# Diffusing More Objects for Semi-Supervised
Domain Adaptation with Less Labeling

Leander van den Heuvel
TNO Netherlands
&Gertjan Burghouts
TNO Netherlands
David W. Zhang
University of Amsterdam
&Gwenn Englebienne
University of Twente
&Sabina B. van Rooij
TNO Netherlands

###### Abstract

For object detection, it is possible to view the prediction of bounding boxes
as a reverse diffusion process. Using a diffusion model, the random bounding
boxes are iteratively refined in a denoising step, conditioned on the image.
We propose a stochastic accumulator function that starts each run with random
bounding boxes and combines the slightly different predictions. We empirically
verify that this improves detection performance. The improved detections are
leveraged on unlabelled images as weighted pseudo-labels for semi-supervised
learning. We evaluate the method on a challenging out-of-domain test set. Our
method brings significant improvements and is on par with human-selected
pseudo-labels, while not requiring any human involvement.

##  1 Introduction

Diffusion models achieved success on a wide range of generative tasks,
including synthetic images, see e.g. recent surveys [2, 15]. Recently, they
have been explored for other tasks, such as image segmentation [1] and object
detection [3]. DiffusionDet [3] formulates object detection as a box denoising
process conditioned on the image input, starting with a set of random bounding
boxes that is iteratively improved (denoised). This is advantageous because it
avoids the necessity of region proposals [4], anchors [8], or learnable
queries [18]. The process of denoising object boxes is stochastic since it
depends on the random initialization of the boxes. This brings the advantage
that each run leads to slightly different outputs. Running the model multiple
times may increase the probability of detecting objects that are difficult to
detect. We hypothesize that this is especially useful when there is a
significant domain gap between the training distribution and test
distribution. In this work we aim to solve the domain gap caused by different
viewpoints: from frontal view (source domain) to aerial view (target domain).
To that end, we propose a stochastic accumulator of the diffusion model’s
outputs to acquire improved object detections in the target domain. We
anticipate that the improved detections are more useful as pseudo-labels for
semi-supervised learning. For that purpose, we reformulate the diffusion
model’s loss function to include unlabeled images and their pseudo-labels.
Since the pseudo-labels are imperfect, we propose a weighted loss that takes
their confidence into account. Our contribution is two-fold: we propose a
stochastic accumulator and a semi-supervised weighted pseudo-label loss. We
show their efficacy on a challenging domain gap from frontal everyday images
(MS-COCO [6]) to aerial images that cover day and night and were taken from
much larger distances (VisDrone [17]). We demonstrate that our method is on
par with human-selected pseudo-labels.

##  2 Related Work

Aiming at label-efficient domain adaptation for object detection, we consider
related work in domain adaptation and semi-supervised object detection. For
domain adaptation, there are unsupervised e.g. [9, 12] and supervised training
procedures e.g. [11, 14]. With supervision in the target domain, adaptation
methods can improve greatly [12], however, supervised training requires
labeled data. Semi-supervised training uses both labeled and unlabeled images
[16]. One approach is the use of pseudo-labels on the unlabeled images. In
Sohn et al. [13] a self-training and the augmentation driven consistency
regularization (STAC) for object detection is proposed. This work introduces a
trained teacher model that generates pseudo-labels, which are used to finetune
a student model. The authors note that class imbalance of the labeled images
could lead to reduced pseudo-label quality. To address this issue, Unbiased
Teacher is proposed by Liu et al. [10], that uses the Focal loss [7] to down-
weigh overly confident pseudo-labels. STAC and Unbiased Teacher show
impressive results for training with fewer labels. However, both papers focus
on in-domain performance improvements using pseudo-labels. We explore semi-
supervised training with pseudo-labels for domain adaptation.

##  3 Method

We extend DiffusionDet [3] (Section 3.1) in two ways. First, we improve its
predictions for a large domain gap, using a stochastic accumulator (Section
3.2). Secondly, we use the improved predictions as weighted pseudo-labels for
semi-supervised learning (Section 3.3).

Figure 1: Our method improves object detection by exploiting the stochastic
nature of the diffusion-based model, thereby accumulating diverse predictions,
and leveraging them as weighted pseudo-labels in semi-supervised learning.

###  3.1 Preliminaries

DiffusionDet [3] trains a neural network $f_{\theta}(z_{t},t)$ to predict a
set of boxes $z_{0}$ from noisy boxes $z_{t}$ by minimizing a combination of
losses including the $\ell_{2}$ loss:
$\mathcal{L}=\frac{1}{2}||f_{\theta}(z_{t},t)-z_{0}||^{2}$. The boxes are a
set of coordinates in the image, $z\in\mathbf{R}^{t\times 4}$, with for each
box a category label. The noisy boxes $z_{t}$ are taken from a forward noise
process that gradually transforms the boxes $z_{0}$ to noisy versions $z_{t}$:
$q(z_{t}|z_{0})=\mathcal{N}((z_{t}|\sqrt{\overline{\alpha_{t}}})z_{0},(1-\overline{\alpha_{t}})\textit{{I}})$,
where $\overline{\alpha_{t}}$ defines the noise schedule. At inference,
$z_{0}$ is predicted from noise $z_{T}$ by $f_{\theta}$ iteratively,
$z_{T}\rightarrow\dots\rightarrow z_{t}\rightarrow\dots\rightarrow z_{0}$. The
network $f_{\theta}(z_{t},t,x)$ is trained to predict $z_{0}$ from noisy boxes
$z_{t}$ for an image $x$. For more details, we refer to [3].

###  3.2 Stochastic accumulator

The noise $z_{T}$ is a random variable. Hence, we can apply $f_{\theta}$ for
an image $x$ with a different draw $i$ of noisy bounding boxes, $z_{T}^{i}$,
leading to a prediction $z_{0}^{i}=f_{\theta}(z_{T}^{i},t,x)$. We can do this
multiple times, each time leading to a (slightly) different prediction,
$\\{z_{0}^{i}\\}_{i\in 1..n}$. The rationale is that the object detection can
be improved by accumulating the model’s outputs over these multiple runs. Our
accumulator function is:

$g_{\theta}(x)\,\,=\,\,\operatorname{nms}(\operatorname{concat}(\\{f_{\theta}(z_{T}^{i},t,x)\\}_{i\in
1..n}))$ (1)

where $\operatorname{concat}(\cdot)$ is the operator that concatenates the
outputs of $f_{\theta}$ in a row-wise manner, and $\operatorname{nms}(\cdot)$
is the non-maximum-suppression operator. The resulting prediction
$(u_{0},w_{0})=g_{\theta}(x)$ is a set of boxes $u$ with respective
confidences $w$. This procedure is illustrated in Figure 1.

In case of large domain gaps, where objects from the train and test set differ
in e.g. viewpoint, size, or background, some objects in the test set may be
difficult to detect. With $g_{\theta}(x)$, we leverage the different,
stochastic outputs of the model between runs, thereby increasing the
probability of finding challenging objects in the test set.

