# TreeGCN-ED: Encoding Point Cloud using a Tree-Structured Graph Network

###### Abstract

Point cloud is one of the widely used techniques for representing and storing
3D geometric data. In the past several methods have been proposed for
processing point clouds. Methods such as PointNet and FoldingNet have shown
promising results for tasks like 3D shape classification and segmentation.
This work proposes a tree structured autoencoder framework to generate robust
embeddings of point clouds by utilizing hierarchical information using graph
convolution. We perform multiple experiments to assess the quality of
embeddings generated by the proposed encoder architecture and visualize the
t-SNE map to highlight its ability to distinguish between different object
classes. We further demonstrate the applicability of the proposed framework in
applications like: 3D point cloud completion and Single image based 3D
reconstruction.

Index Terms—  Graph Convolution, GAN, Autoencoder, Point Cloud, Deep Learning

##  1 Introduction

Encoder-decoder based methods have been widely used for generating information
preserving embeddings for different data modalities. In the past decades,
several encoder-decoder based methods have been proposed for 2D images [1, 2,
3] and have shown impressive results for image compression and filtering tasks
[4, 5]. However, it is challenging to extend these methods for 3D point cloud
data due to its irregular structure as compared to that of images and 3D
voxels. In this work, we focus on designing a deep-learning based encoder-
decoder framework for processing point cloud data.

Several methods have been proposed for encoding-decoding of point cloud data
[6, 7, 8, 9]. PointNet [6] is a pioneer deep-learning-based method for
encoding the point cloud data to lower-dimensional embeddings. These
embeddings carry rich information about the point clouds and can be used for
downstream tasks like segmentation and classification of point clouds. In a
recent work [10], the authors propose a tree-structured decoder which uses the
idea of graph convolution [11] to generate a point cloud using a noise vector
$z\in R^{96}$ sampled from a normal distribution $\mathcal{N}(0,I)$. They
propose aggregating the information from parent nodes at each layer instead of
spatially adjacent nodes to leverage the tree-structured decoder architecture
when applying graph convolution. The unique definition of graph convolution
proposed in [11] highlights the effect of using information from parent nodes
at multiple levels during the aggregation stage.

Inspired by the tree-structured decoder architecture in [10], we propose a
tree-structured encoder and a graph convolution mechanism for down-sampling.
We combine our proposed encoder with the decoder proposed in [10] to create a
complete tree-based encoder-decoder framework called TreeGCN-ED 111Code is
available at: https://github.com/prajwalsingh/TreeGCN-ED for processing point
clouds. To show the effectiveness of proposed framework, we compare its
results with FoldingNet [12] architecture. The results show that TreeGCN-ED
performs better than FoldingNet on two different evaluation metrics - Chamfer
Distance (CD) [13] and Fréchet point cloud distance (FPD) [13]. We also
observe that TreeGCN-ED learns inherent semantic information of the point
cloud and hence, performs semantic segmentation without any explicit training.
This highlights that our encoder generates more information-preserving
embeddings. We also perform ablation studies to determine the effect of
feature embedding dimension and data augmentation on the proposed TreeGCN-ED
network. We further use the learned embeddings in a transfer learning setup
and compare the results for point cloud classification on ModelNet10 and
ModelNet40 datasets [14]. Finally, we demonstrate the applicability of
proposed framework for 3D point cloud completion and single image based 3D
reconstruction.

Contributions. The following are the major contributions of this work.

  * •

A tree-structured encoder to generate robust embeddings for point cloud
processing using graph convolution.

  * •

An autoencoder based framework formed by the proposed encoder with the tree-
based decoder [10] for better point cloud reconstruction.

Fig. 1: TreeGCN-ED Architecture. The encoder part (on the left) consist of
downsampling and graph convolution modules for encoding the input 3D point
cloud into a feature embedding $\psi\in\mathbb{R}^{n}$. The decoder
architecture (on right) is taking the embedding as input from the encoder and
reconstructing the 3D point cloud. Fig. 2: TreeGCN-ED Down-Branching and Up-
Branching Architecture. The down-branching module (on the left) consists of a
fully connected layer followed by max-pooling for feature extraction. Output
of the fully connected layer is divided into $\mathbb{C}$ equal components
that are passed to the max-pooling layer. The up-branching architecture is
responsible for collecting information from the feature embedding of the
ancestors and upsampling. The upsampled feature is passed to graph convolution
layer for further processing.

##  2 RELATED WORK

Point cloud is an important data structure that can be used to store
information about the geometry of any 3D shape. There are various applications
related to point cloud processing. Some of the applications and methods are
point cloud classification and segmentation [6, 15, 16, 17, 18, 19, 20], point
cloud completion [21, 22, 23, 12], point cloud auto-encoder [12, 6, 24, 25,
26, 27, 28, 29], and Generative Adverserial Networks (GANs) for point cloud
[10, 30].

In [6], the authors propose the first end-to-end deep auto-encoder to directly
process point cloud data. The encoder uses 1D CNN and global max-pooling to
extract features of the input point cloud. This makes the model permutation
invariant. The decoder reconstructs the point cloud using a three-layer fully
connected network. FoldingNet [12] model is build up on the idea of PointNet
[6], by proposing an auto-encoder network that uses graph-based method to
learn encoding of a point cloud. Edge based convolution method is proposed in
[20] to learn the local neighborhood as well as global properties of the 3D
shape. In [10], the authors have proposed a deep generative model for 3D point
cloud generation. This method is unique because the authors have used graph
convolution on point cloud data which inherently does not contain any edge
connections.

