# Semantic-aware Generation of Multi-view Portrait Drawings

Biao Ma1 Fei Gao2,3∗ Chang Jiang1 Nannan Wang4 Gang Xu1 1 School of Computer
Science and Technology, Hangzhou Dianzi University
2 Hangzhou Institute of Technology, Xidian University; 3 AiSketcher Tech.
4 ISN State Key Laboratory, Xidian University
{aiartma, jc233<EMAIL_ADDRESS>{fgao<EMAIL_ADDRESS>
###### Abstract

Neural radiance fields (NeRF) based methods have shown amazing performance in
synthesizing 3D-consistent photographic images, but fail to generate multi-
view portrait drawings. The key is that the basic assumption of these methods
– a surface point is consistent when rendered from different views – doesn’t
hold for drawings. In a portrait drawing, the appearance of a facial point may
changes when viewed from different angles. Besides, portrait drawings usually
present little 3D information and suffer from insufficient training data. To
combat this challenge, in this paper, we propose a Semantic-Aware GEnerator
(SAGE) for synthesizing multi-view portrait drawings. Our motivation is that
facial semantic labels are view-consistent and correlate with drawing
techniques. We therefore propose to collaboratively synthesize multi-view
semantic maps and the corresponding portrait drawings. To facilitate training,
we design a semantic-aware domain translator, which generates portrait
drawings based on features of photographic faces. In addition, use data
augmentation via synthesis to mitigate collapsed results. We apply SAGE to
synthesize multi-view portrait drawings in diverse artistic styles.
Experimental results show that SAGE achieves significantly superior or highly
competitive performance, compared to existing 3D-aware image synthesis
methods. The codes are available at https://github.com/AiArt-HDU/SAGE.

##  1 Introduction

3D-aware image synthesis Mildenhall et al. (2021) aims to generate multi-view
consistent images and, to a lesser extent, extract 3D shapes, without
supervision on geometric or multi-view image datasets. Recently, inspired by
the great success of Neural Radiation Fields (NeRF) Mildenhall et al. (2021)
and Generative Adversarial Networks (GANs) Karras et al. (2019), impressive
progress has been achieved in generating multi-view photos as well as detailed
geometries Deng et al. (2022b); Xiang et al. (2022); Chan et al. (2022b).
Besides, several recent methods Zhang et al. (2022); Gu et al. (2021);
Niemeyer and Geiger (2021); Zhou et al. (2021) can also synthesize high
quality artistic images, such as oil-paintings. While marveling at the
impressive results of 3D-aware image synthesis methods, we wish to extend the
style of synthesized images. Unfortunately, the advanced methods all fail to
generate high quality multi-view portrait drawings, e.g. facial line-drawings
(Fig. 1).

Figure 1: Portrait drawings synthesized by MVCGAN Zhang et al. (2022), CIPS-3D
Zhou et al. (2021), and our method, i.e. SAGE. The final column show training
examples in target styles.

There are mainly three reasons for their failure. First, the assumption of
NeRF-based methods – a surface point is consistent when rendered from
different views – doesn’t hold for drawings. Human artists typically use a
sparse set of strokes to represent geometric boundaries, and use diverse
levels of tones to present 3D structures Sousa and Buchanan (1999). As
illustrated in Fig. 2, the boundaries may vary when a face is viewed from
different angles. In other words, the appearance of a facial point may be
inconsistent between different views, in portrait drawings. Second, portrait
drawings usually present sparse information with little 3D structures.
Existing NeRF-based methods produce radiance fields and render images based on
adjacent correlations and stereo correspondence Zhang et al. (2022); Deng et
al. (2022b). As a result, it is not appropriate to directly apply previous
methods for portrait drawing synthesis. Third, previous methods require a
large amount of training data. Unfortunately, it is extremely time-consuming
and laborious for human artists to create adequate portrait drawings.

Figure 2:  In portrait drawings, the appearance of a facial point may be
inconsistent between different views. For example, the nose tip is represented
by strokes in both View 1 and View 2, but is left blank in View 0.

To combat this challenge, in this paper, we propose a Semantic-Aware GEnerator
(SAGE) for synthesizing multi-view portrait drawings. Our motivation is that
facial semantic labels are view-consistent and are highly correlated with the
appearance of portrait drawings. Commonly human artists draw different
semantic areas by using adaptive drawing techniques Sousa and Buchanan (1999).
We therefore collaboratively synthesize multi-view consistent semantic maps
and the corresponding portrait drawings. Besides, we use semantic maps to
guide synthesis of portrait drawings, through semantic-adaptive normalization
Wang et al. (2018). As a result, the synthesized drawings are constrained to
convey facial semantic structures, instead of multi-view consistency.

In addition, NeRF-based modules, including radiation fields production and
volume rendering (VR) Zhang et al. (2022), are still essential for producing
multi-view images. For effective learning of such modules, we propose a two-
stage generation and training strategy to make NeRF-based modules suitable for
synthesizing portrait drawings. First, we train the NeRF modules for
generating multi-view facial photos and semantic maps in parallel. Afterwards,
we use a semantic-adaptive domain translator to synthesizes portrait drawings
from features of photographic faces. The warm-up in the first stage makes
NeRF-based methods suitable for synthesizing portrait drawings. Consequently,
the final generator becomes capable of producing high quality portrait
drawings under different views. Finally, we use data augmentation via
synthesis to obtain adequate training samples, and to mitigate collapsed
results under large pose variants.

We apply our method, SAGE, to synthesize diverse styles of portrait drawings,
including pen-drawings Yi et al. (2019), line-drawings, pencil-drawings Fan et
al. (2022), and oil-paintings Nichol (2016). Experimental results show that
SAGE achieves significantly superior or highly competitive performance,
compared to existing 3D-aware image synthesis methods. Especially, our method
stably generate high quality results across a wide range of viewpoints.

##  2 Related Work

###  2.1 3D-aware Image Synthesis

3D-aware image synthesis aims to explicitly control the camera view of
synthesized images. Recently, numerous NeRF-based methods have been proposed
and achieved impressive progress. For example, to achieve more photorealistic
results, StyleNeRF Gu et al. (2021) uses StyleGAN Karras et al. (2019) to
obtain high-resolution images by up-sampling low-resolution feature maps
through Convolutional Neural Networks (CNNs). StyleSDF Or-El et al. (2022)
employs a signed distance field to obtain low-resolution spatial features and
then produces high-resolution images through up-sampling. GIRAFFE Niemeyer and
Geiger (2021) instead up-samples multi-scale feature maps based on GRAF
Schwarz et al. (2020) to generate high-resolution and multi-view consistent
images. Recently, MVCGAN Zhang et al. (2022) introduces a method to warp
images based on camera matrices, based on pi-GAN Chan et al. (2021). GRAM Deng
et al. (2022b); Xiang et al. (2022) optimizes point sampling and radiance
field learning.

A similar work to ours is CIPS-3D Zhou et al. (2021), which use a NeRF-based
network for view control and a 2D implicit neural representation (INR) network
for generating high-resolution images. CIPS-3D is also trained in two-stages
so as to synthesize artistic portraits, e.g. oil-paintings and cartoon faces.
However, CIPS-3D produces geometric deformations and unpleasant artifacts,
when applied to portrait drawings (Fig. 1). Differently, we propose a novel
two-stage generation framework, i.e. using a domain translator to synthesize
portrait drawings conditioned on photographic features. Besides, we generate
multi-view images based on semantic consistency instead of 3D consistency.
Recently, both FENeRF Sun et al. (2022b) and IDE-3D Sun et al. (2022a)
directly use NeRF to render facial semantics, as well as multi-view images,
for face editing. Differently, we decode both facial semantics and portrait
drawings from photographic features, and use semantics to provide structural
guidance.

Figure 3: Pipeline of our semantic-aware portrait drawing generator, SAGE. The
feature projector $G_{F}$ generates feature maps $\mathbf{F}$ and
normalization parameters $\mathbf{w}$, which control content and viewpoints of
generated faces. In Stage-I, we train the feature projector $G_{F}$, semantic
decoder $G_{s}$, and image decoder $G_{I}$ to enable the generator producing
multi-view facial photos and semantic masks. In Stage-II, we add the portrait
drawing decoder $G_{p}$, and refine all decoders for synthesizing high quality
portrait drawings based on features of photographic faces.

###  2.2 2D Artistic Portrait Drawing Generation

For now, there have been a huge number of methods for generating 2D artistic
portrait drawings. Most existing methods try to translate facial photos to
sketches, e.g. pencil-drawings Wang and Tang (2009); Wang et al. (2017), pen-
drawings Yi et al. (2019), and line-drawings Gao et al. (2020); Yi et al.
(2022). The advanced methods are mainly based on 2D conditional GANs Isola et
al. (2017); Gui et al. (2021) and formulate this task as image-to-image
translation. Researchers have made great efforts to boost the identity-
consistency and texture-realism, by designing new learning architectures Zhang
et al. (2019); Zhu et al. (2021); Zhang et al. (2018), or using auxiliary
geometric information, e.g. facial landmarks Yi et al. (2019); Gao et al.
(2020), semantics Yu et al. (2021); Li et al. (2021), and depth Chan et al.
(2022a). There are also several unsupervised methods for synthesizing 2D
portrait drawings. These methods Noguchi and Harada (2019); Li et al. (2020);
Ojha et al. (2021); Kong et al. (2021) aims at solving the few-shot learning
problem of big generative models (e.g., StyleGAN series Karras et al. (2019))
and verify them on portrait drawings. In this paper, we use a modification of
GENRE Li et al. (2021) to synthesis 2D portrait drawings for data augmentation
(Section 3.4). Besides, the architecture of our domain translator is similar
to GENRE (Section 3.2).

##  3 Method

Our whole model follows the architectures of NeRF-based GANs Zhang et al.
(2022). As shown in Fig. 3, our generator $G$ includes four parts: a NeRF-
based feature projector $G_{F}$, a semantic decoder $G_{s}$, and an image
decoder $G_{I}$ followed by a domain translator $G_{p}$. $G_{p}$ produces
portrait drawings conditioned on input latent code $\mathbf{z}$ and viewpoint
$\mathbf{x}$. Correspondingly, we use three discriminators to judge realism of
semantic maps, facial images, and drawings, respectively.

