# Defect Classification in Additive Manufacturing Using CNN-Based Vision
Processing

Xiao Liu  Dublin City University, Glasnevin, Dublin 9, Ireland
<EMAIL_ADDRESS>Alessandra Mileo and Alan F. Smeaton  Dublin City
University, Glasnevin, Dublin 9, Ireland
<EMAIL_ADDRESS>
###### Abstract

The development of computer vision and in-situ monitoring using visual sensors
allows the collection of large datasets from the additive manufacturing (AM)
process. Such datasets could be used with machine learning techniques to
improve the quality of AM. This paper examines two scenarios: first, using
convolutional neural networks (CNNs) to accurately classify defects in an
image dataset from AM and second, applying active learning techniques to the
developed classification model. This allows the construction of a human-in-
the-loop mechanism to reduce the size of the data required to train and
generate training data.

Keywords: Convolutional neural networks, additive manufacturing, defect
classification, active learning

##  1 Introduction

Large and openly available datasets of annotated images containing up to
millions of training examples such as Pascal VOC [Everingham et al., 2010] are
available to machine learning researchers for many applications. This has
enabled huge improvements in machine learning over recent years. By contrast,
such openly available datasets are not available in the domain of Additive
Manufacturing (AM) or 3D printing because labelled samples are difficult,
expensive, and time-consuming to obtain as shown in [Qin et al., 2022] and
[Wang et al., 2020]. As a result of poor data availability, researches in AM
often have to use only a limited amount of labelled samples for training tasks
before then leveraging a large number of unlabelled image data. Some
researchers have called this the “small data challenge in the big data era”
[Qi and Luo, 2020].

To overcome this challenge, we present a method that applies transfer learning
and fine-tuning on a CNN-based neural network model to achieve accurate
classification of manufacturing defects. This uses a dataset of images of the
melt pool, created from the interaction between a laser and the materials used
in manufacturing, taken during the AM process. Structural defects in the
resulting output can sometimes be detected during manufacture from
observations of the melt pool. Our technique involves using active learning
algorithms to reduce the number of labelled samples required in the training
process. We perform automatic labelling using the model to generate larger
datasets of labelled images from unlabelled samples, for use in training.

##  2 Methods

Transfer learning is a method that performs training a neural network model
using data from a source domain then later applying the trained model to a
target domain that is different from the source. This allows rapid progress in
re-training and significantly reduces the required number of training samples
in the target domain. It is commonly used in computer vision tasks such as
classification to support improved performance in domains which are data-poor.
In recent years, transfer learning has proved to be effective in the task of
defect classification in AM, such as the work presented in [Liu and Mileo,
2021] and [Westphal and Seitz, 2021] where transfer learning and fine-tuning
were applied to the training of CNN based neural network architectures.

Active learning [Settles, 2009] is a technique for labelling data that selects
and prioritises the most informative data points to submit to an annotator for
labelling. Such prioritised data points have the highest potential impact on
the supervised training of a machine learning model, thus accelerating the
training process. The combination of transfer learning and active learning
allows leveraging small amounts of labelled data to improve the performance of
the training process of a deep learning model.

##  3 Classification Experiments in Additive Manufacturing

To investigate the potential for transfer learning and active learning in the
task of defect detection in AM, a case study was carried out using the open
image dataset from [Westphal and Seitz, 2021]. This contains 4,000 images,
manually divided into 2 different defect detection classes in AM. The images
in this dataset are clearly separated into 3 balanced subsets for training
(2,000), testing (1,000) and validation (1,000).

To conduct experiments, we employed a VGG16 based classifier from previous
work which proved to be accurate in the task of defect classification on
images generated from emission monitoring during additive manufacturing [Liu
et al., 2022]. This classifier relies on transfer learning in which 13
convolutional layers from a pre-trained VGG16 model are used for feature
extraction and the weights in these layers had been trained using ImageNet
data. After the convolutional layers, 2 dense layers with ReLU activation
function are added followed by 1 dense layer as the output layer using Sigmoid
as the activation function, since the targeted dataset are divided into 2
classes for binary classification. In the original paper [Westphal and Seitz,
2021], the best classification performance is generated using a VGG16 based
CNN model which is the reason we do not use a more recent model such as
ResNet. We consider that as a baseline for further investigation in this
study.

The tuning of hyperparameters involves adjusting the optimiser, learning rate,
batch size and training epochs. There are 3 optimisers in the test we use
which are Adam, SGD and RMSprop in combination with learning rate in a range
from 10-2 to 10-5. We have also conducted training using different batch sizes
(4, 32, 64) and training epochs (30, 60, 120). The cost function used in all
tests is binary cross entropy. To reduce overfitting, weight regularisers are
added to the 2 dense layers with the ReLU activation function mentioned above.
The weight decay regulariser, also known as L2 regulariser which calculates
the sum of the squared weights, is applied when initialising the keras model.
The tuning of this hyperparameter is in a range from 10-1 to 10-4 and tested
for multiple times until no obvious overfitting issue appears in the training
and validation.

After tuning on hyperparameters for multiple combinations, the best preforming
combinations regarding the 3 types of optimisers are shown in Table 1 together
with classification results on the validation dataset in comparison with the
baseline from [Westphal and Seitz, 2021]. These initial tests were performed
to check how adaptive our approach is on this dataset. The results show that
all 3 optimisers can reach a value around 98% of the validation accuracy and
our classification model is well-adapted to this dataset. The results also
show that for this dataset a smaller batch size used in the training process
such as 4, gives better performance and this can be explained as smaller batch
sizes require more frequent weight updates during training. In turn this can
help the model adjust its parameters more quickly and respond to changes in
the data distribution which increases the model’s ability to adapt to a new
dataset. Finally, although not shown here, accuracy is stable thoughout the
training showing no overfitting.

Table 1: Best performing hyperparameters for each optimiser, performance
results on the validation set. Results marked ‘*’ are updates provided
directly to us by the authors of [Westphal and Seitz, 2021] in response to us
pointing out errors in the original paper. An author correction to the copy of
record is now underway.

| Experiment:
---
Optimiser, learning rate
Batch | Epochs |  | Confusion
---
matrix
Accuracy | Precision | Recall | F1-Score | AUC
Baseline | 64 | 30 | 496 | 4* | 0.977* | 0.992* | 0.963* | 0.977* | 0.993
19 | 481
SGD, lr=0.01 | 4 | 60 | 483 | 17 | 0.979 | 0.967 | 0.992 | 0.979 | 0.998
4 | 496
Adam, lr =0.00001 | 4 | 120 | 490 | 10 | 0.988 | 0.980 | 0.996 | 0.988 | 0.998
2 | 498
RMSprop lr =0.00001 | 4 | 60 | 485 | 15 | 0.982 | 0.971 | 0.994 | 0.982 | 0.997
3 | 497

##  4 Active Learning Experiments in Additive Manufacturing

Having developed a classifier which uses domain transfer across AM image
datasets, we extended training to include active learning applied to further
investigate classification performance during the progression of AL
iterations. The second experiment was performed in a series of steps of (1)
active sample section, (2) query for label, (3) train with queried sample, and
(4) validate for current query iteration. The cycle iterates until a human
supervisor decides to complete the training phase when validation accuracy
achieves a target level.