##  3 METHOD

###  3.1 Tree-GAN

Tree-GAN [10] proposes a deep generative model for 3D point cloud generation.
It uses a branching method to gather information from neighbouring points. The
accumulated information is then distributed to other points using graph
convolution. The point cloud thus generated through this method is implicitly
segmented.

In Tree-GAN [10], a noise vector $z\in\mathbb{R}^{96}$ is sampled from
$\mathcal{N}(0,I)$ and is given as input to the generator network. Each layer
of generator consists of a branching network and a graph convolution layer.
The branching network accumulates the feature vectors from the previous layers
which is then upsampled by the graph convolution layer to generate a new
feature vector for that layer. This is repeated until the point cloud of the
desired dimension $\mathbb{R}^{3\times n}$ is obtained at the output. Note
that the feature vector for the first layer is the noise vector $z$ itself.
The generator and discriminator are trained under WGAN [31].

###  3.2 TreeGCN Based Point Cloud Encoder-Decoder

Fig. 3: Interpolation Result. Illustration of intra-class (on left) and inter-
class (on right) point cloud interpolation.

In this section, we discuss the proposed method for 3D point cloud processing.
The key idea of this approach is inspired from the tree-GAN [10], a deep
learning based model for generating a 3D point cloud from a noise vector. We
use the idea of tree-based graph convolution from [10] to develop an encoder
that extracts rich embeddings to performs well on the unseen point cloud data.
Our model takes a 3D point cloud of size $\mathbb{R}^{n\times 3}$ as input,
then passes it through sequences of graph-based operation to generate encoding
for the point cloud. The generated encoding is then passed through the decoder
network where a sequence of graph-based operations upsample the encoding to
obtain a $\mathbb{R}^{n\times 3}$ point cloud as the output. The complete
network is trained end-to-end by minimizing chamfer loss [13] and is called
TreeGCN-ED.

Fig. 1 represents the proposed architecture of TreeGCN-ED. A
$\mathbb{R}^{n\times 3}$ point cloud is given as input to the model, which
then passes through a down branching network for gathering features from the
ancestors of each node. Fig. 2 shows the down branching network and here, we
are first passing each ancestor to a sequence of fully connected. Then, max
pooling is applied on it to extract dominant features and we pass this feature
to the next network, i.e. graph convolution network. The point cloud
continuously passes through the down branching and graph convolution network
sequence until the desired encoding $\psi$ is obtained. The generated encoding
is given as input to the decoder network. In the decoder, the embedding is
first passed through up branching network. The internal working of up
branching network is shown in Fig. 2, where the encoding is first passed
through the fully connected layer. Then the feature vector is concatenated
with the ancestor information, which can be useful for reconstructing point
clouds. Then the constructed feature is passed on to the graph convolution
network. This process is repeated till the point cloud of size
$\mathbb{R}^{n\times 3}$ is reconstructed. The decoder architecture is similar
to tree-GAN [10]. The overall model is trained in an end-to-end manner using
the chamfer loss function [13].

The branching network is an essential part of the TreeGCN-ED network. It helps
in accumulating information from ancestors for each node. Every ancestor
feature is passed through a fully connected layer at each stage to help the
network learn the relation between a node and its neighbour. This is also
useful because the point cloud does not have edge connections between points.
We use max pooling for selecting important features from the encoded point
cloud. Max pooling has been proved to be a permutation invariant function [6].
We experimented with other pooling functions, such as averaging and adding
feature vectors, but max-pooling works better than other methods. Tree graph
convolution network learns the semantic segmentation of point cloud implicitly
[10].

###  3.3 Loss Function

To train the TreeGCN-ED, we have used the chamfer loss [13] function as it
shows promising results for point cloud based reconstruction [10].

$\begin{aligned} \mathcal{L}_{chamfer}(S_{1},S_{2})&=\sum_{x\in
S_{1}}min_{y\in S_{2}}||x-y||_{2}^{2}+\sum_{y\in S_{2}}min_{x\in
S_{1}}||x-y||_{2}^{2}\end{aligned}$

(1)

In Equation 1, $S_{1}\in\mathbb{R}^{n\times 3}$ and
$S_{2}\in\mathbb{R}^{n\times 3}$ represents two different point clouds. There
are two specific reasons for using this loss function. First, it is
permutation invariant [13]. Second, it penalises the loss function if a point
from one set is not matched with its corresponding nearest neighbour in
another set and vice-versa. This forces the model to learn information
preserving embedding for the point cloud.

###  3.4 Data Preprocessing

To train our model, we have used ShapeNetBenchmarkV0 dataset [22] consisting
of 16 object classes. The dataset is split into training, validation, and
testing as per the standard ratio proposed in [22]. We uniformly sample $2048$
points from the meshes of the ShapeNet dataset [22]. To ensure uniform
sampling of points, we make use of barycentric coordinates for the surface
sampling.