###  3.1 NeRF-based Feature Projection

We first project latent codes and viewpoints to representations of multi-view
faces by NeRF. Specially, we map the latent code $\mathbf{z}$ to parameters
$\mathbf{w}=\\{\mathbf{w}_{c},\mathbf{w}_{s}\\}$ through a mapping network,
$G_{m}$. $\textbf{w}_{c}$ controls content (e.g. identity, structure, and
appearance) of the generated images. $\textbf{w}_{s}$ controls the style of
synthesized portrait drawing. Given a viewpoint $\mathbf{x}$ in space, we map
it to appearance parameters through a multilayer FiLM network Perez et al.
(2018). Afterwards, we use the _Volume Rendering_ (VR) Deng et al. (2022a)
module to produce a facial image and the corresponding feature map. In the
implementation, $\textbf{w}_{c}$ is composed of multiple vectors representing
frequencies and phases, and is fed into the FiLM layers. $\textbf{w}_{s}$
modulate deep features in sematic and image decoders, in the manner of AdaIN
Huang and Belongie (2017).

During training, we follow the settings of MVCGAN Zhang et al. (2022).
Specially, we render two images from primary viewpoint $\mathbf{x}_{pri}$ and
auxiliary viewpoint $\mathbf{x}_{aux}$, respectively. The corresponding
images, $I_{pri}$ and $I_{aux}$, represent the same face but with different
poses. Let $\mathbf{F}_{pri}$ and $\mathbf{F}_{aux}$ be the corresponding
feature representations. Afterwards, $I_{aux}/\mathbf{F}_{aux}$ are
geometrically aligned to $I_{pri}/\mathbf{F}_{pri}$ thorough warping. The
warped image $I_{warp}$ is constrained to be the same as $I_{pri}$ by a image-
level reprojection loss (Eq.5). Besides, the primary feature
$\mathbf{F}_{pri}$ is linearly mixed with the warped feature
$\mathbf{F}_{warp}$. The resulting mixed feature $\mathbf{F}$ is fed into the
following decoders. In the testing stage, the auxiliary position and the
stereo mixup module aren’t needed.

###  3.2 Semantic-aware Portrait Generation

To enable the network successfully produce facial structures, we propose to
decode portrait drawings from features of photographic faces. Besides, we use
2D semantic information to guide the synthesis of portrait drawings.

Semantic and Image Decoders. First, we use a semantic $G_{s}$ and an image
decoder $G_{I}$ to collaboratively synthesize semantic maps and facial photos.
Both decoders follow the same architectures, but with different numbers of
output channels. As shown in Fig. 4, each decoder contains three upsampling
convolutional layers, and progressively generates high-resolution outputs. As
previously mentioned, features over each layer are channel-wisely modulated by
$\mathbf{w}_{s}$ in the manner of AdaIN. Besides, we produce multi-scale
outputs and integrate them together for producing a final output. The image
decoder, $G_{I}$, generates a RGB facial image, i.e.

$\hat{I}^{3\times 256\times 256}=G_{I}(\mathbf{F},\textbf{w}_{s}).$ (1)

The semantic decoder, $G_{s}$, produces the corresponding 19-channel semantic
masks Lee et al. (2020), i.e.

$\hat{S}^{19\times 256\times 256}=G_{s}(\mathbf{F},\textbf{w}_{s}).$ (2)

Semantic-adaptive Domain Translator.  In addition, we use a semantic-adaptive
domain translator, $G_{p}$, to generate portrait drawings based on features of
photographic faces. We design $G_{p}$ following the structure of U-Net Isola
et al. (2017), and use facial semantics $\hat{S}$ to guide synthesis in the
manner of SPADE Park et al. (2019); Li et al. (2021). Since our model is
trained without supervision, the synthesized semantic maps might be incorrect.
We therefore only use SPADE over small-scale layers in $G_{p}$ to control the
major structure of portrait drawings. The process of portrait drawing
synthesis is formulated as:

$\begin{split}\hat{P}^{3\times 256\times
256}=G_{p}(\hat{I},\hat{S}).\end{split}$ (3)

The semantic guidance will enable our model producing natural and distinct
facial structures, as well as semantic-related details (Section 4.4). In
addition, with the semantic modulation module, our method allows minor editing
on portrait drawings (Section 4.5).

Figure 4: Semantic decoder (right) and portrait decoder (left and bottom). The
$\mathrm{tRGB}$ and $\mathrm{tSem}$ module map deep features to a RGB image
and semantic maps by $1\times 1$ convolution, respectively. $\mathrm{UpSmp}$
denotes upsampled interpolation.

###  3.3 Two-stage Training

In preliminary experiments, we train the whole model end-to-end on portrait
drawings. The model produces massy ink patterns, and fails to generate
acceptable facial structures (Section 4.4). As previously analyzed, the
possible reason is that portrait drawings present sparse appearances with
little 3D information. To combat this challenge, we propose to train the model
in two stages. First, we train the feature projector, semantic decoder, and
image decoder, to enable them generating high quality facial structures.
Afterwards, we add the domain translator $G_{p}$, and refine all the decoding
modules for synthesizing multi-view portrait drawings. This training strategy
conveys well with the architecture of our generator.

Training Stage-I. In the first training stage, we use facial photos and their
corresponding semantic maps Lee et al. (2020) to train our model without
$G_{p}$. We use discriminators similar to pi-GAN Chan et al. (2021), and use a
semantic discriminator $D_{s}$ and an image discriminator $D_{I}$ during
training. The loss function of the discriminators are defined as:

$\begin{split}\mathcal{L}_{D_{s}}=&~{}\mathbb{E}_{S\sim\mathcal{S}}[f(D_{s}(S))+\lambda_{1}\lvert\nabla
D_{s}(S)\rvert^{2}]\\\
&+\mathbb{E}_{\bm{z}\sim\mathcal{Z},\bm{x}\sim\mathcal{X}}[f(-D_{s}(\hat{S}))],\\\
\mathcal{L}_{D_{I}}=&~{}\mathbb{E}_{I\sim\mathcal{I}}[f(D_{I}(I))+\lambda_{1}\lvert\nabla
D_{I}(I)\rvert^{2}]\\\
&+\mathbb{E}_{\bm{z}\sim\mathcal{Z},\bm{x}\sim\mathcal{X}}[f(-D_{I}(\hat{I}))],\end{split}$
(4)

where $f(u)=-\log(1+\exp(-u))$; $I$ and $S$ denote facial photos and semantic
maps in the training set $\\{\mathcal{I},\mathcal{S}\\}$. The loss function of
the generator (without $G_{p}$) is:

$\begin{split}\mathcal{L}_{G}^{(1)}=&~{}\mathbb{E}_{\bm{z}\sim\mathcal{Z},\bm{x}\sim\mathcal{X}}[f(-D_{s}(\hat{S}))+f(-D_{I}(\hat{I}))]+\lambda_{2}\mathcal{L}_{rec},\\\
\mathcal{L}_{rec}=&~{}\lambda_{3}\rvert
I_{pri}-I_{aux}\lvert+(1-\lambda_{3})\mathrm{SSIM}(I_{pri},I_{aux}),\end{split}$
(5)

where $\mathrm{SSIM}$ denotes the structural similarity Wang et al. (2004)
between $I_{1}$ and $I_{2}$; $\lambda_{1,2,3}$ are weighting factors.

Training Stage-II. In the second stage, we load the pre-trained model and add
the portrait drawing decoder $G_{p}$ to it. Afterwards, we fix parameters of
$G_{F}$ and refine all the other parts, by using portrait drawings and their
semantic maps $\\{P\sim\mathcal{P},S\sim\mathcal{S}\\}$. Here, we use an
portrait drawing discriminator $D_{p}$ and the previous semantic discriminator
$D_{s}$ during training. The loss functions of $D_{p}$ and $G$ are defined as:

$\begin{split}\mathcal{L}_{D_{p}}=&~{}\mathbb{E}_{P\sim\mathcal{P}}[f(D_{p}(P))+\lambda_{1}\lvert\nabla
D_{p}(P)\rvert^{2}]\\\
&+\mathbb{E}_{\bm{z}\sim\mathcal{Z},\bm{x}\sim\mathcal{X}}[f(-D_{p}(\hat{P}))],\\\
\mathcal{L}_{G}^{(2)}=&~{}\mathbb{E}_{\bm{z}\sim\mathcal{Z},\bm{x}\sim\mathcal{X}}[f(-D_{s}(\hat{S}))+f(-D_{p}(\hat{P}))].\end{split}$
(6)

We remove the image discriminator $D_{I}$ and $\mathcal{L}_{rec}$ in this
stage, so as to eliminate their interference on synthesizing realistic
portrait drawings.

Figure 5: Qualitative comparison on Pen-drawings, Line-drawings, and three
styles of Pencil-drawings Fan et al. (2022), at $256^{2}$ resolution. The left
part show results of CIPS-3D Zhou et al. (2021), the right part show ours.

###  3.4 Data Augmentation via Synthesis

To effectively learn a 3D-aware image synthesis model, it’s critical to
collect a huge number of training images of the same style. This requirement
is almost impossible in our task. It’s time-consuming and laborious for human
artists draw adequate portraits. In existing portrait drawing datasets Fan et
al. (2022); Yi et al. (2019), there are merely hundreds of samples for a given
style. These observations imply the significance of data augmentation for
training. In the implementation, we train a modification of GENRE Li et al.
(2021) to synthesize adequate 2D portrait drawings. Afterwards, we use the
synthesized data to train our model in Stage-II.

Figure 6: Qualitative comparison in synthesizing multi-view oil-paintings at
$256^{2}$ resolution. Figure 7: The average preference percent of CIPS-3D and
our method w.r.t. each style of artistic portrait drawings.

##  4 Experiments and Analysis

###  4.1 Settings

To verify and analyze our method, we conduct thorough experiments by using the
following datasets: Pen-drawings. We conduct experiments on the APDrawing Yi
et al. (2019) dataset, which contains 70 pen-drawings of faces. Line-drawings.
We drew about 800 facial line-drawings ourselves. Pencil-drawings. We apply
our method to the three styles of pencil-drawings on the FS2K Fan et al.
(2022) dataset. Each style contains about 350 training images. Oil paintings.
We randomly select 3,133 oil-paintings of humans from WikiArt Nichol (2016) to
evaluate the generalization capability of our model. Facial Photos. We use the
CelebAMask-HQ Lee et al. (2020) dataset during training stage-I and for data
augmentation. It contains 30,000 facial photos and the corresponding semantic
maps. For Pen-drawings, Line-drawings, and three styles of Pencil-drawings, we
train a 2D GAN by using the corresponding real portrait drawings. Afterwards,
we apply the learned models to the facial photos in CelebAMask-HQ dataset.
Finally, the synthesized portrait drawings are used to train 3D-aware image
synthesis models.

