# Magic-angle helical trilayer graphene

Trithep Devakul<EMAIL_ADDRESS>Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Patrick J. Ledwith
Department of Physics, Harvard University, Cambridge, MA 02138, USA  Li-Qiao
Xia  Department of Physics, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA  Aviram Uri  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Sergio de la Barrera  Department of
Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada
Pablo Jarillo-Herrero  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Liang Fu  Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

We propose helical trilayer graphene (HTG), a helical structure featuring
identical rotation angles $\theta\approx 1.5^{\circ}$ between three
consecutive layers of graphene, as a unique and experimentally accessible
platform for realizing exotic correlated topological states of matter. While
nominally forming a supermoiré (or moiré-of-moiré) structure, we show that HTG
locally relaxes into large regions of a periodic single-moiré structure in
which $C_{2z}$ is broken, giving rise to flat topological bands carrying
valley-Chern numbers $C=\pm(1,-2)$. These bands feature near-ideal quantum
geometry and are isolated from remote bands by a large gap
$E_{\mathrm{gap}}\sim 100$ meV, making HTG a promising platform for
experimental realization of correlated topological states such as integer and
fractional quantum anomalous Hall states in $C=1$ and $2$ bands.

The intricate interplay of topology and strong electronic interactions is one
of the most fascinating and rapidly evolving areas of modern condensed matter
physics. Following the discovery of superconductivity and strong correlations
in twisted bilayer graphene (TBG) [1, 2], moiré materials have risen to the
forefront of both theoretical and experimental condensed matter physics
research as an ideal platform for exploring strongly correlated physics in
topological bands [3]. In the graphene family, significant progress has also
been made in multilayer moiré heterostructures, such as alternating twist
multilayers [4, 5, 6, 7, 8, 9], or single twist multilayers [10, 11, 12, 13,
14, 15, 16, 17] such as twisted monolayer-bilayer graphene. In parallel, moiré
heterostructures based on semiconductor transition metal dichalcogenides (TMD)
have also revealed a trove of complementary physics ranging from generalized
Wigner crystals to topological states [18]. The sheer versatility of the moiré
platform has lead to the experimental realization of an extraordinarily
diverse array of physical phenomena.

In magic-angle TBG, a manifold of nearly flat isolated single-particle bands
enables a unique regime of physics dominated by interactions and band
geometry. Perhaps the most fascinating and direct observations of strongly
correlated topology are the quantum anomalous Hall (QAH)[19, 20, 21, 22] and
fractional Chern insulator (FCI)[23, 24, 25, 26, 27, 28], lattice analogues of
the integer and fractional quantum Hall states driven by intrinsic band
geometry rather than Landau level physics [29, 30, 31, 32, 33, 34, 35, 36,
37]. However, these topological states in TBG are often fragile and
overpowered by competing non-topological states, likely because they require
hBN-alignment [38, 19] or spontaneous breaking of $C_{2z}\mathcal{T}$ symmetry
[39]. The FCI states have thus far only been observed in a substrate aligned
sample and at finite magnetic field $B\sim 5$T[23]. The apparent requirement
of substrate alignment poses a significant experimental challenge that
severely limits reproducibility of strongly correlated topology in the TBG
platform, and it is not clear whether the FCI state can be made stable at zero
field. Very recently, evidence of a zero-field FCI was found in a twisted TMD
homobilayer [40, 41]. It is therefore an important theoretical task to
identify new platforms in which such topological states may appear most
robustly, as well as for the realization of further exotic phases of matter.

We propose “helical trilayer graphene” (HTG), a helical structure featuring
identical rotation angles between three consecutive layers of graphene, as a
promising and experimentally accessible platform for realizing exotic
topological states of matter. As we will elaborate, unrelaxed HTG does not
realize a single periodic moiré superlattice, but instead realizes a
supermoiré (or “moiré-of-moiré”) structure [42, 43, 44]). Nevertheless, we
show that HTG locally relaxes into large regions hosting a single-moiré
structure featuring a periodic honeycomb lattice of the AA stacking regions
(shown in Figs 1a,b). In these regions, which we call h-HTG, $C_{2z}$ is
broken by a fixed lateral shift ${\boldsymbol{d}}=\pm{\boldsymbol{\delta}}$
between the two moiré superlattices. Remarkably, we find that at a magic angle
$\theta\approx 1.5^{\circ}$, the moiré band structure of h-HTG features a pair
of flat, isolated, nearly-degenerate topological bands, shown in Fig2b, with
valley-contrasting Chern numbers $C=\pm(1,-2)$. Since each valley carries a
net Chern number, even the band insulators are topological quantum valley-Hall
states, and valley polarization alone yields a net Chern number. In
particular, these bands feature remarkably uniform charge and Berry curvature
distributions, as well as “near-ideal quantum geometry"[45, 46, 24, 47, 48,
49, 28, 50, 51], making HTG a promising platform for realizing FCI states in
$\absolutevalue{C}=1$ and $2$ Chern bands. Furthermore, the topological flat
band manifold is isolated from remote bands by a very large gap
$E_{\mathrm{gap}}\sim 100$ meV, implying a high degree of stability and
providing a potential route to higher temperature QAH and FCI states.

Zooming out, HTG realizes large regions of h-HTG domains (and its $C_{2z}$
related counterpart), which form a triangular tiling on the supermoiré scale,
as shown in Fig 1a. These domains are large (several hundreds of nanometers)
so the bulk properties of h-HTG are accessible via local probes such as
scanning single-electron transistors [23, 52, 53] and scanning nano
superconducting quantum interference devices [54]. Furthermore, the domain
size may be tuned via heterostrain engineering [55, 56], and with a small
amount of uniform heterostrain ($\approx 0.03\%$), the entire device can relax
into a single domain of h-HTG, providing a route to a quantized Hall response
measurable by transport.

Figure 1:  (a) The relaxed structure of HTG at $\theta=1.5^{\circ}$, where
orange and purple dots show the AA stacking regions of adjacent layer pairs,
and the red background indicates the moiré aperiodicity $A({\boldsymbol{r}})$.
The system relaxes to large triangular domains of h-HTG (and its $C_{2z}$
counterpart, $\overline{\mathrm{h}}$-HTG), a periodic moiré superlattice with
$A({\boldsymbol{r}})\approx 0$, separated by a network of domain walls. (b) A
zoom in to the h-HTG region and a further zoomed in illustration of the atomic
scale structure at high symmetry points. (c) The monolayer graphene BZs for
each layer are shown. In the h-HTG region, the three $K$ points relax onto a
single line and fold to the $\kappa,\gamma,\kappa^{\prime}$ points on the mBZ
as illustrated on the right.

When the domain size is finite, a triangular network of interwoven domain
walls is realized as shown in Fig1a. When the domains are tuned to
incompressible states at integer or fractional filling, including full and
empty filling of the flat bands, the low-energy electronic physics is
dominated by the network of gapless domain walls. This system therefore
provides a natural realization of chiral or counter-propagating edge network
models [57, 58] on the supermoiré scale. Taken together, our work demonstrates
that HTG is a uniquely exciting platform for realizing robust strongly
correlated topology, gapless edge networks, and for exploring their interplay.

