# StyO: Stylize Your Face in Only One-Shot

Bonan Li
University of Chinese Academy of Sciences
<EMAIL_ADDRESS>

Xuecheng Nie
MT Lab, Meitu Inc.
<EMAIL_ADDRESS>

Yinhan Hu
University of Chinese Academy of Sciences
<EMAIL_ADDRESS>

Equal Contribution Zicheng Zhang∗
University of Chinese Academy of Sciences
<EMAIL_ADDRESS>

Congying Han
University of Chinese Academy of Sciences
<EMAIL_ADDRESS>

Tiande Guo
University of Chinese Academy of Sciences
<EMAIL_ADDRESS>

###### Abstract

This paper focuses on face stylization with a single artistic target. Existing
works for this task often fail to retain the source content while achieving
geometry variation. Here, we present a novel StyO model, _i.e_. Stylize the
face in only One-shot, to solve the above problem. In particular, StyO
exploits a disentanglement and recombination strategy. It first disentangles
the content and style of source and target images into identifiers, which are
then recombined in a cross manner to derive the stylized face image. In this
way, StyO decomposes complex images into independent and specific attributes,
and simplifies one-shot face stylization as the combination of different
attributes from input images, thus producing results better matching face
geometry of target image and content of source one. StyO is implemented with
latent diffusion models (LDM) and composed of two key modules: 1) Identifier
Disentanglement Learner (IDL) for disentanglement phase. It represents
identifiers as contrastive text prompts, _i.e_. positive and negative
descriptions. And it introduces a novel triple reconstruction loss to fine-
tune the pre-trained LDM for encoding style and content into corresponding
identifiers; 2) Fine-graind Content Controller (FCC) for recombination phase.
It recombines disentangled identifiers from IDL to form a augmented text
prompt for generating stylized faces. In addition, FCC also constrains the
cross-attention maps of latent and text features to preserve source face
details in results. The extensive evaluation shows that StyO produces high-
quality images on numerous paintings of various styles and outperforms the
current state-of-the-art. Code will be released upon acceptance.

##  1 Introduction

Face stylization aims at automatically creating personalized artistic
portraits from face photographs. It is a fundamental problem in vision and
graphics communities [1, 19, 10, 40] and widely applied in creative industries
such as social media avatars, films, advertising, etc. Recently, one-shot
solution for this task has drawn lots of attention, due to its significant
feature of data-efficiency, which makes the deployment easier in practice.

Existing works [3, 52, 50, 49] consider one-shot face stylization as a style
transfer problem [7], _i.e_., transferring face style of a single target image
into face of the source image, and solve this problem with Generative
Adversarial Networks (GANs) [9, 17]. Despite of their success, we observe that
even state-of-the-art methods still fail to achieve reasonable style (_e.g_.,
geometry variations) while maintain source content (_e.g_., facial and hair
colors), as shown in Figure LABEL:fig:teaser. This is caused by two main
reasons. First, these methods heavily relied on pre-trained GAN models on
large real face dataset [17], thus hard to match styles of faces, especially
geometry changes, from another domain with one-shot guidance, due to the
capability limitation of GANs for producing distribution variations. Second,
they entangle the style and content information of facial images together in
the latent space, making current encoding and inversion techniques difficult
to derive accurate latent code representing the stylized faces. It is
important to address above issues for facilitating usage of relevant
techniques in real world.

Motivated by this, we propose a novel StyO model in this paper for pushing
forward the frontier of the task of Stylizing the face in only One-shot. For
improving the generation variety, StyO exploits Diffusion Denosing
Probabilistic Models (DDPMs) [12, 41, 34] instead of GANs, given the
superiority of DDPM to produce high-quality images in wild range of
distribution. For deriving accurate representation of stylized faces, StyO
leverages a disentanglement and recombination strategy. Specifically, StyO
factorizes the style and content information of source and target images into
different identifiers, then fuse them in a cross manner to form prompts, which
plays as representations of stylized faces to drive DDPMs to generate
reasonable images. In this way, StyO decomposes complex images into
independent and specific attributes, and simplifies one-shot face stylization
as the combination of different attributes from input images, thus producing
results better matching face style of target image and content of source one.

In particular, StyO is implemented with the efficient Latent Diffusion Models
(LDMs) [34]. It is composed of two core modules: 1) Identifier Disentanglement
Learner (IDL). IDL aims to disentangle the style and content of an image into
different identifiers. Here, IDL represents identifiers as text descriptors
for fully leveraging the powerful capability of text-based image generation of
LDMs. Then, IDL defines contrastive disentangled prompt template as text
descriptions for input images with content and style identifiers, _e.g_., “a
drawing with [Source/Target Style Identifier][not Target/Source Style
Identifier] style of [Source/Target Content Identifier][not Target/Source
Content Identifier] of portrait” for source and target images, respectively.
In addition, we introduce an auxiliary prompt template with only source and
target style identifiers to describe an auxiliary image set with same style of
source one. This helps style and content identifiers correctly representing
corresponding attributes of images, and also avoid the risk of relating
style/content information with other words in prompts rather than
corresponding identifiers. Given prompts defined above, StyO builds text-image
pairs to fine-tune the pre-trained LDMs for injecting attributes of images
into identifiers, thus achieving the disentanglement goal. 2) Fine-grained
Content Controller (FCC). FCC aims to recombine style and content identifiers
from IDL to generate stylized face images. Specifically, FCC constructs prompt
template as “a drawing with [Target Style Identifier][not Source Style
Identifier] style of [Source Content Identifier][not Target Content
Identifier] of portrait” to describe stylized faces with style from target and
content from source, which is used as condition to LDMs for deriving
stylization results. However, we find that only with the above reconstructed
prompt will cause the loss of fine-grained details of source image, such as
head pose, hair color, beard style, etc, due to the randomness feature of
diffusion models yielding less control on diversity. To solve this issue, FCC
presents a new manipulation mechanism on attention maps to improve
controllability. This is inspired by the fact that attention maps from cross
attention layers are semantically correlated with texts in prompts. Hence, FCC
extracts attention maps for source content and uses them to replace that for
stylized one, which effectively controls fine-grained details of results. In
addition, FCC proposes to augment prompts by repeating identifiers. This
simple augmentation strategy further successfully advances generation quality.
By this design as shown in Figure 1, StyO can produce faces with suitable
target style while maintain source content.

Extensive experiments show that the proposed StyO model can produce
surprisingly good quality of facial images on various styles in the one-shot
manner. It also outperforms state-of-the-art models quantitatively. Our
contributions are summarized in four folds: First, to the best knowledge, we
are the first to deploy diffusion models for tackling one-shot face
stylization; Second, we present a novel StyO model with disentanglement and
recombination strategy, which effectively solve core issues of prior works;
Third, we present a classical way to encoding different attributes of images
into multiple identifiers; Fourth, our StyO model sets new state-of-the-art
for one-shot face stylization.

Figure 1: Overview of the proposed StyO for one-shot face stylization. StyO
consists of two core modules: Identifier Disentanglement Learner (IDL) and
Fine-grained Content Controller (FCC). IDL learns to disentangle style and
content information of source and target images into different identifiers.
Then, FCC recombine disentangled identifiers for fusing source content and
target style to generate the stylized facial images.

##  2 Related Work

###  2.1 Diffusion model

The previous few years have seen the rise of diffusion models for image
generation, which breaks the long-time dominance of generative adversarial
networks (GANs) [9]. Since [39] first propose the denoising diffusion
probabilistic models (DDPM) [12, 29, 4] to form the image generation as a
Markov Process, the community is inspired to investigate the generative model
from non-Markov process [41], score matching [42], and stochastic differential
equation [43]. Besides, many sampling techniques are proposed to speedup [41,
23, 16] or condition [4, 13] the inference. Recent text-to-image diffusion
models [28, 33, 34, 37] trained on extremely large-scale data have exhibited
remarkable ability to generate diverse, creative images controlled by text
prompts. [5, 36, 27, 18, 11] shows that these models can edit images provided
by users with a text prompt effectively, thereby enabling text-based
stylization. However, compared with the fine-grained details provided by an
artistic exemplar, the text prompt only provides a rough description of
desirable style, thereby it is hard to obtain acceptable results with previous
methods. To our best knowledge, StyO is the first to explore the exemplar-
guided fine-style transfer in context of text-to-image diffusion model.

###  2.2 Face stylization

Face stylization has been widely studied in the field of Neural Style Transfer
(NST), image-to-image translation (I2IT), and GAN adaptation. NST methods
match the feature distribution [7, 22, 31, 21, 15] to transfer artistic
texture. These methods can achieve pleasing results when the artistic
portraits have an intense color or texture pattern, but usually underperform
in styles involving significant geometric distortion of facial features, such
as anime and cartoons. I2IT methods [14, 51, 25, 46, 20] learn a mapping from
a large-scale dataset to translate an input image from a source domain to a
target domain. [2, 8, 38, 26, 10, 40, 24] design specific modules and losses
based on facial knowledge (e.g., landmarks, expression) to facilitate the face
stylization. These methods can achieve reasonable texture and geometric
variation when the training dataset includes more than hundreds of artistic
portraits. However, collecting such extensive images for a specific style is
not non-trivial, thus limiting the usage of I2IT in practice.

Recently, the works about domain adaptation of GANs [30] provide a creative
way to solve the task. Since the pre-trained StyleGAN [17] forms a generative
prior of natural faces, an artistic portrait, which belongs to the proximal
domain of natural face, can guide the adaptation of StyleGAN [52, 6, 3, 50] to
generate artistic portraits. Thereafter, given a face photograph, a latent
code can be found by inversion techniques [47] and fed into adapted StyleGAN
to generate the stylized image. However, the strong facial prior of pre-
trained StyleGAN limits the geometric variation, and compressing face images
into latent code also drops the content partially. Hence these methods perform
not well when implementing artistic images of large geometric deformations and
face photographs out of the facial prior. In contrast, StyO properly utilizes
the potent generative and reconstructive abilities of LDM to alleviate these
problems.