###  3.3 Weighted pseudo-labels for semi-supervised learning

The rationale is to use the enhanced object predictions
$(u_{0},w_{0})=g_{\theta}(x)$ as pseudo-labels in a semi-supervised setup.
This is done to further improve the overall model $g_{\theta}$ on the target
domain, by finetuning its predictor function $f_{\theta}$. We assume that for
some images the objects are labeled, $z_{0}$, while for others no labels are
available. For the unlabeled images, we have the predicted boxes and their
confidence, i.e., $(u_{0},w_{0})$. We reformulate the original loss of
DiffusionDet, which was fully supervised (see Section 3.1), as a loss that
enables semi-supervised learning (SSL) over $m$ labeled images and $l$
unlabelled images:

$\mathcal{L}_{SSL}=\frac{\frac{1}{2}\sum_{k=1}^{m}||f_{\theta}(z_{t}^{k},t)-z_{0}^{k}||^{2}}{m}+\frac{\frac{1}{2}\sum_{k=1}^{l}||f_{\theta}(z_{t},t)-u_{0}^{k}||^{2}\cdot
w^{k}}{\sum_{k=1}^{l}w^{k}}$ (2)

Our semi-supervised approach to reduce the domain gap is illustrated in the
right part of Figure 1.

##  4 Experiments

###  4.1 Setup and domain gap

The source domain is MS-COCO [6] with everyday frontal images. The target
domain is VisDrone [17] with aerial images taken by drones in urban areas.
Images in VisDrone are captured from a greater distance using a drone, leading
to smaller objects, and, in some cases, motion blur. We are interested in a
domain adaptation problem where the classes between the train and test set
overlap. For evaluation, we select overlapping classes between MS-COCO and
VisDrone: car, person (including pedestrian from VisDrone), bus, and truck. A
fragment of an image from VisDrone is shown in Figure 2 where the objects have
an aerial view, are relatively small compared to MS-COCO and contain motion
blur. More examples are shown in Figure 4 (Appendix). The domain gap is
severe, not only in appearance, but also the distribu tion of object sizes is
very different, see Figure 5 (Appendix). Our baselines are YOLOv5 [5] and
DiffusionDet [3] with the standard settings (300 random proposal boxes and 10
reverse diffusion steps). We evaluate the merit of our extensions to
DiffusionDet.

###  4.2 Object detection can be improved by stochasticity

Figure 5 (Appendix) visualizes that the DiffusionDet baseline has problems
with small objects. Small objects are highly prevalent in the target domain,
but they are less frequently predicted by the baseline. This is also concluded
from the performance metrics (mAP) in Table 1, where we evaluate the
performance of various models on different object sizes in the target domain.
The small objects are very difficult to detect: YOLOv5 and DiffusionDet
respectively have a random or almost random performance. The best improvement
over the baseline is achieved by running our accumulator for $n$ = 18 runs
(Equation 1). It increases the mAP from 0.02 to 0.09 for small objects, and
from 0.18 to 0.38 for large objects.

Additionally, we experimented with two other adjustments to the stochastic
input of DiffusionDet: increasing the number of input boxes and reducing the
input box sizes to align with the smaller objects in the target domain. These
adjustments are less effective; reducing the target boxes to half of the size
($f_{box\\_size}$ = 0.5) degrades the results; more boxes (5400) are effective
but not as much as our extension. Here, 5400 boxes correspond to 18 runs with
the 300 default boxes of DiffusionDet. Our stochastic accumulator is effective
in improving performance under the tested large domain gap. Illustrations of
improvements over the baseline are shown for various $n$ in Figure 6
(Appendix). We show that, in line with our hypothesis, the diffusion model’s
stochasticity can be exploited to detect more objects.

object size |  accumulator runs ($n$) (proposed) | random proposal boxes | $f_{box\\_size}$ | YOLOv5
---|---|---|---|---
| 1 | 9 | 18 | 300 | 2700 | 5400 | 1 | 0.75 | 0.5 |
small | 0.02 | 0.08 | 0.09 | 0.02 | 0.02 | 0.03 | 0.02 | 0.00 | 0.01 | 0.00
medium | 0.06 | 0.17 | 0.18 | 0.09 | 0.13 | 0.17 | 0.09 | 0.04 | 0.05 | 0.08
large | 0.18 | 0.35 | 0.38 | 0.18 | 0.34 | 0.37 | 0.18 | 0.13 | 0.15 | 0.29

Table 1: VisDrone object detection performance (mAP). Our accumulator (left)
performs best, with more runs it is more effective, exploiting the
stochasticity of the underlying diffusion model.

###  4.3 Pseudo-labels: weighted loss vs. human-selected

We use the improved predictions as pseudo-labels in semi-supervised learning,
in combination with labels from a few (10 or 50) images. Pseudo-labels may be
noisy, for which we validate the merit of our weighted loss (see Section 3.3).
To provide a strong baseline, we compare against human-verified pseudo-labels
generated from DiffusionDet. The human annotator has selected regions in the
unlabeled images that contain good pseudo-labels, meaning that the boxes are
approximately correct and the class label is correct. We validate how our
weighted usage of pseudo-labels, requiring no human involvement, compares to
the human-verified pseudo-labels. For completeness, we also compare against
the groundtruth labels within the human-verified image regions, to compare the
pseudo-labels with actual labels.

Figure 3 shows that the generated pseudo-labels are effective: all models that
use them (four bars to the right) outperform the models trained only on the GT
images (baseline). This is the case for both 10 and 50 training images. The
pseudo-labels after 18 runs with our accumulator are better than a single run
of DiffusionDet, as expected, given its improved performance (see Section
4.2). Surprisingly, the performance of human-verified pseudo-labels is on par
with real labels in the same image regions. This is unexpected because the
pseudo-labels are imperfect. We have used all pseudo-labels that have a
confidence above 0.5. In our weighted loss, the confidence is taken into
account during the learning. This strategy is effective: it performs on par
(right-most bar) with the human-verified pseudo-labels. Our weighted loss
removes the necessity of manual selection, which is labour intensive, without
a degradation of performance. For a hard case, we visualise of the improved
detections over the standard model, baseline (with finetuning), and the model
finetuned with unverified weighted pseudo-labels in Figure 2, the entire
picture is shown in Figure 7 (Appendix).

Figure 2:  Detections in VisDrone using DiffusionDet shown from left to right:
standard model (300 boxes and 1 run), finetuned with 50 GT samples and 18 runs
(baseline) , baseline with unverified weighted pseudo-labels and 18 runs
Figure 3: Semi-supervised learning is effective for this domain gap, where our
weighted pseudo-label loss is a good performer, with an mAP close to or better
than human-verified labels, while not requiring any human involvement.

##  5 Conclusions

We leveraged the stochasticity of a diffusion model for object detection,
increasing the performance by accumulating variable predictions. For a large
domain gap, we have shown that the improved detections on unlabeled images are
useful for semi-supervised learning. This is especially the case when weighted
according to their confidence with our weighted semi-supervised loss function.