The key ingredient that enables all this richness is lattice relaxation on the
supermoiré scale, an aspect which was not fully incorporated in previous
theoretical studies. Refs [59, 60] focused on the electronic properties of a
different single-moiré superlattice defined by ${\boldsymbol{d}}=0$, which we
find is energetically unfavorable and is minimized in the relaxed structure of
Fig1a. Refs [42, 43] examined the electronic properties of the full unrelaxed
supermoiré structure, thus missing the physics of h-HTG. Various extensions to
higher number of layers have also been explored [61, 62, 63].

This paper is structured as follows: We first introduce the HTG structure and
demonstrate that relaxation favors the formation of a network of large h-HTG
domains. We examine the electronic properties of h-HTG and its symmetries via
an effective continuum model description, revealing the advertised magic
angle, topological flat bands, and large remote band gap. We then study the
model in the “chiral limit”[64], which features exactly flat bands with “ideal
quantum geometry"[45, 46, 24, 47, 48, 49, 28, 50, 51], explaining the origin
of the magic angle. Finally, we examine the features that make h-HTG promising
for the realization of strongly correlated topology, and discuss possible
correlated states at integer and fractional fillings.

##  I Supermoiré reconstruction

We consider the HTG structure consisting of three graphene layers with the
twist configuration $(\theta_{1},\theta_{2},\theta_{3})=(\theta,0,-\theta)$.
In the absence of lattice relaxation, the moiré superlattices of the lower and
upper two graphene layers are themselves misaligned by an angle $\theta$,
which therefore forms a supermoiré structure. This results in a parametric
separation of lengthscales: the atomic lengthscale $a_{0}=2.46$Å is much
smaller than the moiré lengthscale $a_{m}=a_{0}/(2\sin\frac{\theta}{2})$ which
is in turn much smaller than the supermoiré lengthscale
$a_{mm}=a_{0}/(2\sin\frac{\theta}{2})^{2}$.

Because of the large supermoiré lengthscale, lattice relaxation plays a
pivotal role in the physics of HTG and cannot be ignored. This is because even
a small amount of atomic lattice relaxation can result in a magnified effect
on the moiré scale, and hence a doubly magnified effect at the supermoiré
scale. As an analogy, consider the bilayer case. In TBG at $\theta\approx
1.1^{\circ}$, lattice relaxation is minor and typically accounted for by a
phenomenological parameter $\kappa$ known as the chiral ratio. At very small
angles $\theta\lesssim 1^{\circ}$ ($a_{m}\gtrsim 14$ nm), however, lattice
relaxation results in severe moiré lattice reconstruction [65, 66, 67, 68, 69,
70]: the energetically favorable AB and BA stacking regions are enlarged to
form large triangular domains of locally atomically-periodic Bernal stacking
regions at the expense of the energetically unfavorable AA regions. In HTG,
the analogous effect can now occur at the supermoiré scale. Indeed, as we will
now demonstrate, while the moiré scale lattice reconstruction is minor at
$\theta\approx 1.5^{\circ}$, the supermoiré scale $a_{mm}\sim 300-400$nm is
well in the regime of severe supermoiré lattice reconstruction.

We model in-plane lattice relaxation in HTG using the configuration space
method developed in Refs [71, 70]: the total intra- and inter-layer energy,
with parameters extracted from ab initio theory, is minimized in configuration
space which avoids issues associated with real space incommensurability. From
this, we extract a real space map of the local shift field
$\mathbf{u}_{l}(\mathbf{r})$ which indicates the in-plane displacement of the
relaxed structure relative to the unrelaxed structure, for each layer $l$.

In Fig. 1a, we show the AA stacking regions of adjacent layer pairs, labeled
by AA12 and AA23, for the relaxed structure. The dramatic effect of supermoiré
lattice reconstruction is clearly visible by eye: large domains separated by a
triangular network of domain walls. Within each domain, the AA12 and AA23
regions come together to form the two sublattices of a periodic moiré-scale
honeycomb lattice, as shown in Fig1b. These are characterized by a finite
lateral shift ${\boldsymbol{d}}=\pm{\boldsymbol{\delta}}$ (defined later)
between the two moiré sublattices; since ${\boldsymbol{d}}$ is opposite in two
adjacent domains, a domain wall must form between them. Thus, while HTG
nominally forms a supermoiré structure, it is energetically favorable to relax
to large domains of locally periodic regions. We use the term periodic moiré
superlattice to refer to these periodic structures, and specifically those
realized in the upwards and downwards pointing triangular domains as h-HTG and
$\overline{\mathrm{h}}$-HTG, respectively.

On the atomic scale, the high symmetry stacking regions in h-HTG correspond to
AAB, ABA, and BAA stacking regions (Fig1b), while in
$\overline{\mathrm{h}}$-HTG they are ABB, BAB, and BBA. The relaxed structure
therefore completely avoids the energetically costly AAA stacking region. It
is interesting to contrast our results to that of alternating-twist trilayer
graphene, $\theta_{l}=(0,\theta,0)$, where the A-twist-A configuration (which
does contain an AAA region) was shown, using the exact same method and
energetic parameters, to be favorable [72, 73]. Our results demonstrate that
the favorable stacking configuration is not obvious _a priori_ , and depends
on subtle energetic properties.

The periodic structure of h-HTG can be understood from the fact that unrelaxed
HTG is already very close to a periodic moiré superlattice. In Fig. 1c, we
illustrate the monolayer Brillouin zone (BZ) of each graphene layer. The moiré
BZ (mBZ) for each layer pair, the edges of which are determined by
${\boldsymbol{K}}_{2}-{\boldsymbol{K}}_{1}$ and
${\boldsymbol{K}}_{3}-{\boldsymbol{K}}_{2}$, are incommensurate with each
other as they are rotated by a small angle $\pm\theta/2$. However, this
incommensurability can be remedied by a minuscule uniform compression of the
outer graphene layers (and/or dilation of the middle layer) by a factor
$\lambda=\cos(\theta)\approx 0.9997$ for $\theta=1.5^{\circ}$. The result is
that the new ${\boldsymbol{K}}_{l}$ points all lie along a vertical line,
satisfying
${\boldsymbol{K}}_{2}-{\boldsymbol{K}}_{1}={\boldsymbol{K}}_{3}-{\boldsymbol{K}}_{2}$,
and therefore resulting in a periodic moiré superlattice. We define the
commensurate mBZ as shown in Fig1c, in which the ${\boldsymbol{K}}_{1}$,
${\boldsymbol{K}}_{2}$, and ${\boldsymbol{K}}_{3}$ points fold to the
$\kappa$, $\gamma$, and $\kappa^{\prime}$ points, respectively.

To verify that this is the correct picture, we obtain the local twist angle
$\theta_{l}({\boldsymbol{r}})=\theta_{l}+\sin^{-1}[\frac{1}{2}\nabla\crossproduct{\boldsymbol{u}}_{l}({\boldsymbol{r}})]$
and uniform scaling factor
$\lambda_{l}({\boldsymbol{r}})=1+\frac{1}{2}\nabla\cdot{\boldsymbol{u}}_{l}({\boldsymbol{r}})$
of the relaxed HTG structure. We then define the “local moiré aperiodicity”
via
$A({\boldsymbol{r}})\equiv\sum_{l=1,3}|K_{lx}({\boldsymbol{r}})/K_{2x}({\boldsymbol{r}})-1|$,
where
$K_{lx}({\boldsymbol{r}})=K\cos[\theta_{l}({\boldsymbol{r}})]/\lambda_{l}({\boldsymbol{r}})$
is the “local $K_{x}$”, and $K=\frac{4\pi}{3a_{0}}$. $A({\boldsymbol{r}})$ is
zero if all three $K$ points lie on a line and non-zero otherwise. The local
moiré aperiodicity is plotted in the background of Fig1c, which shows that the
large triangular domains have indeed relaxed to the locally periodic structure
with $A({\boldsymbol{r}})\approx 0$. Thus, the physics within each domain is
indeed described by the periodic moiré structure with the mBZ illustrated in
Fig1a.