##  4 EXPERIMENTS AND RESULTS

###  4.1 Training and Comparison of Encoder-Decoder Model

We have used ShapeNetBenchmarkV0 dataset [22], which consists of 16 object
classes, with train-test split officially available along with the dataset.
The dataset is first uniformly sampled for 2048 points and then passed on to
the network. We have trained the complete network using the chamfer distance
[13] as the loss function till convergence. We compare the performance of
TreeGCN-ED with the FoldingNet architecture [12] on the test set of [22]. We
use two different metrics for the evaluation task: Chamfer Distance (CD) [13]
and Fréchet Point Cloud Distance (FPD) [10]. The results are shown in Table 1
on ShapeNetBenchmarkV0 dataset [22]. For fair evaluation, in case of
FoldingNet [12], we have considered the first $2025$ minimum point cloud
distances for calculating CD, owing to the difference in input and output
point cloud size. The quantitative results shown in the table highlight that
our proposed method performs better than FoldingNet [12] for chamfer distance
(CD) as well as Fréchet point cloud distance (FPD).

Models | Metrics | Object Class |
---|---|---|---
Airplane | Bag | Cap | Car | Chair | Earphone | Guitar | Knife | Lamp | Laptop | Motorbike | Mug | Pistol | Rocket | Skateboard | Table | Average
FoldingNet [12] |  | CD
---
FPD
| 0.67
---
11.10
| 3.12
---
87.45
| 2.82
---
117.36
| 1.76
---
28.47
| 1.47
---
12.00
| 3.34
---
152.04
| 0.44
---
19.55
| 0.55
---
19.56
| 2.60
---
45.19
| 1.01
---
11.19
| 1.48
---
33.91
| 2.28
---
40.17
| 1.16
---
30.14
| 0.88
---
32.53
| 1.35
---
47.17
| 1.70
---
24.62
| 1.48
---
44.52
TreeGCN-ED |  | CD
---
FPD
| 0.50
---
5.79
| 1.88
---
21.02
| 1.62
---
16.14
| 1.45
---
9.47
| 1.32
---
7.85
| 1.91
---
51.79
| 0.40
---
13.90
| 0.41
---
14.80
| 1.97
---
21.82
| 0.88
---
2.56
| 1.14
---
14.67
| 1.72
---
12.70
| 0.79
---
9.62
| 0.61
---
23.91
| 0.78
---
13.90
| 1.41
---
13.90
| 1.21
---
11.54

Table 1: Comparison of the efficiency for 3D point cloud encoding-decoding
between our proposed architecture and the FoldingNet [12] model on
ShapeNetBenchmarkV0 dataset [22].

###  4.2 Point Cloud Interpolation

To show that our proposed encoder architecture is learning information rich
embedding, we perform inter-class and intra-class interpolation experiments
between the source and the target point clouds. The interpolation results are
shown in Fig. 3.

The intra-class interpolation results illustrate the ability of our model to
synthesize novel shapes between two given shapes. We observe that the
generated shapes faithfully represent the object class at each interpolation
stage and the interpolation is observed to be very smooth. Similarly, in the
case of inter-class interpolation, we observe a smooth transition of
characteristic class features from one object class to another.

###  4.3 t-SNE Visualization

We use t-SNE [32] plot to show how well our encoder model can generate feature
embedding for each class. We set the perplexity value to 40. Based on the
results of t-SNE plot shown in Fig. 4, the inter-class separation is higher.
This signifies the discriminative capacity of our proposed encoder model.

Fig. 4: The visualization of t-SNE [32] clustering of the feature embeddings
obtained from TreeGCN-ED model.

###  4.4 Ablation Studies

We perform ablation studies to determine how feature embedding $\psi$
dimension and data augmentation affect the ability of TreeGCN-ED to learn a
meaningful feature representation. Four different training regimes are
compared on the ShapeNetCore.v2 test-set [33] which consist of 55 classes. In
Regime 1 and 2, the dimension of feature embedding is fixed to 256 and 512,
respectively, without augmentation. Similarly, in Regime 3 and 4, the
dimension of feature embedding is fixed to 256 and 512, respectively, but with
augmentation. We use ShapeNetCore.v2 dataset [33] to train TreeGCN-ED for all
the four regimes. Table 2 shows the variation in chamfer distance for the
above mentioned regimes. Based on the results, Regime 4 gives the best model
performance.

Dataset | No Augmentation | Rotation Augmentation
---|---|---
$\psi=256$ | $\psi=512$ | $\psi=256$ | $\psi=512$
ShapeNetCore.v2 | 10.90 | 10.07 | 8.82 | 7.88
Table 2: Quantitative results for all four regimes for the task of 3D point
cloud encoding-decoding. Chamfer distance [13] is used as the metric for
comparison.

Furthermore, we also evaluate the efficiency of feature representation
learning of TreeGCN-ED on ModelNet10 and ModelNet40 datasets [14] for all the
four regimes. We follow the same procedure as mentioned in [12] to train a
linear SVM classifier on features extracted from trained TreeGCN-ED for the
ModelNet datasets [14]. Table 3 shows the variation in classification accuracy
for all the four regimes on the test set of ModelNet datasets [14]. Based on
the results, Regime 4 gives the best model performance.