Here we apply a pool-based sampling scenario and an uncertainty sampling query
strategy [Settles, 2009]. This is the most commonly used query strategy to
start generalised sampling on this particular AM dataset. The implementation
of active learning uses Python 3 and Google Colab. During the experiment, a
classifier model is initialised and the optimser chosen is SGD as we found
this gives more stabilised performance in the validation test and has minimal
overfitting even when training is continued long after convergence. While Adam
and RMSprop converge faster, there are larger fluctuations in the validation
and minor overfitting after training reaches convergence. In addition, though
SGD yields a result lower than the other 2 optimisers, it has slightly better
potential that can be improved by applying active learning. During this
experiment, 2,000 training samples were fed to the classifier with a total of
40 queries and for each query 50 samples were actively selected by the
uncertainty sampling query strategy.

The selected and queried samples were assigned a label by an annotator after
which the newly labelled samples were used to fine-tune the classifier to
improve performance. This was evaluated using classification accuracy on the
validation dataset at the end of each query iteration and later we show
results on the test set.

Following the inclusion of active learning, validation accuracy in each query
iteration is shown in Figure 1 where results show that with the aid of active
learning, the model reaches convergence after the 13th query and the value of
validation accuracy is around 98%. More specifically, the calculated mean
value from the 13th to 40th queries is 0.981 with a standard deviation of
0.0246 and a peak of 0.990. This is slightly higher than the result of the SGD
optimiser based model shown in Table 1 and 1% higher than the baseline.
Overall performance after convergence is also relatively stable. Results also
show that the model only needs the first 650 most informative samples to
achieve the best performance which is only 37.5% of the total 2,000 labelled
training data.

Figure 1: classification accuracy on the validation dataset at the end of each
query iteration

This trained model was used to classify the labels on the testing dataset
mentioned in Section 3, which is a balanced dataset consisting of 1,000
samples and the results are shown in Table 2.

Table 2: Predicted results for auto-labeling on the test dataset with P, R and F1 calculated for each of 2 classes | Confusion
---
matrix
Precision | Recall | F1-Score | AUC | Accuracy
487 | 13 | 0.994 | 0.974 | 0.984 | 0.998 | 0.984
3 | 497 | 0.975 | 0.995 | 0.984

##  5 Conclusions

This paper presents an investigation into performance of a computer vision
based classification task on a dataset from the additive manufacturing
process. We use a CNN based classifier in combination with transfer learning
and active learning strategies. We improved the overall validation accuracy to
about 98%. We also conducted experiments to investigate the approximate
minimum number of labelled samples needed to reach convergence in training. In
future work we plan to further investigate the sampling strategies for active
learning especially regarding class imbalance problems. We will involve
approaches from semi-supervised learning to reinforce the labelling and self
training as an extension to the current active learning mechanism.

## References

  * [Everingham et al., 2010] Everingham, M., Van Gool, L., Williams, C. K., Winn, J., and Zisserman, A. (2010).  The Pascal visual object classes (VOC) challenge.  International Journal of Computer Vision, 88:303–338.
  * [Liu and Mileo, 2021] Liu, X. and Mileo, A. (2021).  A deep learning approach to defect detection in additive manufacturing of titanium alloys.  In 2021 International Solid Freeform Fabrication Symposium. University of Texas at Austin.
  * [Liu et al., 2022] Liu, X., Smeaton, A. F., and Mileo, A. (2022).  An adaptive human-in-the-loop approach to emission detection of additive manufacturing processes and active learning with computer vision.  In 2022 IEEE International Conference on Big Data (Big Data), pages 4021–4027. IEEE Computer Society.
  * [Qi and Luo, 2020] Qi, G.-J. and Luo, J. (2020).  Small data challenges in big data era: A survey of recent progress on unsupervised and semi-supervised methods.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(4):2168–2187.
  * [Qin et al., 2022] Qin, J., Hu, F., Liu, Y., Witherell, P., Wang, C. C., Rosen, D. W., Simpson, T. W., Lu, Y., and Tang, Q. (2022).  Research and application of machine learning for additive manufacturing.  Additive Manufacturing, 52:102691.
  * [Settles, 2009] Settles, B. (2009).  Active learning literature survey.  Computer Sciences Technical Report 1648, University of Wisconsin–Madison.
  * [Wang et al., 2020] Wang, C., Tan, X., Tor, S., and Lim, C. (2020).  Machine learning in additive manufacturing: State-of-the-art and perspectives.  Additive Manufacturing, 36:101538.
  * [Westphal and Seitz, 2021] Westphal, E. and Seitz, H. (2021).  A machine learning method for defect detection and visualization in selective laser sintering based on convolutional neural networks.  Additive Manufacturing, 41:101965.


# Enhanced superconductivity and various edge modes in modulated $t$-$J$
chains

Yong-Feng Yang  School of Physical Science and Technology $\&$ Lanzhou Center
for Theoretical Physics $\&$ Key Laboratory of Theoretical Physics of Gansu
Province, Lanzhou University, Lanzhou, Gansu 730000, China  Jing Chen  Center
for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USA
Chen Cheng<EMAIL_ADDRESS>School of Physical Science and Technology
$\&$ Lanzhou Center for Theoretical Physics $\&$ Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
Beijing Computational Science Research Center, Beijing 100094, China  Hong-
Gang Luo<EMAIL_ADDRESS>School of Physical Science and Technology $\&$
Lanzhou Center for Theoretical Physics $\&$ Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
Beijing Computational Science Research Center, Beijing 100094, China

###### Abstract

We numerically investigate the ground state of the extended $t$-$J$
Hamiltonian with periodic local modulations in one dimension by using the
density-matrix renormalization group method. Examining charge and spin
excitation gaps, as well as the pair binding energy, with extrapolated results
to the thermodynamic limit, we obtain a rich ground-state phase diagram
consisting of the metallic state, the superconducting state, the phase
separation, and insulating states at commensurate fillings. Compared to the
homogeneous 1D $t$-$J$ model, the superconductivity is greatly enhanced and
stabilized by the flat-band structure. This superconducting state in quasi-
periodic chains shares similar properties with ladder systems: significant
negative pair binding energy occurs, and the singlet pairing correlation
function dominates with the algebraic decay while the single-particle Green’s
function and spin correlation function decay exponentially. On the other hand,
quasi-periodicity leads to nontrivial topological nature in insulating states,
characterized by different integer Chern numbers at different fillings. Due to
the interplay among the topology, the interaction, and the 1D confinement,
gapless edge modes show strong spin-charge separation and in different regions
can relate to different collective modes, which are the charge of a single
fermion, the magnon, and the singlet-pair. We also find two interaction driven
topological transitions: i) at particle filling $\rho=1/2$, the low-energy
edge excitations change from the magnon to singlet-pair, accompanied with pair
formation in bulk; and ii) at $\rho=3/4$, while the gapless edge mode remains
the charge of a single fermion, there is a gap-closing point and a $\pi$-phase
shift in the quasi-particle spectrum.