In the domain wall region and their intersection, $A({\boldsymbol{r}})>0$ and
is much larger than in the unrelaxed structure, $A_{0}=2|1-\cos\theta|\approx
0.68\times 10^{-3}$. This implies that these regions (which contain the
previously studied ${\boldsymbol{d}}=0$ model [59, 60]) actually relax _away_
from the periodic structure, and therefore appears more locally
quasicrystalline [74].

We remark that, although the moiré period $a_{m}$ and the domain size
(determined by the unrelaxed supermoiré period) $a_{mm}$ considered thus far
are both determined by $\theta$, they can in principle be tuned independently.
This is important as it means that the domain size can be controlled while
keeping the local physics within each domain fixed. By applying a small global
uniform compression to the outer layers via $\lambda<1$, which may be possible
via heterostrain engineering, the domain size
$a_{mm}=a_{0}\lambda/(2|\lambda-\cos\theta|)$ quickly increases and diverges
at $\lambda=\cos\theta$ at which point the entire system is a single domain.
This single domain structure has lower elastic energy density due to the
absence of domain walls, so we speculate that some degree of this may already
occur naturally in finite systems. Finally, we remark that our conclusion
about the relaxed structure are qualitatively insensitive to details such as
the precise ratio of intra- and inter-layer elastic energies, which is a
potential tuning knob in comparing with experiment [73].

##  II Electronic structure

Having established the importance of lattice relaxation on the resulting
supermoiré structure, we now turn to the electronic structure within a h-HTG
domain. Rather than deriving a quantitative electronic model based on the
relaxed structure [75, 76], which would contain many detail-dependent terms,
we instead take an effective approach that captures the essential physics. The
starting point for our analysis is the Bistritzer-MacDonald continuum model
generalized to three layers. For more than two layers we must take into
account the displacements of the two moiré superlattices,
${\boldsymbol{d}}_{t,b}$.

$H_{K}=\begin{pmatrix}-iv{\boldsymbol{\sigma}}_{\theta}\cdot{\boldsymbol{\nabla}}&T({\boldsymbol{r}}-{\boldsymbol{d}}_{t})&0\\\
T^{\dagger}({\boldsymbol{r}}-{\boldsymbol{d}}_{t})&-iv{\boldsymbol{\sigma}}\cdot{\boldsymbol{\nabla}}&T({\boldsymbol{r}}-{\boldsymbol{d}}_{b})\\\
0&T^{\dagger}({\boldsymbol{r}}-{\boldsymbol{d}}_{b})&-iv{\boldsymbol{\sigma}}_{-\theta}\cdot{\boldsymbol{\nabla}}\\\
\end{pmatrix}$ (1)

where
${\boldsymbol{\sigma}}_{\theta}=e^{-i\theta\sigma_{z}}(\sigma_{x},\sigma_{y})$
and

$T({\boldsymbol{r}})=w\begin{pmatrix}\kappa
U_{0}({\boldsymbol{r}})&U_{-1}({\boldsymbol{r}})\\\
U_{1}({\boldsymbol{r}})&\kappa U_{0}({\boldsymbol{r}})\end{pmatrix}$ (2)

is the moiré tunneling between layers, with
$U_{l}({\boldsymbol{r}})=\sum_{n=0}^{2}e^{\frac{2\pi
i}{3}ln}e^{-i{\boldsymbol{q}}_{n}\cdot{\boldsymbol{r}}}$. The tunneling
wavevectors are such that $q_{n,x}+iq_{n,y}=-ik_{\theta}e^{\frac{2\pi
i}{3}n}$, where $k_{\theta}=2K\sin\frac{\theta}{2}$. We will use $v=1.03\times
10^{6}\text{m/s}$ and $w=105$meV, which we believe well models trilayer
graphene at these twist angles capturing some degree of interaction-induced
velocity renormalization [73, 74].

The intra-sublattice tunneling strength is suppressed due to lattice
relaxation and renormalization by $\kappa<1$; while hard to estimate
precisely[28], TBG studies [77, 70, 75, 78, 69, 79, 80, 76, 81, 82] suggest
$\kappa\approx 0.5-0.8$, and we therefore take a conservative estimate
$\kappa=0.7$ for now. The Hamiltonian for the $K^{\prime}$ valley can be
obtained by time reversal symmetry, and spin degeneracy is implied. This model
has a moiré translation symmetry with reciprocal lattice vectors
${\boldsymbol{b}}_{1,2}={\boldsymbol{q}}_{1,2}-{\boldsymbol{q}}_{0}$ and
lattice vectors
${\boldsymbol{a}}_{1,2}=\frac{4\pi}{3k_{\theta}}(\pm\frac{\sqrt{3}}{2},\frac{1}{2})$.
The Bloch periodicity of $l$’th layer is given by
$\psi_{{\boldsymbol{k}},l}({\boldsymbol{r}}+{\boldsymbol{a}})=e^{i({\boldsymbol{k}}-{\boldsymbol{K}}_{l})\cdot{\boldsymbol{a}}}\psi_{{\boldsymbol{k}},l}({\boldsymbol{r}})$,
where ${\boldsymbol{K}}_{1,3}=\mp{\boldsymbol{q}}_{0}+{\boldsymbol{K}}_{2}$
are the $\kappa$ and $\kappa^{\prime}$ points of the mBZ and
${\boldsymbol{K}}_{2}$ is the $\gamma$ point.

Since we may always translate the entire system at the moiré scale, only
${\boldsymbol{d}}={\boldsymbol{d}}_{t}-{\boldsymbol{d}}_{b}$, the offset
between the moiré patterns, affects the spectrum of the continuum Hamiltonian
$H_{K}$. While a generic ${\boldsymbol{d}}$ breaks most crystalline
symmetries, there is an approximate particle-hole-inversion symmetry
$\mathcal{IC}$ which exchanges the top and bottom layers, multiplies the
middle layer by $-1$, takes ${\boldsymbol{r}}\to-{\boldsymbol{r}}$, and
anticommutes with the Hamiltonian. This symmetry is exact if we take
$\sigma_{\pm\theta}\to\sigma$, which is a very good approximation for the
small $\theta$ of interest here, and is easiest to see if one chooses
${\boldsymbol{d}}_{t}=-{\boldsymbol{d}}_{b}$.

Figure 2: (a) The remote band gap $E_{\mathrm{gap}}$ multiplied by
$C_{\mathrm{tot}}=\pm 1$ of $H_{K}$ is shown as a function of
${\boldsymbol{d}}$, the relative offset between the moiré lattices
(illustrated in inset), for $\theta=1.5^{\circ}$ and $\kappa=0.7$. (b) The
moiré band structure for h-HTG, corresponding to
${\boldsymbol{d}}={\boldsymbol{\delta}}$. (c) The density of states for h-HTG
as a function of $\theta$.