Figure 2: Pipeline of the proposed StyO. Left panel: The training phase. StyO
constructs text-image pairs for source, target images and auxiliary image set
with a contrastive disentangled prompt template. Then, StyO fine-tunes a pre-
trained latent diffusion model in the one-shot manner. In this way, StyO
injects attributes of images into different identifiers. Right panel: The
inference phase. Given style and content identifiers, StyO recombines them to
form text prompt to generating stylized face. To maintain fine-grained
details, StyO extracts attention maps for source content and uses them to
replace that for stylized one. This process yields faces with suitable target
style while maintains good source content.

##  3 Method

###  3.1 Preliminaries

Text-to-image diffusion models [34, 37] are a family of generative models to
map a noise latent $\epsilon$ and a text prompt $\mathcal{P}$ to an image
$x_{0}$. In this paper, we focus on the pre-trained latent diffusion model
(LDM) [34], which is widely used as a base model in recent research. LDM first
encodes the image $x^{0}$ into latent space with a pre-trained encoder
$\mathcal{E}$, i.e., $z^{0}=\mathcal{E}(x^{0})$, while $x^{0}$ can be restored
by means of a pre-trained decoder $\mathcal{D}$. It then performs the
destructive and reverse processes in the latent space. Specifically, the
destructive process is known as a fixed-length Markov chain starting from
$z^{0}$ to $z^{T}$ with a Gaussian transition at each step. For each
$t\in\\{0,...,T\\}$, the transition holds that

$z^{t}=\sqrt{\alpha^{t}}z^{0}+\sqrt{1-\alpha^{t}}\epsilon,\ \epsilon\sim
N(0,I),$ (1)

where $\\{\alpha^{t}\\}_{t=0}^{T}$ is a monotonic, strictly decreasing
sequence, in which $\alpha^{0}=1$ and $\alpha^{T}$ vanishes as $T$ increases.
This ensures $z^{T}$ roughly follows an isotropic Gaussian distribution. In
contrast, the reverse process involves sequential Gaussian sampling given by

$z^{t-1}=\mu(z^{t},\epsilon_{\theta},t)+\sigma(t)w,\ w\sim N(0,I).$ (2)

Here, $\mu$ and $\sigma$ are pre-defined functions to compute parameters of
Gaussian distribution. $\mu$ is parameterized by $\epsilon_{\theta}$ that is a
time-conditioned UNet [35] equipped with attention mechanism and trained to
achieve the objective

$\mathop{\mathrm{min}}_{\theta}{\mathbb{E}}_{z^{0},\epsilon,t}||\epsilon-\epsilon_{\theta}(z^{t},t,\gamma)||^{2}_{2}.$
(3)

$\gamma$ is the text embedding of $\mathcal{P}$ mapped by a frozen model
$\phi$, _e.g_., the CLIP text encoder [32]. The reverse process ensures the
final sample $z^{0}$ follows the distribution of latents, and the generated
image aligns with the prompt.

###  3.2 Framework of StyO

Given a source image $x_{src}$ in natural face domain, and a target image
$x_{tgt}$ in artistic portrait domain, our objective is to synthesize an image
with the source content as well as the target style. Building upon the state-
of-the-art LDM [34], we propose a novel approach named StyO, which consists
identifier disentanglement learner (IDL) and fine-grained content controller
(FCC) to tackle the task. In general, StyO learns disentangled identifier
representing content and style via IDL, and enables the retention of fine-
grained content details through FCC when inferring stylized image with the
prompt comprising recombined identifiers. A detailed sketch of our proposed
StyO is illustrated in Figure 2.

####  3.2.1 Identifier Disentanglement Learner

In order to extract the content and style information from given images, we
design a novel contrastive disentangled prompt template with positive-negative
identifiers and information descriptors to construct the text label for each
image. These image-text pairs will be used to fine-tune LDM with a triple
reconstruction loss, such that the content and style are encoded into
identifiers.

##### Contrastive Disentangled Prompt Template

As shown in Figure 2, we first label each image with a novel contrastive
disentangled prompt template: “a drawing with [$S^{+}$][$S^{-}$] style of
[$C^{+}$][$C^{-}$] portrait”, where “style” and “portrait” are the information
descriptor, “[$S^{+}$]”/“[$S^{-}$]” is a positive/negative unique identifier
linked to the style, and “[$C^{+}$]”/“[$C^{-}$]” is a positive/negative unique
identifier linked to the content of subject. Specifically, the positive
identifiers are expected to hold semantic meaning corresponding to the image.
With regard to the source/target image, we use “[$S_{src}$]”/“[$S_{tgt}$]” and
“[$C_{src}$]”/“[$C_{tgt}$]” for “[$S^{+}$]” and “[$C^{+}$]”, respectively. In
contrast, the negative identifier is the nomenclature proposed to represent
elements that are not presented in the image. We cooperate the negator,
_e.g_., “not”, “without” or “except”, with above positive identifiers to
construct negative identifiers. For instance, we set “[$S^{-}$]” as “[$not\
S_{tgt}$]”, and “[$C^{-}$] as “[$not\ C_{tgt}$]” for the negative identifiers
in the prompt of source image. Such design of contrastive prompt highlights
the significant visual disparity between the source and target images, and
will facilitate the disentangled learning by the effective exploitation of the
limited given data.

In addition to the task-relevant data $x_{src}$ and $x_{tgt}$, we introduce a
no-cost auxiliary image set $x_{aux}$ to further enhance the disentanglement,
in which about 200 natural faces are randomly sampled from FFHQ [17] and
labeled with a same prompt “a drawing with [$S_{src}$][$not\ S_{tgt}$] style
of portrait”. The contrast between prompts of $x_{aux}$ and $x_{src}$ helps to
encode source content into identifiers “[$C^{+}$]”/“[$C^{-}$]” instead of the
descriptor “portrait”, and strengthen the style distinction between $x_{src}$
and $x_{tgt}$. To sum up, we list prompts corresponding to different images or
set as:

  * •

Prompt of source image ($\mathcal{P}_{src}$) : “a drawing with
[$S_{src}$][$not\ S_{tgt}$] style of [$C_{src}$][$not\ C_{tgt}$] portrait”.

  * •

Prompt of target image ($\mathcal{P}_{tgt}$): “a drawing with
[$S_{tgt}$][$not\ S_{src}$] style of [$C_{tgt}$][$not\ C_{src}$] portrait”.

  * •

Prompt of auxiliary image set ($\mathcal{P}_{aux}$): “a drawing with
[$S_{src}$][$not\ S_{tgt}$] style of portrait”.

Note that, different from [36] using rare words in T5-XXL tokenizer, we simply
use firearms (_e.g_., “sks”, “aug” and “ak47”) as identifiers and also achieve
phenomenal results.

##### Triple Reconstruction Loss

Fine-tuning of diffusion models is a powerful technique relevant to many use
cases, _e.g_., image editing [44, 37, 36] and image-to-image translation [45].
To make the identifiers work as intended, we opt for this simple approach with
the text-image pair $\\{(\mathcal{P}_{i},x_{i})\\}_{i\in\\{src,tgt,set\\}}$.
We tune the text-to-image model with a triple reconstruction loss, thereby
injecting different information to individual identifiers:

$\displaystyle\mathop{\mathrm{min}}_{{\theta}}{\mathbb{E}}_{z_{src},z_{tgt},z_{aux},\epsilon,t}||\epsilon-\epsilon_{\theta}(z_{src}^{t},t,\gamma_{src})||$
(4)
$\displaystyle+||\epsilon-\epsilon_{\theta}(z_{tgt}^{t},t,\gamma_{tgt})||+||\epsilon-\epsilon_{\theta}(z_{aux}^{t},t,\gamma_{aux})||,$

where $z_{src}=\mathcal{E}(x_{src})$, $\gamma_{src}=\phi(\mathcal{P}_{src})$,
$z_{src}^{t}$ is the noisy latent according to Eq. (1), and same goes for
other symbols. By implementing IDL, we can obtain a tuned model and
disentangled identifiers biased to given images. See more implementation
details in Section 4.

####  3.2.2 Fine-grained Content Controller

The proposed IDL allows for the effective fusion of source identity and
artistic style via the recombination of identifiers:

  * •

Prompt of stylized portrait ($\mathcal{P}_{sty}$): “a drawing with
[$S_{tgt}$][$not\ S_{src}$] style of [$C_{src}$][$not\ C_{tgt}$] portrait”

Nonetheless, as depicted in Figure 2, the synthesis that is highly correlated
with the source face, exhibits noticeable diversity in terms of poses and some
facial features from source image. Although this diversity can offer
additional customization options, it would be desirable for the model to
consistently produce artistic portraits that accurately capture the source
face. To alleviate this problem, we introduce a Fine-grained Content
Controller (FCC) module that further transfers delicate details of source
image into syntheses. In technique, FCC devises the cross attention control
and augments text prompt to ensure content attention maps of the stylized
portrait correspond closely to those of the source face, as depicted in the
right panel of Figure 2.

##### Cross Attention Control

LDM utilizes a cross attention mechanism to modify the latent features based
on the conditional text features. Here we slightly abuse some notations: given
the text embedding $\gamma\in\mathbb{R}^{s\times d_{1}}$, latent features
$z\in\mathbb{R}^{hw\times d_{2}}$, where $s$ denotes the number of tokens, $h$
and $w$ are the height and width of latent feature, and $d_{1}$ and $d_{2}$
are the dimensions of features, respectively. The cross attention first
computes query $Q=zW^{q}$, key $K=\gamma W^{k}$, and value $V=\gamma W^{v}$.
Then latent features are updated by

$M(z,\gamma)=\mathop{softmax}\left(\frac{QK^{T}}{\sqrt{d}}\right),$ (5)
$\text{Attn}(z,\gamma)=M(z,\gamma)V,$ (6)

where $d$ is the dimension of key and query feature.