## References

  * Baranchuk et al. [2021] Dmitry Baranchuk, Andrey Voynov, Ivan Rubachev, Valentin Khrulkov, and Artem Babenko.  Label-efficient semantic segmentation with diffusion models.  In _International Conference on Learning Representations_ , 2021.
  * Cao et al. [2022] Hanqun Cao, Cheng Tan, Zhangyang Gao, Yilun Xu, Guangyong Chen, Pheng-Ann Heng, and Stan Z Li.  A survey on generative diffusion model.  _arXiv preprint arXiv:2209.02646_ , 2022.
  * Chen et al. [2023] Shoufa Chen, Peize Sun, Yibing Song, and Ping Luo.  DiffusionDet Diffusion Model for Object Detection.  _Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV2023)_ , 2023.  URL https://github.com/ShoufaChen/.
  * He et al. [2017] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick.  Mask r-cnn.  In _Proceedings of the IEEE international conference on computer vision_ , pages 2961–2969, 2017.
  * Jocher [2022] Glenn Jocher.  ultralytics/yolov5: v7.0 - YOLOv5 SOTA Realtime Instance Segmentation (v7.0).  _Https://Github.Com/Ultralytics/Yolov5/Tree/V7.0_ , page 10, 11 2022.  doi: 10.5281/ZENODO.7347926.  URL https://zenodo.org/record/7347926https://ultralytics.com/.
  * Lin et al. [2014] Tsung Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C. Lawrence Zitnick.  Microsoft COCO: Common objects in context.  _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , 8693 LNCS(PART 5):740–755, 2014.  ISSN 16113349.  doi: 10.1007/978-3-319-10602-1{\\_}48.
  * Lin et al. [2017a] Tsung Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollar.  Focal Loss for Dense Object Detection.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 42(2):318–327, 8 2017a.  ISSN 19393539.  doi: 10.1109/TPAMI.2018.2858826.  URL https://arxiv.org/abs/1708.02002v2.
  * Lin et al. [2017b] Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár.  Focal loss for dense object detection.  In _Proceedings of the IEEE international conference on computer vision_ , pages 2980–2988, 2017b.
  * Liu et al. [2022] Xiaofeng Liu, Chaehwa Yoo, Fangxu Xing, Hyejin Oh, Georges El Fakhri, Je-Won Kang, Jonghye Woo, and C C Jay Kuo.  Deep Unsupervised Domain Adaptation: A Review of Recent Advances and Perspectives.  _APSIPA Transactions on Signal and Information Processing_ , 1:1–48, 2022.
  * Liu et al. [2021] Yen-Cheng Liu, Chih-Yao Ma, Zijian He, Chia-Wen Kuo, Kan Chen, Peizhao Zhang, Bichen Wu, Zsolt Kira, and Peter Vajda.  Unbiased Teacher for Semi-Supervised Object Detection.  _ICLR 2021_ , pages 1–17, 2 2021.  URL https://github.com/facebookresearch/unbiased-teacher.https://arxiv.org/abs/2102.09480v1.
  * Motiian et al. [2017] Saeid Motiian, Marco Piccirilli, Donald A Adjeroh, and Gianfranco Doretto.  Unified Deep Supervised Domain Adaptation and Generalization.  In _IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017_ , pages 5716–5726. 2017.
  * Oza et al. [2021] Poojan Oza, Vishwanath A Sindagi, Vishal M Patel, and Senior Member.  Unsupervised Domain Adaptation of Object Detectors: A Survey.  2021\.
  * Sohn et al. [2020] Kihyuk Sohn, Zizhao Zhang, Chun-Liang Li, Han Zhang, Chen-Yu Lee, and Tomas Pfister.  A Simple Semi-Supervised Learning Framework for Object Detection.  5 2020.  URL https://arxiv.org/abs/2005.04757v2.
  * Wang and Deng [2018] Mei Wang and Weihong Deng.  Deep visual domain adaptation: A survey.  _Neurocomputing_ , 312:135–153, 10 2018.  ISSN 0925-2312.  doi: 10.1016/J.NEUCOM.2018.05.083.
  * Yang et al. [2022a] Ruihan Yang, Prakhar Srivastava, and Stephan Mandt.  Diffusion probabilistic modeling for video generation.  _arXiv preprint arXiv:2203.09481_ , 2022a.
  * Yang et al. [2022b] Xiangli Yang, Zixing Song, Irwin King, and Zenglin Xu.  A Survey on Deep Semi-Supervised Learning.  _IEEE Transactions on Knowledge and Data Engineering_ , pages 1–20, 2022b.  ISSN 1041-4347.  doi: 10.1109/tkde.2022.3220219.
  * Zhu et al. [2020a] Pengfei Zhu, Longyin Wen, Dawei Du, Xiao Bian, Heng Fan, Qinghua Hu, and Haibin Ling.  Detection and Tracking Meet Drones Challenge.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 44(11):7380–7399, 1 2020a.  ISSN 19393539.  doi: 10.1109/TPAMI.2021.3119563.  URL https://arxiv.org/abs/2001.06303v3.
  * Zhu et al. [2020b] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang, and Jifeng Dai.  Deformable detr: Deformable transformers for end-to-end object detection.  In _International Conference on Learning Representations_ , 2020b.

## Appendix

(a) Source domain: MS-COCO [6]

(b) Target domain: VisDrone [17]

Figure 4: Examples from the source (a) and target (b) domains. The target
domain is very different, with changes in viewpoint, day and night, and
distance to the objects.

(a) Source domain: MS-COCO [6]

(b) Target domain: VisDrone [17]

Figure 5: The groundtruth object sizes are distributed very differently for
the source (a) and target (b) domains. The distribution of predictions by the
DiffusionDet baseline do not align well with the groundtruth.

(a) DiffusionDet

(b) Stochastic accumulator, $n$ = 2

(c) Stochastic accumulator, $n$ = 3

(d) Stochastic accumulator, $n$ = 18

Figure 6: Improving DiffusionDet with our stochastic accumulator, with various
runs $n$ (Equation 1).

(a) DiffusionDet, standard configuration (1 run, 300 boxes)

(b) DiffusionDet, 18 runs and finetuned on 50 GT samples

(c) DiffusionDet, 18 runs and finetuned on 50 GT samples in combination with
weighted semi-supervised loss

Figure 7: Improving the finetuning with our weighted semi-supervised loss
using pseudo-labels on unlabelled images taking into account their
confidences.