In Fig2a, we show the remote band gap, defined as the minimum of the gap
between the second and first conduction or valence bands, as a function of
${\boldsymbol{d}}$ for $\theta=1.5^{\circ}$. For special shifts such as
${\boldsymbol{d}}=0$, or along high symmetry lines, the remote band gap is
forced to be zero [59, 43]. For generic shifts, however, the remote band gap
is non-zero and maximized for shifts at the corners of the moiré unit cell:
${\boldsymbol{d}}=\pm{\boldsymbol{\delta}}=\pm\frac{1}{3}({\boldsymbol{a}}_{2}-{\boldsymbol{a}}_{1})$.
Computing the total Chern number of the first conduction and valence bands, we
find $C_{\mathrm{tot}}=\mp 1$ in the regions smoothly connected to the high
symmetry $\pm{\boldsymbol{\delta}}$ points. The corresponding bands in the
$K^{\prime}$ valley have opposite Chern numbers by time-reversal symmetry. The
band structure at ${\boldsymbol{d}}={\boldsymbol{\delta}}$, shown in Fig 2b,
demonstrates the existence of two isolated nearly-flat bands carrying net
topology.

We now focus on the properties of the h-HTG periodic moiré superlattice,
obtained by setting ${\boldsymbol{d}}={\boldsymbol{\delta}}$. The
$\overline{\mathrm{h}}$-HTG model, obtained by setting
${\boldsymbol{d}}=-{\boldsymbol{\delta}}$, is related by
$C_{2z}\mathcal{T}=\sigma_{x}\mathcal{K}$, where $\mathcal{K}$ is complex
conjugation, which takes ${\boldsymbol{r}}\to-{\boldsymbol{r}}$ and leaves
${\boldsymbol{k}}$ invariant. At this special value of ${\boldsymbol{d}}$ the
model has additional symmetries. Because ${\boldsymbol{\delta}}$, as the
corner of the unit cell, is a $C_{3z}$ invariant point, the resulting moiré
superlattice is $C_{3z}$ symmetric. Furthermore, because
${\boldsymbol{\delta}}\to-{\boldsymbol{\delta}}$ under $x\to-x$, the model is
additionally symmetric under $C_{2y}$, which exchanges the top and bottom
layers and also exchanges valleys. The antiunitary mirror symmetry
$C_{2y}\mathcal{T}$ acts within a valley.

Fig2c shows the density of states (DOS) of h-HTG as a function of $\theta$.
The most prominent feature is the appearance of the advertised magic angle at
$\theta\approx 1.5^{\circ}$, where two topological flat bands appear at the
charge neutrality point. At the magic angle, the DOS exhibits a sharp peak
($\mathrm{DOS}>20$eV-1nm-2), the remote band gap is large
$E_{\mathrm{gap}}\approx 85$meV, and the dispersion, half the total bandwidth
of both bands, is small $W\approx 15$meV. These values should be contrasted
with a typical interaction scale; this can be obtained by scaling up the
$20-30$meV estimate[2, 83] of TBG interactions by $\approx 1.5$, due to the
larger angle, yielding the range $30-45$meV. This ordering of energy scales is
ideal for exploring strongly correlated topology, in which the large
$E_{\mathrm{gap}}$ essentially “locks in” the quantum geometry of the flat
band manifold, within which interactions are dominant.

##  III Chiral limit

The origin of these flat bands can be understood from the chiral model[64],
obtained by setting $\kappa=0$, which we now analyze in detail. Chiral models,
with exactly flat bands[64, 84, 85, 86, 87, 88, 89, 90, 91, 92], also motivate
a “sublattice-Chern basis”[93, 94, 49], useful for all $\kappa$, where the
flat bands have Chern number $C=(1,-2)$. The chiral model enables strong
coupling approaches[49] to several types of correlated insulating states
including generalized ferromagnets, topological charge density waves, and
fractional Chern insulators.

To write an explicit form of the chiral Hamiltonian, we choose
${\boldsymbol{d}}_{t}=-{\boldsymbol{d}}_{b}=-{\boldsymbol{\delta}}$ such that
${\boldsymbol{d}}=-2{\boldsymbol{\delta}}\equiv{\boldsymbol{\delta}}$ and
$U_{l}({\boldsymbol{r}}-{\boldsymbol{d}}_{t,b})=U_{l\pm 1}({\boldsymbol{r}})$.
We therefore have, in the basis where
$\sigma_{z}=\text{diag}(1,1,1,-1,-1,-1)$,

$\displaystyle H_{K}$
$\displaystyle=vk_{\theta}\begin{pmatrix}0&D^{\dagger}\\\ D&0\end{pmatrix},$
(3) $\displaystyle D$
$\displaystyle=\begin{pmatrix}-2ie^{i\zeta}\overline{\partial}&\alpha
U_{-1}({\boldsymbol{r}})&0\\\ \alpha
U_{0}(-{\boldsymbol{r}})&-2i\overline{\partial}&\alpha
U_{0}({\boldsymbol{r}})\\\ 0&\alpha
U_{-1}(-{\boldsymbol{r}})&-2ie^{-i\zeta}\overline{\partial}\end{pmatrix}.$

Here we have nondimensionalized the Hamiltonian using
${\boldsymbol{r}}\to{\boldsymbol{r}}k_{\theta}$,
$\overline{\partial}\to\overline{\partial}/k_{\theta}$ where
$\overline{\partial}=\frac{1}{2}(\partial_{x}+i\partial_{y})$. Nominally
$\zeta=\theta$, but it can be instructive to imagine tuning it independently;
none of our conclusions depend on its precise value.

As we tune the dimensionless tunneling strength $\alpha=w/vk_{\theta}\sim
1/\theta$, we find a sequence of magic angles, listed in Table 1, at which we
obtain _exactly_ flat bands at zero energy, seen by the vanishing bandwidth in
Fig3a. Due to the chiral symmetry of the Hamiltonian $\\{H,\sigma_{z}\\}=0$,
we may label zero modes by their eigenvalue under $\sigma_{z}$, such that the
flat bands correspond to zero modes of $D$ polarized on the A sublattice and
of $D^{\dagger}$ on the B sublattice. As shown in Fig3b, odd parity magic
angles have two flat bands per spin per valley, while even-parity magic angles
have four together with a dispersive Dirac cone at $\Gamma$ (for $\zeta=0$).
Interestingly, the even magic angle dispersive cone is gapped out by
$\zeta\neq 0$ but the four exactly flat bands remain. The distinction between
even and odd magic angles, together with ratios between magic $\alpha$ that do
not match those of TBG, suggest that the magic angles here do not descend from
those of TBG. This is in contrast to the chiral magic angles of twisted
chirally stacked multilayers[46, 4], alternating twist multilayers[4], and the
${\boldsymbol{d}}=0$ periodic HTG [60], which can all be related to TBG. A
detailed understanding of the mathematical structure of this model is an
interesting subject beyond the scope of this work.

Let us focus on the first magic angle $\alpha_{1}\approx 0.377+O(\zeta)$ which
is the most experimentally relevant. Here we obtain two exactly flat bands;
the A sublattice band has $C_{A}=1$ and the B sublattice band has $C_{B}=-2$.