###  4.2 Training Details

In the pre-training phase of the model, we set $\lambda_{1}=0.1$,
$\lambda_{2}=1$, and $\lambda_{3}=0.25$. In the second phase of training, the
parameters remain unchanged. We use the Adam optimizer and set $\beta_{1}=0$
and $\beta_{2}=0.9$. At $64^{2}$ resolution, the learning rate of generator is
$6\times 10^{-5}$, the learning rate of discriminators $2\times 10^{-4}$, and
batch size 36. At $128^{2}$ resolution, the learning rate of generator is
$5\times 10^{-5}$ and batch size 24. At $256^{2}$ resolution, the learning
rate of generator is $3\times 10^{-5}$, the learning rate of discriminators
$1\times 10^{-4}$, and batch size 24. We use a single GeForce RTX3090 to train
our model.

Table 1: Quantitative comparison between SAGE and advanced methods, including CIPS-3D Zhou et al. (2021) and MVCGAN Zhang et al. (2022). Smaller values of both SIFID and SWD indicate better quality of synthesised images. The best criteria are highlighted in bold. | Pen-drawing | Line-drawing | Pencil-drawing Style0 | Pencil-drawing Style1 | Pencil-drawing Style2 | Oil-paintings
---|---|---|---|---|---|---
| SIFID | SWD | SIFID | SWD | SIFID | SWD | SIFID | SWD | SIFID | SWD | SIFID | SWD
CIPS-3D Zhou et al. (2021) | 3.51 | 86.9 | 3.31 | 91.3 | 5.49 | 40.7 | 5.12 | 25.1 | 5.33 | 24.9 | 4.86 | 14.0
MVCGAN Zhang et al. (2022) | 4.19 | 59.8 | 3.69 | 99.1 | 5.76 | 48.6 | 5.15 | 24.7 | 5.29 | 25.5 | 4.56 | 36.3
SAGE (Ours) | 3.07 | 38.5 | 3.04 | 89.8 | 5.16 | 34.0 | 5.13 | 19.7 | 5.11 | 19.9 | 4.71 | 26.5
Figure 8: Curves of SIFID values across different views. The $x$-axis denotes
the pose variants sequentially corresponding to Fig. 5 and Fig. 6. The
$y$-axis denotes values of SIFID.  Figure 9: Results of ablation study.

###  4.3 Comparison with SOTAs

We compare SAGE with two SOTA methods, i.e. CIPS-3D Zhou et al. (2021) and
MVCGAN Zhang et al. (2022). We use their official implementations provided by
the authors, and conduct experiments under the same settings as ours.

Qualitative comparison on portrait-drawings. As previously presented in Fig.
1, MVCGAN fails to generate acceptable portrait drawings. Fig. 5 shows the
portrait drawings generated by CIPS-3D and SAGE. Here, we show seven views of
each sample. Obviously, our results show more natural structures than CIPS-3D.
The synthesized drawings of CIPS-3D, especially the pen-drawings and pencil-
drawings, present geometric deformations. Besides, CIPS-3D produces
unappealing artifacts under large pose variants. In contrast, our method,
SAGE, generates high quality portrait drawings across all the poses and all
the styles. Specially, our synthesized portrait drawings present distinct
facial structures with realistic strokes and textures.

Quantitative comparison on portrait-drawings. We further evaluate the quality
of synthesized images by using the Fréchet Inception Distances (FID) Heusel et
al. (2017) and sliced Wasserstein distance (SWD) Karras et al. (2017). For
each style of portrait drawing, we use $4K$ generated images and $4K$ real
images for computing FID and SWD. As shown in Table1, our method achieves
significantly lower values of both FID and SWD than CIPS-3D and MVCGAN; except
that the SIFID value is slightly higher than CIPS-3D on Pencil-drawing Style
1. Such comparison results are consistent with Fig. 5, and further demonstrate
the superiority of our method in generating multi-view portrait drawings.

Comparison on oil-paintings. Since both MVCGAN and CIPS-3D perform well on
synthesizing oil-paintings, we additionally compare with them on the WikiArt
dataset. Fig. 6 shows the results of multi-view oil-paintings synthesis.
Obviously, CIPS-3D produces geometric deformations and unappealing artifacts.
MVCGAN produces reasonable paintings but with blurring details. In contrast,
our results present the best quality, in terms of either facial structures or
painting textures. Table 1 shows the SIFID and SWD values computed from $1500$
generated paintings and $1500$ real ones. Our method achieves the best SIFID
value, but shows inferiority to CIPS-3D in terms of SWD. Based on all the
comparison results, we can safely conclude that our method is superior and at
least highly competitive to SOTAs in synthesizing multi-view oil-paintings.

Stability across pose variants. We finally analyze the stability of the three
models in generating multi-view images. Specially, we compute the SIFID values
corresponding to different viewpoints/poses. Fig. 8 shows that the SIFID
values of either CIPS-3D or MVCGAN present dramatically fluctuate in most
cases. Such fluctuations indicate the quality of generated portrait drawings
changes dramatically with pose variants. In contrast, our method achieves the
best and most stable performance in general, across all the six styles. In
other words, our method consistently produces high quality multi-view portrait
drawings. This comparison result is consistent with Fig. 5 and Fig. 6.

User study. We conducted a series of user study. Specially, we adopt the
stimulus-comparison method following ITU-R BT.500-12. We randomly select 600
generated comparison pairs in total, i.e. 100 pairs for each style. Each pair
is evaluated by 37 participants. Finally, we collect 22,200 preference labels.
Fig.7 shows the average preference percent of CIPS-3D and SAGE, w.r.t. each
style. Obviously, over $92\%$ of participants think the portraits generated by
SAGE are better than CIPS-3D, across all the styles. Such results demonstrate
that our method is significantly better than CIPS-3D in synthesizing multi-
view artistic portrait drawings.

###  4.4 Ablation Study

We conduct a series of ablation experiments to validate the significance of
our proposed techniques, including two-stage training, the domain translator
$G_{p}$, and the use of semantic masks, i.e. SPADE. We build several model
variants by progressively using these techniques on our base model. The
experiments are mainly conducted on the APDrawing dataset. The corresponding
results are shown in Fig. 9 and Table 2.

Two-stage training strategy. We compare end-to-end training (End2End) with
two-stage training on the base model. As shown in Fig. 9, end-to-end training
leads to messy facial structures. In contrast, our two-stage training enables
the base model producing natural facial structures. Besides, two-stage
training dramatically decreases the SIFID and SWD values. Such significant
performance improvement demonstrate our motivation of using two-stage
training.

Domain translator $G_{p}$. We further evaluate the role of domain translator.
We here build a model variant by removing the SPADE modules from $G_{p}$. the
architecture of pristine U-Net. As shown in Fig. 9, using a domain translator,
even U-Net, dramatically boosts the quality of generated pen-drawings.
Correspondingly, Table 2 shows that using U-Net alone significantly decreases
both SIFID and SWD values. These results verify our motivation of using a
domain translator to decode portrait drawings from features of facial photos.
In this way, our generator can produce distinct facial structures and realism
artistic patterns.

Semantic guidance. Finally, we analyze the role of semantics in $G_{p}$. As
shown in Fig.9, SPADE improves clarity and continuity of facial semantic
boundaries. Correspondingly, SPADE further decreases SIFID and SWD; and our
full model achieves the best performance among all the model variants. To
further verify the significance of facial semantics, we conduct experiments on
line-drawing synthesis. The last two columns of Fig.9 show the corresponding
results. Obviously, without the guidance of facial semantics, the model
generate chaotic lines. In contrast, our full model generate distinctly better
line-drawings.

These results demonstrate that semantic guidance are crucial for generating
multi-style portrait drawings. From one hand, semantic masks enable the domain
translator producing distinct facial structures. For the other hand, facial
semantics are highly correlated with the drawing techniques, human artists
used during the creation process.

Table 2: Quantitative results of ablation experiments about the training strategy and the domain translator $G_{p}$, on pen-drawing synthesis. Training | $G_{p}$ |  |
---|---|---|---
End2End | Two-Stage | U-Net | SPADE | SIFID | SWD
✔ |  |  |  | 7.36 | 92.78
| ✔ |  |  | 4.29 | 65.83
| ✔ | ✔ |  | 3.18 | 43.27
| ✔ | ✔ | ✔ | 3.07 | 38.45
Figure 10: Semantic editing on synthesized portrait drawings. Figure 11: Style
transfer from line-drawings to pen-drawings.  Figure 12: Result of identity
interpolation.

###  4.5 Applications

Semantic editing. Recall that we use semantic maps in the domain translator
$G_{p}$ to control facial structures of portrait drawings. As a result, our
model allows for editing of portrait drawings, to a certain extent. In other
words, if we modify the input semantics of $G_{p}$ slightly, the synthesized
portrait drawing will change accordingly. As shown in Fig. 10, when we change
the semantic masks of teeth or eyes, the corresponding areas in synthesized
images change accordingly.

Style transfer. In our generator, the content and style information are
disentangled and stored in $\mathbf{F}$ and $\mathbf{w}_{s}$, respectively. It
is thus possible for us to change a portrait drawing to another style without
changing the facial content. To this end, we put a latent code
${\mathbf{z}_{1}}$ into the learned line-drawing model to get the content
feature $\mathbf{F}$; and put another latent code ${\mathbf{z}_{2}}$ into the
learned pen-drawing model to get the style vector $\mathbf{w}_{s}$.
Afterwards, we use $\mathbf{F}$ and $\mathbf{w}_{s}$ for decoding a pen-
drawing. As shown in Fig. 11, the line-drawings are transferred to pen-
drawings, while preserving facial semantic content.