# Gate-tunable topological superconductivity in a supramolecular electron spin
lattice

Rémy Pawlak,1∗† Jung-Ching Liu,1† Chao Li,1† Richard Hess,1† Hongyan Chen,2
Carl Drechsel,1, Ping Zhou,3 Robert Häner,3 Ulrich Aschauer,3,4
Thilo Glatzel,1 Silvio Decurtins,3 Daniel Loss,1 Jelena Klinovaja,1
Shi-Xia Liu,3∗ Wulf Wulfhekel,2 & Ernst Meyer1

1Department of Physics, University of Basel, Klingelbergstrasse 82,
4056 Basel, Switzerland
2Physikalisches Institut, Karlsruhe Institute of Technology,
Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany
3Department of Chemistry, Biochemistry and Pharmaceutical Sciences,
University of Bern, Freiestrasse 3, 3012 Bern, Switzerland
4Department of Chemistry and Physics of Materials, University of Salzburg,
Jakob-Haringer-Strasse 2A, 5020 Salzburg, Austria

†These authors equally contributed;
∗To whom correspondence should be addressed;
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
> Topological superconductivity emerges in chains or arrays of magnetic atoms
> coupled to a superconductor. However, the external controllability of such
> systems with gate voltages is crucial for their future implementation in a
> topological quantum computer. Here we showcase the supramolecular assembly
> of radical molecules on Pb(111), whose discharge is controlled by the tip of
> a scanning tunneling microscope. Charged molecules carry a spin-1/2 state,
> as confirmed by observing Yu-Shiba-Rusinov in-gap states by tunneling
> spectroscopy at millikelvin temperature. Low energy modes are localized at
> island boundaries with an exponential decay towards the interior, whose
> spectral signature is consistent with Majorana modes protected by mirror
> symmetry. Our results open up a vast playground for the synthesis of gate-
> tunable organic topological superconductors.

### One sentence summary:

Radical molecules placed on superconducting Pb(111) represent a novel platform
for topological superconductivity

### Keywords:

Topological crystalline superconductor, tetraazapyrene radicals, scanning
tunneling microscopy, atomic force microscopy, molecular quantum dot, Yu-
Shiba-Rusinov states

## Introduction

Majorana zero modes (MZM) in topological superconductors obey non-Abelian
statistics and are considered as the most promising building blocks for
constructing topological qubits (?). Topological superconductivity (TS) can be
obtained in hybrid systems by coupling semiconducting nanowires with strong
spin-orbit interaction (?, ?), ferromagnetic atomic chains (?, ?, ?, ?, ?, ?),
or magnetic islands (?, ?, ?, ?) to a $s$-wave superconductor. One signature
of MZMs is a zero-energy conductance peak, measured with a scanning tunneling
microscope (STM) or in transport experiments. This zero-energy conductance
peak appears close to the zero-dimensional boundary separating the trivial
from the topological region. Higher dimensional boundaries instead, lead to
the formation of propagating edge states, namely chiral Majorana modes. Since
local disorder can close the gap of non-trivial phase by severely affecting
the proximitized superconducting states (?, ?, ?), assigning Majorana modes
requires a fundamental understanding of the system parameters with high
spectral resolution down to the atomic level (?, ?, ?, ?).

The ability of STM to create and probe lattices with single-atom precision
using manipulation techniques has offered unique opportunities for realizing
designer quantum materials at the atomic scale (?). With the spectral
resolution of scanning tunneling spectroscopy (STS), the detection of Yu-
Shiba-Rusinov (YSR) states arising from magnetic moments on a superconductor
(?, ?, ?) has revealed how important surface coordination (?), interatomic
coupling (?, ?, ?, ?), or magnetic anisotropy (?) are to the formation of a
complex topological phase diagram. Moreover, the experimental analysis of
these YSR bands in atomic structures has revealed the emergence of topological
non-trivial phases by probing localized zero-bias peaks consistent with their
topological origin, and thus attributed to MZMs (?, ?, ?, ?, ?, ?, ?, ?, ?,
?).

Beyond densely-packed atomic structures governed by nearest neighbor exchange
interaction (?), dilute spin chains or two-dimensional ”Shiba” lattices are
also an exciting platform for the emergence of topological superconductivity
(?, ?, ?, ?, ?, ?, ?, ?, ?, ?). By increasing the spacing $a$ between magnetic
impurities while keeping it smaller than the coherence length of the
superconductor $\xi$, the YSR band formation remains possible by the in-gap
state hybridization over a large number of nearest neighbors mediated by
Ruderman–Kittel–Kasuya–Yosida interaction (?). In such a regime (i.e. $k_{\rm
F}a$ $\gg$ 1 with $k_{\rm F}$ being the Fermi wave-vector), two-dimensional
ferromagnetic Shiba lattices are predicted to exhibit a rich phase diagram
with a large number of phases with high Chern numbers $\mathcal{C}$ (?, ?, ?),
where chiral MZMs are localized at edges of the island with an exponential
decay towards the island’s interior. Recently, topological phases protected by
spatial symmetries have been proposed to occur in a rich variety of
topological crystalline superconductors (?, ?, ?). Using STM, the first
attempt to build and to probe such atomic lattices showed interesting
signatures of edge modes consistent with a mirror-symmetry-protected
topological superconductor (?).

While the physics of YSR states and edge states can be addressed by tunneling
spectroscopy, the control over the chemical potential near these artificial
structures, an essential prerequisite for future applications to tune the
system with external gate voltages from trivial to topological, remains an
open issue. Inspired by recent works on the charge-state control of organic
molecules using the electric field of an STM tip (?, ?, ?), our work explores
the experimental realization of a two-dimensional spin lattice using the
supramolecular assembly of gate-tunable radical molecules on superconducting
Pb(111). This system could not only serve as a unique starting point for
investigating the interplay of a prototypical array of electron spins with a
superconductor but also provides a general playground for discovering
crystalline topological superconductivity in metal-free supramolecular
network-superconductor hybrids.

## Results and Discussion

###  Supramolecular assembly of TBTAP molecules on superconducting Pb(111).

As precursor, we used the 4,5,9,10-tetrabromo-1,3,6,8-tetraazapyrene (TBTAP)
molecule (Fig. 1A) consisting of an electron acceptor tetraazapyrene backbone
equipped with four peripheral bromine atoms (?). We recently showed that
TBTAP∙- radicals with a 1/2 spin state retain a single electron on Ag(111)
without using a decoupling layer (?). To obtain large supramolecular domains
of more than 100 nanometers in diameter, TBTAP molecules were sublimed in
ultra high vacuum on a Pb(111) substrate kept at about 200 K (?) (Fig. 1B). A
densely packed rectangular network of lattice parameter $a_{1}$ = 12.3 Å and
$b_{1}$ = 17.2 Å (arrows in Fig. 1C), is observed by STM as alternating dark
and bright rows. The corresponding image obtained by atomic force microscopy
(AFM) (Fig. 1D) shows each Br atom bound to the TAP backbone as bright
protrusion allowing us to assign the exact molecule position in the array (see
models in Fig. 1D). Similar to STM imaging, two AFM contrasts are observed as
a function of the considered rows denoted in the following as charged (c)
(dashed line) and neutral (n) (dotted line), which will be discussed later.

Using density functional theory (DFT) (?), we relaxed the TBTAP network on
Pb(111) (Fig. 1E, supplementary text and fig. S1). The assembly is in registry
with the Pb(111) surface in agreement with the experimental data ($a_{1}$ =
12.1 Å and $b_{1}$ = 17.5 Å , fig. S1). Molecules are stabilized by a
combination of halogen bonds between Br atoms (C–Br…Br–C) and TAP units
(C–N…Br–C). Dashed and dotted lines correspond to rows of charged (c) and
neutral (n) molecules, respectively. Molecules lie flat in a plane 3.4 Å above
the surface, suggesting that the variation of STM/AFM contrasts between
neighboring rows is due to the coexistence of two molecule charge states in
the network (?), rather than a difference in relative height (see fig. S1).