To understand the emergence of flat bands and their Chern numbers we begin
with the 3 Dirac cones associated with the $\alpha=0$ decoupled limit. These
cones are protected and pinned to zero energy by chiral symmetry, pinned to
the $\kappa,\gamma,\kappa^{\prime}$ points by $C_{3}$ symmetry, and all have
positive chirality. The net chirality of three implies that the flat bands
obtained by gapping all three cones with a $\sigma_{z}$ mass results in bands
that differ in Chern number by $C_{A}-C_{B}=3$ [50]. The low-energy bands must
therefore have a net topology $C_{A}+C_{B}\neq 0$.

Figure 3:  (a) The dispersion $W$ and remote gap $E_{\mathrm{gap}}$ for the chiral model with $\zeta=0$, as a function of the dimensionless tunneling parameter $\alpha$. (b) The band structure at $\alpha_{1}$ (left) and $\alpha_{2}$ (right). (c) The charge density $n({\boldsymbol{r}})$ and Berry curvature $F({\boldsymbol{k}})$ in the Chern basis for the $|C|=1,2$ bands, where $A_{uc}$ and $A_{BZ}$ are the moiré unit cell and mBZ areas respectively. (d) The evolution of $W$, Berry curvature deviation $\delta F$, and trace condition violation $\overline{T}$, as a function of $\kappa$ in the Chern basis, and corresponding TBG values for comparison.  $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{5}$ | $\alpha_{6}$ | $\alpha_{7}$ | $\alpha_{8}$ | $\alpha_{9}$
---|---|---|---|---|---|---|---|---
0.377 | 1.297 | 1.755 | 2.414 | 2.991 | 3.628 | 4.213 | 4.840 | 5.430
Table 1: List of magic angles for the chiral model with $\zeta=0$.

We now analytically derive the exact flatness and Chern numbers at the first
magic angle. Let us focus on the $A$ sublattice, $\psi=\psi_{A}$. From the
decoupled limit, we find that the $C_{3z}$ representation of the zero mode
$\gamma$-point wavefunction is such that $\psi_{\gamma
1,3}({\boldsymbol{r}}=0)=0$ but $\psi_{\gamma 2}(0)$ is in general nonzero. In
a $C_{2y}\mathcal{T}$ symmetric gauge, $\psi_{\gamma 2}(0)$ is a signed real
number, and it is natural for it to cross through zero [49, 92]; we find that
it does at $\alpha=\alpha_{1}$. Such a crossing point is stable to $C_{3z}$
and $C_{2y}\mathcal{T}$ preserving perturbations, and leads to exactly flat
bands. At the crossing point, the entire $\gamma$ point zero-mode wavefunction
vanishes at ${\boldsymbol{r}}=0$. We may therefore write[64, 84]

$\psi_{{\boldsymbol{k}}}({\boldsymbol{r}})=e^{\frac{i}{2}\overline{k}z}\frac{\sigma(z+ik)}{\sigma(z)}\psi_{\gamma}({\boldsymbol{r}})$
(4)

as a zero mode wavefunction at wavevector ${\boldsymbol{k}}$, measured from
the $\gamma$ point for concreteness. Here, $\sigma(z)=\sigma(z|a_{1},a_{2})$
is the (modified [95]) Weierstrass sigma function which satisfies
$\sigma(-z)=-\sigma(z)$ and
$\sigma(z+a_{1,2})=-e^{\frac{1}{2}\overline{a}_{1,2}\left(z+\frac{a_{1,2}}{2}\right)}$.
The pole associated with the zero of the sigma function is cancelled by the
zero of $\psi_{\gamma}$. Here we have used the complex number notation
$k=k_{x}+ik_{y}$, $z=x+iy$, and $a=a_{x}+ia_{y}$. We note that (4) may be
interpreted as the wavefunction of a Dirac particle moving in an effective
inhomogeneous magnetic field of $2\pi$ flux per unit cell, where $B_{\rm
eff}({\boldsymbol{r}})=\nabla^{2}\log\frac{\absolutevalue{\sigma(z)}}{||\psi_{\gamma}({\boldsymbol{r}})||}$[24,
48], up to an unimportant ${\boldsymbol{k}}$-independent normalized layer
vector.

The fact that $\psi$ has a single $k$-space zero, for each ${\boldsymbol{r}}$,
of positive winding implies that the band has $C_{A}=1$ (there is a winding by
$2\pi$ around the mBZ)[48]. It is also possible to compute the Chern number
from the $k$-space quasiperiodicity of (4) [49, 48]. Since $D$ and
$D^{\dagger}$ have the same singular values when there is no external magnetic
flux[91], the B sublattice must also have an exact band of zero modes from
which $C_{A}-C_{B}=3$ implies $C_{B}=-2$.

In Fig3c, we show the charge density $n({\boldsymbol{r}})$ and Berry curvature
$F({\boldsymbol{k}})$ for these bands. The charge distribution for both bands
are remarkably uniform. This can be motivated from the fact that TBG has a
singly peaked charge density at the AA region: since h-HTG contains two such
AA sites, AA12 and AA23, the charge density has two peaks at these locations
for each layer pair and is overall much more uniform. We also find that the
Berry curvature distribution for both bands feature a multi-peak structure in
momentum space, and are relatively uniform (with the A sublattice $C=1$ band
being extremely uniform). These are both important features reminiscent of the
lowest Landau level which persist to larger $\kappa$, as we discuss later.

All the Chern basis bands, labeled by valley, spin, and sublattice, are
explicitly summarized in Table 2.

Band | h-HTG |  $\overline{\mathrm{h}}$-HTG
---|---|---
$(K,s,A)$ | $C=1$ | $C=2$
$(K,s,B)$ | $C=-2$ | $C=-1$
$(K^{\prime},s,A)$ | $C=-1$ | $C=-2$
$(K^{\prime},s,B)$ | $C=2$ | $C=1$
Table 2: Table of Chern basis bands labeled by (valley, spin, sublattice),
showing their Chern numbers, in the h-HTG and $C_{2z}$-related
$\overline{\mathrm{h}}$-HTG structure.

##  IV Correlated states

We now discuss the correlated physics and band geometry of these bands. At
integer filling, generalized quantum Hall ferromagnets, obtained by filling
any combination of bands within the Chern basis (Table 2), are very
energetically competitive; they are exact eigenstates in the chiral limit, and
exact zero energy ground states when bare and Hartree dispersions can be
neglected[96, 93]. At filling $\nu=\pm 3$, measured relative to charge
neutrality such that the empty and full flat bands have $\nu=\mp 4$
respectively, the generalized ferromagnetic state necessarily carries non-zero
Chern number, though we will later discuss other states which avoid this
restriction. Coherence between bands of differing Chern number is heavily
penalized because such order parameters must have vortices in the Brillouin
Zone, equal in number to the difference in Chern number[21]. Since no two
Chern numbers are the same (including the other valley we have $C=\pm 1,\mp
2$), we therefore do not expect intervalley coherence (IVC) in this system.
Indeed, while TBG has an approximate $\mathrm{U}(4)\times\mathrm{U}(4)$
symmetry[93, 97, 98] that can rotate between IVC and valley diagonal orders,
we have the
$\mathrm{U}(2)\times\mathrm{U}(2)\times\mathrm{U}(2)\times\mathrm{U}(2)$
subgroup consisting of spin and charge rotations for each sublattice and each
valley. For nonzero bare dispersion, or outside the chiral limit, this
symmetry is broken to $\mathrm{U}(2)\times\mathrm{U}(2)$ consisting of spin
and charge rotations in each valley. A rich phase diagram of spontaneous
symmetry breaking has been observed in TBG [99, 100, 52], and we expect
similar physics to arise here. In this sense, the Chern basis is meaningful
and important even outside the chiral limit [93, 101, 49]; to access it one
can diagonalize the band projected sublattice operator
$\Gamma_{\alpha\beta}=\bra{u_{{\boldsymbol{k}}\beta}}\sigma_{z}\ket{u_{{\boldsymbol{k}}\alpha}}$,
where $\ket{u_{{\boldsymbol{k}}\alpha}}$ is the Bloch wavefunction at
wavevector ${\boldsymbol{k}}$ associated to band $\alpha$.