Identity interpolation. We also perform identity interpolation experiments on
SAGE. Given two synthesized images, we perform linear interpolation in the
content space $\textbf{w}_{c}\sim\mathcal{W}_{c}$ and viewpoint space
$\mathbf{x}\sim\mathcal{X}$. Fig. 12 shows interpolation results on Pen-
drawings. The smooth transition in pose and appearance implies that SAGE
allows precise control on both facial identity and pose.

##  5 Conclusion

We propose a novel method, SAGE, for generating multi-view portrait drawings.
SAGE is designed to collaboratively synthesize facial photos, semantic masks,
and portrait drawings. Extensive experiments and a series of ablation study
are conducted on six styles of artistic portraits. SAGE stably shows
impressive performance in generating high quality portrait drawings, and
outperforms previous 3D-aware image synthesis methods. In the future, we are
interested in synthesizing multi-view portrait drawings conditioned on a
single photo. One possible way is to incorporate GAN-inversion and few-shot
learning techniques.

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# Cluster quantum Monte Carlo study of two-dimensional
weakly-coupled frustrated trimer antiferromagnets

Lukas Weber  Nils Caci  Stefan Wessel  Institute for Theoretical Solid State
Physics, RWTH Aachen University, JARA Fundamentals of Future Information
Technology, and
JARA Center for Simulation and Data Science, 52056 Aachen, Germany

###### Abstract

We report results from spin trimer-based cluster quantum Monte Carlo
simulations for the thermodynamic properties of two-dimensional frustrated
quantum antiferromagnets that are composed of weakly-coupled three-spin
(trimer) clusters. In particular, we consider the spin-1/2 kagome lattice with
a strong breathing distortion, and the triangle-square lattice model proposed
previously for the cuprate La4Cu3MoO12. For both cases, we demonstrate that an
appropriately chosen trimer-based computational basis allows us to
significantly reduce the quantum Monte Carlo sign problem down to the low-
temperature regime. Besides exploring the thermodynamic behavior for the
triangle-square lattice model we also assess a mean field theory-based
prediction for the onset of chiral order. For the breathing distorted kagome
lattice model, we observe a robust two-peak structure in the specific heat,
both in the quantum spin liquid and the lattice-nematic regimes.

##  I Introduction

Triangular clusters form one of the most basic building blocks of frustrated
magnets in two dimensions Starykh (2015), including the paradigmatic
Heisenberg antiferromagnets on the triangular or kagome lattice. In particular
the $S=1/2$ Heisenberg antiferromagnet on the kagome lattice, which can be
interpreted as a a lattice of corner-sharing triangles, has attracted
considerable interest as a promising system for realizing a quantum spin
liquid Savary and Balents (2016); Yan  _et al._ (2011a); Depenbrock  _et al._
(2012); Jiang  _et al._ (2012); He  _et al._ (2017a) in its groundstate. In
most candidate materials, the magnetic exchange couplings do however not
realize a perfect kagome lattice and are instead distorted Norman (2016),
which can have significant impact on the low temperature physics, e.g., by
stabilizing magnetic order Schnyder  _et al._ (2008), precluding the formation
of a quantum spin liquid.

This is, however, not true for all kinds of distortions, as seen for example
in the “breathing-distorted” kagome lattice, where upward and downward
triangles both remain equilateral but inequivalent to each other (Fig. 1(a))
Mila (1998); Aidoudi  _et al._ (2011); Honecker  _et al._ (2011). Even under
strong such breathing distortion, a quantum spin-liquid phase of the kagome
lattice is expected to be stable Repellin  _et al._ (2017); Jahromi  _et al._
(2020). Such a distortion is indeed observed in the spin-1/2 vanadium
oxyfluoride [NH4]2[C7H14N][V7O6F18], which shows no sign of order or spin
freezing down to low temperatures Aidoudi  _et al._ (2011); Clark  _et al._
(2013); Orain  _et al._ (2017). It is thus important to accurately examine the
ground state and low-temperature thermodynamic behavior of such distorted
model systems.

Figure 1: (a) The breathing kagome lattice with intratrimer bonds
$J_{\triangle}$ along the upwards triangles and intertrimer bonds
$J_{\bigtriangledown}$ along the downwards triangles. (b) The triangle-square
lattice with the intratrimer bonds $J_{1}$, $J_{2}$, and $J_{3}$ as well as
the intertrimer bonds $J$. The $x$ ($y$) lattice direction points to the right
(upwards).

Unbiased numerical studies of both the undistorted and the breathing distorted
kagome lattice have so far been based mainly on exact diagonalization (ED)
calculations Yan  _et al._ (2011a); Sugiura and Shimizu (2013); Läuchli  _et
al._ (2019); Schnack  _et al._ (2018) on (relatively) small clusters, tensor-
network methods such as the density matrix renormalization group (DMRG) Yan
_et al._ (2011b); Depenbrock  _et al._ (2012); Repellin  _et al._ (2017); He
_et al._ (2017b) or infinite projected-entangled pair states (iPEPS) and
related approaches Liao  _et al._ (2017); Chen  _et al._ (2018); Jahromi  _et
al._ (2020), which do however suffer in the case of DMRG from finite-
circumference cylinder geometries and in the case of iPEPS from limited
accessible bond dimensions. Unbiased quantum Monte Carlo methods such as the
stochastic series expansion (SSE) Sandvik and Kurkijärvi (1991); Sandvik
(1992, 1999); Syljuåsen and Sandvik (2002); Alet  _et al._ (2005) typically
suffer from the negative sign problem Henelius and Sandvik (2000); Troyer and
Wiese (2005) in the face of frustration. Averting this basis-dependent sign
problem is however possible for a set of special models by performing a local
change of basis. A notable example for this is the class of highly-frustrated
Heisenberg antiferromagnets on lattices with “fully frustrated” interactions,
where changing to a suitable cluster basis removes the sign-problem Nakamura
(1998); Alet  _et al._ (2016); Honecker  _et al._ (2016); Ng and Yang (2017);
Stapmanns  _et al._ (2018); Weber  _et al._ (2022).

In this work, we show how a suitable cluster basis can also be beneficial in
the case of weakly coupled trimer clusters. Here, the sign problem is in
general still present but can be reduced to a degree where unbiased
calculations of the thermodynamic quantities become possible down to the low-
temperature regime. We will perform these calculations for the $S=1/2$
Heisenberg antiferromagnet on two basic lattices of coupled trimers,
highlighting the strengths and shortcomings of this approach. First, we will
consider a square lattice of triangle clusters (the triangle-square lattice,
Fig. 1(b)), where the triangular clusters are arranged on a bipartite parent
lattice so that the frustration predominantly arises within the clusters
themselves. This model Hamiltonian has been put forward Wessel and Haas
(2001); Wang (2001) to describe the frustrated quantum magnetism observed in
the cuprate-based compound La4Cu3MoO12. Second, we will then return to the
kagome lattice with strong breathing distortion (Fig. 1(a)) as a more
challenging example where also the interaction between trimers takes place on
a highly frustrated triangular parent lattice Repellin  _et al._ (2017);
Jahromi  _et al._ (2020).

The remainder of this article is structured as follows. In Section II we
outline the details of the different cluster bases we employ in our
simulation. In Section III, we present our results for the triangle-square
lattice, which are contrasted in Section IV to those for the breathing-
distorted kagome lattice. Finally, in Section V, we draw a conclusion on our
results and provide an outlook.

##  II Cluster bases

In this paper, we employ and compare three different computational bases that
correspond to the eigenbasis of different physical single-trimer operators.
The first is the conventional single-spin $S_{z}$-basis Sandvik and Kurkijärvi
(1991), in which the $S^{z}$ component of each trimer spin is diagonal,

$S^{z}_{\Delta,\alpha}\ket{m_{1},m_{2},m_{3}}=m_{\alpha}\ket{m_{1},m_{2},m_{3}},\quad\alpha=1,2,3.$

The second is the dimer basis, $\ket{l_{12},m_{12},m_{3}}$, which diagonalizes
the total spin,
$\mathbf{S}_{\Delta,12}^{2}=(\mathbf{S}_{\Delta,1}+\mathbf{S}_{\Delta,2})^{2}$,
and magnetization, $S_{\Delta,12}^{z}=S^{z}_{\Delta,1}+S^{z}_{\Delta,2}$,
along one bond of the trimer (taken here to be the one connecting
$\mathbf{S}_{1}$ and $\mathbf{S}_{2}$) and leaves the third spin untouched,

$\displaystyle\ket{1,1,m_{3}}$
$\displaystyle=\ket{\uparrow,\uparrow,m_{3}},~{}\ket{1,-1,m_{3}}=\ket{\downarrow,\downarrow,m_{3}},$
$\displaystyle\ket{1,0,m_{3}}$
$\displaystyle=(\ket{\uparrow,\downarrow}+\ket{\downarrow,\uparrow})\otimes\ket{m_{3}}/\sqrt{2},$
$\displaystyle\ket{0,0,m_{3}}$
$\displaystyle=(\ket{\uparrow,\downarrow}-\ket{\downarrow,\uparrow})\otimes\ket{m_{3}}/\sqrt{2}.$
(1)

Finally we consider the trimer basis, where the total spin,
$\mathbf{S}_{\Delta}^{2}=(\sum_{\alpha=1}^{3}\mathbf{S}_{\Delta,\alpha})^{2}$,
and total magnetization,
$S^{z}_{\Delta}=\sum_{\alpha=1}^{3}S_{\Delta,\alpha}^{z}$ of the full trimer
are diagonal. In addition to these two, a third operator is needed to
completely distinguish the 8 states on the trimer. This operator is not
uniquely determined, but can be readily constructed from symmetry
considerations. First, choosing an SU(2) symmetric operator ensures that it
has a common eigenbasis with $\mathbf{S}^{2}_{\Delta}$ and $S^{z}_{\Delta}$.
Second, by additionally requiring time-reversal symmetry, its eigenbasis (and
thus the trimer basis) can be chosen to be real-valued with respect to the
single-spin $S^{z}$ basis. Therefore, the Hamiltonian, which is also real-
valued in the $S^{z}$ basis, will not acquire a phase problem by
transformation to the trimer basis 111Even with a complex basis, one can, in
principle, retain real-valued weights within the SSE and related algorithms
due to pairs of configurations with cancelling imaginary parts Hen (2021). In
this work, however, we concentrate on real-valued bases.. The simplest way of
fulfilling these condition is to use one of the dimer total spin operators,
such as $(\mathbf{S}_{1}+\mathbf{S}_{2})^{2}$, which we do in the following.
Therefore, for the trimer basis $\ket{l_{\Delta},m_{\Delta},l_{\Delta,12}}$,
we have three quantum numbers and