Hertz et al. [11] find that the attention map $M$ in Eq. (5) has a strong
influence on the spatial layout of syntheses. In the face domain, we are
inspired to control the pose and facial details of stylized portraits by
constraining the attention mask of syntheses to that of source image.
Specifically, considering the feed-forward process of LDM prompted by
$\mathcal{P}_{src}$ and $\mathcal{P}_{sty}$, their latent features and text
features are ($z_{sty}$, $z_{src}$) and ($\gamma_{sty}$,$\gamma_{src}$),
respectively. We modify the cross attention to a condition form:

$\displaystyle\text{Attn}(z_{sty},\gamma_{sty};z_{src},\gamma_{src})=M_{ctr}V_{sty},$
(7) $\displaystyle\text{where}\ M_{ctr}^{i}=M_{src}^{i}\ \text{if}\
i\in\text{Content Index}\ \text{else}\ M_{sty}^{i}.$

$M_{sty}$ and $M_{src}$ are the attention maps computed by the corresponding
text feature and latent feature, and the superscript $i$ denotes the $i$-th
column of matrix, and the Content Index includes the indexes of “[$C_{src}$]”,
“[$not\ C_{tgt}$]” and “portrait” in $\mathcal{P}_{src}$. Note that the tokens
of $\mathcal{P}_{src}$ and $\mathcal{P}_{sty}$ have a same length, thus FCC
indeed swaps the attention maps of those content identifiers to pull the
content of the stylized portrait towards source image.

Figure 3: Qualitative comparison of our StyO with state-of-the-arts. Better
see in color and 2x zoom.

##### Augmented Text Prompt

The text prompt is a critical factor in controlling the content of syntheses
for LDM, and improving its quality usually enhances the semantic details of
syntheses directly. Leveraging the disentangled identifiers obtained from IDL,
we introduce a augmented text prompt that offers greater customization and
control over stylized portraits. Given a identifier “[C]” and
$n\in\mathbb{Z}^{+}$, we define the augmented identifier “[C]”$*n$ as
repeating the identifier $n$ times. Augmented text prompts
$\mathcal{P}^{aug}_{src}$ and $\mathcal{P}^{aug}_{sty}$ comprises augmented
identifiers:

  * •

Augmented prompt of source face ($\mathcal{P}^{aug}_{src}$):
“a drawing with ([$S_{src}$][$not\ S_{tgt}$])*$n_{s}$ style of
([$C_{src}$][$not\ C_{tgt}$])*$n_{c}$ portrait”.

  * •

Augmented prompt of stylized portrait ($\mathcal{P}^{aug}_{sty}$):
“a drawing with ([$S_{tgt}$][$not\ S_{src}$])*$n_{s}$ style of
([$C_{src}$][$not\ C_{tgt}$])*$n_{c}$ portrait”.

Instead of $\mathcal{P}_{src}$ and $\mathcal{P}_{sty}$, we use prompt
$\mathcal{P}^{aug}_{src}$ to reconstruct the source face and
$\mathcal{P}^{aug}_{sty}$ to generate the stylized portrait with the aid of
cross attention control. It is important to note that two hyper-parameters,
$n_{s}$ and $n_{c}$, are introduced to control the degree of stylization and
identity preservation. Compared with the single identifier as used during
training, a balance of $n_{s}$ and $n_{c}$ will result in better effects.
Actually, it is not complicated to determine the optimal values for the two
hyper-parameters. Through experiments, we find setting $n_{s}$ and $n_{c}$ to
small values ($\leq 3$) is adequate to produce satisfactory results.

##### Inference with FCC

On this basis, the process of generation with FCC can be summarized as
follows: 1) We carry out destructive process Eq. (1) to obtain the noisy
latent $z_{src}^{T}$ of source image. Note that compared with random Gaussian
noise, $z_{src}^{T}$ provides a better prior to restore $x_{src}$. 2) Then we
perform the reverse diffusion starting from $z_{src}^{T}$ and prompted by
$\mathcal{P}^{aug}_{src}$. Meanwhile their attention maps $M_{sty}$ in all
cross attention layers are recorded as Eq. (5). 3) Once again, we perform the
reverse diffusion starting from $z_{src}^{T}$ and prompted by
$\mathcal{P}^{aug}_{sty}$ to synthesize the stylized portrait, while it
implements the conditional cross attention as Eq. (7) to fetch fine-grained
details from source image.

Figure 4: Ablation of Fine-grained Content Controller. Please zoom in for better visual effects. Table 1: Quantitative comparison of our StyO and state-of-the-arts based on the user study. See details in text. Model |  Identity$\uparrow$ |  Geometry$\uparrow$ |  Texture$\uparrow$
---|---|---|---
JoJoGAN [3] | 0.27 | 0.15 | 0.24
GOGA [50] | 0.31 | 0.16 | 0.27
CTDM [48] | 0.06 | 0.28 | 0.14
Ours | 0.36 | 0.41 | 0.35

##  4 Experiments

StyO utilizes IDL to fine-tune LDM, and generates stylized portrait by means
of FCC. In Section 4.1, we conduct both qualitative and quantitative
experiments to compare our method with previous state-of-the-art methods of
one-shot face stylization. And in Section 4.2, we perform the ablation study
to demonstrate the effectiveness of the proposed IDL and FCC.

##### Implementation Details

We adopt LDM111https://github.com/CompVis/stable-diffusion [34] pre-trained on
large-scale dataset LAION-5B [LAION-5B] as our base model. For IDL, the style
positive identifier “[$S_{src}$]”/“[$S_{tgt}$]” of source/target is set as
“ak47”/ “aug”, and content positive identifier “[$C_{src}$]”/“[$C_{tgt}$]” of
source/target is set as “sks”/ “m4a1”. We use Adam [adam] with learning rate
1e-6 to optimize the weights of UNet in LDM with $\sim$400 iterations. This
process takes $\sim$10 minutes on a single NVIDIA Tesla V100. For inference
with FCC, we empirically set $n_{s}=3$ and $n_{c}=1$ in augmented text prompt,
and others are the same as the default hyper-parameters of LDM, e.g., the
conditional text prompt is set to the null word, and the scale guidance is
$7.5$. DDIM scheduler [41] with 100 steps is applied to speed up the sampling.
We observe that these parameters generally work well in most cases, proving
the generalizability of StyO. We also point that better results may be
obtained with a customized setting, e.g., more iterations for IDL, or a larger
guidance scale for inference.

###  4.1 Comparison with SOTA methods

##### Baselines

We compare StyO with two recent representative state-of-the-arts of the task
JoJoGAN [3] and GOGA [50], which tune the pre-trained StyleGAN to the target
domain and stylize faces via GAN inversion [47]. We also report the results
from a concurrent diffusion-based stylization method CTDM [48] to show the
advantage of StyO. All these methods are implemented by their official codes.
The artistic portraits are token from AAHQ dataset [blengan], while the
natural faces employ photos of celebrities.

##### Qualitative Comparison

Figure 3 shows the qualitative results. The top row images are the source
natural faces and the leftmost images are the target artistic portraits. From
the results, we can draw the following conclusions: 1) Our method not only
achieves facial identity consistency between the source and output images, but
also highly preserves local details such as hair and eye color, which is a
significant improvement over other methods. In contrast, other methods often
fails to generate images that are faithful to the identity information of
source images (_e.g_., the 2nd, 5th and 8th-10th columns) and transfers
unexpected colors in some cases (_e.g_., the 4th, 5th and 7th rows). 2) Our
method captures the desired geometric variations in the target image, such as
big eyes and round face. Instead of using GAN based generative model, we use
potential text-image diffusion model to successfully capture the geometric
variation in reference image and inject it to unique identifier. Obviously,
other methods can not associate big eyes and round faces as consistent
geometric variations and smoothly introduce both of these into the source
image (_e.g_., the 6th and 7th rows). (3) Our method produces syntheses that
flexibly acquire vivid local stroke characteristics and overall appearance,
resulting in high-quality images with sharp and realistic high-frequency
details. In contrast, the results produced by other methods tend to be fuzzy.
While our approach has a tiny probability of poor style transfer (e.g., in the
4th row, 9th columns hair color is mistakenly transferred to the source
image), experiments show that our method outperforms competitors in most cases
and provides significant visual advantages.

Figure 5: Ablation of Contrastive Disentangled Prompt Template. The arrows
indicate the changes made to the current prompt template by sequentially
plusing positive identifiers, negative identifiers, and auxiliary image set.
Figure 6: Ablation of hyperparameter in augmented text prompt. Please zoom in
for better visual effects.

##### Quantitative Comparison

As portrait stylization is often regarded as a subjective task, we resort to
user studies to evaluate which method generates results that are most favored
by humans. We conducted three user studies on the results in terms of the
identity, geometric and texture. In the first study, participants are asked to
select the stylized images best preserve the identity of source image. In the
second study, participants are asked to point out the best stylized images
with reasonable exaggerated geometric deformation. In the third study, we
encourage participants to touch the image with the finest texture and less
distorted artifacts. We receive 150 answers on 18 source-target pairs in total
for each study. As shown in Table 1, over 35% of our results are selected as
the best in both three metrics, which proves a significant advantage in
stylization.

###  4.2 Ablation Study

##### Effect of Contrastive Disentangled Prompt Template

Our proposed contrastive disentangled prompt template acts as a powerful tool
to disentangle and extract style and content information of given images. In
contrast, a prompt template that only contains positive identifiers, e.g., the
prompt used in Dreambooth [36], faces two challenging issues: 1) Over-fitting
to source or target image. 2) Failed to decouple style and content
information. To better characterize the claims, we fine-tune LDM with three
different configurations, positive identifiers, positive-negative identifiers,
and our positive-negative identifiers plus the auxiliary image set. As shown
in Figure 5, the results generated by positive identifier overfit to source
image and often generates “haloing” artifacts around faces. Adding negative
identifiers tend to alleviate this problem and generate acceptable images.
Nevertheless, it may fail to capture the local stroke characteristics and
generate sharp details, and there still exists entanglement between style and
content. In comparison, the full prompt template achieves the best performance
in terms of style effect and content preservation.

##### Effect of Fine-grained Content Controller

To verify the effectiveness of the proposed FCC module, we perform the
evaluation by inferring artistic portrait without it. As shown in Figure 4,
StyO without FCC will generate diverse artistic portraits that maintains
source identity, due to the content identifiers also store the content
information of source image. This characteristic allows users to have more
choices, such as different poses, different hairstyles and eye colors. With
the introduction of FCC module, it is obvious that the diversity of syntheses
is significantly reduced and the identity fidelity to source image is also
improved.