Zero mode bands of chiral Hamiltonians of the form (3) have “ideal quantum
geometry"[45, 46, 24, 47, 48, 49, 28, 50, 51] for fractional Chern insulators
in a sense that we now describe. Because $D$ only depends on antiholomorphic
derivatives, the zero mode band of $D$ maps to itself under multiplication by
$z=x+iy$; we have $zP=PzP$ where $P$ is the projector onto the band and
$z=x+iy$ can be thought of as a vortex operator; this condition is referred to
as “vortexability”[45]; vortices may be added while remaining within the band
of interest. This condition may be iterated to replace $z$ with any
holomorphic function $f(z)$. In momentum space, vortexability is equivalent to
the ability to choose a gauge where the wavefunctions
$u_{k}=e^{-i{\boldsymbol{k}}\cdot{\boldsymbol{r}}}\psi_{{\boldsymbol{k}}}$ are
holomorphic in $k_{x}+ik_{y}$ [102, 103, 104, 105, 106, 45]. It is also
equivalent to the momentum space “trace condition"[36, 31, 29, 107], the
saturation of the inequality $\overline{T}=\int d^{2}{\boldsymbol{k}}(\tr
g_{\text{FS}}({\boldsymbol{k}})-|F({\boldsymbol{k}})|)\geq 0$, where
$g_{\text{FS}}$ is the Fubini-Study metric and $F({\boldsymbol{k}})$ is the
Berry curvature. We say a system has ideal quantum geometry if
$\overline{T}=0$.

Ideal quantum geometry is intimately related to fractional Chern insulator
ground states; we may begin with an ordinary many-body state $\ket{\Psi_{0}}$,
e.g. the fully filled state, and create the state [46, 45, 47]

$\ket{\Psi_{2s}}=\prod_{i<j}(z_{i}-z_{j})^{2s}\ket{\Psi_{0}}$ (5)

which lies entirely within the band of interest due to the vortexability
condition. This construction generalizes that of the $\nu=1/(2s+1)$ Laughlin
state but also applies to bands with $C>1$. If the band is flat and the band-
projected interaction is sufficiently short-ranged and normal ordered with
respect to an empty “vacuum”, then $\ket{\Psi_{2s}}$ is the unique ground
state at its filling factor[108, 109, 24, 45, 48].

The previously mentioned charge density and Berry curvature homogeneity
further help with stability to long-ranged interactions, which can be
motivated by analogy with the lowest Landau level [36, 107, 110].
Additionally, the interaction generated Hartree dispersion obtained by
integrating out already-filled bands is much smaller if charge density is
peaked at more than one point[50]. This is indeed the case here, especially
relative to TBG where an AA-peaked charge density leads to a strong Hartree
dispersion that works against FCIs [28]. Moving away from the chiral limit, we
show the evolution of various geometric indicators in Fig3d. We first observe
the dispersion $W$ increasing with $\kappa$, and we identify the optimal magic
angle $\alpha_{1}^{\mathrm{opt}}(\kappa)$ by the minimum in $W$. We show the
Berry curvature deviation $\delta F=\left(\int
d{\boldsymbol{k}}[\frac{1}{2\pi}F({\boldsymbol{k}})-C]^{2}\right)^{\frac{1}{2}}$
and trace condition violation $\overline{T}$ of the two bands in the Chern
basis, at $\alpha_{1}^{\mathrm{opt}}(\kappa)$, as a function of $\kappa$,
along with the corresponding values calculated for TBG. We find that $\delta
F$ shows remarkably weak dependence on $\kappa$, and is significantly lower
than TBG for both bands at realistic values of $\kappa$. Turning to
$\overline{T}$, we find that the $|C|=2$ ($1$) band is uniformly more (less)
ideal than TBG, but are all of similar magnitude. Overall, these favorable
quantum geometric indicators are highly suggestive of an FCI ground state at
fractional filling, and call for a detailed numerical analysis.

Many correlated states have been predicted for $C=2$ bands, from the starting
point of ideal geometry [47, 111]. By doubling the unit cell, the $C=2$ band
may be split [112, 113, 114] into two $C=1$ bands that are each individually
vortexable [47, 111] and related by translation symmetry; linear combinations
of these $C=1$ subbands lead to a nearly-degenerate manifold of charge and
spin density waves that can occur at half-integer fillings [47, 111]. These
states are guaranteed to be stabilized in the limit of short-ranged
interactions when the bands have ideal geometry, similar to FCIs, but are
numerically present for realistic parameters as well [27]. A $C=1$ insulator
of this nature at half-integer filling was observed in twisted monolayer-
bilayer graphene [10]. We expect that this could be the case in h-HTG as well.
By filling two of the $|C|=1$ sub-bands, say in opposite valleys, it is
possible to also obtain integer filling states with unexpected properties,
such as a $\nu=3$ insulator with net Chern number zero. At fractional filling,
there are a variety of fractional Chern insulating states that have been
proposed in higher Chern bands [112, 115, 116, 117, 118, 119, 120, 121, 122,
114, 123]. Many can be constructed from (5) using different parent states
$\ket{\Psi_{0}}$. We refer readers to Refs. [47, 111] for more details.

We briefly discuss the incommensurate Kekulé spiral state, which was recently
found to be important in strained TBG [124, 125, 126, 127]. We expect that
such incommensurate orders are less likely in h-HTG since their energetics
appear to rely on a large, peaked, Hartree dispersion. The large Hartree
dispersion originates from particular features of TBG wavefunctions, such as a
single peak of the charge density per unit cell, that are not shared by h-HTG.
Instead, we expect the previously discussed topological states to be favored.

We have highlighted that the topological nature and nearly ideal quantum
geometry of h-HTG lead to a wide variety of potential correlated states, from
generalized quantum Hall ferromagnets to topological density waves to
fractional Chern insulators. Due to the proximity to the $\kappa=0$ chiral
limit, there is a pathway towards understanding the competition between these
various correlated states. The investigation of the detailed energetic
competition of such correlated states is an important subject for future
works.