###  Electrical control of molecule’s charge state in the assembly.

To confirm this, we compared dI/dV point-spectra of TBTAP molecules located in
c and n rows, respectively (Fig. 2B). Molecule c (blue spectra) shows a strong
resonance D at $V_{\rm D}$ $\approx$ 1 V, which is absent for molecule n (red
spectra). The D resonance is assigned to a charge-state transition induced by
the local electric field of the tip from the anionic TBTAP∙- molecule to its
neutral TBTAP0 counterpart (?, ?, ?). Without gating, radical TBTAP∙-
molecules are obtained by the transfer of one electron from the surface to the
lowest unoccupied molecular orbitals (LUMO) (?, ?), leading to the LUMO
splitting into a singly-unoccupied molecular orbital (SUMO) and a singly-
occupied molecular orbital (SOMO) (see supplementary text, figs. S2-S4).
Charging events expected as a dip in dI/dV spectra for negative voltages were
not observed for both type of molecules.

Figure 2C shows a constant-height dI/dV map acquired at the threshold voltage
$V_{\rm D}$. Rings/dots of high conductance centered to molecules c are the
hallmark of a successful discharge, which also indicates the spatial position
of the electron in TBTAP∙- molecules prior to its removal. Using the double-
junction tunneling barrier (DJTB) model (?, ?, ?), the efficiency with which
the tip locally discharges nearby molecules is characterized by the lever arm
$\mathcal{L}$, which at first approximation linearly depends on the tip-sample
voltage $V_{\rm S}$ and its position with respect to the molecule
(supplementary text, figs. S5-S6). Figure 2D shows a dI/dV cross-section
acquired across n-c-n rows (plain line in Fig. 2A). Discharging rings are
absent along n rows since no charge can be extracted from neutral TBTAP0. The
discharge parabola is centered to the TBTAP∙- with its bottom ($\approx$ 0.9
V) corresponding to the resonance. Due to the linear voltage-dependency of
$\mathcal{L}$ the parabola branches expand with increasing $V_{\rm S}$, thus
reflecting the increase in size of rings in dI/dV maps with increasing
voltages (Figs. 2F-I, see supplementary text).

At fixed $V_{\rm S}\geq V_{\rm D}$, discharging rings form a superlattice of
parameter $a_{2}$ = 20 Å and $b_{2}$ = 39.7 Å and rotated by 30° with respect
to the molecular lattice observed in dI/dV mapping (Fig. 2C). Their diameters
vary between neighboring TBTAP∙- as the result of a local modulation of the
resonance, estimated to $\approx$ 150 mV by comparing the bottom of each
parabola (dashed line in Fig. 2E). For $V_{\rm S}$ $\geq$ 1.1 V and when the
tip is located between two neighboring molecules, the parabolas start to merge
promoting the removal of two electrons $2e$ from the neighboring molecules
(region $e$ $\geq$ 1). Accordingly, increasing $V_{\rm S}$ in a series of
spatial dI/dV maps leads to ring expansion (Fig. 2G) followed by their
coalescence (Fig. 2H). In contrast to a simple superposition of rings expected
for non-interacting quantum dots, their fusion as observed in Figs. 2H-I
indicates a cascade discharge along c rows and thus a manifestation of the
electron correlation in the supramolecular assembly (?) (see supplementary
text, fig. S6).

###  Yu-Shiba-Rusinov bound states of radical molecules.

Radical TBTAP∙- on Pb(111) feature a S = 1/2 ground state with a strong spin-
polarization according to DFT calculations (Fig. 3A). We probed YSR bound
states of radical TBTAP∙- in the middle of a molecular island using tunneling
spectroscopy with a metallic tip at $T$ = 35 mK (Fig. 3B) (?). Figure 3C
compares dI/dV spectra of three representative TBTAP∙- molecules marked in
Fig. 3B (black spectra, Fig. 3C) with that of Pb(111) (blue) and a neutral
TBTAP0 (red). For the last two, a hard gap centered to $E_{\rm F}$ and framed
by the two coherence peaks of Pb(111) (?) is systematically measured without
in-gap states. Each TBTAP∙- spectrum additionally shows one pair of YSR states
at energies $\varepsilon_{\alpha}$ = $\pm$ 460 µeV, $\varepsilon_{\beta}$ =
$\pm$ 720 µeV and $\varepsilon_{\gamma}$ = $\pm$ 940 µeV (dotted lines),
resulting from the spin-1/2 nature of radical TBTAP∙-. By applying an out-of-
plane magnetic field of 0.5 T, we also quenched the superconductivity state to
probe the Kondo resonance (figs. S8) and estimated its Kondo temperature
$T_{\rm K}$ to be 10.3 K (figs. S7) (?). Electron-like and hole-like wave-
functions of TBTAP∙- were also probed by d$I$/d$V$ mapping at the
$\varepsilon_{\alpha}^{\pm}$, $\varepsilon_{\beta}^{\pm}$ and
$\varepsilon_{\gamma}^{\pm}$ energies (Figs. 3E-G), respectively. The typical
donut-shape is similar to the spin density map (Fig. 3A), while their energies
depends on the molecule positions in the assembly. We also infer that the
shift of the YSR states to higher energies as compared to that of the isolated
molecule (fig. S9) and their spatial distribution points to a coupling of the
quasi-particle excitations within the supramolecular network (?). Figure 3G
shows a dI/dV cross-section across the island (red arrow of Fig. 3D), where
white dotted lines refer to the $\varepsilon_{\alpha,\beta,\gamma}^{\pm}$ YSR
energies. Broader resonances near $E_{\it F}$ coexist with the YSR peaks as
marked by black arrows in Figs. 3C and G. These low-energy modes (LEM) are
systematically observed with the highest magnitude for molecules aligned along
the white dashed lines marked in the zero-energy dI/dV map of Fig. 3D.

###  Spectral signature and localization length of low-energy modes near an
island edge.

The intrinsic electron-hole symmetry of zero-energy modes, imposed by the
Bogoliubov quasi-particle character, can be probed by tunneling spectroscopy
using superconducting STM tips ($\Delta_{\rm T}$ = 1.35 meV is the
superconducting pairing energy of the tip). Experimentally, a zero-energy peak
appears in dI/dV spectra as a pair of peaks of equal amplitude shifted from
zero to the finite voltages eV = $\pm$ $\Delta_{\rm T}$, while the
superconducting edge is observed at $\pm$ ($\Delta_{\rm T}$ \+ $\Delta_{\rm
S}$) = $\pm$ 2.7 meV ($\Delta_{\rm S}$ = 1.35 meV is the superconducting gap
of the substrate). Using bulk Pb tips at $T$ = 1 K (Fig. 4) (?), we confirmed
the presence of YSR in-gap states at the TBTAP∙- locations by dI/dV point-
spectra (Fig. 4B) acquired along seven TBTAP∙- molecules of a c row (Fig. 4A,
fig. S12). Due to the larger thermal broadening of $\approx$ 90-100 µeV at 1K,
the accurate assignment of the $\varepsilon_{\alpha,\beta,\gamma}$ energies is
less evident than that of the millikelvin measurements since these peaks merge
into a single resonance found at eV = $\pm(\Delta_{\rm T}$ \+
$\varepsilon_{\alpha,\beta,\gamma}$) $\approx$ $\pm$ 2.1 meV. Note also that
the YSR states are always accompanied by a broader resonance near zero-energy
(i.e. $\pm$ $\Delta_{T}$) which is the fingerprint of the low-energy modes
using superconducting tips.