$\displaystyle\ket{3/2,+3/2,1}$
$\displaystyle=\ket{\uparrow\uparrow\uparrow},\quad\ket{3/2,-3/2,1}=\ket{\downarrow\downarrow\downarrow},$
$\displaystyle\ket{3/2,+1/2,1}$
$\displaystyle=(\ket{\uparrow\uparrow\downarrow}+\ket{\uparrow\downarrow\uparrow}+\ket{\downarrow\uparrow\uparrow})/\sqrt{3},$
$\displaystyle\ket{3/2,-1/2,1}$
$\displaystyle=(\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\uparrow\downarrow\downarrow})/\sqrt{3},$
$\displaystyle\ket{1/2,+1/2,0}$
$\displaystyle=(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})\otimes\ket{\uparrow}/\sqrt{2},$
$\displaystyle\ket{1/2,-1/2,0}$
$\displaystyle=(\ket{\downarrow\uparrow}-\ket{\uparrow\downarrow})\otimes\ket{\downarrow}/\sqrt{2},$
$\displaystyle\ket{1/2,+1/2,1}$
$\displaystyle=(\ket{\uparrow\downarrow\uparrow}+\ket{\downarrow\uparrow\uparrow}-2\ket{\uparrow\uparrow\downarrow})/\sqrt{6},$
$\displaystyle\ket{1/2,-1/2,1}$
$\displaystyle=(\ket{\downarrow\uparrow\downarrow}+\ket{\uparrow\downarrow\downarrow}-2\ket{\downarrow\downarrow\uparrow})/\sqrt{6}.$
(2)

For the Hamiltonian of uncoupled trimers, the single-spin basis has a sign
problem. The dimer basis is sign-free under the condition that the decoupled
trimers are mirror symmetric perpendicular to the dimer it singles out. By
contrast, the trimer basis can be made sign-free in any case by choosing an
appropriate third operator so that it coincides with the eigenbasis of a
single trimer.

Once an intertrimer coupling is introduced, in general, all three bases are
subject to the sign problem. The severity of this sign problem depends on the
model studied and can also differ largely between different bases. For
example, for the specific case of the fully-frustrated trilayer (FFTL) model
considered in Ref. Weber  _et al._ (2022), the sign problem is completely
eliminated only in the trimer basis (the FFTL is obtained upon adding in Fig.
1(b) couplings of strength $J$ between all spins belonging to nearest-neighbor
trimers). In the following, we will show how, in contrast to the single-spin
basis, the dimer and trimer bases remain useful in the regime of weak
intertrimer coupling and allow us to resolve the thermodynamics of the two
trimer magnets that we examine here.

##  III Triangle-square lattice

In this section, we first consider the triangle-square lattice model of
coupled spin trimers. In this system, the trimers are arranged on a biparite
square lattice, so that the magnetic frustration resides within the individial
trimers. The ground state phase diagram of this model has been analyzed based
on ED and mean-field theory Wessel and Haas (2001); Wang (2001). As detailed
further in Sec. III.3, in the weak $J^{\prime}$-regime, the ground state phase
diagram is composed of various magnetically ordered regimes, depending on the
relative strength of the three intratrimer couplings. In the following we
consider finite-size systems with periodic boundary conditions, where the
number of spins relates via $N=3L^{3}$ to the linear system size $L$ of the
square lattice.

###  III.1 Sign

Figure 2: Average configuration sign, $\langle\mathrm{sign}\rangle$, of the
$L=12$ triangle-square lattice in different computational bases for different
values of the intratrimer couplings $J_{1/2/3}$ and the intertrimer coupling
$J$. In panels (a–d) solid lines show the dimer and trimer basis using the
total spin of the $\mathbf{S}_{1}$-$\mathbf{S}_{2}$ dimer as a quantum number.
In panel (c), the dashed lines show rotated versions of these bases using the
$\mathbf{S}_{2}$-$\mathbf{S}_{3}$ dimer, allowing for an improved sign in this
case.

We start our discussion of the triangle-square lattice by comparing the
average configuration sign, $\langle\mathrm{sign}\rangle$ Henelius and Sandvik
(2000); Troyer and Wiese (2005); Honecker  _et al._ (2016), for the three
different bases and at different points in the parameter space of the
intradimer couplings. (Fig. 2). Generally, the average sign decays to zero as
temperature goes to zero, but the scale of this decay depends on the basis.
For the single-spin basis it is set by the dominant of the intratrimer
couplings $J_{1}$, $J_{2}$, $J_{3}$. For weakly coupled trimers (Fig. 2(a–c)),
the dimer and trimer bases greatly outperform the single spin basis, retaining
a robust sign down to $T\sim J$. Along the symmetric line $J_{1}=J_{2}$, the
dimer basis is slightly more favorable than the trimer basis and off the
symmetric line (Fig. 2(c)), the trimer basis is favorable. In the special case
where $J_{1}\neq J_{2}$ but $J_{2}=J_{3}$ a rotated version of the dimer and
trimer basis, using the $S_{2}$-$S_{3}$ instead of the $S_{1}$-$S_{2}$ dimer,
can be used to further enhance the average sign (Fig. 2). For stronger
couplings (Fig. 2(d)), the trimer basis becomes less effective than the single
spin basis while the dimer basis retains a robust sign down to $T\sim J$ (in
practice, our SSE simulations remain feasible for
$\langle\mathrm{sign}\rangle$ larger than $O(10^{-2})$).

This behavior can be intuitively understood by considering the ways in which
the triangle-square lattice can be extended to eliminate the sign-problem in
the respective bases. For the single-spin basis, no such extension exists. The
only way to remove the sign-problem is to remove couplings – both inter- and
intratrimer – until there is no more geometric frustration. For the dimer
basis, if $J_{1}=J_{2}$, the model becomes sign-free upon adding further
interactions of strength $J$ among all spins that belong to neighboring
horizontal dimers so that that the model turns into rows of fully frustrated
two-leg ladders interlaced by rows of single spin-1/2s. In the trimer basis,
the model can be made sign-free for any values of $J_{1}$, $J_{2}$, and
$J_{3}$ by extending it to the FFTL model Weber  _et al._ (2022), however
doing so requires more additional couplings than in the case of the dimer
basis.

Going back to the original triangle-square lattice, we may assume that at
temperatures $T$ that are large compared to the energy scale of these
modifications, the system should behave similarly to the sign-free system. In
particular, the average sign should stay finite. At lower temperatures, where
these differences become important, the sign will, however, drop towards zero.
For the single-spin basis, the energy scale needed to make the model sign-free
is that of the intratrimer couplings. For the dimer basis, the scale is the
bigger of $|J_{1}-J_{2}|$ (to make the model symmetric) and $J$. For the
trimer basis it is $J$, although with a larger prefactor since in the extended
model, more bonds are added. Thus, unless the mirror symmetry is broken, the
dimer basis is superior. After having examined the behavior of the average
computational sign, we next consider the thermodynamic behavior of the
triangle-square lattice.

###  III.2 Thermodynamics

In the following, we will investigate the thermodynamics of the weakly coupled
triangle-square lattice in the regime $T\gtrsim J$, choosing for different
points in the phase diagram the best-performing basis according to the
preceding analysis. We will commence by looking at the magnetic
susceptibility, followed by the specific heat of the system, at the three
characteristic sets of intratrimer couplings already considered in the above
analysis of the average sign.

Figure 3: The magnetic susceptibility $\chi$ of the $L=12$ triangle-square
lattice, multiplied by the temperature $T$ as a function of temperature for
different values of the intra- and intertrimer couplings. The insets show a
magnification of the data in the low temperature region. For comparison, the
exact solutions for decoupled trimers ($J=0$) and $L=2$ are drawn.

The magnetic susceptibility $\chi$ (Fig. 3) shows that at weak intertrimer
coupling $J\ll J_{1},J_{2},J_{3}$, the physics at high temperatures is well
described by an ensemble of decoupled trimers ($J=0$). In this ensemble, there
is a crossover from the high-temperature Curie law $T\chi=1/4$ to a lower-
temperature regime, where the spins on each trimer combine into one effective
spin-1/2, corresponding to a $T\chi=1/12$ Curie law. At even lower
temperatures, the uncoupled-trimer approximation breaks down as the
intertrimer interactions couple the effective spin-1/2s, eventually leading to
different AFM orders compatible with the square parent lattice of the trimers
Wessel and Haas (2001); Wang (2001). This crossover happens at different
temperatures, depending on the intratrimer couplings, with
$(J_{1},J_{2})/J_{3}=(0.5,1.0)$ being the most robust to interactions out of
those studied here. With increasing intertrimer interaction $J$, deviations
from the decoupled-trimer form are also visible at higher temperatures. We
note that while the exact solution for $L=2$ captures this behavior
qualitatively, it significantly differs from the $L=12$ QMC results at low
temperatures. This indicates an increased correlation length at low
temperatures in the triangle-square lattice. At the same time, this deviation
confirms that the QMC approach does perform well beyond the temperature regime
of trivial few-trimer physics.

Figure 4: The specific heat $C$ of the $L=12$ triangle-square lattice, divided
by the temperature $T$ as a function of temperature for different values of
the intra- and intertrimer couplings. Black lines show the exact solution for
$J=0$.

The specific heat $C$ provides a different perspective on this situation (Fig.
4). Here, at high temperatures, the data is again well described by the
decoupled case (showing even less deviations than $\chi$ at $J/J_{3}=0.1$). We
note that the area under the $C/T$ curves shown in Fig. 4 corresponds to the
released entropy of the model and a deviation from the value $\log 2$ per spin
signifies a residual extensive ground state degeneracy. For $J=0$ this is
indeed the case as the ground state is a product state of doublets (or
quartets in the special case $J_{1}=J_{2}=J_{3}$). At low temperatures, the
intertrimer interactions will in general lift this degeneracy leading to an
additional release of entropy. For $J/J_{3}=0.01$, we cannot resolve this
regime due to the severity of the sign-problem at such low temperatures. At
$J/J_{3}=0.1$, the corresponding temperature scale is higher and we clearly
resolve a low-temperature peak in the case $(J_{1},J_{2})/J_{3}=(1,1)$
corresponding to the lifting of the quartet degeneracy.