##### Hyper-parameters in Augmented Text Prompt

The ablation in Figure 6 analyzes the visual results by varying the number of
style identifiers $n_{s}$ and the number of content identifiers $n_{c}$ in
FCC. Here, we find that with the increase of $n_{s}$, the style of syntheses
gradually change from photorealistic to the animated. Similarly, a larger
$n_{c}$ will force the output image to “overfit” to the source image.
Actually, there is a trade-off between content faithfulness and style
similarity related to hyper-parameter $n_{s}$ and $n_{c}$. Note that in
practice, we can experiment with various hyper-parameters to synthesize
multiple artistic portraits, allowing the user to choose among different
levels of artistic effects.

##  5 Conclusions

In this paper, we present a novel StyO model to effectively solve the one-shot
face stylization. StyO exploits a disentanglement and recombination strategy.
For this purpose, StyO introduces two new modeuls, _i.e_., Identifier
Disentanglement Learner (IDL) and Fine-grained Content Controller (FCC). IDL
aims to disentangle style and content attributes of images into different
identifiers. To achieve this goal, IDL proposes a novel contrastive prompt
template for describing source and target images. Given this, IDL constructs
text-image pairs for source, target and auxiliary image set to fine-tune the
pre-trained latent diffusion models. This successfully disentangles style and
content information into corresponding identifiers. Then, FCC recombines style
and content identifiers to form prompts describing the stylization faces,
which plays as conditions to generate images. FCC also presents a manipulation
mechanism on attention maps to enhance fine-grained details. Extensive results
show that our StyO model can generate high-quality stylization results in one-
shot manner, clearly outperforming existing stat-of-the-art methods.

In addition, we show that image attributes can be disentangled into text
identifiers with appropriate design. This inspires to extend style and content
to other attributes, _e.g_., pose, lighting, foreground and background etc.
This will further facilitate the challenging text-based image editing task. We
think the proposed StyO model presents a good starting point for followup
researches.

##### Limitations

Although the proposed StyO model is data-efficiency by utilizing the one-shot
learning framework, it is not time-efficiency due to the requirements for
finetuning a pre-trained latent diffusion model, which costs about 10 minutes
on a single GPU. This problem is also faced by other one- or few-shot learning
methods relied on diffusion model, such as DreamArtist and Dreambooth. We
believe it is useful and interesting to accelerate one- and few-shot diffusion
model, and will leave this as future work.

## References

  * [1] Ergun Akleman.  Making caricatures with morphing.  CGIT, 1997.
  * [2] Kaidi Cao, Jing Liao, and Lu Yuan.  Carigans: Unpaired photo-to-caricature translation.  arXiv preprint arXiv:1811.00222, 2018.
  * [3] Min Jin Chong and David Forsyth.  Jojogan: One shot face stylization.  In ECCV, pages 128–152. Springer, 2022.
  * [4] Prafulla Dhariwal and Alexander Nichol.  Diffusion models beat gans on image synthesis.  In NeurIPS, pages 8780–8794, 2021.
  * [5] Rinon Gal, Yuval Alaluf, Yuval Atzmon, Or Patashnik, Amit H Bermano, Gal Chechik, and Daniel Cohen-Or.  An image is worth one word: Personalizing text-to-image generation using textual inversion.  arXiv preprint arXiv:2208.01618, 2022.
  * [6] Rinon Gal, Or Patashnik, Haggai Maron, Amit H Bermano, Gal Chechik, and Daniel Cohen-Or.  Stylegan-nada: Clip-guided domain adaptation of image generators.  ACM Transactions on Graphics, 41(4):1–13, 2022.
  * [7] Leon A Gatys, Alexander S Ecker, and Matthias Bethge.  Image style transfer using convolutional neural networks.  In CVPR, pages 2414–2423, 2016.
  * [8] Julia Gong, Yannick Hold-Geoffroy, and Jingwan Lu.  Autotoon: Automatic geometric warping for face cartoon generation.  In WACV, pages 360–369, 2020.
  * [9] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron C. Courville, and Yoshua Bengio.  Generative adversarial nets.  In NeurIPS, 2014.
  * [10] Fangzhou Han, Shuquan Ye, Mingming He, Menglei Chai, and Jing Liao.  Exemplar-based 3d portrait stylization.  IEEE Transactions on Visualization and Computer Graphics, 29:1371–1383, 2021.
  * [11] Amir Hertz, Ron Mokady, Jay Tenenbaum, Kfir Aberman, Yael Pritch, and Daniel Cohen-Or.  Prompt-to-prompt image editing with cross attention control.  arXiv preprint arXiv:2208.01626, 2022.
  * [12] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  In NeurIPS, pages 6840–6851, 2020.
  * [13] Jonathan Ho and Tim Salimans.  Classifier-free diffusion guidance.  arXiv preprint arXiv:2207.12598, 2022.
  * [14] Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros.  Image-to-image translation with conditional adversarial networks.  In CVPR, pages 1125–1134, 2017.
  * [15] Nikolai Kalischek, Jan Dirk Wegner, and Konrad Schindler.  In the light of feature distributions: moment matching for neural style transfer.  In CVPR, 2021.
  * [16] Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine.  Elucidating the design space of diffusion-based generative models.  arXiv preprint arXiv:2206.00364, 2022.
  * [17] Tero Karras, Samuli Laine, and Timo Aila.  A style-based generator architecture for generative adversarial networks.  In CVPR, pages 4401–4410, 2019.
  * [18] Bahjat Kawar, Shiran Zada, Oran Lang, Omer Tov, Huiwen Chang, Tali Dekel, Inbar Mosseri, and Michal Irani.  Imagic: Text-based real image editing with diffusion models.  arXiv preprint arXiv:2210.09276, 2022.
  * [19] Junho Kim, Minjae Kim, Hyeonwoo Kang, and Kwanghee Lee.  U-gat-it: Unsupervised generative attentional networks with adaptive layer-instance normalization for image-to-image translation.  arXiv preprint arXiv:1907.10830, 2019.
  * [20] Junho Kim, Minjae Kim, Hyeonwoo Kang, and Kwanghee Lee.  U-gat-it: Unsupervised generative attentional networks with adaptive layer-instance normalization for image-to-image translation.  arXiv preprint arXiv:1907.10830, 2019.
  * [21] Nicholas I. Kolkin, Jason Salavon, and Gregory Shakhnarovich.  Style transfer by relaxed optimal transport and self-similarity.  In CVPR, 2019.
  * [22] Yanghao Li, Naiyan Wang, Jiaying Liu, and Xiaodi Hou.  Demystifying neural style transfer.  arXiv preprint arXiv:1701.01036, 2017.
  * [23] Luping Liu, Yi Ren, Zhijie Lin, and Zhou Zhao.  Pseudo numerical methods for diffusion models on manifolds.  arXiv preprint arXiv:2202.09778, 2022.
  * [24] Mingcong Liu, Qiang Li, Zekui Qin, Guoxin Zhang, Pengfei Wan, and Wen Zheng.  Blendgan: Implicitly gan blending for arbitrary stylized face generation.  In NeurIPS, pages 29710–29722, 2021.
  * [25] Ming-Yu Liu, Thomas Breuel, and Jan Kautz.  Unsupervised image-to-image translation networks.  In NeurIPS, 2017.
  * [26] Yifang Men, Yuan Yao, Miaomiao Cui, Zhouhui Lian, and Xuansong Xie.  Dct-net: domain-calibrated translation for portrait stylization.  ACM Transactions on Graphics, 41(4):1–9, 2022.
  * [27] Ron Mokady, Amir Hertz, Kfir Aberman, Yael Pritch, and Daniel Cohen-Or.  Null-text inversion for editing real images using guided diffusion models.  arXiv preprint arXiv:2211.09794, 2022.
  * [28] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen.  Glide: Towards photorealistic image generation and editing with text-guided diffusion models.  arXiv preprint arXiv:2112.10741, 2021.
  * [29] Alexander Quinn Nichol and Prafulla Dhariwal.  Improved denoising diffusion probabilistic models.  In ICML, pages 8162–8171, 2021.
  * [30] Utkarsh Ojha, Yijun Li, Jingwan Lu, Alexei A Efros, Yong Jae Lee, Eli Shechtman, and Richard Zhang.  Few-shot image generation via cross-domain correspondence.  In CVPR, pages 10743–10752, 2021.
  * [31] Li Pan, Lei Zhao, Duanqing Xu, and Dongming Lu.  Optimal transport of deep feature for image style transfer.  In MM, 2019.
  * [32] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  In ICML, pages 8748–8763, 2021.
  * [33] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen.  Hierarchical text-conditional image generation with clip latents.  arXiv preprint arXiv:2204.06125, 2022.
  * [34] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.  High-resolution image synthesis with latent diffusion models.  In CVPR, pages 10684–10695, 2022.
  * [35] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.  U-net: Convolutional networks for biomedical image segmentation.  In MICCAI, pages 234–241. Springer, 2015.
  * [36] Nataniel Ruiz, Yuanzhen Li, Varun Jampani, Yael Pritch, Michael Rubinstein, and Kfir Aberman.  Dreambooth: Fine tuning text-to-image diffusion models for subject-driven generation.  arXiv preprint arXiv:2208.12242, 2022.
  * [37] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al.  Photorealistic text-to-image diffusion models with deep language understanding.  arXiv preprint arXiv:2205.11487, 2022.
  * [38] Yichun Shi, Debayan Deb, and Anil K Jain.  Warpgan: Automatic caricature generation.  In CVPR, pages 10762–10771, 2019.
  * [39] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli.  Deep unsupervised learning using nonequilibrium thermodynamics.  In ICML, pages 2256–2265, 2015.
  * [40] Guoxian Song, Linjie Luo, Jing Liu, Wan-Chun Ma, Chun-Pong Lai, Chuanxia Zheng, and Tat-Jen Cham.  Agilegan: stylizing portraits by inversion-consistent transfer learning.  ACM Trans. Graph., 40:117:1–117:13, 2021.
  * [41] Jiaming Song, Chenlin Meng, and Stefano Ermon.  Denoising diffusion implicit models.  arXiv preprint arXiv:2010.02502, 2020.
  * [42] Yang Song and Stefano Ermon.  Improved techniques for training score-based generative models.  In NeurIPS, pages 12438–12448, 2020.
  * [43] Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole.  Score-based generative modeling through stochastic differential equations.  arXiv preprint arXiv:2011.13456, 2020.
  * [44] Dani Valevski, Matan Kalman, Yossi Matias, and Yaniv Leviathan.  Unitune: Text-driven image editing by fine tuning an image generation model on a single image.  arXiv preprint arXiv:2210.09477, 2022.
  * [45] Tengfei Wang, Ting Zhang, Bo Zhang, Hao Ouyang, Dong Chen, Qifeng Chen, and Fang Wen.  Pretraining is all you need for image-to-image translation.  arXiv preprint arXiv:2205.12952, 2022.
  * [46] Ting-Chun Wang, Ming-Yu Liu, Jun-Yan Zhu, Andrew Tao, Jan Kautz, and Bryan Catanzaro.  High-resolution image synthesis and semantic manipulation with conditional gans.  In CVPR, 2018.
  * [47] Weihao Xia, Yulun Zhang, Yujiu Yang, Jing-Hao Xue, Bolei Zhou, and Ming-Hsuan Yang.  Gan inversion: A survey.  arXiv preprint arXiv:2101.05278, 2021.
  * [48] Yuxin Zhang, Nisha Huang, Fan Tang, Haibin Huang, Chongyang Ma, Weiming Dong, and Changsheng Xu.  Inversion-based creativity transfer with diffusion models.  arXiv preprint arXiv:2211.13203, 2022.
  * [49] Yabo Zhang, Mingshuai Yao, Yuxiang Wei, Zhilong Ji, Jinfeng Bai, and Wangmeng Zuo.  Towards diverse and faithful one-shot adaption of generative adversarial networks.  ArXiv, abs/2207.08736, 2022.
  * [50] Zicheng Zhang, Yinglu Liu, Congying Han, Tiande Guo, Ting Yao, and Tao Mei.  Generalized one-shot domain adaption of generative adversarial networks.  arXiv preprint arXiv:2209.03665, 2022.
  * [51] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros.  Unpaired image-to-image translation using cycle-consistent adversarial networks.  In ICCV, pages 2223–2232, 2017.
  * [52] Peihao Zhu, Rameen Abdal, John Femiani, and Peter Wonka.  Mind the gap: Domain gap control for single shot domain adaptation for generative adversarial networks.  arXiv preprint arXiv:2110.08398, 2021.