The constant-height dI/dV maps of Figs. 4D and E compare the spatial
distribution near the edge of an island of the hole-like wavefunctions
extracted at $\varepsilon^{+}$ with the LEM wavefunctions at +$\Delta_{T}$.
While the DOS at the YSR energy is homogeneous along the c rows, the LEM lines
emerge from the ferromagnetic edge schematized in the model of Fig. 4F. They
propagate along the direction rotated by 60° with respect to the edge
corresponding to a ferromagnetic direction of the spin structure. Figure 4G
shows a dI/dV(V,X) cross-section acquired along one LEM line marked by a blue
dashed line in Fig. 4E. All sub-gap excitations now appear at zero energy with
equal amplitudes between electron-like and hole-like regions (Fig. 4H). This
observation, in stark contrast with the strong intrinsic electron-hole
asymmetry of the YSR resonances (Fig. 4B), underlines the zero-energy
character of these edge modes.

We next characterize the LEM localization length (Fig. 4I) by comparing
dI/dV(X) profiles along the LEM line (blue) with that obtained at the YSR
energy (gray) (see also supplementary text, fig. S13). In contrast to the
continuous DOS at $\varepsilon^{+}$, the LEM wavefunction has a maximum
amplitude at the border of the island ($X$ = 0) and decays towards the
interior but without completely vanishing. In Fig. 4I, we estimated the
experimental decay of the edge mode by fitting its envelope (dashed line) with
a function $f(x)$ composed of two exponents representing the short $\xi_{1}$ =
3 nm and the long $\xi_{2}$ = 110 nm localization length (?). We explain it by
considering the TBTAP∙- network as a lattice of spin-1/2 impurities with long-
range YSR overlap coupled to a superconductor, as reported in References (?,
?).

###  Theoretical analysis.

To further rationalize our findings, we used a tight binding model on a
rectangular spin lattice similar to one introduced by Soldini et al. (?) in
order to describe the spatial-symmetry-protected topological order of an
antiferromagnet-superconductor hybrid structure (?). As suggested by our
tunneling measurements (fig. S10), we assume an antiferromagnetic ordering of
the spin (schematized by red and blue arrows in Fig. 5A and Fig. 1E), along
the lattice imposed by the TBTAP molecular assembly (dashed line). Based on
our STM observations (Fig. 1B and Fig. 4C), we construct a prototypical
”Shiba” island mimicking the supramolecular network boundaries by considering
only ferromagnetic edges along the [110] directions with respect to the
molecular lattice (red line in Fig. 5A). Figure 5B shows the calculated zero-
energy LDOS map of the system, which demonstrates the formation of edge modes
in agreement with reference (?). Theoretical LDOS spectra are plotted in Figs.
5C and D for two edge positions marked by green and red squares in Fig. 5B,
respectively. These edge modes have two typical spectral signatures consisting
of either two peaks of equal amplitude split from zero energy (green in Fig.
5C) or three resonances centered to zero energy (red in Fig. 5C). They are
both framed by a topological gap ($\pm\Delta_{\rm Top}$) extracted at the
center of the island (yellow spectra in Figs. 5C-D) as well as the
superconducting gap at $\pm\Delta_{\rm S}$ (black spectra). Importantly, the
45°-edges of the antiferromagnetic island respect the underlying spatial
symmetries, namely mirror symmetries, (?), such that a gapped topological
crystalline superconducting phase with topological edge states can form. The
experimental spectroscopic signatures of the LEM (Figs. 5E and F), acquired
near an edge at 50 mK with a metallic tip (see figs. S14 and S15), is in good
agreement with the theoretical predictions of these topological edge modes.

## Conclusion and outlook

In conclusion, we demonstrate the formation of an extended array of electron
spins on superconducting Pb(111) through the supramolecular assembly of
organic radicals. Occupied by a single electron transferred from the
substrate, radical molecules are in a spin-1/2 ground state confirmed by
probing one pair of Yu-Shiba-Rusinov in-gap states in differential conductance
spectra. In the two-dimensional supramolecular assembly, spectroscopic
signatures of low energy modes (LEM) are observed in tunneling spectroscopy
near edges of the island with a long decay towards the interior. Using both
metallic and superconducting tips, we characterized the near zero-energy
character of this resonance, its intrinsic particle-hole symmetry, the
localization length as well as site-dependent spectral signatures. Altogether,
these key features confirmed by theory are consistent with the emergence of
topological non-trivial modes in an antiferromagnet-superconductor hybrid
structure that can be assigned to Majorana modes (?).

Such a spatial-symmetry-protected topological superconductor has a complex
phase diagram which crucially depends on the edge terminations of the system
boundaries (fig. S10) as well as the lattice parameter $a$ (i.e. hopping
parameters $t$) (?). In fig. S10 (see supplementary text), we also explored
disorders or alternative boundaries that can break the mirror symmetry of the
system. Since $a$ depends on the molecular spacing, future work could explore
the design of the precursor’s side groups to access variable sizes and lattice
symmetries on alternative superconducting platforms (?). Importantly, our
findings demonstrate the reversible control of the charge (spin) in radical
molecules by the local electric field of the tip, opening interesting avenues
for the fine-tuning of the system with external gate voltages. Creating local
charge defects in the molecular assembly using probe chemistry (?) might also
allow to investigate the effect of disorder on the topological phase as well
as the topological edge modes. Overall, our work constitutes key advances in
designing gate-tunable organic topological superconductors by the self-
assembly of organic metal-free molecules in proximity to a superconducting
substrate.