##  I Introduction

The $t$-$J$ Hamiltonian Zhang and Rice (1988) is one of the most important
theoretical models in strongly correlated systems, especially for its role as
a canonical model for studying high-temperature superconductors Dagotto
(1994); Keimer  _et al._ (2015). While the initial interest of the model
resides in its two-dimensional (2D) realization, the one-dimensional (1D)
$t$-$J$ model attracts many investigations as it shares some signatures with
the 2D case like the spin-gap and the superconducting phase Ogata and Fukuyama
(2008); Moreno  _et al._ (2011). However, the superconducting state in the 1D
chain supports a very weak spin gap and pair binding, which vanishes for
physically relevant values of $J$ in real materials Moreno  _et al._ (2011);
Zhu  _et al._ (2014). Compared to the pure 1D case, in a higher dimension,
even-leg $t$-$J$ ladders show stronger evidence of superconductivity with
significant binding energy Dagotto  _et al._ (1992); Hayward  _et al._ (1995);
Hayward and Poilblanc (1996); Poilblanc  _et al._ (2003); White and Scalapino
(1997); Zhu  _et al._ (2014) and have been extensively studied Cheng  _et al._
(2018); Jiang and Kivelson (2021); Gong  _et al._ (2021); Jiang  _et al._
(2021). In a recent work Reja  _et al._ (2016), the substantial negative
binding energy has been demonstrated in 1D coupled $t$-$J$ segments for
physically relevant values of exchange $J$ and hole doping. This work provides
new possibilities in superconducting material design and raises whether the
enhanced superconductivity is pervasive for general quasi-periodic systems.

The 1D quasi-periodic superlattice has been attracting continuous
investigations in the last decade, partially because of its nontrivial
topological nature Lang  _et al._ (2012); Kraus  _et al._ (2012); Xu  _et al._
(2013); Hu  _et al._ (2016); Zhu  _et al._ (2013); Kuno  _et al._ (2017);
Stenzel  _et al._ (2019). In a noninteracting setting, the essential physics
behind it is well understood, that these 1D quasi-periodic models can be
mapped to the 2D Harper-Hofstadter model Harper (1955); Hofstadter (1976)
which is used to describe the quantum Hall effect von Klitzing (2017). In the
presence of interactions, there occur more unique phenomena such as fractional
topological insulators Xu  _et al._ (2013); Hu  _et al._ (2016) and
topological Mott insulators Zhu  _et al._ (2013); Kuno  _et al._ (2017);
Stenzel  _et al._ (2019). Moreover, in the spinful fermionic Hubbard chain,
the spin-charge separation due to the confined geometry can also affect
topological properties, leading to the phase transition with the low-energy
edge excitations change from spin-1/2 fermionic single-particle modes to
spin-1 bosonic collective modes at specific fillings Hu  _et al._ (2019).

The quasi-periodic $t$-$J$ Hamiltonian in 1D superlattices has plentiful
elements supporting pregnant physics, which has not been systematically
explored. The interplay among topology, interaction, spin-charge separation,
and pairing would bring interesting phenomena in perspectives from both bulk
and edge properties, such as the enhanced superconductivity and various
topological edge modes. In parallel, the development of ultracold gases in
optical lattices within the last two decades provides a new way to explore the
physical phenomena described by theoretical model Hamiltonians Bloch  _et al._
(2008); Esslinger (2010); Ni  _et al._ (2008); Park  _et al._ (2015); De Marco
_et al._ (2019), including quasi-periodic models Tai  _et al._ (2017) and
extended $t$-$J$ models Gorshkov  _et al._ (2011a, b). The experimental
setting with remarkable control and tunability has also brought more
flexibilities to the theoretical model, which can be studied with sufficient
motivation in a broader range of parameters and in different extended forms
Gorshkov  _et al._ (2011a, b); Cheng  _et al._ (2015); Fazzini  _et al._
(2019). The quasi-periodic $t$-$J$ Hamiltonian, which has not yet been
explicitly proposed in optical lattices, has the potential to be realized in
future experiments Jepsen  _et al._ (2020). Motivated by various
possibilities, in this work, we systematically investigate the $t$-$J$ model
in 1D superlattices with quasi-periodic modulations in a wide range of
parameters. We aim to draw the ground-state phase diagram of the modulated
$t$-$J$ chain and unveil its nontrivial topological nature from numerically
exact results.

The remainder of the paper is organized as follows. In Sec. II, we introduce
the quasi-periodic $t$-$J$ model, the numerical method, and the way to
characterize the bulk and topological properties. In Sec. III, we summarize
our main results by the ground-state phase diagram and introduce the bulk
state properties, especially the enhanced superconductivity. In Sec. IV we
focus on the topological nature of insulating states and demonstrate various
gapless edge modes corresponding to different quasi-particles. Finally, the
summary and discussion are made in Sec. V.

##  II Model and Methodology

We investigate the extended $t$-$J$ model with site-dependent couplings and
interactions,

$\displaystyle{\cal H}=$
$\displaystyle\sum_{i\sigma}t_{i}\left(\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{i+1,\sigma}+\mathrm{H.c.}\right)$
$\displaystyle+\sum_{i}J_{i}\left(\vec{S}_{i}\cdot\vec{S}_{i+1}-\frac{1}{4}\hat{n}_{i}\hat{n}_{i+1}\right),$
(1)

where $\hat{c}_{i,\sigma}^{\dagger}$ ($\hat{c}_{i,\sigma}$) creates
(annihilates) a fermion with spin $\sigma=\uparrow,\downarrow$ at site $i$.
Here
$\vec{S}_{i}=\frac{1}{2}\sum_{\alpha\beta}\hat{c}_{i,\alpha}^{\dagger}\vec{\sigma}_{\alpha,\beta}\hat{c}_{i,\beta}$
is the spin-1/2 operator with Pauli matrices $\vec{\sigma}$, and
$\hat{n}_{i}=\sum_{\sigma}\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{i,\sigma}$ is
the density operator. The local modulation is introduced by the site-dependent
offset $\delta t_{i}$ on the hopping for the local site $i$ as

$\displaystyle\delta t_{i}$ $\displaystyle=\lambda\cos\left(2\pi
i/p+\phi\right).$ (2)

Here $\lambda$ is the amplitude of the modulation, $p$ is the size of the unit
cell, and $\phi$ is the phase factor. These kinds of modulations are very
common in researching topological insulators in 1D optical superlatticesLang
_et al._ (2012); Hu  _et al._ (2014); Guo (2016); Stenzel  _et al._ (2019); Hu
_et al._ (2019). Taking into account the $t$-$J$ model is derived from the
Hubbard model in the large $U$ limit, we restrict values of parameters to keep
$J=4t^{2}/U$ everywhere in real space and therefore set local parameters as

$\displaystyle t_{i}$ $\displaystyle=-t+\delta t_{i},$ $\displaystyle J_{i}$
$\displaystyle=Jt^{2}_{i}/t^{2}.$ (3)

To avoid possible ground-state localization Aubry and Andre (1980); Sokoloff
(1980), we consider the commensurate lattice with integer $p$. Specifically,
we focus on $p=4$ in this work, and other finite even values behave similar
phenomena. For convenience, we set $t=1$ as the energy unit and fixed the
modulation amplitude $\lambda=0.2$. The model is then systematically
investigated at different particle fillings $\rho$ and exchange interaction
$J$.