Dataset | No Augmentation | Rotation Augmentation
---|---|---
$\psi=256$ | $\psi=512$ | $\psi=256$ | $\psi=512$
ModelNet10 | 0.83 | 0.83 | 0.85 | 0.85
ModelNet40 | 0.71 | 0.72 | 0.73 | 0.73
Table 3: Quantitative results for all four regimes for the task of 3D point
cloud classification using transfer learning on ModelNet10 and ModelNet40
dataset [14].

It can be easily argued that the tree-GAN [10] decoder itself is enough for
point cloud processing at hand. However, to establish the need and examine the
strength of the proposed encoder, we perform an additional experiment by
replacing it with the PointNet [6] encoder to train the complete network on
ShapeNetBenchmarkV0 dataset [22]. We observed that the average CD is $8.65$
with the PointNet encoder and $1.21$ with the proposed encoder. This clearly
establishes the efficacy of the proposed encoder.

###  4.5 Applications

We showcase two potential applications of our proposed method: 3D point cloud
completion and single image based 3D reconstruction.

####  4.5.1 3D Point Cloud Completion

The task of 3D point cloud completion is to reconstruct the incomplete input
point cloud. To achieve this, we train TreeGCN-ED on the Completion 3D
benchmark dataset [21, 22, 23] and perform a qualitative evaluation on the
officially available test set. The qualitative results are shown in Fig. 5.
Since the ground truth for the test set is not available, we only perform
qualitative analysis.

Fig. 5: Qualitative results on the test set of Completion 3D benchmark dataset
[21, 22, 23].

####  4.5.2 Single Image Based 3D Reconstruction

3D reconstruction from a single image is an ill-posed problem. This problem
arises due to the ambiguity involved in the occluded part of the object, which
is not visible in the image. We attempt to solve this problem using TreeGCN-ED
architecture. We train the TreeGCN-ED model on 16 different classes of the
ShapeNetBenchmarkV0 dataset [22] till convergence. Later, we replace the
encoder of TreeGCN-ED with a CNN based architecture to extract image features.
We freeze the trained weights of the decoder and train the image encoder
network end-to-end for 3D reconstruction. We use Chamfer Distance (CD) [13] as
the loss function. We use the synthesized images available in
ShapeNetBenchmarkV0 dataset [22] to train the single image to 3D shape
reconstruction model. The qualitative results are shown in Fig. 6.

Fig. 6: Qualitative results for single image to 3D reconstruction on the test
set of ShapeNetBenchmarkV0 dataset [22].

##  5 CONCLUSION

In this work, we propose a tree-structured graph convolution-based encoder
architecture and combine it with the decoder of tree-GAN to create a complete
tree-structured encoder-decoder model for processing 3D point cloud data. The
experimental results of our proposed architecture highlight the effectiveness
of the encoder model in learning information-rich features. We also showcase
that TreeGCN-ED can be used for the task of point cloud completion and single
image based 3D reconstruction.

##  6 Acknowledgments

This research is supported by SERB MATRICS and SERB IMPRINT-2 grants. Also, we
would like to thank Ashish Tiwari and Dhananjay Singh for their constructive
and valuable feedback.

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# Quantum anomalous Hall crystal at fractional filling of moiré superlattices

D. N. Sheng<EMAIL_ADDRESS>Department of Physics and Astronomy,
California State University Northridge, Northridge, California 91330, USA  A.
P. Reddy<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  A. Abouelkomsan
<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  E. J. Bergholtz
<EMAIL_ADDRESS>Department of Physics, Stockholm University,
AlbaNova University Center, 106 91 Stockholm, Sweden  L. Fu<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA

###### Abstract

We predict the emergence a state of matter with intertwined ferromagnetism,
charge order and topology in fractionally filled moiré superlattice bands.
Remarkably, these quantum anomalous Hall crystals exhibit a quantized integer
Hall conductance that is different than expected from the filling and Chern
number of the band. Microscopic calculations show that this phase is robustly
favored at half-filling ($\nu=1/2$) at larger twist angles of the twisted
semiconductor bilayer $t$MoTe2.

Moiré superlattices provide an exciting venue for realizing correlated and
topological quantum states due to strong interaction effect in minibands [1,
2]. As a common manifestation of electron correlation, a variety of symmetry
breaking phenomena have been observed in graphene and semiconductor moiré
materials. These include superconductivity [3], spin/orbital ferromagnetism
[4, 5] and Wigner crystallization [6, 7], which spontaneously break gauge
symmetry, flavor symmetry and lattice translation symmetry respectively.
Another exciting aspect of moiré physics lies in topological quantum
phenomena, which are characterized by quantized transport and thermodynamics
responses. These include the quantum Hall effect arising from graphene
Hofstadter band at strong magnetic field [8, 9], and the recently observed
quantum spin Hall effect in semiconductor moiré band at zero field [10, 11].

Moiré quantum matter may also exhibit intertwined symmetry breaking and
topological phenomena. One example is the quantum anomalous Hall (QAH) states
observed in graphene and semiconductor moiré superlattices exhibit integer
quantized anomalous Hall effect at zero magnetic field [12, 13, 14]. Even more
remarkably, the fractional quantum anomalous Hall (FQAH) states have recently
been observed for the first time in twisted semiconductor bilayer $t$MoTe2 and
pentalayer rhombohedral graphene/boron nitride moiré heterostructure [15, 16,
17, 18, 19].