## Acknowledgments

Funding: E.M. and R.P. acknowledge funding from the Swiss Nanoscience
Institute (SNI) and the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (ULTRADISS grant
agreement No 834402 and supports as a part of NCCR SPIN, a National Centre of
Competence (or Excellence) in Research, funded by the SNF (grant number
51NF40-180604). E.M., T.G. and S.-X.L. acknowledge the Sinergia Project funded
by the SNF (CRSII5_213533). E.M., T.G. and R.P. acknowledge the SNF grant
(200020_188445). T.G. acknowledges the FET-Open program (Q-AFM grant agreement
No 828966) of the European Commission. S.-X. L. acknowledges the grant from
the SNF (200021_204053). J.-C.L. acknowledges funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement number 847471. U.A. acknowledges funding by
the SNF Professorship (Grant No. PP00P2 187185/2). R.H. acknowledges the
European Union’s Horizon 2020 research and innovation programme under Grant
Agreement No. 862046 and the ERC grant under Grant Agreement No.757725.
Calculations were performed on UBELIX (http://www.id.unibe.ch/hpc), the HPC
cluster at the University of Bern. C.L. acknowledges the Georg H. Endress
Foundation for financial support. W. W. gratefully acknowledge financial
support from the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) through the Collaborative Research Centre ”4f for Future” (CRC
1573, project number 471424360) project B2. R.P. acknowledges Ruslan Temirov
for the fruitful discussions on the improvement of the spectral resolution of
the Basel microscope.

Authors contributions: R.P., S.-X.L., S.D and E.M. conceived the experiments.
P.Z., Ro.H., S.-X.L. and S.D. synthesized the materials. C.D., C.L., and R.P.
performed the STM/AFM measurements at 1 K. J.-C.L., H.C. and W.W. performed
the millikelvin measurements. U.A. performed DFT calculations. Ri.H., D.L. and
J.K. performed tight-binding calculations. R.P. analyzed the data and wrote
the manuscript. All authors discussed on the results and revised the
manuscript.

Competing interests: The authors declare no competing financial interests.

Data and materials availability: Data in formats other than those presented
within this paper are available from the corresponding authors upon reasonable
request.

## Supplementary materials

Materials and Methods
Supplementary Text
Figs. S1 to S15
References (50-69+)

## References

  * 1. C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * 2. V. Mourik, et al., Science 336, 1003 (2012).
  * 3. R. M. Lutchyn, et al., Nat. Rev. Mat. 3, 52 (2018).
  * 4. S. Nadj-Perge, et al., Science 346, 602 (2014).
  * 5. M. Ruby, et al., Phys. Rev. Lett. 115, 197204 (2015).
  * 6. R. Pawlak, et al., npj Quant. Info. 2, 1 (2016).
  * 7. B. E. Feldman, et al., Nat. Phys. 13, 286 (2017).
  * 8. H. Kim, et al., Sci. Adv. 4, eaar5251 (2018).
  * 9. L. Schneider, et al., Nat. Nanotechnol. 17, 384 (2022).
  * 10. G. C. Ménard, et al., Nat. Comm. 8, 2040 (2017).
  * 11. A. Palacio-Morales, et al., Sci. Adv. 5, eaav6600 (2019).
  * 12. Z. Wang, et al., Science 367, 104 (2020).
  * 13. S. Kezilebieke, et al., Nature 588, 424 (2020).
  * 14. C. Reeg, O. Dmytruk, D. Chevallier, D. Loss, J. Klinovaja, Phys. Rev. B 98, 245407 (2018).
  * 15. E. Prada, et al., Nat. Rev. Phys. 2, 575 (2020).
  * 16. R. Hess, H. F. Legg, D. Loss, J. Klinovaja, Phys. Rev. B 106, 104503 (2022).
  * 17. L. Schneider, et al., Nat. Comm. 11, 4707 (2020).
  * 18. H. Ding, et al., Proc. Nat. Acad. Sci. 118, e2024837118 (2021).
  * 19. F. Küster, et al., Proc. Nat. Acad. Sci. 119, e2210589119 (2022).
  * 20. E. Liebhaber, et al., Nat. Comm. 13, 2160 (2022).
  * 21. A. A. Khajetoorians, D. Wegner, A. F. Otte, I. Swart, Nat. Rev. Phys. 1, 703 (2019).
  * 22. L. Yu, Acta Phys. Sin. 21, 75 (1965).
  * 23. H. Shiba, Progr. Theor. Phys. 40, 435 (1968).
  * 24. A. I. Rusinov, JETP Lett. 9, 85 (1969).
  * 25. F. Friedrich, R. Boshuis, M. Bode, A. Odobesko, Phys. Rev. B 103, 235437 (2021).
  * 26. N. Hatter, B. W. Heinrich, M. Ruby, J. I. Pascual, K. J. Franke, Nat. Comm. 6, 8988 (2015).
  * 27. S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, A. Yazdani, Phys. Rev. B 88, 020407 (2013).
  * 28. J. Klinovaja, P. Stano, A. Yazdani, D. Loss, Phys. Rev. Lett. 111, 186805 (2013).
  * 29. F. Pientka, L. I. Glazman, F. von Oppen, Phys. Rev. B 88, 155420 (2013).
  * 30. B. Braunecker, P. Simon, Phys. Rev. Lett. 111, 147202 (2013).
  * 31. S. Hoffman, J. Klinovaja, T. Meng, D. Loss, Phys. Rev. B 92, 125422 (2015).
  * 32. J. Röntynen, T. Ojanen, Phys. Rev. Lett. 114, 236803 (2015).
  * 33. J. Li, et al., Nat. Comm. 7, 12297 (2016).
  * 34. S. Rachel, E. Mascot, S. Cocklin, M. Vojta, D. K. Morr, Phys. Rev. B 96, 205131 (2017).
  * 35. R. Pawlak, S. Hoffman, J. Klinovaja, D. Loss, E. Meyer, Prog. Part. Nucl. 107, 1 (2019).
  * 36. B. Jäck, Y. Xie, A. Yazdani, Nat. Rev. Phys. 3, 541 (2021).
  * 37. M. O. Soldini, et al., Nat. Phys. pp. 1745–24811–7 (2023).
  * 38. A. P. Schnyder, P. M. R. Brydon, J. Phys.: Cond. Matt. 27, 243201 (2015).
  * 39. F. Schindler, et al., Sci. Adv. 4, eaat0346 (2018).
  * 40. M. Geier, L. Trifunovic, M. Hoskam, P. W. Brouwer, Phys. Rev. B 97, 205135 (2018).
  * 41. I. Fernández-Torrente, D. Kreikemeyer-Lorenzo, A. Stróżecka, K. J. Franke, J. I. Pascual, Phys. Rev. Lett. 108, 036801 (2012).
  * 42. N. Kocić, et al., Nano Lett. 15, 4406 (2015).
  * 43. C. Li, et al., Nat. Comm. 14, 5956 (2023).
  * 44. P. Zhou, et al., Chem. Comm. 57, 12972 (2021).
  * 45. Materials and methods are available as supplementary materials.
  * 46. H. Li, et al., Nature Phys. 17, 1114 (2021).
  * 47. M. Ruby, B. W. Heinrich, J. I. Pascual, K. J. Franke, Phys. Rev. Lett. 114, 157001 (2015).
  * 48. J.-C. Liu, et al., ACS Materials Lett. 5, 1083 (2023).
  * 49. S. Kawai, et al., Sci. Adv. 6, eaay8913 (2020).

