that when
substituting $u_{\bm{k}}=N_{\bm{k}}\tilde{u}_{k}$ into this definition, the
derivatives of $N_{\bm{k}}$ from the first and the second part of Eq.(45)
cancel out, which leads to

$\eta_{ab}(\bm{k})={\braket{\partial_{k_{a}}\tilde{u}_{k}}{\partial_{k_{b}}\tilde{u}_{k}}}-{\braket{\partial_{k_{a}}\tilde{u}_{k}}{\tilde{u}_{k}}\braket{\tilde{u}_{k}}{\partial_{k_{b}}\tilde{u}_{k}}}.$
(46)

Therefore, the factor $N_{\bm{k}}$, although depending on $\bm{k}$, does not
determine the properties of the ideal flatband.

We close by noting that the wavefunction in Landau gauge can be readily
obtained by applying the gauge transformation, namely,

$\displaystyle\psi_{L}(x,y)$ $\displaystyle=\psi_{S}(x,y)e^{-ixy/2}$ (47)
$\displaystyle=e^{i\frac{\bm{k}\cdot\bm{a}_{1}}{a_{1}}z}\frac{a_{1}e^{-\frac{y^{2}}{2}-\frac{|k|^{2}+k^{2}}{4}}}{\vartheta_{1}^{\prime}(0,\eta)}{\vartheta_{1}\left(\frac{z+ik}{a_{1}},\omega\right)}.$

####  I.2.2 Laughlin wavefunction

Using sigma function, we can write the ansatz for the many-body wavefunction
similar to that in Eq.(24), but since we will be using sigma functions, we
write the ansatz as

$\Psi(\\{z_{i}\\})=e^{-\sum_{i}\frac{|z_{i}|^{2}}{4}}F(Z)\prod_{i<j}g(z_{i}-z_{j}),$
(48)

where $g(z)$ is now given by

$g(z)=[\sigma_{L}(z)]^{m}$ (49)

and

$\sigma_{L}(z)=\frac{L_{1}}{\pi}e^{\frac{L_{1}^{*}z^{2}}{4N_{s}L_{1}}}\frac{\vartheta_{1}\left(\frac{z}{L_{1}},\tilde{\eta}\right)}{\vartheta^{\prime}(0,\tilde{\eta})}.$
(50)

Similar to Eq.(37),

$\sigma_{L}(z+L_{1,2})=-e^{\frac{L_{1,2}^{*}}{2N_{s}}(z+L_{1,2}/2)}\sigma(z).$
(51)

Clearly $g(z)$ still scales as $z^{m}$ when $z\to 0$. Then it is easy to see,
similar to Eq.(29), we now have

$\displaystyle\prod_{j}g(L_{1}-z_{j})$
$\displaystyle=(-1)^{N_{s}-m}e^{\frac{|L_{1}|^{2}}{4N_{s}}(N_{s}-m)-\frac{mL_{1}^{*}}{2N_{s}}Z}\prod_{j}g(-z_{j}),$
(52) $\displaystyle\prod_{j}g(L_{2}-z_{j})$
$\displaystyle=(-1)^{N_{s}-m}e^{\frac{|L_{2}|^{2}}{4N_{s}}(N_{s}-m)-\frac{mL_{2}^{*}}{2N_{s}}Z}\prod_{j}g(-z_{j}).$

Accordingly, the periodic boundary condition on the many-body wave function,
when applied to one of the many particles, leads to the following constraints
for $F(Z)$, namely

$\displaystyle\frac{F(Z+L_{1})}{F(Z)}$
$\displaystyle=(-1)^{-N_{s}+m}e^{\frac{mL_{1}^{*}}{2N_{s}}(Z+\frac{L_{1}}{2})}e^{i\phi_{1}},$
(53) $\displaystyle\frac{F(Z+L_{2})}{F(Z)}$
$\displaystyle=(-1)^{-N_{s}+m}e^{\frac{mL_{2}^{*}}{2N_{s}}(Z+\frac{L_{2}}{2})}e^{i\phi_{2}}.$

These equations are solved by assuming the following general form

$F(Z)=e^{iKZ}\prod_{\nu=1}^{m}\sigma_{L}(Z-iZ_{\nu}).$ (54)

Introducing $Z_{0}=\sum_{\nu=1}^{m}Z_{\nu}$, then the parameter $K$ is
determined via

$\displaystyle KL_{1}$ $\displaystyle=\frac{L_{1}^{*}Z_{0}}{2N_{s}}+\pi
N_{s}+\phi_{1}+2n_{1}\pi$ (55) $\displaystyle KL_{2}$
$\displaystyle=\frac{L_{2}^{*}Z_{0}}{2N_{s}}+\pi N_{s}+\phi_{2}+2n_{2}\pi$

with $n_{1},n_{2}\in\mathbb{Z}$. It is also straightforward to see that under
translation operation

$t(\bm{L}_{1})t(\bm{L}_{2})\Psi(\\{z_{i}\\})=t(\bm{L}_{2})t(\bm{L}_{1})\Psi(\\{z_{i}\\})e^{i(\bm{L}_{1}\times\bm{L}_{2})\cdot\hat{z}}.$
(56)

##  II B. Laughlin wavefunction for a generic ideal flatband

From above we see the single particle LLL wavefunction (in symmetric gauge)
can be conveniently written as