The system described by Eq. (1) has $U(1)$ symmetry with conserved total
particle number $N_{\sigma}=\sum_{i}\hat{n}_{i,\sigma}$ for spin species
$\sigma$. In practice, we employ the density-matrix renormalization group
(DMRG) method White (1992, 1993) and numerically compute the $m$th eigenstate
$\Psi_{m}(L,N_{\uparrow},N_{\downarrow})$ of energy
$E_{m}(L,N_{\uparrow},N_{\downarrow})$ with fixed good quantum numbers. For
convenience, we adopt shortened forms $\Psi_{m}(N_{\uparrow},N_{\downarrow})$
and $E_{m}(N_{\uparrow},N_{\downarrow})$ when the information of the system
size $L$ is clear. We dynamically Legeza  _et al._ (2003) adopt DMRG many-body
states to make sure the maximum truncation error is of the order of $10^{-8}$.
The number of DMRG kept states is up to 1000 in most calculations but can be
very large in some rare cases, for example, up to 8000 in computing the first
excited state with degeneracy. Open boundary conditions (OBCs) and periodic
boundary conditions (PBCs) are used in different calculations for different
purposes. Conclusions are made according to the extrapolated results to the
thermodynamic limit.

###  II.1 Energy criteria for bulk state

We adopt several energy criteria to characterize the phase and the ground-
state phase diagram. The conducting and insulating bulk state can be
determined by the charge excitation gap

$\displaystyle\Delta E_{C}=$
$\displaystyle[E_{0}(N_{\uparrow}+1,N_{\downarrow}+1)+E_{0}(N_{\uparrow}-1,N_{\downarrow}-1)$
$\displaystyle-2E_{0}(N_{\uparrow},N_{\downarrow})]/2,$ (4)

and the zero (nonzero) charge gap indicates the continuous (discrete) charge
excitation. The spin gap is defined as the excitation energy from a singlet to
a triplet state

$\displaystyle\Delta
E_{S}=E_{0}(N_{\uparrow}+1,N_{\downarrow}-1)-E_{0}(N_{\uparrow},N_{\downarrow}),$
(5)

which distinguishes the gapped and gapless spin excitation. Another important
energetic criterion is the binding energy

$\displaystyle\Delta E_{B}=$ $\displaystyle
E_{0}(N_{\uparrow}+1,N_{\downarrow}+1)+E_{0}(N_{\uparrow},N_{\downarrow})$
$\displaystyle-2E_{0}(N_{\uparrow}+1,N_{\downarrow}),$ (6)

which compares the energy of two interacting particles (or holes, depending on
the filling) with that of two noninteracting ones, and $\Delta E_{B}<0$
indicates a tendency toward pair formation. Note that all these energy
criteria are meaningful only in the thermodynamic limit.

###  II.2 Correlation functions and structure factors

To further characterize different phases, we compute correlation functions and
correspondence structure factors. Despite this a quasi-periodic system, we
adopt correlations between sites instead of unit cells since two essentially
give the same physics Reja  _et al._ (2016) (see Appendix A for more
information). With the correlation function in a generic form $X_{ij}$ between
site $i$ and $j$, one can extract the correlation decay

$\displaystyle X(r)=\frac{1}{\cal N}\sum_{|i-j|=r}X_{ij},$ (7)

and the structure factor

$\displaystyle
X(k)=\frac{1}{L}\sum_{i,j=1}^{L}\mathrm{e}^{\mathrm{i}k(i-j)}X_{ij}.$ (8)

Here $\cal N$ is the total number of pairs $\\{i,j\\}$ satisfying $|i-j|=r$,
and $k$ is the momentum. The superconducting order can be captured by the
singlet-pairing correlation function defined as

$\displaystyle P_{ij}^{S}=\langle\Delta^{\dagger}_{i}\Delta_{j}\rangle,$ (9)

where

$\displaystyle\Delta_{i}=\frac{1}{\sqrt{2}}\left(\hat{c}_{i,\uparrow}\hat{c}_{i+1,\downarrow}-\hat{c}_{i,\downarrow}\hat{c}_{i+1,\uparrow}\right)$
(10)

is the annihilation operator for the singlet pair on two nearest sites. We are
also interested in the density-density correlation function

$\displaystyle
N_{ij}=\langle\hat{n}_{i}\hat{n}_{j}\rangle-\langle\hat{n}_{i}\rangle\langle\hat{n}_{j}\rangle,$
(11)

the spin-spin correlation function

$\displaystyle S_{ij}=\langle\hat{S}^{z}_{i}\hat{S}^{z}_{j}\rangle,$ (12)

and the single-particle Green’s function

$\displaystyle
G^{\sigma}_{ij}=\langle\hat{c}_{i,\sigma}\hat{c}^{\dagger}_{j,\sigma}\rangle.$
(13)

###  II.3 topological invariant

In the gapped state, we characterize the nontrivial topology by the Chern
number

$\displaystyle
C=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\int_{0}^{2\pi}d\theta~{}F(\phi,\theta)$
(14)

defined in the 2D parameter space ($\phi,\theta$), where
$F(\phi,\theta)=\mathrm{Im}\left(\langle\frac{\partial\Psi}{\partial\phi}|\frac{\partial\Psi}{\partial\theta}\rangle-\langle\frac{\partial\Psi}{\partial\theta}|\frac{\partial\Psi}{\partial\phi}\rangle\right)$
is the Berry curvature, and $\theta$ is the phase factor for twisted boundary
conditions Niu  _et al._ (1985); Xiao  _et al._ (2010). Specifically, we
impose twisted boundary conditions via the replacement
$\hat{c}_{j,\sigma}\rightarrow e^{\mathrm{i}\delta\theta
j}\hat{c}_{j,\sigma}$, where $\delta\theta=\theta/L$ is the phase gradient. We
compute the Chern number for insulating states using the method in Ref. Fukui
_et al._ (2005); Varney  _et al._ (2011) on a $12\times 12$ discrete grid,
using both exact wavefunctions from the exact diagonalization method for
smaller system size ($L=16$) and matrix-product-state wavefunctions from DMRG
for a larger one ($L=64$). These two methods give the same result.

##  III Ground-state phase diagram and Enhanced superconductivity

Figure 1: The ground-state phase diagram consists of the metallic phase (M),
superconducting phase (SC), phase separation (PS), and different topological
insulating regions (TI). For the topological insulating regions, the gapless
quasi-particle excitation is labeled as well. The gray region of $\rho=0$
($\rho=1$) denotes the particle empty (full) state. The phase diagram is drawn
in the $\mu$-$J$ ($\rho$-$J$) plane in the main (inset) panel. Here we use
$\phi=0$ for all calculations and results are extrapolated to the
thermodynamic limit.  Figure 2: Energy criteria for metallic, superconducting,
and insulating phases: (a) the charge gap, (b) the binding energy, and (c) the
spin gap as functions of $1/L$ for different $J$s, with filled symbols for
results in the finite systems and empty symbols for the extrapolated result in
the limit $1/L\rightarrow 0$. (d) Extrapolated results in the thermodynamic
limit as a function of $J$. Here we use data of $L=32$, $64$, $96$, and $128$
with OBCs, at the filling $\rho=1/4$. Panels (a-c) share the same legend. Here
and after, error bars for extrapolated results denote the uncertainty from the
least square polynomial fitting.