# Direct Observation of an Incommensurate Charge Density Wave in the
BiS2-based Superconductor NdO1-xFxBiS2

Jooseop Lee1,2 Masanori Nagao3 Yoshikazu Mizuguchi4 Jacob Ruff2 1CALDES,
Institute for Basic Science, Pohang 37673, Republic of Korea  2CHESS, Cornell
University, Ithaca, New York 14853, USA  3University of Yamanashi, Kofu,
Yamanashi 400-8511, Japan  4Department of Physics, Tokyo Metropolitan
University, Hachioji, Tokyo 192-0397, Japan

###### Abstract

The nature of superconductivity in BiS2-based superconductors has been
controversial while ab-initio calculations proposed this system in close
proximity to a charge-density-wave (CDW) phase. Using high-energy high-flux
X-ray diffraction, we reveal an intrinsic and long-range CDW phase coexisting
with superconductivity in NdO1-xFxBiS2 superconductor ($x$ = 0.37 and 0.3).
The CDW wavevector in NdO0.63F0.37BiS2 correspond QCDW = (0.17, 0.17, 0.5) and
is associated with transverse atomic displacements. Interestingly, this
wavevector does not match theoretical expectations based on either phonon
softening or Fermi surface nesting. In NdO0.7F0.3BiS2, where the
superconducting transition temperature is highest, the CDW satellites are
slightly broader and weaker compared to NdO0.63F0.37BiS2, possibly suggesting
the competition with the superconductivity. Lastly, we measure a thermal
diffuse scattering across the superconducting transition temperature and find
no meaningful changes. Our result suggests the importance of understanding CDW
which might hold a key to the superconductivity in the BiS2-based
superconductor.

##  I INTRODUCTION

The BiS2-based superconductors have many similarities with the unconventional
cuprates and iron-based superconductors First ; MizuRev ; DeguRev ; YaziciRev
; UsuiRev ; TeraRev . Their layered structure is low dimensional composed of
BiS2 superconducting layers and Ln2O2 (Ln: Lanthanide) blocking layers
alternating along the c-axis. Moreover, Bi ions responsible for the
superconductivity are arranged in a square lattice. On the other hand, Bi is
not magnetic and shows 6px,y orbital characters, which have less electron
correlation compared with 3d orbitals UsuiPRB . The highest transition
temperature reported is relatively low $\approx$ 11.5 K HighestTc .

The nature of superconductivity in BiS2-based superconductors, therefore, has
been controversial whether it can be considered as a conventional phonon-
mediated BCS superconductor or another family of non-BCS superconductors that
opens a new route to unconventional superconductivity. Density functional
theory (DFT) calculations have predicted a significant electron-phonon (e-ph)
coupling Yildirim ; WanPRB ; LiEPL ; FengJAP , that was later supported by
Raman experiment RamanPRB . Meanwhile, many ab initio studies also find strong
Fermi surface nesting (FSN) near optimal doping Yildirim ; WanPRB ; UsuiPRB
that can lead to unconventional pairing mechanisms T2 ; T8 ; T9 ; T10 ; T15 ;
T28 ; T29 . Inelastic neutron scattering on polycrystalline samples JLEE2013
and Raman studies RamanSST  indicate that the possible e-ph coupling can be
much weaker than theoretically expected. Yet, a momentum-resolved phonon study
on a single-crystalline sample is still lacking.

Interestingly, theoretical studies predict negative energy phonon modes
possibly related to the FSN that leads to ferroelectric or charge-density-wave
(CDW) phases Yildirim ; WanPRB . According to the ab-inito calculation, in
LaOBiS2, there is a ferroelectric soft phonon at the zone center, whose
shallow potential curve prevents the development of static long-range order.
In the optimally doped LaO0.5F0.5BiS2, on the other hand, unstable phonon
branches are expected along the $\bm{q}$ = ($\delta$, $\delta$, 0) causing the
in-plane distortion of the BiS2 plane, especially the shift of S atoms by
about 0.16 $\rm{\AA}$ away from the high-symmetry position. Total energy
analysis finds the structure with the lowest energy to be a $\sqrt{2}$ x
$2\sqrt{2}$ CDW superstructure with sinusoidally distorted one-dimensional
Bi-S chains.

The relation between CDW and superconductivity has been hotly debated as CDW
turns out to be a ubiquitous ground state found in the phase diagram of many
unconventional superconductors, for example, cuprate or transition-metal
dichalcogenides (TMDC) superconductors Dean ; Kis . In BiS2-based
superconductors, Scanning Tunneling Microscopy (STM) studies asserted a
modulation in the real-space electronic features that are characterized, for
example, by a checkerboard stripe CheckerboardSTM ; STM2017 ; STM2018 . These
results, however, poses a question if the observed CDW is an intrinsic bulk
property especially since they also found a large spectroscopic gap of about
40 meV contrary to a bulk metallic nature CheckerboardSTM .

Moreover, it has been known from the early stage that the superconducting
transition temperature in BiS2-based superconductors can be greatly enhanced
under pressure or by pressure annealing HighestTc ; HPEnhance . First-
principles energy calculations show that there is a family of polytypes
nearly-degenerate in energy within $\approx$ 1 meV PolyDFT ; PolyARPES  and
x-ray diffraction measurement found different structures coexisting in the
same batch of crystals PolyXRD . In-plane dynamic charge fluctuations and
local distortions have been claimed from neutron and x-ray diffraction and the
pair density function analysis Anushika19 ; AnushikaNd17 ; AnushikaLa17 ;
Anushika15 . These findings all point to a crucial role of structural
instability in understanding BiS2-based systems.

Here, we report a long-range CDW ground state that coexists with
superconductivity in NdO0.63F0.37BiS2 observed using high-energy and high-flux
x-ray diffraction. The CDW phase onsets around $\approx$ 120 K with a
wavevector QCDW = (0.17, 0.17, 0.5). This wavevector is distinct from the
values proposed from either phonon calculation or Fermi surface evaluation and
calls attention to the interlayer couplings neglected in the previous studies.
A detailed study on the x-ray reflection pattern shows that the CDW satellite
peaks are stronger along one of the diagonal directions in the reciprocal (H,
K, 0.5) plane, suggesting one-dimensional and diagonal atomic modulation that
breaks 4-fold symmetry. In the NdO0.7F0.3BiS2 with optimal electron doping
NdDiscovery , while a similar satellite diffraction pattern is observed, the
CDW peaks are broader and weaker suggesting the competition between
superconductivity and charge density wave in this system. Also, the
relationship between phonon and superconductivity is investigated by x-ray
diffuse scattering measurement. No changes in the phonon frequency across the
superconducting transition temperature, Tc $\approx$ 5 K, are observed.

##  II METHOD

###  II.1 SAMPLE PREPARATION

The single-crystal samples of NdO1-xFxBiS2 (x = 0.3 and 0.37) were prepared as
described in the Ref First ; Sample2012 ; Sample2013 . Samples were
synthesized using high temperature CsCl/KCl flux method in a vacuum-sealed
quartz tube with the starting materials of Nd2S3, Bi, Bi2S3, Bi2O3, BiF3,
CsCl, and KCl at approximately 616 ∘C. The single crystals are flat and plate-
like with the c-axis perpendicular to the growth direction which can be easily
exfoliated due to a Van der Waals gap between the BiS2 planes CheckerboardSTM
. The typical sample size is about 2 mm x 2 mm x 0.05 mm. From resistivity
measurements, the transition temperature was determined to be 5.1 K for x =
0.3 and 5.0 K for x = 0.37. The compositions were determined by electron probe
microanalysis (EPMA).