$\psi(\bm{r})=e^{-\frac{|z|^{2}}{4}}e^{-\frac{|k|^{2}}{4}}\left[e^{i\frac{k^{*}z}{2}}\sigma\left(z+i\frac{A}{2\pi}k\right)\right],$
(57)

where we re-introduced $A=2\pi l_{B}^{2}$ but in our convention $l_{B}=1$ it
reduces to $A=2\pi$. It contains three part. The first one is the factor
$e^{-\frac{|z|^{2}}{4}}$ which depends only on $\bm{r}$ and makes sure the
wavefunction decays at large distances. The second term
$e^{-\frac{|k|^{2}}{4}}$ is a factor which depends solely on $\bm{k}$. The the
last term inside $[...]$ has the nice property that it becomes a holomorphic
function in $k$ when multiplied by the factor $e^{-i\bm{k}\cdot\bm{r}}$, as

$\bm{k}\cdot\bm{r}=\frac{1}{2}\left(k^{*}z+z^{*}k\right).$ (58)

The manybody Laughlin wavefunction is given by

$\displaystyle\Psi(\\{z_{i}\\})=$ $\displaystyle
e^{iKZ}\left(\prod_{i=1}^{N_{e}}e^{-\frac{|z_{i}|^{2}}{4}}\right)\prod_{\nu=1}^{m}\sigma_{L}(Z-iZ_{\nu})\times$
(59) $\displaystyle\prod_{i<j}\left[\sigma_{L}(z_{i}-z_{j})\right]^{m},$

with $K$ and $Z_{0}=\sum_{\nu}Z_{\nu}$ satisfying Eq.(55).

In fact, as suggested in Ref.[83], any ideal flatband wavefunction with Chern
number $\mathcal{C}=1$ can be written in a way similar to Eq.(57), namely

$\psi_{\text{IFB}}(\bm{r})=F(\bm{r})N(\bm{k})\left[e^{i\frac{k^{*}z}{2}}\sigma\left(z+i\frac{A}{2\pi}k\right)\right]$
(60)

where $F(\bm{r})$ depends on the lattice details, and $N(\bm{k})$ is some
normalization factor which is less important as we already see in Eq.(45) and
(46).

As an example, now let’s come back to the magic angle chiral limit flatband
wavefunctoin for, say $\mathcal{C}=+1$,

$\displaystyle\chi_{+,\bm{k}}(\bm{r})$
$\displaystyle=e^{i\bm{k}\cdot\bm{a}_{1}\frac{z-z_{0}}{a_{1}}}\frac{\vartheta_{1}\left(\frac{z-z_{0}}{a_{1}}-\frac{k}{b_{2}},\eta\right)}{\vartheta_{1}\left(\frac{z-z_{0}}{a_{1}},\eta\right)}\chi_{+}^{\kappa}(\bm{r})$
(61)
$\displaystyle=e^{i\bm{k}\cdot\bm{a}_{1}\frac{z-z_{0}}{a_{1}}}\frac{\vartheta_{1}\left(\frac{z-z_{0}+i\frac{A}{2\pi}k}{a_{1}},\eta\right)}{\vartheta_{1}\left(\frac{z-z_{0}}{a_{1}},\eta\right)}\chi_{+}^{\kappa}(\bm{r}).$

It is more convenient to work with the wavefunction with origin shifted to
$\bm{r}_{0}$, so we define a new
$\tilde{\chi}_{+,\bm{k}}(\bm{r})=\chi_{+,\bm{k}}(\bm{r}+\bm{r}_{0})$. After
some manipulation we rewrite it is as

$\displaystyle\tilde{\chi}_{+,\bm{k}}(\bm{r})=$
$\displaystyle\frac{\chi_{+}^{\kappa}(\bm{r}+\bm{r}_{0})e^{-\frac{\pi
a_{1}^{*}z^{2}}{2Aa_{1}}}}{\vartheta_{1}\left(\frac{z}{a_{1}},\eta\right)}\frac{\pi\vartheta^{\prime}(0,\eta)}{a_{1}}e^{\frac{Aa_{1}^{*}k^{2}}{8\pi
a_{1}}}\times$ (62)
$\displaystyle\left[e^{i\frac{k^{*}z}{2}}\sigma\left(z+i\frac{A}{2\pi}k\right)\right].$

From this expression we can easily read off the factors $F(\bm{r})$ and
$N(\bm{r})$ introduced in Eq.(60) in this case.

As a directly generalization of Eq.(59), the manybody wavefunction ansatz can
be written by modifying $F(\bm{r})$ accordingly. Therefore, for the magic
angle chiral limit flatband, the manybody Laughlin wavefunction ansatz is
given by

$\displaystyle\Psi_{\text{IFB}}(\\{z_{i}\\})=$ $\displaystyle
e^{iKZ}\left(\prod_{i=1}^{N_{e}}\frac{\chi_{+}^{\kappa}(\bm{r}_{i}+\bm{r}_{0})e^{-\frac{\pi
a_{1}^{*}z_{i}^{2}}{2Aa_{1}}}}{\vartheta_{1}\left(\frac{z_{i}}{a_{1}},\eta\right)}\right)\times$
(63)
$\displaystyle\prod_{\nu=1}^{m}\sigma_{L}(Z-iZ_{\nu})\prod_{i<j}\left[\sigma_{L}(z_{i}-z_{j})\right]^{m}.$

Likewise, by using the quasiperiodic properties of $\sigma_{L}$ and
$\vartheta_{1}$, it is easy to see the values of $K$ and
$Z_{0}=\sum_{\nu}Z_{\nu}$ must satisfy

$\displaystyle e^{iKL_{1}}$ $\displaystyle=(-1)^{N_{s}+N_{1}}e^{i\frac{\pi
L_{1}^{*}Z_{0}}{N_{s}A}+i\phi_{1}},$ (64) $\displaystyle e^{iKL_{2}}$
$\displaystyle=(-1)^{N_{s}+N_{2}}e^{i\frac{\pi
L_{2}^{*}Z_{0}}{N_{s}A}+i\phi_{2}},$

in order to meet the periodic boundary conditions, which are similar to
Eq.(55).