In twisted semiconductors, the existence of QAH and FQAH states was
theoretically predicted [20, 21, 22, 23, 24] as a consequence of spin
ferromagnetism spontaneously breaking time reversal symmetry as well as
topological moiré band with spin-contrasting Chern numbers [25]. At fractional
band filling, ferromagnetism lifts the spin degeneracy, and Coulomb
interaction drives spontaneously spin polarized electrons in Chern band into a
fractional Chern insulator state—the lattice analog of the fractional quantum
Hall state [26, 27, 28, 29]. Importantly, ferromagnetism and topological order
both arise from the same electrons in moiré band, and therefore are strongly
intertwined.

In this work, we present a novel type of topological states intertwined with
both time-reversal and translation symmetry breaking in moiré superlattices.
Specifically, we uncover the emergence of integer QAH effect in a commensurate
charge density wave (CDW) state at fractional filling of the moiré unit cell.
Focusing on twisted semiconductor bilayer $t$MoTe2, by exact diagonalization
study and many-body Chern number calculation, we find a novel QAH state with
$2\times 2$ CDW at half filling ($\nu=1/2$) in a previously unexplored twist
angle range. This state features a quantized Hall conductance $\sigma_{xy}=1$
in unit of $e^{2}/h$, and is therefore distinguished from all observed QAH
states in moiré materials in that the Hall conductance differs from the Chern
number of the underlying band multiplied by the filling factor (as in FQH
states in a Landau level): $\sigma_{xy}\neq\nu C_{\rm band}$. The existence of
QAH effect in a CDW state defines a novel quantum phase of matter that
intertwines ferromagnetism, charge order and topology. Such state was recently
termed a QAH-crystal [30].

Historically, the possible coexistence of quantized Hall effect and charge
density wave was theoretically considered for two-dimensional electrons in a
strong magnetic field [31]. Recent advances on moiré materials have stimulated
interest in QAH-crystals. In magic-angle graphene aligned with boron nitride,
a variety of finite-field Chern insulators with integer $\sigma_{xy}$ were
found at fractional fillings of the $C_{\rm band}=1$ moiré band [32]. In
twisted monolayer-bilayer graphene that hosts $C_{\rm band}=\pm 2$ moiré
bands, the Hall resistance at half band filling approaches $h/e^{2}$ above a
small field of $2$ T [33]. These observations are consistent with “symmetry
breaking Chern insulators” that break lattice translation symmetry and exhibit
$\sigma_{xy}=\nu C_{\rm band}$. Following these experimental advances,
theoretical studies have explored the possibility of QAH-crystals in twisted
multilayer graphene at half-integer fillings [34, 35] as well as semiconductor
moiré materials at $\nu=2/3$ [36, 37, 30]. Despite significant progress, the
(zero-field) QAH-crystal has not yet been experimentally found in any system,
and to our knowledge, the possibility of QAH-crystals at $\nu=1/2$ with
$\sigma_{xy}\neq\nu C_{\rm band}$ has not been considered before.

Twisted bilayer semiconductors host time-reversed pairs of topological moiré
bands, which feature opposite Chern numbers in opposite spin/valley sectors:
$C_{\uparrow}=-C_{\downarrow}=1$ [25]. At small twist angle, the lowest hole
band is sufficiently narrow, which favors strong interaction effect [20].
Previous theoretical studies on fractional-filling states have focused on
$\theta<4^{\circ}$ where FQAH states have been experimentally observed [38,
39, 40, 41, 37, 42, 43, 44].

As a function of twist angle, the dispersion and wavefunction of low-energy
moiré bands change considerably. At small twist angles, the density of states
is mostly concentrated at MX and XM stacking sites of each moiré unit cell,
which collectively form a honeycomb lattice, thus leading to an effective
Kane-Mele tight binding model description [25]. At larger twist angles, the
local density of states is more uniform, but peaked at the triangular lattice
of MM stacking sites where interlayer tunneling is strongest [38, 45].
Moreover, the bandwidth changes non-monotonously on the twist angle: it
reaches a minimum at a magic angle and increases away from it [20, 46]. The
twist-angle-dependent character of the lowest band’s wavefunction and
dispersion play an important role in the competition between crystalline and
liquid states at fractional fillings such as $\nu=1/3$ and $2/3$ [38, 43]. In
the following we shall explore their consequence at $\nu=1/2$.

Figure 1: The spin gap $\Delta_{s}$ for $N_{s}=16$ system as a function of the
spin flip $\Delta S_{z}=S_{zmax}-S_{z}$ for twist angles $\theta=4.0$ and 4.5.
In the inset, we show $\Delta_{s}$ for $N_{s}=16,20$ and $24$ with
$\theta=4.5$ at $\Delta S_{z}=1$.