In summary, in this section, we computed the thermodynamics of the triangle-
square lattice, resolving different temperature regimes where different
couplings of the Hamiltonian start to play a role. In the next step, we will
take a closer look at the low-temperature regime and check various predictions
of ground-state order in the triangle-square lattice.

###  III.3 Ground-state orders

As mentioned in the previous section, at low temperatures, each trimer of the
triangle-square lattice forms an effective spin-1/2, or in other words, only
the lowest energy doublet contributes to the physics. In a leading-order
perturbative expansion in the weak intertrimer coupling $J$, an effective
Heisenberg model

$H_{\text{eff}}=\sum_{\Delta}J_{x}^{\text{eff}}\mathbf{s}_{\Delta}\cdot\mathbf{s}_{\Delta+\hat{x}}+J_{y}^{\text{eff}}\mathbf{s}_{\Delta}\cdot\mathbf{s}_{\Delta+\hat{y}}$
(3)

for these effective spins, denoted
$\mathbf{s}_{\Delta}=P_{\Delta}\,\mathbf{S}_{\Delta}\,P_{\Delta}$, can be
derived Wang (2001), where $P_{\Delta}$ is a projector to the lowest energy
doublet on trimer ${\Delta}$. Because the trimers themselves are arranged on a
square parent lattice, this effective low-energy model is no longer frustrated
and its ground state phase-diagram is well known to display magnetic order at
different wave vectors $\mathbf{Q}$ depending on the signs of
$J^{\text{eff}}_{x/y}$, which in turn depend on the values of $J_{1/2/3}$
Wessel and Haas (2001); Wang (2001).

The case $J_{1}=J_{2}=J_{3}$ forms an exception in this analysis since the
low-energy subspace consists of two degenerate SU(2) doublets. In addition to
an effective SU(2) spin $\mathbf{s}_{\Delta}$, this gives rise to a pseudospin
low-energy degree of freedom $\boldsymbol{\tau}$ distinguishing the two
doublets. The effective Hamiltonian then becomes

$H_{\text{eff}}^{\prime}=J\sum_{\Delta}\mathbf{s}_{\Delta}\cdot\mathbf{s}_{\Delta+\hat{x}}A_{\Delta,\Delta+\hat{x}}+\mathbf{s}_{\Delta}\cdot\mathbf{s}_{\Delta+\hat{y}}A_{\Delta,\Delta+\hat{y}},$
(4)

where the operators $A_{\Delta,\Delta^{\prime}}$ act on the pseudospins (given
in detail in Ref. Wang (2001) and further below). The ground state of this
model is not well known, but ED and mean-field theory suggest the formation of
$(\pi,\pi)$ order around $J_{1}=J_{2}=J_{3}$. A sketch of the complete phase
diagram as obtained in Ref. Wessel and Haas (2001) is shown in Fig. 5.

Figure 5: Illustration of the magnetic phase diagram of the triangle-square
lattice model obtained from ED in Ref. Wessel and Haas (2001). Depending on
the intratrimer coupling ratios, the effective spin-1/2 degrees of freedom per
trimer form magnetic order along different wave vectors $\mathbf{Q}$. The
extent of the $(\pi,\pi)$ phase depends on the magnitude of $J$, shrinking to
the singular point $J_{1}=J_{2}=J_{3}$ as $J\rightarrow 0$; the illustration
corresponds to $J\approx 0.1J_{3}$.

In the following, we will check the validity of this phase diagram for finite
but small intertrimer couplings. To this end, we compute the magnetic
structure factor,

$S(\mathbf{Q})=\frac{1}{L^{2}}\sum_{\Delta,\Delta^{\prime}}e^{i\mathbf{Q}\cdot(\mathbf{R}_{\Delta}-\mathbf{R}_{\Delta^{\prime}})}\braket{\mathbf{S}_{\Delta}\cdot\mathbf{S}_{\Delta^{\prime}}},$
(5)

with the unit cell positions $\mathbf{R}_{\Delta}$, while simultaneously
scaling the system size $L$ and the temperature $T=J_{3}/2L$ to resolve the
onset of the different magnetic ground state orders. Since the sign problem is
least severe along the symmetric line $J_{1}=J_{2}$, we concentrate on this
case in the following analysis.

Figure 6: Finite-size and temperature scaling ($T=J_{3}/2L$) of the magnetic
structure factor $S(\mathbf{Q})$ of the triangle-square lattice along the
symmetric line ($J_{1}=J_{2}$, $J/J_{3}=0.1$, see Fig. 5). The system size
increases with the opacity of the lines.

A scan of the structure factor at different possible ordering wave vectors
(Fig. 6) reveals that along the symmetric line, the $(\pi,0)$ and $(\pi,\pi)$
orders are most prevalent, with the former dominating at high $J_{1}=J_{2}$
and the latter taking over in the regime around $J_{1}=J_{2}=J_{3}$. At lower
$J_{1}=J_{2}$, the competition between different orders is very close.
However, out of the three structure factors, the one belonging to $(\pi,0)$ is
the only one growing with system size in this region, suggesting that it will
dominate in the thermodynamic limit. These findings are in agreement with the
phase diagram obtained using ED, perturbation theory and mean-field theory. In
particular, the close competition at low $J_{1}=J_{2}$ can be explained by a
weak effective coupling $J_{y}^{\text{eff}}$ that only appears at higher
orders perturbatively in the intertrimer coupling $J$ Wang (2001).

As we saw, most of the phase diagram of the triangle-square lattice can be
understood in terms of the effective spin-1/2 degrees of freedom formed by
each trimer. So far, however, we have not considered the pseudospin degree of
freedom at the point $J_{1}=J_{2}=J_{3}$. This degree of freedom has a similar
role as the third quantum number we had to add in the trimer basis in Section
II in that it lives in the subspace of fixed $l_{\Delta}$ and $m_{\Delta}$. In
Section II, we used the eigenbasis of the operator
$(\mathbf{S}_{\Delta,1}+\mathbf{S}_{\Delta,2})^{2}$ to map out this part of
the trimer Hilbert space while retaining a real basis. Without this
constraint, it is however often useful to use a basis that makes the symmetry
of the $J_{1}=J_{2}=J_{3}$ trimer explicit. One such basis is the eigenbasis
of the chirality operator Subrahmanyam (1995)

$\tau^{z}_{\Delta}:=\frac{\sqrt{3}}{4}\mathbf{S}_{\Delta,1}\cdot\left(\mathbf{S}_{\Delta,2}\times\mathbf{S}_{\Delta,3}\right)=:\ket{R}\\!\bra{R}-\ket{L}\\!\bra{L},$
(6)

with

$\displaystyle\ket{L,\uparrow}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\ket{\uparrow\uparrow\downarrow}+\omega\ket{\uparrow\downarrow\uparrow}+\omega^{*}\ket{\downarrow\uparrow\uparrow}\right),$
(7) $\displaystyle\ket{L,\downarrow}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\ket{\downarrow\downarrow\uparrow}+\omega\ket{\downarrow\uparrow\downarrow}+\omega^{*}\ket{\uparrow\downarrow\downarrow}\right),$
(8) $\displaystyle\ket{R,\uparrow}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\ket{\uparrow\uparrow\downarrow}+\omega^{*}\ket{\uparrow\downarrow\uparrow}+\omega\ket{\downarrow\uparrow\uparrow}\right),$
(9) $\displaystyle\ket{R,\downarrow}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\ket{\downarrow\downarrow\uparrow}+\omega^{*}\ket{\downarrow\uparrow\downarrow}+\omega\ket{\uparrow\downarrow\downarrow}\right),$
(10)

and $\omega=e^{i2\pi/3}$. Writing the effective Hamiltonian of Eq. 4 in this
basis yields Wang (2001)

$\displaystyle H_{\text{eff}}=\frac{J}{9}\sum_{\Delta}$ $\displaystyle
A^{l}_{\Delta}A^{r}_{\Delta+\hat{x}}\,\mathbf{S}_{\Delta}\cdot\mathbf{S}_{\Delta+\hat{x}}$
$\displaystyle+$ $\displaystyle
A^{d}_{\Delta}A^{u}_{\Delta+\hat{y}}\,\mathbf{S}_{\Delta}\cdot\mathbf{S}_{\Delta+\hat{y}},$
(11)

where 222Because our spin numbering convention differs from Ref. Wang (2001),
we pick up an extra phase $\omega$ here, simplifying the Hamiltonian slightly.

$\displaystyle A^{l}_{\Delta}$
$\displaystyle=1-2\omega^{*}\tau^{+}_{\Delta}-2\omega\tau^{-}_{\Delta},$
$\displaystyle=1+\tau_{\Delta}^{x}+\sqrt{3}\tau_{\Delta}^{y},$ (12)
$\displaystyle A^{r}_{\Delta}$
$\displaystyle=1-2\omega\tau^{+}_{\Delta}-2\omega^{*}\tau^{-}_{\Delta},$
$\displaystyle=1+\tau_{\Delta}^{x}-\sqrt{3}\tau_{\Delta}^{y}$ (13)
$\displaystyle A^{d}_{\Delta}$ $\displaystyle=1-2\tau^{x}_{\Delta},$ (14)
$\displaystyle A^{u}_{\Delta}$ $\displaystyle=2+2\tau^{x}_{\Delta}.$ (15)

act on the pseudospin degree of freedom via the operators
$\tau^{+}=\ket{L}\\!\bra{R}$ and $\tau^{-}=(\tau^{+})^{\dagger}$. Note that
the shape of the $A$ operators is strongly constrained by the symmetries of
the triangle-square lattice. The horizontal mirror symmetry $\Sigma$,
represented by $\tau^{x}$ (skipping the trimer index $\Delta$) on each trimer
implies

$\Sigma A^{l}\Sigma^{-1}=A^{r},\quad\Sigma A^{d}\Sigma^{-1}=A^{d},\quad\Sigma
A^{u}\Sigma^{-1}=A^{u}.$ (16)

Similarly, the time-reversal symmetry $\Theta$, given by $K\tau^{x}$, where
$K$ is the complex conjugation operator in the chirality basis, leads to

$\displaystyle\Theta A^{l}\Theta^{-1}$ $\displaystyle=A^{l},$
$\displaystyle\Theta A^{r}\Theta^{-1}$ $\displaystyle=A^{r},$ (17)
$\displaystyle\Theta A^{d}\Theta^{-1}$ $\displaystyle=A^{d},$
$\displaystyle\Theta A^{u}\Theta^{-1}$ $\displaystyle=A^{u}.$ (18)

At low temperatures, the pseudospin degree of freedom may spontaneously break
either of these symmetries, forming a chiral or lattice nematic state. The
former was indeed proposed based on mean-field theory results for
$H_{\text{eff}}^{\prime}$ Wang (2001). In contrast to the magnetic order which
constitutes the breaking of a continuous symmetry, the pseudospin can break
its discrete symmetries also at finite temperature.