##  V Discussion

Having discussed extensively the rich correlated physics of h-HTG, we now
briefly discuss the global physics when the domain size is finite. In this
case, HTG separates into large domains of h-HTG and
$\overline{\mathrm{h}}$-HTG, related by the valley-preserving
$C_{2z}\mathcal{T}$ transformation which flips the sign of all Chern numbers.
When the domains are tuned to incompressible fillings, the low energy physics
is dominated by the network of domain wall states which provide a realization
of Chalker-Coddington type network models [57, 58]. At the full or empty band
insulating states $\nu=\pm 4$, stabilized by the $E_{\mathrm{gap}}\sim 100$meV
remote gap, each domain is a quantum valley Hall state with net valley-Chern
number that is opposite in adjacent domains: we therefore expect the
appearance of gapless edge modes counter-propagating along the network of
domain walls. Similar physics of counter-propagating edge networks has been
actively explored in marginally twisted $\theta\ll 1^{\circ}$ TBG [128, 129,
130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143], where a
strong vertical displacement field is needed to gap the AB and BA domains. In
HTG, the large intrinsic band gap eliminates the need for a displacement
field. More interesting possibilities also arise here due to the correlated
topological states. When the system is valley polarized into QAH domains,
adjacent domains carry opposite Chern numbers thus realizing a network of
gapless chiral edge modes. Alternatively, adjacent domains may have opposite
valley polarizations; the competition between these possibilities depends on
detailed domain wall energetics [144] and may be probed by an out-of-plane
magnetic field. This situation is similar to the Chern mosaic in hBN-aligned
magic-angle TBG [145, 54]. At fractional filling, an even more exotic
possibility is a network of FCI edge states. A detailed theory of the emergent
gapless networks and their interplay with the correlated topological states in
HTG is an important subject for future works.

While we have mainly discussed HTG in the context of TBG, flat Chern bands
have also been studied in a variety of other $C_{2z}$-breaking graphene
structures such as twisted chirally-stacked multilayers[146, 147, 148, 149,
150, 151, 15, 46, 152, 47, 111] and periodically strained graphene[153, 154,
155, 156, 50]. Twisted monolayer-bilayer graphene in particular also has
$C=\pm(1,-2)$. All of these systems have natural “chiral limits"[64] where the
band-wavefunctions have ideal quantum geometry for fractional Chern insulators
[46, 152, 47, 111, 50]. However, for monolayer-bilayer and bilayer-bilayer the
exact chiral limit requires ignoring trigonal warping, and in its presence a
large displacement field must be added to flatten the bands and reveal
correlated states in experiment [15, 14, 12, 11, 13, 10, 17, 16]. In total,
these realities likely impair the ideal quantum geometry for fractional Chern
insulators in these systems. While strained graphene’s chiral-flat
$\absolutevalue{C}=1$ band is more robust [50], realizing the strain requires
nanorod engineering [157] or buckling over a $C_{2z}$-breaking substrate
[158]; the latter has only been achieved with the metallic
$\text{NbSe}_{\text{2}}$ [158], which precludes tuning the density with a
gate. Furthermore, HTG is unique in its supermoiré scale domain walls and
gapless edge-modes between $C_{2z}\mathcal{T}$ related topological states.

We have therefore demonstrated that HTG is a unique and exciting platform for
realizing strongly correlated topological states, without the need for
substrate alignment. While unrelaxed HTG forms a moiré-of-moiré pattern, we
have shown that lattice relaxation favors the formation of h-HTG, a periodic
$C_{2z}$-breaking single-moiré superlattice. We identify the relevant
continuum model description for h-HTG and identify a magic angle at which a
pair of flat topological bands with near-ideal quantum geometry isolated by a
large remote band gap emerges. These flat bands can be traced back to a chiral
limit, which enables a controlled starting point for a strong coupling
approach to correlated topological states. Our work lays the foundation for
future theoretical and experimental studies of the strongly correlated
topological physics in this platform.

###### Acknowledgements.

We thank Ziyan Zhu for useful discussions and for collaboration on a related
project. TD thanks Yves Kwan for helpful discussions. PJL thanks Eslam Khalaf,
Ashvin Vishwanath, Daniel Parker, Junkai Dong, Grigory Tarnopolsky, and Qiang
Gao for collaborations on related projects. This work was supported by the Air
Force Office of Scientific Research (AFOSR) under award FA9550-22-1-0432. This
work was partially supported by the Army Research Office MURI W911NF2120147,
the 2DMAGIC MURI FA9550-19-1-0390, the National Science Foundation
(DMR-1809802), the STC Center for Integrated Quantum Materials (NSF grant no.
DMR-1231319), and the Gordon and Betty Moore Foundation’s EPiQS Initiative
through grant GBMF9463 to PJH. AU acknowledges support from the MIT Pappalardo
Fellowship and from the VATAT Outstanding Postdoctoral Fellowship in Quantum
Science and Technology.

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# Astronomy Data, Virtual Observatory and Education

Priya Hasan & S N Hasan  Maulana Azad National Urdu University, Hyderabad,
India
email<EMAIL_ADDRESS>
(2021)

###### Abstract

We shall present with examples how analysis of astronomy data can be used for
an educational purpose to train users in methods of data analysis, statistics,
programming skills and research problems. Special reference will be made to
our IAU-OAD project ‘Astronomy from Archival Data’ where we are in the process
of building a repository of instructional videos and reading material for
undergraduate and postgraduate students as well as interested participants.
Virtual Observatory tools will also be discussed and applied. As this is an
ongoing project, by the time of the conference we will have the projects and
work done by students included in our presentation. The material produced can
be freely used by the community.

###### keywords:

astronomical data bases, sociology of astronomy

††volume: 367††journal: Education and Heritage in the era of Big Data in
Astronomy††editors: R.M. Ros, B. Garcia, S. Gullberg, J. Moldon & P. Rojo,
eds.

##  1 Introduction

Virtual learning is a learning-teaching experience that is technology and
communication enabled in a synchronous (web-conferencing) or asynchronous
(self-paced) manner. Teaching is online and the teacher and learners are
physically separated (in terms of place, time, or both). One of the biggest
challenges to educators during the covid pandemic was virtual teaching which
became the sole mode of communication. Virtual teaching has its advantages
namely, convenience of access, wide global reach, inclusive content and
collaborations across the globe. However, it has it’s disadvantages too:
essentially the lack of face-to-face interaction and the human element. The
present talk describes our experiences with virtual teaching of astronomy with
the help of astronomy data and Virtual Observatory tools.

In the early April 2020, we collaborated with Jana Vigyan Vedika (JVV)
Telangana which is a member organisation of the All Indian People Science
Network (AIPSN) and a member of the National Council for Science and
Technology Communication (NCSTC) in India. The NCSTC is a scientific programme
of the Government of India for the popularisation of science, dissemination of
scientific knowledge and inculcation of scientific temper. JVV is an effective
Science Forum in the state and has a very good reach in rural and urban
Telangana.

We started off with a Six-Day Online Astronomy Course from the 18-23 May 2020
from 4:30- 6:30 pm IST for School Teachers and interested students. We
conducted a quiz at the end of the course and gave certificates to
participants who scored above a threshold mark. We had 475 registrations and
many were given certificates with JVV.