###  II.2 X-RAY DIFFRACTION

We used a high-energy and high-flux synchrotron x-ray at the A2 beamline at
Cornell High Energy Synchrotron Source (CHESS). For the x = 0.3 sample,
incident energy, Ei, of 41.75 keV and Pilatus 300 area detector were used, and
the sample was cooled with a closed-cycle refrigerator. For the x = 0.37
sample, an Ei = 39.80 keV was used with other conditions identical. The beam
was slit down to a typical size of 100 x 100 $\mu$m. All measurements were
performed in transmission geometry taking advantage of the high flux at high
incident energies at CHESS. Typical reciprocal volume maps are obtained by
combining two sets of 800 images at different two-theta angles with each image
having an increment of theta of 0.025∘ and counting time 10 seconds.

##  III RESULTS

Figure 1: Single-crystal x-ray diffraction of NdO0.63F0.37BiS2 obtained at A2,
CHESS with Ei = 39.80 keV. (a) (H, K) reciprocal space map at L = 0.5 measured
at the base temperature of 4 K. Miller indices are all in the reciprocal
lattice unit (r.l.u.). Expanded view of satellite reflections at 4 K around
(b) $\bm{q}$ = (4, 2) and (c) $\bm{q}$ = (4, 0), and the corresponding regions
have been marked by yellow squares in (a). Same figures at an elevated
temperature of 120 K, i.e., around (d) $\bm{q}$ = (4, 2) and (e) $\bm{q}$ =
(4, 0). Intensity curves measured at various temperature along (f) ($\delta$,
$\delta$)- and (g) ($\delta$, -$\delta$)- directions at $\bm{q}$ = (4, 0) and
along (h) ($\delta$, $\delta$)- and (i) ($\delta$, -$\delta$)- directions at
$\bm{q}$ = (4, 2) showing the effect of unidirectional atomic displacement in
diagonal directions. (j) L-dependence of the CDW satellite peaks at (H, K) =
(4-0.17, 0.17). For all 1D curves, fits to Gaussian are shown in sold lines
except (h) where there is no peak. In fitting the L-dependence, the width of
all Gaussian peaks were tied to a same value.

A high-energy single-crystal x-ray diffraction results of NdO0.63F0.37BiS2 are
shown in Fig. 1. Figure 1 (a) shows the intensity map of the reciprocal space
plane (H, K) at L = 0.5 measured at 4 K. While weak in intensity, we can
clearly observe sharp satellite peaks away from the Bragg peak positions with
$\delta\bm{q}$ = (0.17, 0.17). Fig. 1 (b) and (c) shows a magnified view
around $\bm{q}$ = (4, 2) and (4, 0), respectively, for instance. The Bragg
reflection at the integer position is surrounded by four satellite reflections
which are marked with dashed green circles to make it clear. These satellite
peaks indicate a periodic lattice modulation that most likely results from a
CDW.

It is intriguing to note that there are lines of diffuse scattering in
diagonal directions, reminiscent of the unstable phonon modes along the entire
$\Gamma$-M direction suggested by ab-initio calculations Yildirim . See the
zoom-in views Fig. 1 (d) and (e) again around $\bm{q}$ = (4, 2) and (4, 0),
respectively, measured at 120 K where the thermal diffuse scattering intensity
is stronger due to the Bose-factor. The lines of diffuse scattering seem to be
enhanced on the line from $\Gamma$ to satellite peaks. While the unstable
phonon modes are merely an artifact from the calculation, the soft phonon
branch along the entire line should have an important physical origin Yildirim
, for example, an extended collapse of phonon modes due to electron-phonon
coupling that condenses into CDW at low temperature as in TMDC Weber . Further
energy-resolved phonon dispersion measurement will be needed to reach a firm
conclusion on the origin of the thermal diffuse scattering line.

Moreover, we notice differences in the intensities of the satellite peaks. For
instance, as shown in Fig. 1 (b), satellite reflections around $\bm{q}$ = (4,
2) are much stronger along ($\delta$, -$\delta$) direction than along the
other diagonal ($\delta$, $\delta$) direction. Yet, the CDW peaks around (4,
0) all have comparable intensities, as in Fig. 1 (c).

The intensity of CDW satellite reflections can be expressed as Overhauser ;
JLEE2020

$I(\bm{q})\propto\Big{|}\sum_{\alpha}(\bm{q}\cdot\bm{\epsilon}_{\alpha})f_{\alpha}(\bm{q}\pm\bm{Q}_{\rm{CDW}})\Big{|}^{2}\delta(\bm{q}-\bm{G}\pm\bm{Q}_{\rm{CDW}}).$
(1)

Here, $\bm{\epsilon}_{\alpha}$ and $f_{\alpha}$ are the atomic displacement
and the structure factor of atom $\alpha$, respectively, while $\bm{G}$ is the
reciprocal lattice vector. Considering the
$(\bm{q}\cdot\bm{\epsilon}_{\alpha})$ prefactor, we conclude that the
displacement of atom responsible for the CDW is almost transverse and along
diagonal direction: along ($\delta$, -$\delta$) direction for QCDW = (0.17,
0.17) and along ($\delta$, $\delta$) direction for QCDW = (0.17, -0.17). The
diffraction pattern observed should be understood as an overlay of two
domains, breaking 4-fold symmetry. This unidirectional modulation is
consistent with the quasi-1D nature expected from small mixing between 6px and
6py orbitals of Bi ions UsuiPRB .

Figure 1 (f)-(i) show the intensity curves at various temperatures along
diagonal directions around two different Bragg peaks: $\bm{q}$ = (4, 0) for
Fig. 1 (f) and (g) and $\bm{q}$ = (4, 2) for Fig. 1 (h) and (i). At all
temperatures, the satellite reflections in Fig. 1 (h) has negligible
intensities compared with Fig. 1 (i) due to near-orthogonality between
$\bm{q}$ and $\bm{\epsilon}_{\alpha}$ while the CDW reflections around (4, 0)
all have similar intensities, Fig. 1 (f) and (g). From the temperature
dependence of the CDW peaks showing a rapid decrease around 120 K, we expect
the onset temperature should be a bit above, but close to 120 K. Future study
on the detailed temperature dependence will be required to pin down the CDW
onset temperature and the order of phase transition.

A CDW phase has been conjectured from previous STM studies on BiS2-based
superconductors CheckerboardSTM ; STM2017 ; STM2018 , though there have been
concerns about possible surface effects and/or sample inhomogeneities. Our
x-ray data, from the sharp width of satellite peaks, demonstrate that the CDW
is indeed a long-range order. Furthermore, the transmission geometry ensures
that the CDW is a bulk intrinsic property, rather than a surface
reconstruction. The length of the wavevector, 0.172$\sqrt{2}a^{*}$,
corresponds to about 3.97$a$ (=2.81$\sqrt{2}a$) in real space, consistent with
STM results reporting 3$a$ $\sim$ 6$a$ in-plane superstructure CheckerboardSTM
; STM2017 ; STM2018 .

Figure 2: (a) NdO0.7F0.3BiS2 (H, K, 0.5) reciprocal space map measured at 4.3
K (b) The same map where the positions of reflections from the 3 x 3 unit cell
indexed with blue squares. The corresponding reciprocal basis vectors are
shown in yellow arrows. (c) A zoom-in view around $\bm{q}$ = (4, 2) which
shows the CDW satellite peaks marked with green dashed circles. The
contribution from the inhomogeneous phase has been removed for clarity.
Intensity curves of the CDW satellite $\bm{q}$ = (4 +$\delta$, -$\delta$) (d)
along H and (e) along K for NdO0.7F0.3BiS2. For comparison, similar curves are
shown (f) along H and (g) along K for NdO0.63F0.37BiS2. (h) L-dependence at
$\bm{q}$ = (4 +$\delta$, -$\delta$). For all 1D curves, fits to Gaussian are
shown in sold lines. In fitting the L-dependence, the widths of all Gaussian
peaks were tied to the same value and an extra Gaussian was added to account
for a hump, possibly from the imperfect crystallinity.

To check the L-dependence of the CDW satellite peaks, we made an intensity
plot along L-direction at (H, K) = (4 - 0.17, 0.17) as shown in Fig. 1 (j).
Surprisingly, the intensity is peaked at every half-integer position
indicating doubling of the unit cell along the c-axis. The width of the peak
is fitted with 0.33(1) r.l.u. corresponding to a correlation length of
$\approx$ 3 unit cell. Though the broad width in Q-space suggests a weak and
short-ranged interlayer correlation, the peak structure is still well-defined.
It is worthy to note that previous phonon studies focused on the instabilities
along q = ($\delta$, $\delta$, 0) direction leading to a CDW of 2$\sqrt{2}$ x
$\sqrt{2}$ x 1 superstructure Yildirim ; WanPRB . DFT calculation also pointed
to FSN along the same direction assuming the 2D character of the Fermi surface
Yildirim ; WanPRB ; UsuiPRB . Our data, however, reveals that there is a clear
L-dependence in the CDW reflections, indicating interlayer couplings in
NdO0.63F0.37BiS2 superconductors, which places an important constraint on
future studies on the phonon, CDW, and superconductivity in BiS2-based
superconductors.

We repeated a similar x-ray diffraction measurement on NdO0.7F0.3BiS2 sample
to figure out the doping dependence of CDW. Figure 2 (a) shows the scattering
intensity map of (H, K) plane at L = 0.5 measured at 4.3 K. At a first glance,
we find unexpected numerous weak reflections at non-integer positions. We
notice that the positions of these reflections are not correlated with the
main Bragg peaks as expected for a CDW, and therefore cannot be from a
superstructure formation. It turns out that they are indexed with a reciprocal
basis vectors
$\bm{a}_{1}^{*}=(\cos(60^{\circ}),~{}\sin(60^{\circ}))\bm{a}^{*}/3$ and
$\bm{a}_{2}^{*}=(\cos(60^{\circ}),~{}\sin(-60^{\circ}))\bm{a}^{*}/3$
($\bm{a}^{*}$: the original a-axis reciprocal lattice vector), which are
marked with blue squares in Fig. 2 (b). This new phase must be closely linked
to the original structure considering that the lattice parameters are exactly
3 times the original ones and have a precise 60-degree angle with respect to
the $\bm{a}^{*}$. The hexagonal (or trigonal) reflections disappear when we
decrease the beam size to tens of microns and scan other surface areas. It,
therefore, might be induced from inhomogeneity in the sample, for example, by
uneven doping concentration, defect, vacancy, local strain, etc. Rough
estimation reveals that the volume fraction of the 3 $\times$ 3 phase is about
0.71 $\pm$ 0.08 $\%$, negligible enough to ensure that the composition is
close to the nominal average one SuppleVol . The L-dependence is not obvious
in this sample due to the imperfect crystallinity along the c-axis.