We summarize our main result by the ground-state phase diagram in Fig. 1,
which contains the metallic phase, the superconducting phase, the phase
separation, and different topological insulating states. To emphasis the
gapped phases, in the main panel, we plot the phase diagram in the grand
canonical ensemble in the presence of a uniform chemical potential $\mu$. The
ground-state energy of the grand ensemble Hamiltonian

$\displaystyle{\cal H}_{\mu}={\cal H}+\mu\sum_{i}\hat{n}_{i}$ (15)

can be easily obtained by
$E_{0}(N_{\uparrow},N_{\downarrow},\mu)=E_{0}(N_{\uparrow},N_{\downarrow})+\mu
N$, with $N=\sum_{\sigma}N_{\sigma}$ is the total number of particles.
Insulating phases ($\Delta E_{C}>0$) lie in commensurate fillings, i.e.,
$\rho=1/4$, $1/2$, and $3/4$, where the particle filling is defined as
$\rho=N/L$. Boundaries between the metallic and superconducting state can be
obtained from the binding energy $\Delta E_{B}$. The negative inverse
compressibility determines the boundary entering the phase separation (see
Appdendix B). In this work, we restrict ourselves to the case with
$N_{\uparrow}=N_{\downarrow}$ and $\rho\in(0,1)$. The system at $\rho=1$,
where the Hamiltonian in Eq. (1) degenerates to a quasi-periodic Heisenberg
chain, which also shows nontrivial topological properties and has been
investigated in previous literatures Hu  _et al._ (2014, 2015). In this
section, we focus on the bulk state characterization with $\phi=0$ in all
calculations, and the topological nature of insulating states is discussed in
Sec. IV.

More specifically, we display in Fig. 2 energy criteria for bulk phases by
taking the filling $\rho=1/4$ as an example. The finite size value of the
charge gap $\Delta E_{C}$ and its extrapolation by a second-order polynomial
fitting is shown in Fig. 2(a). In the thermodynamic limit, there exists a
finite charge gap at small $J$s, and the gap closes for larger values of $J$.
This indicates a transition from the charge insulating to the conducting
state. Similarly, the finite size extrapolation of the binding energy $\Delta
E_{B}$ and the spin gap $\Delta E_{S}$ is shown in Fig. 2(b) and (c)
respectively. By plotting these energy criteria together in Fig. 2(d), one can
tell bulk phases and transitions at this filling immediately. At small $J$s,
the system behaves as a charge insulating state with the gapless spin
excitation. At intermediate exchange interactions, the system is in the
conducting state of unpaired fermions ($\Delta E_{B}=0$), dubbed the metallic
phase. At larger $J$s, while the charge excitation remains gapless,
significant negative binding energy occurs. In this conducting state of paired
fermions, the binding energy $\Delta E_{B}$ and the spin gap $\Delta E_{S}$
have the same magnitude, showing evidence for the singlet-paired
superconductivity. Compared to the homogeneous $t$-$J$ chain, the spin gap is
greatly enhanced in the modulated setting. For example, while we have $\Delta
E_{S}\approx 0.32$ at $J=2.5$ in Fig. 2(d), the spin gap is around $0.01$ for
the same $J$ and $\rho$ in the standard $t$-$J$ model Moreno  _et al._ (2011).

Figure 3: (a) Energy criteria as a function of $J$ at $\rho=1/8$. (b) Spin gap
$\Delta E_{S}$ as a function of $\rho$ for typical $J$s. Extrapolated results
in the thermodynamic limit are obtained from data of $L=32$, $64$, $96$, and
$128$ with OBCs. Gary solid line in (b) denotes the spin gap at $J=2.5$ for
the standard $t$-$J$ chain with data extracted from Ref. Moreno  _et al._
(2011).

At incommensurate fillings ($p\times\rho$ is not an integer), the system is
always in the conducting phase (before phase separation) with gapless charge
excitation. Taking $\rho=1/8$ in Fig. 3(a) as an example, there is a
transition from the metallic state to the superconducting state with singlet
pairs. We further display the spin gap versus density $\rho$ in Fig. 3(b) in
the superconducting state and find that the spin gap for the same $J$ almost
does not decay as particle density $\rho$ increases for $\rho<1/2$. This
phenomenon is in sharp contrast to the homogeneous case, where $\Delta E_{S}$
dies off very rapidly, as denoted by the gray solid line in Fig. 3(b). The
strongly enhanced superconductivity in this work lies in the very flat band
structure in the ground-state phase diagram 1, which is in agreement with the
argument that the fermionic pairing in flat bands would lead to more robust
pairs and higher critical temperatures Heikkilä and Volovik (2016); Mondaini
_et al._ (2018). However, at fillings $\rho>1/2$, the spin gap becomes less
and less significant and finally disappears around $\rho\approx 11/16$.
Present results for the generally modulated $t$-$J$ chain do not support
superconductivity at physical-relevant parameters in materials (low-doping and
very small $J$), which is different from what was found in $t$-$J$ segments
Reja  _et al._ (2016).

Figure 4: Energy criteria for different typical fillings in the thermodynamic
limit at (a) $\rho=1/2$ and (b) $\rho=3/4$. Extrapolated results in the
thermodynamic limit are obtained from data of $L=32$, $64$, $96$ and $128$
with OBCs for $\rho=1/2$, and $L=32$, $48$, $64$, $80$, $96$ with PBCs for
$\rho=3/4$. All panels share the same legend.

At commensurate fillings, one can observe apparent charge gaps, but
corresponding insulating states show different features. At $\rho=1/4$, the
insulating state at small $J$s has gapless spin excitation and no pair
formation, as discussed in the previous text and shown in Fig. 2. At
$\rho=1/2$, the spin gap is always nonzero, and $\Delta E_{B}$ is finite only
at $J>2$, as shown in Fig. 4(a). Things are more complicated for $\rho=3/4$ in
Fig. 4(b), where appear two disconnected insulating states with a gap-closing
point around $J\approx 0.85$. At this special filling with $J>0.85$, the
energy criteria using OBCs and PBCs are different at $\phi=0$. In contrast,
different boundary conditions give the same result in the thermodynamic limit
in all other fillings. Therefore, we adopt PBCs for the ground-state phase
diagram at $\rho=3/4$ and explain the underlying physics in Sec. IV.3.
Different bulk properties in these different insulating states essentially
affect their topological nature and edge excitations, which will be discussed
in detail in the next section.

###  III.1 Density-density correlations and the Luttinger parameter

Figure 5: Structure factor $N(k)$ at $\rho=1/4$. The black dashed line denotes
$N(k)/(k/\pi)=1$. Here $L=160$ and OBCs are adopted.

In this subsection, we follow the standard way of the Luttiger liquid process
to review bulk phases by analyzing the density-density correlation function
and the Luttiger parameter Giamarchi ; Moreno  _et al._ (2011). The latter is
confirmed to be valid for quasi-periodic lattices even using correlations
between sites Silva-Valencia  _et al._ (2002); Reja  _et al._ (2016). The
Luttiger parameter can be determined by the structure factor $N(k)$ at the
small momentum $k$, as

$\displaystyle K_{\rho}=N(k)/(k/\pi),{k\rightarrow 0},$ (16)

where $N(k)$ is the structure factor of the density-density correlation
function, as shown in Fig. 5, we display $N(k)$ at the filling $\rho=1/4$,
where one can see metallic, superconducting, and insulating phases. At a small
interaction $J=0.5$, where the system is in the insulating phase and the
charge excitation is gapped (see Fig. 2), $N(k)$ shows quadratic behavior at
small momentum $k$. In both gapless phases for charge excitations, $N(k)$ has
a clear linear behavior at small momentum, and $K_{\rho}$ is smaller (larger)
than $1$ in the metallic (superconducting) phase. At the critical point $J=2$,
the Luttiger parameter $K_{\rho}$ is close to 1. Results from the structure
factor $N(k)$ and the Luttiger parameter agree with the ground-state phase
diagram by energy criteria. Note that Fig. 5 shows results for a finite system
with $L=160$. Although we believe that structure factors and energy criteria
must give the same conclusion in the thermodynamic limit, one should pay
careful attention to the finite size effect.