To check these possibilities, we compute the correlation functions of the
chirality pseudospin $\tau^{z}$, which is odd under time reversal,

$C^{z}(\mathbf{r})=\braket{\tau^{z}_{0}\tau^{z}_{\mathbf{r}}},$ (19)

and the operator
$\tau^{y}=\frac{2}{\sqrt{3}}(\mathbf{S}_{1}-\mathbf{S}_{2})\cdot\mathbf{S}_{3}$,
which is odd under horizontal reflections,

$C^{y}(\mathbf{r})=\braket{\tau^{y}_{0}\tau^{y}_{\mathbf{r}}}.$ (20)

Since we cannot directly simulate in the chirality basis, both of these
correlation functions contain operators that are off-diagonal in our
computational basis. Nevertheless, it is possible to compute them on-the-fly
during the loop update. In Appendix A we show how such on-the-fly measurements
can be performed within the SSE loop update also in the presence of a sign
problem.

Having identified $\tau^{z}$ and $\tau^{y}$ in terms of constituent spin
operators, we note that $\tau^{x}=(\mathbf{S}_{1}+\mathbf{S}_{2})^{2}-1$. This
highlights that the chirality basis and the computational “$l_{\Delta,12}$”
trimer basis introduced in Section II are related by a simple unitary
pseudospin rotation. Consequently, in the $l_{\Delta,12}$-trimer basis, the
correlation functions $C^{z}$ and $C^{y}$ can still be expressed by simple
(albeit different) Pauli matrices,

$\displaystyle\tau^{z}$
$\displaystyle=\ket{1/2}\\!\bra{1/2}_{l_{\Delta}}\otimes\mathbf{1}_{m_{\Delta}}\otimes
i(\ket{1}\\!\bra{0}-\ket{0}\\!\bra{1})_{l_{\Delta,12}},$ (21)
$\displaystyle\tau^{y}$
$\displaystyle=\ket{1/2}\\!\bra{1/2}_{l_{\Delta}}\otimes\mathbf{1}_{m_{\Delta}}\otimes(\ket{1}\\!\bra{0}+\ket{0}\\!\bra{1})_{l_{\Delta,12}}.$
(22)

For this reason, we prefer the trimer basis over the dimer computational basis
for this calculation even though it has a slightly reduced average sign. In
the dimer basis, calculating $C^{z}$ and $C^{y}$ in a similar way is in
principle possible, but with much more complicated matrix elements due to the
split of the Hilbert space of $\mathbf{S}_{\Delta,1}$,
$\mathbf{S}_{\Delta,2}$, and $\mathbf{S}_{\Delta,3}$ constituting the spin
chirality into two distinct computational cluster-basis sites.

Figure 7: Chirality correlation functions $C^{z}$ and $C^{y}$ for the
triangle-square lattice for $J_{1}=J_{2}=J_{3}$, $J/J_{3}=0.1$ at
$T/J_{3}=0.05$. Panels (a) and (b) show the respective correlations in both
the $x$ and $y$ lattice directions while panel (c) compares them along the
dominant $x$ direction.

As seen from the QMC data shown for $C^{z}(\mathbf{r})$ (top left panel a) and
$C^{y}(\mathbf{r})$ (top right panel b) in Fig. 7, we observe a rapid decay of
these correlations along both lattice directions, i.e., for
$\mathbf{r}\propto\hat{\mathbf{x}}$ ($x$ direction) and
$\mathbf{r}\propto\hat{\mathbf{y}}$ ($y$ direction). The chirality
correlations exhibit a finite value only among nearest neighbor trimers, and
are essentially zero beyond $r>2$ (the value of $C^{y}(\mathbf{r})$ at
$\mathbf{r}=2\,\hat{\mathbf{x}}$ is also suppressed with increasing system
size $L$, cf. the bottom panel c). We therefore do not obtain indication for
the presence of any further symmetry breaking induced by the chirality degree
of freedom Wang (2001) within the accessible temperature regime of our QMC
approach.

##  IV Breathing kagome lattice

After having considered a system of coupled trimers, for which the
superlattice of trimers is bipartite, we next consider the case of the
(breathing) kagome lattice, which is formed by a (non-bipartite) triangular
lattice of coupled trimers. In this system there is thus an additional source
of magnetic frustration beyond the one introduced by the antiferromagnetic
intratrimer couplings. We again consider finite-size systems with periodic
boundary conditions, and also in this case the number of spins relates via
$N=3L^{3}$ to the linear system size $L$ of the triangular lattice.

###  IV.1 Sign

Figure 8: Average configuration sign, $\langle\mathrm{sign}\rangle$, of the
$L=12$ breathing kagome lattice in different computational bases for different
breathing distortions $J_{\bigtriangledown}/J_{\triangle}$.

For the kagome lattice with breathing distortion, we also start by comparing
the average configuration sign for the computational bases introduced in
Section II (Fig. 8). For weak intertrimer couplings $J_{\bigtriangledown}$,
the dimer and trimer bases again strongly outperform the single-spin basis
retaining a finite sign down to $T\sim J_{\bigtriangledown}$, while at
stronger couplings this advantage fades away. In contrast to the triangle-
square lattice, however, at weak intertrimer couplings, the sign is nearly
identical between the dimer and trimer basis for
$J_{\bigtriangledown}/J_{\triangle}\lesssim 0.1$, with a slight advantage of
the trimer basis in the low-$T$ regime. At larger $J_{\bigtriangledown}$, the
dimer basis has a slightly higher average sign, but eventually falls below the
sign of the single-spin basis. Therefore, moving forward, we will focus on the
weak coupling regime, i.e. the regime of high breathing distortion, and use
the trimer computational basis.

###  IV.2 Thermodynamics

Figure 9: The magnetic susceptibility $\chi$ of the breathing kagome lattice
multiplied by the temperature $T$ as a function of $T$ for different breathing
distortions $J_{\bigtriangledown}/J_{\triangle}$. For
$J_{\bigtriangledown}=0$, the ED solution for decoupled trimers is shown. For
finite $J_{\bigtriangledown}$, the dashed lines show $L=2$ ED data. Figure 10:
The specific heat $C$ of the breathing kagome lattice as a function of
temperature $T$ for different breathing distortions
$J_{\bigtriangledown}/J_{\triangle}$. For $J_{\bigtriangledown}=0$ the ED
solution for decoupled trimers is shown. For finite $J_{\bigtriangledown}$,
the dashed lines show $L=2$ ED data.

In analogy to the triangle-square lattice, we compute the magnetic
susceptibility $\chi$ (Fig. 9) and the specific heat $C$ (Fig. 10). At high
temperatures and weak intertrimer couplings, these observables converge to the
same decoupled trimer limit as the triangle-square lattice. In the
susceptibility (Fig. 9), the effective spin-1/2 plateau at low temperatures
again vanishes with increasing intertrimer interactions. The specific heat
shows the onset of a low temperature peak associated with the lifting of
degeneracies by the intertrimer interactions. In general, due to the sign
problem it is challenging to reach the low-temperature regime $T\sim
J_{\bigtriangledown}$, but in our accessible temperature range both the
susceptibility and the specific heat are well described by ED data for $L=2$.
This is in contrast to the triangle-square lattice for which at similar
temperatures, significant deviations from the $L=2$ data were found due to the
onset of magnetic correlations.

Figure 11: Magnetization per site $M$ of the breathing kagome lattice as a
function of magnetic field $h$ for different breathing distortions
$J_{\bigtriangledown}/J_{\triangle}$, taken at $T=0.083J_{\triangle}$.

In addition to these observables, we furthermore computed the magnetization
(per site) $M=\frac{1}{M}\sum_{i}S^{z}_{i}$ at a finite magnetic field $h$
introduced to the Hamiltonian,

$H^{\prime}=H-h\sum_{\Delta}S_{\Delta}^{z}\,,$ (23)

which shows two extended plateaus (Fig. 11) corresponding to the
magnetizations per trimer $\braket{m_{\Delta}}=1/2$ and
$\braket{m_{\Delta}}=3/2$, respectively Honecker  _et al._ (2011). At the
boundaries of these plateaus, the competition between different magnetization
sectors renders the energetic contributions due to the small intertrimer
coupling more important. In these regions the magnetization displays smooth
crossovers, where at our lowest accessible temperature of around
$T/J_{\triangle}=0.083$, we do not resolve signs of further plateaus. In
general, increasing $J_{\bigtriangledown}$ at fixed $T/J_{\triangle}$ leads to
stronger frustration and thus to an overall decrease of the magnetization.

Previous works have found a quantum phase transition from a nematic state
breaking the lattice rotation symmetry to a spin liquid state at weak
intertrimer coupling in the range of
$J_{\bigtriangledown}/J_{\triangle}\approx 0.05$ Repellin  _et al._ (2017).
While we see no signatures of such a transition in the thermodynamic behavior
presented until now, measuring the associated nematicity order parameter and
its correlation function is in principle possible within our approach,
allowing for the direct detection of such a phase, if it persists to high
enough temperatures. In Appendix B, we show the absence of a lattice nematic
state for $T/J_{\triangle}\gtrsim 0.083$, limiting a finite-temperature
lattice nematic phase, if it persits, to lower temperatures than those
accessible to our QMC approach.

##  V Conclusion

In this article we computed the thermodynamics of two models of weakly coupled
Heisenberg trimers using quantum Monte Carlo simulations in different cluster
bases. In both the triangle-square lattice and the breathing kagome lattice,
these cluster bases allowed us to resolve temperatures down to the order of
the weak intertrimer coupling, greatly outperforming the single-spin basis.