Ruling out the contribution from the 3 x 3 phase, we find the satellite
reflections from the CDW as in Fig. 2 (c) around $\bm{q}$ = (4, 2). Again, the
polarization dependence of the CDW satellites is clear and the atomic
displacements should be unidirectional and along diagonal directions. The
L-dependence is also peaked at half-integer positions consistent with x = 0.37
system.

Furthermore, we notice that the CDW satellite peaks are slightly broader and
weaker compared to those of NdO0.63F0.37BiS2. Figure 2 (d) and (e) show cuts
along the H and K direction, respectively. The FWHM widths estimated from
Gaussian fits are 0.018(2) r.l.u. along H and 0.034(2) r.l.u. along K. These
values can be compared with those from x = 0.37 sample, 0.014(1) r.l.u. and
0.017(1) r.l.u. along H and K as shown in Fig. 2 (f) and (g), respectively.
The intensity ratio with respect to the nearby Bragg peak, $\bm{q}$ = (4, 2),
is 1/1597(197) compared to the ratio of 1/869(91) for x = 0.37. We estimated
the domain size of the CDW Domain1 ; Domain2  and obtained 181(8) $\AA$ for x
= 0.37 and 104(9) $\AA$ for x = 0.3 SuppleCorr . The competition of CDW and
SC, demonstrated by shorter CDW domain size and weaker reflections for higher
superconducting transition temperature, is very similar to what is observed in
cuprates Cuprate1 ; Cuprate2 . Considering that x = 0.3 is the optimal doping
level with the highest Tc among NdO1-xFxBiS2NdDiscovery , there could be
possible competition between the superconductivity and charge density wave,
though they coexist in current doping range.

Figure 3: (a) Thermal diffuse scattering intensity map of NdO0.7F0.3BiS2 at
(H, K, 0) plane across the Tc. (b) Intensity curves along ($\delta$,
-$\delta$) and ($\delta$, 0) directions around $\bm{q}$ = (4, 1) measured at
various temperatures.

To elucidate the relationship between phonon and superconductivity in the
BiS2-based superconductors, we examined the phonon dispersion by observing a
thermal diffuse scattering (TDS), a diffuse scattering process by lattice
thermal vibration. For a phonon-mediated conventional superconductor, there
can be a phonon frequency softening and/or line-width broadening at a momentum
transfer $\bm{q}$ = 2$\bm{k}_{F}$ across the superconducting phase transition
Kohn1 ; Kohn2 ; Kohn3 ; Kohn4 ; Kohn5 ; Kohn5 ; Kohn6 .

To the 1st order, the intensity of TDS by unpolarized x-ray at a momentum
transfer $\bm{q}$ can be represented as TDS

$\begin{split}I(\bm{q})\propto&\sum_{j=1}^{3n}\frac{1}{\omega_{\bm{q},j}}\coth(\frac{\hbar\omega_{\bm{q},j}}{2k_{B}T})\\\
&\times\Big{|}\sum_{s=1}^{n}\frac{f_{s}}{\sqrt{\mu_{s}}}e^{-M_{s}}(\bm{q}\cdot\bm{e}_{\bm{q},j,s})e^{-i\bm{q}\cdot\tau_{s}}\Big{|}^{2},\end{split}$
(2)

where the subscripts $s$ and $j$ represent the atom in a unit cell and the
phonon modes, respectively. Here, $\omega$, $\bm{e}$, $f$, $\mu$, $e^{-M}$,
and $\tau$ refer to phonon frequency, eigenvector for the phonon mode, atomic
form factor, atomic mass, the Debye-Waller factor, and basis vector in real
space, respectively. From this equation, we can tell that the main
contribution to the TDS intensity will be from the low lying phonon modes,
where significant electron-phonon couplings have been estimated in BiS2-based
superconductors. In addition, from the $\coth$-dependence with $1/T$, we
expect the thermal change of intensity will be negligible in the small
temperature range around Tc.

Figure 3 (a) shows some representative intensity maps in the (H, K, 0) plane
measured at 4.3, 5.5, 8.0, and 10 K. We were not able to observe any
meaningful $\bm{q}$-dependent or T-dependent changes in the TDS. For more
careful inspection, we plot intensity curves along several directions, for
example, along diagonal ($\delta$, -$\delta$) and horizontal ($\delta$, 0)
directions in Fig. 3 (b). No clear and consistent change is found across Tc
while some anomalous kink can be attributed to the 3 x 3 phase. It should be
noted that TDS intensity is an energy-integrated measurement and it is always
possible that a small thermal population at low temperature and/or not-enough
x-ray flux may hinder the observation of a small phonon anomaly. With these
limitations in mind, we conclude it is likely that the the change of phonon
frequency across the superconducting phase transition is small to be detected.

##  IV DISCUSSION

Here we report that NdO1-xFxBiS2 (x = 0.37 and 0.3) develops a CDW, a long-
sought structural ground state in the BiS2-based superconductors that coexist
with superconductivity. The origin of a CDW is an interesting and often
challenging question. It can be from FSN as in simple Pierels system,
$\bm{q}$-dependent electron-phonon coupling as in some TMDC, or electron-
electron correlation in cuprates Plummer . While the QCDW seems to be far from
the FSN vectors known so far, electron-phonon coupling calculations were not
successful to predict the wavevector yet. More careful studies considering the
interlayer couplings and even electronic correlations could be required.

The discrepancy between the calculations and measurements might be from the
structural instability to the monoclinic phase in the BiS2-based system. At
high pressure, LaO1-xFxBiS2 undergoes a structural phase transition to P21/m
and yields higher Tc HPEnhance . According to the first-principle
calculations, in the monoclinic phase, the magnitude of the interlayer hopping
becomes much larger stabilizing the electronic state Bilayer ; SuzukiRev . Our
intensity map at (H, K, 0) in the Supplementary show that the x-ray
reflections no longer satisfy the P4/nmm extinction conditions, probably due
to a monoclinic phase transition as observed in some other BiS2-based systems
Monoclinic . The enhanced interlayer hopping in the monoclinic structure,
therefore, can be responsible for the L-dependence of our CDW wavevector. We
claim that the interlayer coupling overlooked so far in the phonon and band
calculations can be essential to properly describe the electron-phonon
coupling, FSN, CDW, and the nature of superconductivity in BiS2-based
superconductors.

Lastly, the role of CDW in superconductivity has been an active research
subject since Fröhlich tried to explain superconductivity with sliding of CDW,
a concept conceived even before BCS formalism Frohlich . While there is
accumulating evidence that a CDW could be a ubiquitous phase in unconventional
superconductors Dean ; Kis , the relation with superconductivity remains
elusive. Here, we suggested that the charge density wave and superconductivity
might be in competition while coexisting over a certain doping range in the
phase diagram. Like lattice vibrations in a conventional superconductor,
charge/orbital fluctuation near the quantum critical point is argued to be
capable of mediating superconductivity. Considering the absence of momentum-
and temperature- dependent anomalies in the phonon spectrum across Tc, we
cannot exclude the unconventional pairing mechanism. The study on the
interplay between CDW and superconductivity in BiS2-based superconductors, by
changing external perturbations like doping or pressure will be a fascinating
topic that requires lots of future attention.

##  V CONCLUSION

In summary, we found an intrinsic and long-range CDW phase in a NdO1-xFxBiS2
(x = 0.37 and 0.3) superconductor by x-ray diffraction measurements. The CDW
wavevector is QCDW $\approx$ (0.17, 0.17, 0.5) with clear L-dependence and the
atomic displacements are transverse and in diagonal directions. The wavevector
does not match theoretical estimations from phonon softening or FSN and
therefore places an important constraint on forthcoming studies. We measured
TDS across the Tc and could not find meaningful $\bm{q}$\- and T- dependent
changes.

##  VI ACKNOWLEDGMENTS

We thanks D. Louca and C. Park for fruitful discussions. Cornell High Energy
Synchrotron Source was supported by the NSF and NIH/National Institute of
General Medical Sciences via NSF award no. DMR-1332208.