###  III.2 Pairing correlations and the structure factor

In the ladder geometry, the $t$-$J$ Hamiltonian can support superconductivity
with the significant spin gap, negative binding energy, and the dominant
singlet pairing correlation Hayward  _et al._ (1995); Cheng  _et al._ (2018).
While in this subsection, we focus on the correlations and the relevant
elementary excitations in the superconducting phase. As shown in Fig. 6, we
display correlations at $\rho=1/8$ as an example. While both the pairing and
density-density correlation functions have the power-law decay, $P^{S}(r)$
decays much slower than $N(r)$. The spin-spin correlation and the single-
particle Green’s function, which correspond to spin and fermion excitations,
both decay exponentially. This result agrees with the occurrence of the spin
gap and negative binding energy in the superconducting state.

Figure 6:  Correlation functions in the superconducting states at $J=2.25$ and
$\rho=1/8$: (a) Density-density and pairing correlation functions in the log-
log scale, and (b) spin-spin correlation function and single-particle Green’s
function in the linear-log scale. The black dashed line in (a) depicts the
power-law fit of the pairing correlation function, with $K_{\rho}=1.4$ for
this finite size. Here $L=160$ and OBCs are adopted.

In the spin-gapped superconducting state, the pairing structure factor
$P^{S}(k)$ at zero momentum was shown to be diverging in the thermodynamic
limit Moreno  _et al._ (2011). Here we present $P^{S}(k)$ in Fig. 7(a) as a
function of $k$, in both the metallic and superconducting phase for
comparison. In the superconducting phase, the pairing structure factor has a
sharp peak at $k=0$, and $P^{S}(k=0)$ increases as the system size $L$
increases. In contrast, despite a broad maximum at zero momentum, $P^{S}(k)$
is much smaller in the metallic phase and almost independent of the system
size. More Specifically, by displaying the size dependence of $P^{S}(k=0)$
explicitly in Fig. 7(b) and (c), it is clear that $P^{S}(k=0)$ has a power-law
growth as $L$ increases. However, while the number of pairs at zero momentum
given by $P^{S}(k=0)$ diverges, the density of pairs given by $P^{S}(k=0)/L$
is infinitely small in the thermodynamic limit. In the quasi-periodic $t$-$J$
chain, we only have the quasi-long-range superconducting order, which does not
go beyond the ladder geometry.

Figure 7: (a) Structure factor $P^{S}(k)$ of singlet pairing correlations as a
function of momentum $k$ in the superconducting ($J=2.25$) and metallic
($J=1.75$) state at $\rho=1/8$ for different $L$. The system size dependence
of $P^{S}(k=0)$ is shown in (b) and (c), where the blue dashed line depicts
the power-law fit. Panels (b) and (c) share the same legend.

##  IV insulating states and topological nature

The system described by Eq. (1) has several insulating regions at commensurate
fillings, as discussed in Sec. III. In this section, we reinvestigate these
gapped phases from the point of view of their topological nature. Before going
into details, we emphasize that the interaction plays a critical role in the
modulated $t$-$J$ chain. For example, we observe insulating states at
$\rho=q\times p$ for any integer $q$ smaller than $p$, while spinful fermions
in the noninteracting setting show insulating behaviors only at even $q$s, in
the absence of the external Zeeman field. In all insulating regions, our
calculations give the nonzero topological Chern number. Furthermore, the Chern
number is different at different fillings, that is, $C=1$ at $\rho=1/4$, $C=2$
at $\rho=1/2$ and $C=-1$ at $\rho=3/4$. Therefore, in the following, we
discuss these topological insulating states at each different filling in one
subsection.

###  IV.1 $\rho=1/4$

The system behaves as charge insulating states at densities $\rho=1/4$ with
small exchange $J$s, as shown in the ground-state phase diagram in Fig. 1. In
this region, particles do not form singlet pairs ($\Delta E_{B}=0$) and the
spin excitation is gapless ($\Delta E_{S}=0$). The quasi-particle excitation,
which is gapped in bulk but gapless at edges, if it exists, can correspond to
the single-particle spin-1/2 fermionic mode. Along this line, we define the
single-particle fermionic excitation energy as the ground-state energy
difference by adding a single fermion

$\displaystyle\delta
E_{f}=E_{0}(N_{\uparrow},N_{\downarrow}+1)-E_{0}(N_{\uparrow},N_{\downarrow}),$
(17)

and the corresponding onsite charge (spin) excitation
$\delta_{f}\langle\hat{n}_{i}\rangle$
($\delta_{f}\langle\hat{S}^{z}_{i}\rangle$) can be defined as differences
between these two ground states. Here we consider adding a spin-down fermion,
and the case for adding a spin-up one is symmetric and obvious.

Figure 8:  (a) The quasi-particle (single fermion) spectrum as a function of
the phase factor $\phi$ near the filling $\rho=1/4$. (b) The corresponding
onsite charge and spin difference for the two adjacent many-body ground
states, at parameters labeled by the black cross in (a). Here results are from
calculations of the system with $J=0.5$, $L=160$ and OBCs. All panels share
the same legend.

As a routine in characterizing the bulk-edge correspondence for 1D topological
insulators, we display the quasi-particle spectrum as a function of the phase
factor $\phi$ in Fig. 8(a), in which we show four typical quasi-particle
levels and nearby lower (upper) levels with smaller (larger) $N_{\downarrow}$
are in the continuous band. While there is a significant gap at $\phi=0$, the
top level in the lower band lifts as $\phi$ evolves and meets the upper band
at $\phi=\pi$. In Fig. 8(b) and (c), we plot the low-energy charge and spin
excitations, respectively, by taking the in-gap state at $0.7\pi$ as an
example. The well-localized charge accumulation at one end of the chain can be
observed, but the spin excitation shows no edge-related anomaly. This strong
spin-charge separation in the bulk-edge correspondence is essentially
different from the noninteracting case, where there exists no quasi-particle
that carries one charge but no spin. In this strongly interacted insulating
region, the spin-1/2 single fermion contributes the energy for the band
touching. However, only the excitation of charge degree of freedom is related
to the edge mode, and its spin degree of freedom merges into the bulk.

###  IV.2 $\rho=1/2$

In the noninteracting case, a spinless-fermion chain with quasi-periodic
modulations described in Eq. (2) has insulating states at commensurate
fillings $\rho=q/p$ with an integer $q<p$ Lang  _et al._ (2012); Kraus  _et
al._ (2012), which results in naive charge and spin gaps (nonzero $\Delta
E_{C}$ and $\Delta E_{S}$) at $\rho=2q/p$ for the noninteracting spinful
fermions. In the interacting setting, the charge and spin gap survive at
$\rho=1/2$ in modulated $t$-$J$ chains, as shown in Fig. 4(b). However, the
exchange interaction $J$ further divides this gapped phase into two regions
according to whether singlet-pairs are formed in the ground state. In the
following, we demonstrate that the bulk property also affects the topological
nature and the gapless edge modes are different in the two regions.
Consequently, there is an interaction-driven transition in the topological
aspect as well.