For the triangle-square lattice, we further picked up the signatures of
different magnetic orders expected to form in the ground state. We also showed
that correlations related to the additional chirality degree of freedom for
the case of equal intratrimer couplings can be efficiently accessed using the
trimer basis. However, no additional symmetry breaking as observed within the
accessibly temperature range.

The thermodynamic data for the breathing kagome lattice was found to be well-
described by a simple $L=2$ cluster throughout our accessible temperature
range. In particular, we observe the presence of a two-peak structure in the
specific heat in both the lattice-nematic and the quantum spin liquid regime
of the breathing kagome lattice. Whereas some studies also reported a two-peak
structure in the undistored kagome lattice
($J_{\bigtriangledown}=J_{\triangle}$) Misguich and Bernu (2005); Shimokawa
and Kawamura (2016), recent work concludes instead in favor of a single peak
with a pronounced low-$T$ shoulder in the undistored kagome lattice Sugiura
and Shimizu (2013); Chen  _et al._ (2018); Schnack  _et al._ (2018). It would
therefore be interesting to further examine the evolution of the low-
temperature specific heat upon increasing $J_{\bigtriangledown}$ from the
strong breathing regime that we can access by QMC into the weak breathing
region, including the undistored limit.

###### Acknowledgements.

We thank Andreas Honecker and Jürgen Schnack for useful discussions. We
acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through Grant No. WE/3649/4- 2 of the program FOR 1807
and through project RTG 1995, and the IT Center at RWTH Aachen University and
the JSC Jülich for access to computing time through JARA-HPC.

##  Appendix A Measuring correlation functions

In the $S^{z}$ basis formulation of the SSE it is possible to measure the
offdiagonal spin correlation functions $\braket{S^{+}_{i}S^{-}_{j}}$ by
treating the (directed) loop update as an extended ensemble where theses
operators are part of the configuration Dorneich and Troyer (2001); Alet  _et
al._ (2005). This approach can be directly generalized to correlations
$\braket{A_{I}B_{J}}$ where $A_{I}$ and $B_{I}$ are local operators on cell
(i.e., trimer) $I$: In particular, these measurements can also be performed in
the presence of signed configurations, as we will show in the following, based
on the notation introduced in Ref. Weber  _et al._ (2022). If we choose a loop
action (out of a total of $N_{\text{actions}}$) and the entry cell (out of a
total of $N_{\text{cells}}$) at random, the extended ensemble takes the form

$\displaystyle
Z_{\text{ext}}=\sum_{\begin{subarray}{c}n,\\{b_{n}\\},\sigma_{0}\\\
a_{h},p_{h},a_{t},p_{t}\end{subarray}}\frac{1}{N_{\text{actions}}N_{\text{cells}}}\Big{|}\frac{(-\beta)^{n}}{n!}$
(24) $\displaystyle\\!\times$
$\displaystyle\braket{\sigma_{0}}{h_{b_{1}}\cdots}{\sigma_{p_{h}}}\braket{a_{h}(\sigma_{p_{h}})}{\cdots}{\sigma_{p_{t}}}\braket{a_{t}(\sigma_{p_{t}})}{\cdots
h_{b_{n}}}{\sigma_{0}}\Big{|}.$

Here, in addition to the expansion order $n$, operator string $\\{b_{n}\\}$
and state $\sigma_{0}$ the configuration contains the position $p$ and action
$a$ of the loop’s head and tail. Furthermore, the system Hamiltonian $H$ has
been decomposed in terms of operators $h_{b}$ that each connect two cells
Weber  _et al._ (2022) (they correspond to the bond operators in the
conventional site-basis formulation of the SSE Sandvik and Kurkijärvi (1991)).
The $a_{t}$, $p_{t}$, $a_{h}$, and $p_{h}$ determine the position and type of
two discontinuities that appear in the string of nonbranching operators
Dorneich and Troyer (2001); Alet  _et al._ (2005).

We want to measure equal-time correlation functions between general state,
i.e.,

$C_{x_{i}y_{i},x_{j}y_{j}}=\frac{\operatorname{Tr}\left[\ket{x_{i}}\\!\bra{y_{i}}\otimes\ket{x_{j}}\\!\bra{y_{j}}e^{-\beta
H}\right]}{Z}.$ (25)

If the local states $x\neq y$, there is a loop action $a$ so that $y=a(x)$ and
we can express the correlations as an expectation value in the extended
ensemble

$C_{x_{i}y_{i},x_{j}y_{j}}=N_{\text{actions}}N_{\text{cells}}\frac{\braket{\text{sign}\times
P(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}}{\braket{\text{sign}\times
Q(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}}$ (26)

with two projectors

$\displaystyle P(a_{h},p_{h},a_{t},p_{t})$
$\displaystyle=\begin{cases}1,&\text{singularities match
}x_{i}y_{i},x_{j}y_{j}\\\ 0,&\text{else}\end{cases}$ (27) $\displaystyle
Q(a_{h},p_{h},a_{t},p_{t})$
$\displaystyle=\begin{cases}1,&a_{h}=a_{t}^{-1}\text{ and }p_{h}=p_{t}\\\
0,&\text{else}\end{cases}$ (28)

i.e., $Q$ filters out closed-loop configurations that are also part of the
regular ensemble. Therefore, the denominator can be simplified to

$\braket{\text{sign}\cdot
Q(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}=\braket{\text{sign}}\braket{Q(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}.$
(29)

In the numerator, we write the sign as a difference of projectors $P_{+}$
($P_{-}$) on the positive (negative) signed configurations

$\displaystyle\braket{\text{sign}\cdot
P(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}$
$\displaystyle=\braket{P_{+}P(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}$ (30)
$\displaystyle-\braket{P_{-}P(a_{h},p_{h},a_{t},p_{t})}_{\text{ext}}.$

Analogously to the usual case Alet  _et al._ (2005); Dorneich and Troyer
(2001), we can write the ratio of the two probabilities $\braket{P}$ and
$\braket{Q}$ as mean counts per loop. Thus,

$\displaystyle C_{x_{i}y_{i},x_{j}y_{j}}$
$\displaystyle=N_{\text{actions}}N_{\text{cells}}$ (31)
$\displaystyle\times\frac{\braket{n_{+}(a_{h},p_{h},a_{t},p_{t})-n_{-}(a_{h},p_{h},a_{t},p_{t})}}{\braket{\text{sign}}}$
$\displaystyle=\frac{\braket{n(a_{h},p_{h},a_{t},p_{t})}}{\braket{\text{sign}}}$
(32)

where $n(a_{h},p_{h},a_{t},p_{t})$ is the signed count of loop steps with
$P(a_{h},p_{h},a_{t},p_{t})=1$. Arbitrary correlation functions of cell-local
operators can now be written as sum of the $C_{x_{i}y_{i},x_{j}y_{j}}$. If
they are offdiagonal ($(x_{i},y_{i})\neq(x_{j},y_{j})$), they can be measured
as described in this section. If they are diagonal, they can be measured like
every other diagonal observable Sandvik and Kurkijärvi (1991); Sandvik (1992).

##  Appendix B Lattice nematic response in the breathing kagome lattice

Figure 12: The nematic order parameter $\phi$ in the $L=6$ breathing kagome
lattice as a function of the intertrimer coupling $J_{3}$ detuned from the
C3-symmetric point $J_{3}=J_{\triangle}$ for different intertrimer couplings
$J_{\bigtriangledown}$.

At low $J_{\bigtriangledown}/J_{\triangle}\lesssim 0.05$ and $T=0$, the kagome
lattice AFM was previously found to spontaneously break the C3 lattice
rotational symmetry while leaving the translational, spin rotational, and a
lattice reflection symmetry intact Repellin  _et al._ (2017); Jahromi  _et
al._ (2020). Such a “lattice nematic” phase is expected to show a singular
response to a small C3 breaking perturbation, which can be detected, e.g.,
using an order parameter defined as

$\phi=\frac{1}{L^{2}}\sum_{\Delta}\sum_{n=1}^{3}e^{2\pi
i(n-1)/3}\braket{\mathbf{S}_{\Delta,n}\cdot\mathbf{S}_{\Delta,n+1}}.$ (33)

In the C3-symmetric case, this quantity is exactly zero.

Previous studies have focused on the ground state phase-diagram, but since the
lattice nematic phase breaks a discrete symmetry it may persist up to a finite
temperature $T_{c}$. As $J_{\bigtriangledown}$ approaches zero, at any fixed
temperature, the partition function of the model should approach the one of
simple decoupled trimers, in agreement with our findings in the main text.
Decoupled trimers cannot, of course, form lattice nematic order, so the
critical temperature $T_{c}$ of the nematic transition is expected to vanish
with $J_{\bigtriangledown}$. Nevertheless, at finite $J_{\bigtriangledown}$, a
finite $T_{c}$ may be observable in the temperature range accessible with our
method, which is what we investigate in this section.

We will introduce a small symmetry-breaking perturbation by detuning one of
the intratrimer couplings, denoted as $J_{3}$ (in analogy to the triangle-
square lattice notation) while leaving the other two intratrimer couplings at
the value $J_{\triangle}$. We can then monitor the evolution of the order
parameter upon tuning across the isotropic point $J_{3}=J_{\triangle}$. Using
the residual lattice reflection symmetry in this setup, the order parameter
$\phi$ can be expressed as

$\phi=\frac{1}{L^{2}}\left(\Braket{\sum_{\Delta}l_{\Delta,12}}-\Braket{\sum_{\Delta}l_{\Delta,23}}\right),$
(34)

Here, we computed the two terms as diagonal observables in two separate
simulations in appropriately rotated trimer bases.

The results for $\phi$ even at our lowest accessible temperature show a smooth
dependence both on $J_{3}/J_{\triangle}$ and
$J_{\bigtriangledown}/J_{\triangle}=0.010-0.075$ (Fig. 12). In particular, for
all values of $J_{\bigtriangledown}/J_{\triangle}$, the data in Fig. Fig. 12
smoothly tend towards zero upon approaching the isotropic point
$J_{3}=J_{\triangle}$. We therefore conclude that a finite-temperature lattice
nematic phase, if it persists, is limited to lower temperatures than
accessible to our QMC approach.

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