Fig. 1: Supramolecular assembly of radical TBTAP∙- molecules on Pb(111). (A),
Chemical structure of the 4,5,9,10-tetrabromo-1,3,6,8-tetraazapyrene (TBTAP)
molecule. (B), STM overview of the supramolecular assembly on Pb(111) ($V_{\rm
s}$ = -0.5 V, $I_{\rm t}$ = 1 pA). (C), Close-up STM image of the molecular
lattice ($V_{\rm s}$ = 0.8 V, $I_{\rm t}$ = 0.8 pA). The unit cell is marked
with arrows. (D), Corresponding AFM image acquired with Br-terminated tip. The
molecular lattice consists of rows of charged (c) and neutral (n) TBTAP
molecules. Red, blue, white and black colors of the ball-and-stick models
refer to bromide, nitrogen, hydrogen and carbon atoms. (E), Perspective view
of the TBTAP assembly on Pb(111) optimized by DFT calculations (fig. S1).
Dashed and dotted lines correspond to rows of charged (c) and neutral (n)
molecules, respectively. Red and blue arrows represent the spin lattice of the
TBTAP assembly.

Fig. 2: Electron superlattice of the supramolecular assembly. (A), STM image
of the molecular assembly ($V_{\rm s}$ = 0.8 V, $I_{\rm t}$ = 50 pA). (B),
Single-point dI/dV spectra of c (blue) and n (red) molecules. The D resonance
peak corresponds to the tip-induced discharge event (Set-points : 100 pA, -0.8
V, lock-in parameters: $f$ = 611 Hz, $A_{\rm mod}$ = 20 mV). (C) dI/dV mapping
showing discharge rings centered to each molecule of c rows. (D), dI/dV cross-
section across n-c-n rows (plain line in A) showing the electron localization.
(E), dI/dV cross-section across five TBTAP∙- molecules (dashed line in A). The
dashed line corresponds to multiple discharge events ($e$ is the number of
removed charges) induced by tip gating. (F-I), Series of dI/dV mapping for
increasing tip-sample voltage $V_{\rm S}$ showing the expansion of the ring
diameter and cascade discharge for $V_{\rm s}$ $\geq$ 1.15 V. Scale bars are 1
nm.

Fig. 3: Yu-Shiba-Rusinov states of TBATP.- molecules. (A), Spin density map of
the TBTAP∙- molecule on Pb(111) calculated by DFT. (B) dI/dV map of the
TBTAP∙- assembly acquired at $T$ = 35 mK with a metallic tip for $V_{\rm S}$ =
1.28 meV (scale bar is 1 nm, lock-in parameters: $f$ = 3.28 kHz, $A_{\rm mod}$
= 40 µV, tunneling parameters: $I_{\rm T}$ = 200 pA, $V_{\rm S}$ = 5 mV.).
(C), Exemplary dI/dV point-spectra of Pb(111) (red) and a neutral TBTAP0
(blue). Black curves are three representative spectra of TBTAP∙- molecules
marked in B. Spectra are shifted by 1 µS for clarity. Pairs of YSR resonances
at $\varepsilon_{\alpha}$ = $\pm$ 460 µeV, $\varepsilon_{\beta}$ = $\pm$ 720
µeV and $\varepsilon_{\gamma}$ = $\pm$ 940 µeV are molecule-dependent along
the $c$ row. The black arrows mark the low-energy modes near $E_{\rm F}$. (D),
Near zero-energy dI/dV map ($\varepsilon$ = +100 µeV) showing the spatial
distribution of the LEM lines (white dashed lines). (E-F) Series of dI/dV maps
extracted at energies of $\pm$ $\varepsilon_{\alpha}$, $\pm$
$\varepsilon_{\beta}$ and $\pm$ $\varepsilon_{\gamma}$, respectively. (G),
dI/dV(V,Y) cross-section acquired along the red arrow of D. White dotted lines
refer to $\pm$ $\varepsilon_{\alpha}$, $\pm$ $\varepsilon_{\beta}$ and $\pm$
$\varepsilon_{\gamma}$ energies. Black dashed lines show the superconducting
edge $\pm$ $\Delta$ while black arrows mark the LEM.

Fig. 4: Tunneling spectroscopy with superconducting tips of the LEM near an
edge . (A), dI/dV cross-section acquired along seven molecules of a c row
(black line in C) (lock-in parameters: $f$ = 611 Hz, $A_{\rm mod}$ = 25 µV,
tunneling parameters: $I_{\rm T}$ = 300 pA, $V_{\rm S}$ = 5mV). White dotted
lines are the $\varepsilon^{\pm}_{\alpha,\beta,\gamma}$ energies. Dashed lines
correspond to $\pm$ ($\Delta_{\rm T}$ \+ $\Delta_{\rm S}$) and $\pm$
$\Delta_{\rm T}$, respectively. (B), dI/dV spectra extracted at the red, blue
and black arrows of A. (C), STM topographic image of and edge and (D-E), dI/dV
maps acquired near the island edge of the LEM wavefunction and the
$\varepsilon^{+}_{\alpha,\beta,\gamma}$ energy. Scale bar is 1 nm. (F),
Schematic of spin structure superimposed on top of the LEM wavefunction. Red
and blue dots refer to spin up and spin down, respectively. LEM lines are
rotated by 6° as compared to the $[1\bar{1}0]$ directions of Ag(111) (dark
lines). (G), dI/dV cross-section across the edge taken along one LEM line
(blue dashed line of C). (H), dI/dV spectra at the blue, red and black arrows
of G. The LEM appears as a pair of peaks of equal magnitude at $\pm\Delta_{\rm
T}$. (I), Extracted decay length of the LEM wavefunction (blue) as compared to
the $\varepsilon^{+}_{\alpha,\beta,\gamma}$ energy (gray) along the blue
dashed line of C. The island border is set at $X$ = 0 nm. The dashed line is a
fit of the edge mode envelope with two exponential functions that estimate a
short ($\xi_{1}$ = 3 nm) and a long ($\xi_{2}$ = 110 nm) localization length.

Fig. 5: Comparison of LEM theoretical modeling with experimental data. (A),
Antiferromagnetic spin lattice (dashed) formed by the supramolecular assembly
(red, spin up; blue, spin down). The molecular lattice imposes different
hopping strengths $t_{\rm x}$ and $t_{\rm y}$ in $x$ and $y$ directions. Red
dashed line is the assumed 45°-edge with respect to the molecular lattice.
(B), Spatial LDOS map of the lattice at zero energy showing the emergence of
edge modes. (C-D), Theoretical LDOS($\omega$) spectra of the LEM extracted at
edges (green and orange squares in A) as compared to the pristine Pb spectrum
(black) and the spectrum at the center of the island (yellow). Dashed and
dotted lines refer to the superconducting ($\pm$ $\Delta_{\rm S}$) and
topological gaps ($\pm$ $\Delta_{\rm Top}$), respectively. (E-F), Experimental
dI/dV spectra of the edge modes acquired at 50 mK with a metallic tip (lock-in
parameters: $f$ = 3.28 kHz, $A_{\rm mod}$ = 10 µV, tunneling parameters:
$I_{\rm T}$ = 200 pA,$V_{\rm S}$ = 5mV.).