Ferromagentism Using exact diagonalization method, we first study $t$MoTe2 at
the filling of $\nu=1/2$ holes per moiré unit cell. We consider a finite
system with lattice vectors ${\bf L}_{1},{\bf L}_{2}$, with the number of
Moirè unit cells $N_{1},N_{2}$ along these two directions, respectively. We
mainly focus on twist angle $\theta\gtrsim 4^{\circ}$ where the lowest hole
band is more dispersive and its Berry curvature is less uniform [38]. These
features motivate us to explore the possibility of new quantum states beyond
those of the lowest Landau level. Since the numbers of spin-$\uparrow$ and
$\downarrow$ holes—denoted as $N_{\uparrow}$ and $N_{\downarrow}$
respectively—are separately conserved, we perform ED calculation for all spin
configurations $(N_{\uparrow},N_{\downarrow})$ with hole number
$N_{h}=N_{\uparrow}+N_{\downarrow}=\nu N_{s}$, where $N_{s}=N_{1}\times N_{2}$
is the number of unit cells in the system.
$S_{z}=(N_{\uparrow}-N_{\downarrow})/2$ denotes the total spin $S_{z}$. Our ED
calculation is carried out within the Hilbert space of the time-reversed pair
of lowest band.

Using the continuum model Hamiltonian in Ref.[38] with parameters for
$t$MoTe2, we compare energies of lowest states in different spin sectors and
determine the ground state spin polarization. The spin gap
$\Delta_{s}=E_{min}(S_{z})-E_{min}(S_{zmax})$ as a function of spin-flip
$\Delta S_{z}=S_{zmax}-S_{z}$ is shown in Fig. 1 for the twist angle range of
our interest, $\theta=4^{\circ}-4.5^{\circ}$, where a clear gap is shown for
all spin sectors for $N_{s}=16$. Such a gap may extend to larger $\theta$ (see
SM). Furthermore, as show in the inset of Fig. 1, spin gaps of $N_{s}=16$, 20
and 24 site clusters are robust, and a finite-size scaling with system size
$N_{s}$ shows that the spin gap remains finite for large system limit. Thus
our results demonstrate that under Coulomb interaction with dielectric
constant $\epsilon=10$, the ground state at $\nu=1/2$ is fully spin polarized
over a wide range of twist angles, up to at least $\theta=4.5^{\circ}$. Since
the underlying band structure is strongly spin-dependent due to spin-valley
locking, the system lacks $SU(2)$ spin-rotational symmetry, thus leading to a
finite spin gap in the ferromagnetic ground state [22, 38, 42]. In the rest of
this work, we perform ED calculations within the Hilbert space of
spin-$\uparrow$ holes in the lowest Chern band, which allows us to access much
larger system sizes up to 32 sites.

Charge density wave We explore the evolution of the ground state and low-lying
states of $t$MoTe2 at $\nu=1/2$ with the twist angle. Consistent with previous
studies [40, 41], the ground state is a composite Fermi liquid (CFL) for a
wide range of twist angles. Fig. 2 shows the many-body energy spectra as a
function of the momentum ${\bf k}=K_{1}{\bf T}_{1}+K_{2}{\bf T}_{2}$ with
${\bf T}_{1},{\bf T}_{2}$ as unit vectors of crystal momentum for
$\theta=3.5$, 4.0 and 4.5 with $N_{s}=28$ cluster and $\epsilon=10$. We find a
pair of quasi-degenerate ground states with center-of-mass momentum at $M$,
similar to the energy spectrum of the CFL state in half-filled lowest Landau
level at $\theta=3.5$. As the twist angle further increases, however, a level
crossing between CFL ground states and higher-energy states occurs around
$\theta=4.0$, indicating a quantum phase transition into a new phase at larger
twist angle.

Figure 2: Energy spectra evolution from CFL to candidate Hall crystal states
for (a) $\theta=3.5$, (b) $\theta=4.0$, (c) $\theta=4.5$ with $N_{s}=28$ and
(d) $\theta=4.5$, $N_{s}=32$ at filling number $\nu=1/2$.

At $\theta=4.5^{\circ}$, on both $N_{s}=28$ and $32$ clusters, we find four
nearly-degenerate ground states with center-of-mass momentum at $\Gamma$ and
three $M$ points, which are separated by an energy gap from higher-energy
states. To understand the nature of this ground state, we study its projected
density structure factors

$\displaystyle\begin{split}S(\bm{q})=\frac{\langle{\bar{\rho}}(-\bm{q}){\bar{\rho}}(\bm{q})\rangle}{N_{s}}-\frac{\langle{\bar{\rho}}(0)\rangle^{2}}{N_{s}}\delta_{\bm{q},\bm{0}}\end{split}$
(1)

and the pair correlation functions

$\displaystyle g(\bm{r},\bm{r}^{\prime})=\frac{\langle
n(\bm{r})n(\bm{r}^{\prime})\rangle-\delta(\bm{r}-\bm{r}^{\prime})\langle
n(\bm{r})\rangle}{(N_{h}/A)^{2}}$ (2)

Here ${\bar{\rho}}_{\bm{q}}=P\sum_{i}e^{-i\bm{q}\cdot\bm{r}_{i}}P$ is the
projected density fluctuation operator where $P$ is a projector onto the first
moiré hole miniband. The local density operator is
$n(\bm{r})=\sum_{l}\psi^{{\dagger}}_{l}(\bm{r})\psi_{l}(\bm{r})$ where
$\psi^{{\dagger}}_{l}(\bm{r})$ creates a hole at position $\bm{r}$ in layer
$l$. $A$ is the area of the system. We also take an average over Moiré unit
cell so that $g(\bm{r}-\bm{r^{\prime}})$ only depends on
$\bm{r}-\bm{r^{\prime}}$.