## References

  * (1) K. Deguchi, Y. Takano, and Y. Mizuguchi, Sci. Technol. Adv. Mater. 13, 054303 (2012).
  * (2) Y. Mizuguchi, H. Fujihisa, Y. Gotoh, K. Suzuki, H. Usui, K. Kuroki, S. Demura, Y. Takano, H. Izawa, and O. Miura, Phys. Rev. B 86, 220510(R) (2012).
  * (3) K. Deguchi, Y. Takano, and Y. Mizuguchi, Sci. Technol. Adv. Mater. 13, 054303 (2012).
  * (4) D. Yazici, I. Jeon, B. D. White, and M. B. Maple, Physica C 514, 218 (2015).
  * (5) H. Usui and K. Kuroki, Nov. Supercond. Mater. 1, 50 (2015).
  * (6) K. Terashima, T. Wakita, and T. Yokoya, J. Phys. Soc. Jpn. 88, 041006 (2019).
  * (7) H. Usui, K. Suzuki, and K. Kuroki, Phys. Rev. B 86, 220501(R) (2012).
  * (8) Y. Mizuguchi, T. Hiroi, J. Kajitani, H. Takatsu, H. Kadowaki, and O. Miura, J. Phys. Soc. Jpn. 83, 053704 (2014).
  * (9) T. Yildirim, Phys. Rev. B 87, 020506(R) (2013).
  * (10) X. Wan, H. -C. Ding, S. Y. Savrasov, and C. -G. Duan, Phys. Rev. B 87, 115124 (2013).
  * (11) B. Li, Z. W. Xing, and G. Q. Huang, EPL 101, 47002 (2013).
  * (12) Y. Feng, H. -C Ding, Y. Du, X. Wan, B. Wang, S. Y. Savrasov, and C. -G. Duan, J. Appl. Phys. 115, 233901 (2014).
  * (13) S. F. Wu, P. Richard, X. B. Wang, C. S. Lian, S. M. Nie, J. T. Wang, N. L. Wang, and H. Ding, Phys. Rev. B 90, 054519 (2014).
  * (14) T. Zhou and Z. D. Wang, J. Supercond. Nov. Magn. 26, 2735 (2013).
  * (15) Y. Liang, X. Wu, W. -F. Tsai, and J. Hu, Frontiers of Physics 9, 194 (2014).
  * (16) K. Suzuki, H. Usui, and K. Kuroki, Physics Procedia 45, 21 (2013).
  * (17) G. B. Martins, A. Moreo, and E. Dagotto, Phys. Rev. B 87, 081102(R) (2013).
  * (18) Y. Yang, W. S. Wang, Y. Y. Xiang, Z. Z. Li, and Q. H. Wang, Phys. Rev. B 88, 094519 (2013).
  * (19) C. Morice, R. Akashi, T. Koretsune, S. S. Saxena, and R. Arita, Phys. Rev. B 95, 180505 (2017).
  * (20) K. Suzuki, H. Usui, K. Kuroki, and H. Ikeda, Phys. Rev. B 96, 024513 (2017).
  * (21) J. Lee, M. B. Stone, A. Huq, T. Yildirim, G. Ehlers, Y. Mizuguchi, O. Miura, Y. Takano, K. Deguchi, S. Demura, and S.-H. Lee, Phys. Rev. B 87, 205134 (2013).
  * (22) Y. Tian, A. Zhang, J. Ji, J. Liu, X. Zhu, H.-H. Wen, F. Jin, X. Ma, and Q.-M. Zhang, Supercond. Sci. Technol. 29, 015007 (2016).
  * (23) H. Miao, R. Fumagalli, M. Rossi, J. Lorenzana, G. Seibold, F. Yakhou-Harris, K. Kummer, N. B. Brookes, G. D. Gu, L. Braicovich, G. Ghiringhelli, and M. P. M. Dean, Phys. Rev. X 9, 031042 (2019).
  * (24) S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, Nat. Rev. Mater. 2, 17033 (2017).
  * (25) T. Machida, Y. Fujisawa, M. Nagao, S. Demura, K. Deguchi, Y. Mizuguchi, Y. Takano, and H. Sakata, J. Phys. Soc. Jpn. 83, 113701 (2014).
  * (26) S. Demura, N. Ishida, Y. Fujisawa, and H. Sakata, J. Phys. Soc. Jpn. 86, 113701 (2017).
  * (27) N. Ishida, S. Demura, Y. Fujisawa, S. Ohta, K. Miyata, and H. Sakata, J. Phys. Conf. Ser. 1054, 012001 (2018).
  * (28) T. Tomita, M. Ebata, H. Soeda, H. Takahashi, H. Fujihisa, Y. Gotoh, Y. Mizuguchi, H. Izawa, O. Miura, S. Demura, K. Deguchi, and Y. Takano, J. Phys. Soc. Jpn. 83, 063704 (2014).
  * (29) Q. Liu, X. Zhang, and A. Zunger, Phys. Rev. B 93, 174119 (2016).
  * (30) X. Zhou, Q. Liu, J. A. Waugh, H. Li, T. Nummy, X. Zhang, X. Zhu, G. Cao, A. Zunger, and D. S. Dessau, Phys. Rev. B 95, 075118 (2017).
  * (31) Y. Chen, Y. Cui, A. Pham, Y. Wang, M. M. Bhadbhade, R. Wang, Y. Su, H. Hu, Z. Wen, C. Cheng, T. T. Tan, S. Li, and Y. Zhao, J. Mater. Chem. C 7, 586 (2019).
  * (32) A. Athauda and D. Louca, J. Phys. Soc. Jpn. 88, 041004 (2019).
  * (33) A. Athauda, Y. Mizuguchi, M. Nagao, J. Neuefeind, and D. Louca, J. Phys. Soc. Jpn. 86, 124718 (2017).
  * (34) A. Athauda, C. Hoffmann, S. Aswartham, J. Terzic, G. Cao, X. Zhu, Y. Ren, and D. Louca, J. Phys. Soc. Jpn. 86, 054701 (2017).
  * (35) A. Athauda, J. Yang, S.-H. Lee, Y. Mizuguchi, K. Deguchi, Y. Takano, O. Miura, and D. Louca, Phys. Rev. B 91, 144112 (2015).
  * (36) S. Demura, Y. Mizuguchi, K. Deguchi, H. Okazaki, H. Hara, T. Watanabe, S. J. Denholme, M. Fujioka, T. Ozaki, H. Fujihisa, Y. Gotoh, O. Miura, T. Yamaguchi, H. Takeya, and Y. Takano, J. Phys. Soc. Jpn. 82, 033708 (2013).
  * (37) Y. Mizuguchi, S. Demura, K. Deguchi, Y. Takano, H. Fujihisa, Y. Gotoh, H. Izawa, and O. Miura, J. Phys. Soc. Jpn. 81, 114725 (2012).
  * (38) S. Demura, Y. Mizuguchi, K. Deguchi, H. Okazaki, H. Hara, T. Watanabe, S. J. Denholme, M. Fujioka, T. Ozaki, H. Fujihisa, Y. Gotoh, O. Miura, T. Yamaguchi, H. Takeya, and Y. Takano, J. Phys. Soc. Jpn. 82, 033708 (2013).
  * (39) F. Weber, S. Rosenkranz, J.-P. Castellan, R. Osborn, R. Hott, R. Heid, K.-P. Bohnen, T. Egami, A. H. Said, and D. Reznik, Phys. Rev. Lett. 107, 107403 (2011).
  * (40) G. F. Giuliani and A. W. Overhauser, Phys. Rev. B 26, 1660 (1982).
  * (41) J. Lee , K. Prokes, S. Park, I. Zaliznyak, S. Dissanayake, M. Matsuda, M. Frontzek, S. Stoupin, G. L. Chappell, R. E. Baumbach, C. Park, J. A. Mydosh, G. E. Granroth, and J. P. C. Ruff, Phys. Rev. B 102, 041112(R) (2020).
  * (42) See Supplementary Material for more discussion on the volume fraction estimation.
  * (43) N. Poccia, M. Fratini, A. Ricci, G. Campi, L. Barba, A. Vittorini-Orgeas, G. Bianconi, G. Aeppli, and A. Bianconi, Nat. Mater. 10, 733 (2011).
  * (44) N. Poccia, S. Y. F. Zhao, H. Yoo, X. Huang, H. Yan, Y. S. Chu, R. Zhong, G. Gu, C. Mazzoli, K. Watanabe, T. Taniguchi, G. Campi, V. M. Vinokur, and P. Kim, Phys. Rev. Mater. 4, 114007 (2020).
  * (45) See Supplementary Material for more discussion on the relationship between CDW domain size and superconductivity.
  * (46) G. Campi, A. Bianconi, N. Poccia, G. Bianconi, L. Barba, G. Arrighetti, D. Innocenti, J. Karpinski, N. D. Zhigadlo, S. M. Kazakov, M. Burghammer, M. v. Zimmermann, M. Sprung, and A. Ricci, Nature 525, 359 (2015).
  * (47) N Poccia, A. Ricci, G. Campi, M. Fratini, A. Puri, D. Di Gioacchino, A. Marcelli, M. Reynolds, M. Burghammer, N. L. Saini, G. Aeppli, and A. Bianconi, Proc. Natl. Acad. Sci. U.S.A. 109, 15685 (2012).
  * (48) M. E. Flatté, Phys. Rev. B 50, 1190 (1994).
  * (49) H. Kawano, H. Yoshizawa, H. Takeya, and K. Kadowaki, Phys. Rev. Lett. 77, 4628 (1996).
  * (50) P. Aynajian, T. Keller, L. Boeri, S. M. Shapiro, K. Habicht, and B. Keimer, Science 319, 1509 (2008).
  * (51) A. D. Christianson, M. D. Lumsden, O. Delaire, M. B. Stone, D. L. Abernathy, M. A. McGuire, A. S. Sefat, R. Jin, B. C. Sales, D. Mandrus, E. D. Mun, P. C. Canfield, J. Y. Y. Lin, M. Lucas, M. Kresch, J. B. Keith, B. Fultz, E. A. Goremychkin, and R. J. McQueeney, Phys. Rev. Lett. 101, 157004 (2008).
  * (52) D. Phelan, J. N. Millican, E. L. Thomas, J. B. Leão, Y. Qiu, and R. Paul, Phys. Rev. B 79, 014519 (2009).
  * (53) R. Chaudhury and M. P. Das, J. Phys. Condens. Matter 25, 122202 (2013).
  * (54) M. Holt, Z. Wu, H. Hong, P. Zschack, P. Jemian, J. Tischler, H. Chen, and T.-C. Chiang, Phys. Rev. Lett. 83, 3317 (1999).
  * (55) Xuetao Zhu, Jiandong Guo, Jiandi Zhang, and E. W. Plummer, Advances in Physics: X 2, 622 (2017).
  * (56) M. Ochi, R. Akashi, and K. Kuroki, J. Phys. Soc. Jpn. 85, 094705 (2016).
  * (57) K. Suzuki, H. Usui, K. Kuroki, T. Nomoto, K. Hattori, and H. Ikeda, J. Phys. Soc. Jpn. 88, 041008 (2019).
  * (58) R. Sagayama, H. Sagayama, R. Kumai, Y. Murakami, T. Asano, J. Kajitani, R. Higashinaka, T. D. Matsuda, and Y. Aokim, J. Phys. Soc. Jpn. 84, 123703 (2015).
  * (59) H. Frohlich, Proc. R. Soc. Lon. Ser-A 223, 296 (1954).