Figure 9:  (a) The spin gap $\Delta E_{S}$ and the neutral excitation energy
$\delta E_{\mathrm{n}e}^{(1)}$ as a function of $\phi$ at the filling
$\rho=1/2$. The corresponding low-energy (b) charge and (c) spin excitations
in real space, at parameters labeled by the black cross in (a). Here results
are from calculations of the system with $J=1.0$, $L=64$ and OBCs. All panels
share the same legend.

The binding energy is zero at small exchange interactions ($J<2$), with no
indication of pair formation. If repeating procedures in Sec. IV.1 by
considering the single fermion as the gapless edge modes, one cannot find
level crossing in the quasi-particle spectrum as in Fig. 8(a). In the region
with both gapped charge and spin excitations, it is useful to examine the
neutral excitation to the first excited state to determine which one is the
minimum. In Fig. 9(a), we compare the spin-flip energy and the neutral
excitation energy $\delta E^{(1)}_{\mathrm{n}e}=E_{1}-E_{0}$, and find the
surprisingly exact match between them for all values of the phase factor
$\phi\in[0,2\pi]$. At $\phi=0$, both excitations are gapped, and the two gaps
close around $\phi=\pi$. In Fig. 9, we further display the low-energy charge
(spin) excitations in real space, which can be defined as the difference of
the corresponding observable between two many-body states. The low-energy
charge excitation shows no anomaly for both the spin-flip and the neutral
excitation, as shown in Fig. 9(b). In contrast, in Fig. 9(c), one can observe
clear spin accumulations at one end of the chain when flipping a spin. Spin
accumulation occurs at both ends of the chain for the neutral excitation to
keep the total spin in the z-direction as $0$. Moreover, at one end of the
chain, where the spin accumulation is observed for the spin-flipping, the spin
distribution shows great agreement with the neutral excitations. These results
confirm the gapless edge modes related to the magnon excitation, which carries
integer spin without charge degree of freedom.

Figure 10:  (a) The quasi-particle (singlet pair) spectrum as a function of
the phase factor $\phi$ near the filling $\rho=1/2$. (b) The corresponding
onsite charge and spin difference for the two adjacent many-body ground
states, at parameters labeled by the black cross in (a). Here results are from
calculations of the system with $J=2.5$, $L=160$ and OBCs. All panels share
the same legend.

When the interaction $J$ is above a critical $J_{c}\approx 2$, the negative
binding energy occurs, and fermions tend to form singlet pairs. Intuitively,
it is interesting to consider the possibility of another kind of bosonic edge
mode attributed to singlet pairs. We define the pairing excitation energy

$\displaystyle\delta
E_{p}=E_{0}(N_{\uparrow}+1,N_{\downarrow}+1)-E_{0}(N_{\uparrow},N_{\downarrow}),$
(18)

and present the corresponding quasi-particle spectrum for a typical $J=2.5$ in
Fig. 10(a). The quasi-particle spectrum has a considerable gap at $\phi=0$,
and the top level of the lower band rises as $\phi$ increases and meets the
upper band around $\phi\approx\pi$. Following similar procedures in the
previous text, we display the low-energy charge and spin excitations of the
in-gap state at $\phi=0.6\pi$ in Fig. 10(b) and (c), respectively. As
expected, charge accumulations occur at one end of the chain, and there is no
spin anomaly in real space. We further compare the low-energy charge
excitation by adding a singlet-pair to which of the neutral excitation in the
main panel of Fig. 11. It is impossible to see the exact match between the two
curves at the finite system size since they correspond to excitations with
different particle number altering. However, charge accumulations at the end
of the chain show great agreement between two kinds of low-energy excitations,
and the total charge altering within the peak (roughly six sites) is about 2.
Considering that there is no anomaly in the spin channel, one can conclude
that singlet-pairs contribute to the gapless edge mode.

Figure 11:  The low-energy charge (main panel) and spin (inset) excitations in
real space, at parameters labeled by the black cross in Fig. 10(a). Here
results are from calculations of the system with $J=2.5$, $L=160$ and OBCs.

###  IV.3 $\rho=3/4$

There are two insulating regions at $\rho=3/4$ separated by a gap closing-
point around $J_{c}\approx 0.85$, as shown in the ground-state phase diagram 1
and Fig. 4(d). The spin gap and the binding energy are zero in both regions,
which rules out the magnon and singlet pairs as gapless edge modes. When $J$
is smaller than the gap closing point, the bulk-edge correspondence is similar
to the case at $\rho=1/4$ as described in Sec. IV.1. Therefore, we skip the
repeated description and move to the case for larger exchange interactions in
the following.

Figure 12:  (a) The quasi-particle (single fermion) spectrum as a function of
the phase factor $\phi$ near the filling $\rho=3/4$, the inset displays the
finite size extrapolation ($L=64,96,128,160,240$) of $\delta
E_{f}(N_{\uparrow},N_{\downarrow})$ at $\phi=0$. (b) The corresponding onsite
charge and spin difference for the two adjacent many-body ground state, at
parameters labeled by the black cross in (a). Here results are from
calculations of the system with $J=1.5$ and OBCs. All panels share the same
legend.

Since the two insulating regions share similar properties in bulk, we again
assume the single fermion as the elementary edge mode for $J>0.85$. Although
the quasi-particle spectrum in Fig. 12(a) gives a very different first
impression, the overall bulk-edge correspondence is substantially the same as
what is shown in Fig. 8 by shifting the phase factor by $\pi$. However, there
is a small remnant gap at $\phi=0$, which is supposed to be the band-touching
point. We attribute this bad connection to the finite size effect and display
the finite size extrapolation in the inset of Fig. 12(a), in which the remnant
gap eventually closes as the system approaches the thermodynamic limit.
Therefore, we believe the quasi-particle corresponds to the gapless edge mode
is the same in both insulating regions at $\rho=3/4$, and is the same to the
case of $\rho=1/4$.

Figure 13:  (a) The effective hopping
$\langle\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i+1,\uparrow}\rangle$ at 4
adjacent bonds as a function of exchange interaction $J$. The input hopping
(b), the effective hopping at $J=0.5$ (d) and $J=1.5$ (e) on same adjacent
bonds. (c) The effective hopping of the noninteracting spinless fermion chain
with the same modulation. In all panels we have $\phi=0$, $L=96$ and PBCs.

There is a remaining issue at this filling that why there is a $\pi$ phase
shift in the quasi-particle spectrum across the gap closing point between bulk
insulating states, while these two regions have the same Chern number and same
edge modes. To answer this question, we turn to the local energy bonds of the
ground state, which is demonstrated to be able to detect topological phase
transitions in various models possessing symmetry-protected topological phases
or long-range topological orders Yu  _et al._ (2020). For the quasi-periodic
system with period $p=4$, we examine the effective hopping
$\langle\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i+1,\uparrow}\rangle$ on 4
different bonds in one unit cell in the middle of the chain. As displayed in
Fig. 13(a), effective hopping bonds as functions of $J$ already show signals
of a transition at $J\approx 0.85$, which agrees with the position the charge
gap closes [see Fig. 4(d)]. We further explicitly compare
$\langle\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i+1,\uparrow}\rangle$ with the
magnitude of the input hopping $|t_{i}|$, and which of the noninteracting
fermions. For the spinless fermions with the same modulation, in the
noninteracting setting, the effective hopping must follow the wave pattern of
the input hopping parameters, as verified by the data in Fig. 13(b) and (c).
In presence of interactions, the profile of
$\langle\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i+1,\uparrow}\rangle$ also
agrees with the input $t_{i}$ in most cases, as shown in Fig. 13(d) for
$J=0.5$ and $\rho=3/4$ as an example. However, for $J$ above the critical
point, the wave pattern shifted by a phase of $\pi$, as shown in Fig. 13(e).
This $\pi$-phase shift of the effective hopping accompanied by the bulk-state
gap closing directly leads to a $\pi$-phase shift in the quasi-particle
spectrum.