As shown in Fig. 3, $S({\bm{q}})$ shows prominent peaks at three $M$ points
and the peak intensity increases with the system size from $N_{s}=28$ to 32,
which indicates the emergence of CDW order in the thermodynamic limit. To
illustrate the real space order, we show pair correlation function $g(\bm{r})$
(see Fig. 3), which demonstrates a $2\times 2$ CDW consistent with the three
$M$-point peaks in $S({\bm{q}})$. Strong peaks in $g(\bm{r})$ are found on
$1/4$ of MM sites on the moiré superlattice, where the local density of states
of the moiré band is high. Importantly, the ordered $2\times 2$ structure in
the pair correlation extends throughout the entire system of $N_{s}=32$
sites—the largest size accessible in our ED study, and similar results are
found for $N_{s}=24$ and $28$. Notably, even though the 32 site cluster
geometry is not threefold symmetric as the 28 site cluster is, the pair
correlation function here still shows the $2\times 2$ CDW pattern. This
strongly suggests that the CDW state at $\nu=1/2$ has a bidirectional $2\times
2$ ordered structure, rather than a unidirectional $2\times 1$ stripe. By
performing similar calculations for various twist angles, we find that for
$\epsilon=10$, the $2\times 2$ CDW state at $\nu=1/2$ exists for twist angles
in the range $4.0\lesssim\theta\lesssim 5.0$.

Figure 3: (a) Density structure factors for $N_{s}=28$ and 32; (b) Pair
correlation function for $N_{s}=32$ at filling number $\nu=1/2$ at twist angle
$\theta=4.5$.

Many-body Chern number To probe the topological order of this CDW state, we
further calculate the many-body Chern number as an integral invariant of many-
body wavefunction over twist boundary condition. The Chern number is defined
as,

$\mathcal{C}={i\over 2\pi}\int
d\phi_{1}d\phi_{2}\\{\langle{\partial\psi\over\partial\phi_{1}}|{\partial\psi\over\partial\phi_{2}}\rangle-c.c.\\},$
(3)

where $|\psi\rangle$ is the many-body state and the integral is over the
$2\pi\times 2\pi$ boundary phase space with twist boundary phases $\phi_{1}$
and $\phi_{2}$ along lattice vectors $\bm{L}_{1}$ and $\bm{L}_{2}$ directions,
playing the role of magnetic fluxes. The twist angles shift the kinetic
momentum of each particle by
$\frac{\phi_{1}}{2\pi}{\bm{T}_{1}}+\frac{\phi_{2}}{2\pi}{\bm{T}_{2}}$. We
discretize the boundary phase space into at least $12\times 12$ square meshes
and numerically obtain total Berry phase summing over each square. We
calculate the consecutive wave function overlaps
$\langle\psi(\phi_{1},\phi_{2})|\psi(\phi^{\prime}_{1},\phi^{\prime}_{2})\rangle$
around each square, which gives the Berry phase from both the single particle
basis states of the Moirè band and the many-body contribution (see SM for more
details).

For CFL in Landau levels at $\nu=1/2$, many-body states carry fluctuating
Chern numbers, with two degenerating states sharing a quantized Chern number
$C=1$. This pair of ground states are connected by inserting one flux quantum.
In our system at $\nu=1/2$, these two states remain in the same momentum
sector due to the commensurate condition between $N_{h}$ and $N_{1},N_{2}$ for
our clusters $N_{s}=24-32$. Thus, we identify the Chern numbers for a wide
range of twist angle for each member of nearly-degenerate ground states, as
well as the first excited state in the same momentum sector to detect
different characterization of topological nature for both CFL and crystal
state. As shown in Fig. 4 for a system of $N_{s}=28$ sites, we demonstrate a
topological quantum phase transition between two phases with different Chern
numbers $C_{ave}=1/2$, and $1$ distinguishing the CFL phase from a QAH phase
with $\sigma_{xy}=1$ at larger $\theta\gtrsim 4.0$ side. The transition
between these two phases is accompanied by a much enhanced fluctuating Chern
numbers around the twist angle $\theta_{c}\sim 3.7-4.0$. We also note that the
$C_{ave}=1/2$ for CFL is geometry dependent, and more state average may be
required for other less symmetric clusters.

Remarkably, this topological phase transition from a compressible CFL phase to
an incompressible QAH phase coincides with the symmetry breaking transition,
i.e., the emergence of $2\times 2$ CDW order as seen from the structure factor
and pair correlation. We have also found that, for the QAH crystal phase, both
the ground state and the 1st excited state have the same integer quantized
$C=1$ demonstrating the robust of the topological quantization. Near the phase
boundary at $\theta=4.0$, the ground state already has $C=1$ (see additional
results for $N_{s}=24-32$ in the SM). Combining all these findings, we
conclude that at larger twist angle, the ground state of $t$MoTe2 at $\nu=1/2$
is a QAH-crystal that exhibits spontaneous ferromagnetism, $2\times 2$ CDW
order, and quantized anomalous Hall effect $\sigma_{xy}=1$. This state
exemplifies a topological quantum phase possessing multiple symmetry-breaking
order parameters.