In May 2020, the International Astronomical Union-Office of Astronomy for
Development (IAU-OAD) issued an Extraordinary Call for Proposals that could
‘use astronomy in any form to help mitigate some of the impacts caused by
COVID-19’. We proposed a project ‘Astronomy from Archival Data’ which involved
Educational Projects for Under-Graduate and Post-Graduate Science Students
([Possel(2020), Possel, 2020]). The project trained students to use the high-
quality astronomy data from various facilities. Participants were shown step-
by-step techniques of accessing and analysing astronomy data from the
internet, introduced to virtual observatory tools and programmimg techniques,
and supported to formulate and develop projects. The training included special
sessions on report writing, publishing and presenting in scientific
journals111More info at: https://shristiastro.com/astronomy-from-archival-
data/.

Almost 1000 students from 25 countries registered for the course, which was
encouraging and which made it clear that students clearly require this kind of
skill training. Figure 1 shows a mosaic of (1) Our website (2) Our YouTube
channel (3) A zoom session in progress (4) One of the recorded sessions.

##  2 Modus Operandi

Although virtual teaching has been there for a while, this time it required
more thought and planning. With the funds we has from OAD, we first set up a
website design and content. To enable interaction, we used Social Media
(Facebook), Telegram as well as regular email. At regular intervals we planned
Homework, Quizzes and Polls so that we could get feedback from our
participants.

The project consisted of three phases. The first was made up of live zoom
sessions on Saturdays and Sundays from 10-11:30 am IST with lectures and
interactive hands-on sessions with us and a variety of speakers from India and
abroad on Virtual Observatory Tools, Data Archives and Science Cases. All the
sessions were recorded and posted online. We now have over 60 video recordings
online on YouTube. Reading material, quizzes, data sources, analysis
techniques, etc. are available on our website
(https://shristiastro.com/astronomy-from-archival-data/) as well as on
YouTube. The sessions ran from August to the end of November, 2020. The second
phase was in December, when students could complete watching videos and select
projects and interact with mentors. The third phase will be till the second
half of February to give students time to work on projects of their choice.
Students will present their results in the last week of February, 2021.

Apart from us, we had a wide variety of speakers like Mark Taylor, Luisa
Rebull, Deborah Baines, Tim Hamilton, Ajit Kembhavi, Avinash Deshpande, Sushan
Konar, Kaustubh Waghmare, Eeshan Hasan etc. to cover a wide range of junior
and senior researchers. The talks ranged from Basics of Python, Astropy,
TOPCAT ([Taylor (2003), Taylor, 2003]), Aladin, ESASky, Machine Learning and
Coding and how to use them. We also had talks on Science Cases for projects
and some hand-held sessions on projects.

Figure 1 shows a Zoom Session in progress: TOPCAT with Mark Taylor.

Figure 1: A Zoom Session in progress TOPCAT with Mark Taylor

##  3 Projects

Students were free to opt for one or more projects of their choice. Mentors
have been having sessions with their groups to individually address issues and
work on the projects. The list of projects offered are as follows:

  * •

Exoplanets (Tim Hamilton/Priya Hasan)

This project involves analysis of Kepler light curves to derive parameters for
Exoplanets including distance to the star, mass, radius and hence density.
This and the next three projects were explained in a step-by-step procedure by
Tim Hamilton and is available on our YouTube website. Enthusiastic
participants are encouraged to try their skills with Transiting Exoplanet
Survey Satellite (TESS) data.

  * •

Globular Clusters Photometry (Tim Hamilton/Priya Hasan)

This is an exercise in doing Aperture Photometry on images of globular
clusters from the SDSS data. Color-magnitude diagrams for the globular cluster
are constructed.

  * •

Globular Clusters Color-Magnitude Diagrams: Finding the ages and distances of
Milky Way globular clusters (Tim Hamilton/Priya Hasan)

For a sample 11 globular clusters, participants had to plot the color-
magnitude diagram showing the cluster data and the best-fitting isochrone
together.

  * •

Gravitational Lensing (Tim Hamilton/Priya Hasan)

For a given list, extracting close-in image of each gravitational lens system,
showing the lens and the Einstein Ring is to be done. Students need to
calculate the total mass, stellar mass, mass of dark matter and dark matter as
a fraction of total mass.

  * •

The Radio Pulsar-Magnetar Connection (Sushan Konar)

This project involved a comparison of the intrinsic properties of radio-
pulsars and magnetars using available data.

  * •

Star clusters with Gaia (TOPCAT) (Priya Hasan)

Extracting Gaia data and plotting the color-magnitude diagram using TOPCAT.

  * •

Python and Gaia

Using python code with Gaia data (Priya Hasan)

  * •

The spatial distribution of pulsars and the spiral structure of the galaxy
(Avinash Deshpande)

  * •

Galaxy Morphology (S N Hasan)

Studying galaxy morphology using decomposition of luminosity profiles using
GALFIT ([Peng (2010), Peng, 2010])

  * •

Others

Participants were free to propose their own projects based on the material
covered.

We hope that some of these students may even use this for applications in PhD
programs/Graduate school as well as a start to serious research careers.

##  4 Lessons Learnt

This project is close to completion. The Internal Virtual Observatory Alliance
(IVOA) supports this project. Participants will present their projects and
their presentations will be added to the repository 222As a team, we run
Shristi Astronomy (https://shristiastro.com/). Details of our activities are
on the website and the reader is encouraged to have a look. All the sessions
have been recorded and are available on youtube
https://www.youtube.com/c/shristiastronomy.

In the course of our sessions we learned a few lessons which we would like to
share.

  * •

Programs like this are very useful to students since it shows them important
steps in data analysis and techniques which are generally never taught as part
of a course. To get the true flavour of a research career, it is very useful
if students try their hand on some projects.

  * •

Not all who register participate and hence there has to be constant monitoring
of the participants. Interaction via social media, email, etc has to be
maintained to keep in tune with issues faced by participants. This can be very
time consuming as well as difficult.

  * •

Planning of the talks has to be done carefully since most potential
participants had their own online sessions on. We selected weekends since most
students/participants do not have classes then. As time progressed, classes as
well as examinations were on which made it difficult for students/participants
to attend our classes.

  * •

There are various issues like internet connectivity problems, power as well as
overlapping sessions. Hence all our activities were recorded and made
available on YouTube so that participants could attend at their convenience.

  * •

A few of the participants were from non-English speaking countries, hence we
enabled subtitles on our videos for their benefit.

  * •

It is good to have a variety of speakers ranging from students, research
scholars, post-doctoral fellows and junior and senior faculty since it can
match the level of the participant and it makes the environment more
inclusive.

  * •

While sessions were on, gradually many students had regular classes and exams
due to which the numbers dropped. Since the repository we created is available
online, we hope that students can access the material at their convenience.

We have received approval for a new IAU-OAD project ‘AstroSprint’ which will
be a follow-up of this project in the form of three workshops in the year
2021. The first will still be online and we hope the next two in July and
November 2021 wgich will be offline where we can have interactions in-person.
We think this was a very timely and purposeful initiative and it was a very
good learning exercise. The support of the IAU-OAD is appreciated. We hope to
continue with this work of developing a repository of instructive videos and
study material that can be used for astronomy research.

## References

  * [Peng (2010)] Peng, C. Y., Ho, L. C., Impey, C. D. and Rix, H. 2010, AJ, 139, 6
  * [Possel(2020)] Possel, M. 2020, The Open Journal of Astrophysics, 3, 1
  * [Taylor (2003)] Taylor M. B. 2005, ASPC, 347, 29