##  V summary and discussion

We systematically investigate the ground-state properties of the extended
$t$-$J$ model with modulated hopping and interaction. The system has a rich
ground-state phase diagram consisting of the metallic state, the
superconducting state, the phase separation, and different insulating states
with nontrivial topological nature. The superconducting state is characterized
by the significant singlet spin gap and negative binding energy, the dominant
pairing correlation decay, and the divergent paring structure factor at zero
momentum. The spin gap and the magnitude of binding energy have been
dramatically enlarged in the superconducting phase, which can be attributed to
the flat-band structure of the quasi-periodic superlattice. This enhancement
of the superconductivity, which was first proposed in coupled $t$-$J$ segments
Reja  _et al._ (2016), has been confirmed in a more general quasi-periodic
geometry. However, here in this work, the superconducting region in the
$\rho$-$J$ plane is not significantly expanded compared to the homogeneous
$t$-$J$ chain Moreno  _et al._ (2011) and is still far from $J\approx 1/3$ in
actual materials.

The interplay among the quasi-periodicity, the interaction, and the 1D
confinement leads to various topological edge modes that manifest an extreme
spin-charge separation nature. At $\rho=1/4$ and $3/4$, while the spin-1/2
fermion contributes the energy of the gapless edge mode, only the charge
degree of freedom manifests the localized excitation at the end. However, the
spin excitation completely merges in bulk. At $\rho=1/2$, we demonstrate two
bosonic edges modes (related to the magnon and the singlet pair) and the
corresponding interaction-driven transition between them. There is also a
topological transition at $\rho=3/4$ accompanied by the $\pi$-phase shift in
bulk’s effective hopping wave pattern.

Our investigation has shown pregnant physics in the 1D $t$-$J$ model in the
presence of off-diagonal modulations, with the fixed unit cell size and
modulation magnitude. Moreover, there are still plenty of interesting issues
to be explored in the modulated spinful fermionic chains. For example, is it
possible to find the gapless magnon modes at the edges in the enhanced
superconducting state with the significant spin gap? We do not find this new
kind of topological superconductivity in the present work with the current
setting and parameters, but we cannot rule out the possibility. On the other
hand, we expect the rich physics in the modulated fermionic system can
stimulate experimental studies in the platform of cold atoms Jepsen  _et al._
(2020) or the quantum computing devices Arute  _et al._ (2020).

## Acknowledgement

We thank Z.-S. Zhou for helpful discussions in numerics. This research was
supported by the National Natural Science Foundation of China (grant nos.
11904145, 12174167, 11834005, and 12047501). The computations were partially
performed in the Tianhe-2JK at the Beijing Computational Science Research
Center (CSRC).

##  Appendix A Correlations between sites & unit cells

In the main paper we adopt the correlation function $X_{ij}$ between sites
$\\{i,j\\}$. Although it is demonstrated that the correlation function defined
between unit cells gives the same Luttinger parameter Reja  _et al._ (2016);
Silva-Valencia  _et al._ (2002), in this section, we explicitly compare these
two forms of correlations. The correlation function between unit cells can be
easily extracted from which between sites as:

$\displaystyle\tilde{X}_{\alpha,\beta}=\sum_{i\in\alpha,j\in\beta}X_{ij},$
(19)

where $\alpha$ and $\beta$ are indices of unit cells. Similar to Eq. (7) and
(8), we define the correlation decay

$\displaystyle\tilde{X}(R)=\frac{1}{\cal\tilde{N}}\sum_{|\alpha-\beta|=\tilde{r}}\tilde{X}_{\alpha\beta},$
(20)

and the structure factor

$\displaystyle\tilde{X}(k)=\frac{1}{\tilde{L}}\sum_{\alpha,\beta=1}^{\tilde{L}}\mathrm{e}^{\mathrm{i}\tilde{k}(\alpha-\beta)}\tilde{X}_{\alpha\beta}.$
(21)

Here $\tilde{L}=L/p$ is the number of unit cells, and $\tilde{r}$
($\tilde{k}$) is the distance in the real (momentum) space based on the unit
cell. By the straightforward mapping $\tilde{r}p\rightarrow r$ and
$\tilde{k}/p\rightarrow k$, we can directly compare the correlation and the
corresponding structure factor based on sites and unit cells. In Fig. 14, we
present typical numerical results for the density-density and pairing
correlation function, as well as corresponding structure factors. After being
rescaled, curves extracted from correlations between sites and unit cells show
very close behaviors. Differences in local details can be attributed to the
fact that the correlation based on unit cells neglects terms within the same
unit cell.

Figure 14:  Direct comparison between correlations based on sites and unit
cells. a) and c) The real space decay of the density-density correlation
function and the pairing correlation function. b) and d) Corresponding
structure factors. Here we use data for $J=2.25$ and $L=128$ at $\rho=1/8$
with OBCs.

##  Appendix B Phase separation

For the $t$-$J$ model, large exchange interaction $J$ leads to the unstable
phase separation Moreno  _et al._ (2011), which can be detected by the inverse
compressibility

$\displaystyle\kappa^{-1}(\rho)$
$\displaystyle=\rho^{2}\frac{\partial^{2}e_{0}(\rho)}{\partial\rho^{2}}$
$\displaystyle\approx\rho^{2}\frac{e_{0}(\rho+\Delta\rho)+e_{0}(\rho-\Delta\rho)-2e_{0}(\rho)}{\Delta\rho^{2}},$
(22)

where $e_{0}(\rho)=E_{0}/L$ is the ground-state energy per site. In this work,
we take not extra calculations for the phase separation state, since which is
not our main focus. Instead, we adopt a special $\Delta\rho=2/L$ and therefore
can extract $\kappa^{-1}$ from the charge gap $\Delta E_{C}$ as

$\displaystyle\kappa^{-1}(N/L)\approx\frac{N^{2}}{2L}\Delta E_{C}(L,N).$ (23)

The estimated $\kappa^{-1}$ from Eq. (23) is displayed in Fig. 15(a), for
$\rho=1/4$ as an example. For small $J$s, The inverse compressibility
increases as the system size increases, which ensures the system is away from
phase separation with positive $\kappa^{-1}$. Roughly at $J\geq 2.75$, the
inverse compressibility shows less system size dependency and approaches $0$.
This estimated critical point can be verified by the density profile shown in
Fig. 15(b). For an exchange interaction slightly larger than the critical
point, the system separates into particle-rich and hole-rich regions in real
space, and the density distribution loses mirror symmetry because of massive
degeneracy. By taking the estimation in Eq. (23) at different densities, we
obtain the phase boundary in the ground-state phase diagram 1 in the main
paper.

Figure 15:  a) The inverse compressibility $\kappa^{-1}$ as a function of $J$
at $\rho=1/4$ for different system size $L$. b) The charge distribution in the
conducting state ($J=2.5$) and the phase separation ($J=3.0$) with $L=64$.
Here OBCs are adopted.

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