Figure 4: Chern numbers averaged over each member of nearly-degenerate ground
states, as well as the first excited state in the same momentum sector, versus
$\theta$ for $N_{s}=28$ system at filling $\nu=1/2$. The $C_{ave}$ for
$\Gamma$ point is only included at $\theta>3.5$, where the energy of the state
in this sector drops and becomes one of the ground state at larger $\theta$.

The existence of a $2\times 2$ CDW at $\nu=1/2$ with a quantized anomalous
Hall conductance $\sigma_{xy}=1$ is unexpected and seems counter-intuitive.
For comparison, we note that CDW states were previously found at $\nu=1/2$ in
semiconductor moiré superlattices with topologically trivial bands [6, 7].
However, those CDW states are generalized Wigner crystals with $2\times 1$
stripe order, which host one charge per enlarged unit cell. Such Wigner
crystals arise from strong Coulomb interaction between charges in a triangular
lattice and are insulating states with zero Hall conductivity. Our QAH-crystal
state in half-filled topological band of $t$MoTe2 is clearly different.

Figure 5: The momentum space occupation number obtained from exact
diagonalization for $N_{s}=28$ at $\theta=4.5$. (b) The energy band splitting
and Chern numbers in the phenomenological model with weak potential
$V_{0}=0.1$.

To better understand the origin of the QAH-crystal state, we calculate its
momentum space occupation number $n({\mathbf{k}})$ at $\theta=4.5^{\circ}$,
shown in Fig. 5(a) for $N_{s}=28$ site cluster. Notably, $n({\mathbf{k}})$ is
highly nonuniform: it is close to $0$ near $\gamma$ (hole band maxima) and
close to $1$ near $\kappa,\kappa^{\prime}$ (hole band mimima). This behavior
contrasts with the nearly uniform momentum distribution previously found in
the CFL state at smaller twist angles [40], but shares more similarity with
$n({\mathbf{k}})$ of the noninteracting metal at $\nu=1/2$. This motivates us
to introduce a phenomenological “mean-field” description for the $2\times 2$
CDW state. Our mean-field Hamiltonian consists of the non-interacting
continuum model of t$\text{MoTe}_{2}$ that describes the kinetic energy of
holes, as well as an effective potential with $2\times 2$ periodicity that
represents the CDW order parameter: $V_{2\times
2}(\mathbf{r})=-V_{0}\sum_{i=1}^{6}e^{i\mathbf{M}_{i}\cdot\mathbf{r}}$ where
$\mathbf{M}_{i}$ are the three $M$ points. Due to the enlarged real-space
periodicity, the original hole band folds onto four minibands. We find that a
small $V_{0}$ is sufficient to induce CDW gaps, resultin in an insulating
ground state at $\nu=1/2$. The CDW-reconstructed band structure is shown in
Fig 5(b). Moreover, we find that the lowest four minibands of holes carry
Chern numbers $C=1$ , $C=0$ , $C=-1$ and $C=1$ respectively. At half filling
$\nu=1/2$ , corresponding to completely filling the lowest two minibands, our
system with $2\times 2$ CDW order has a ground-state Chern number $C=1$, and
therefore is a QAH-crystal with $\sigma_{xy}=1$.

To summarize, our main finding is a QAH-crystal phase at $\nu=1/2$ in $t$MoTe2
at larger twist angles than hitertho studied. This state is interesting for a
number of reasons. First, it exhibits a quantized Hall conductance that is
unexpected from the filling factor [47]: $\sigma_{xy}\neq\nu C_{\rm band}$,
unlike all observed QAH states. Second, it exhibits a $2\times 2$
bidirectional CDW order rather than the $2\times 1$ stripe order observed or
proposed in other moiré materials [7, 6, 33]. Third, as a function of twist
angle, there appears to be a direct transition between CFL and QAH-crystal
(although the possibility of a narrow intermediate phase cannot be excluded).

Our work raises new questions for future investigation. While our study is
focused on $t$MoTe2 at zero displacement field, it will be interesting to
explore the possibility of inducing crystal and QAH-crystal states by applying
a displacement field. The nature of the CFL to QAH-crystal phase transition
calls for theoretical understanding. Last but not the least, other physical
realizations of QAH- and FQAH-crystals in moiré superlattices [30, 48, 49,
50], as well as the theoretical possibility of QAH-crystals without lattice
effect [51], remain to explored.

###### Acknowledgements.

Acknowledgments.— We thank Xiaodong Xu for insightful discussions. D.N.S. was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences
under Grant No. DE-FG02-06ER46305. The work at Massachusetts Institute of
Technology was supported by the Air Force Office of Scientific Research
(AFOSR) under Award No. FA9550-22-1-0432 and benefited from computing
resources provided by the MIT SuperCloud and Lincoln Laboratory Supercomputing
Center. A.A was supported by the Knut and Alice Wallenberg Foundation (KAW
2022.0348). E.J.B. was supported by the Swedish Research Council (grant
2018-00313), Knut and Alice Wallenberg Foundation (2018.0460) and the Göran
Gustafsson Foundation for Research in Natural Sciences and Medicine. L. F. was
partly supported by the Simons Investigator Award from the Simons Foundation.

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