# Generated Faces in the Wild: Quantitative Comparison of Stable Diffusion,
Midjourney and DALL-E 2

Ali Borji
Quintic AI, San Francisco, CA
<EMAIL_ADDRESS>
###### Abstract

The field of image synthesis has made great strides in the last couple of
years. Recent models are capable of generating images with astonishing
quality. Fine-grained evaluation of these models on some interesting
categories such as faces is still missing. Here, we conduct a quantitative
comparison of three popular systems including Stable Diffusion, Midjourney,
and DALL-E 2 in their ability to generate photorealistic faces in the wild. We
find that Stable Diffusion generates better faces than the other systems,
according to the FID score. We also introduce a dataset of generated faces in
the wild dubbed GFW, including a total of 15,076 faces. Furthermore, we hope
that our study spurs further research in assessing the generative models and
improving them. Data and code are available at data and code, respectively.

##  1 Introduction

The field of image synthesis has made great strides in the last couple of
years. Variations of Generative Adversarial Networks (GAN) [4] and Variational
Autoencoders (VAE) [8] were the first to generate high quality images. Recent
diffusion-based models trained on massive datasets have transcended progress
and have attracted a lot of attention both among AI scientists and the public.
Several blog posts (_e.g._here, here, and here) and articles (_e.g._ [9]) have
offered anecdotal evaluations. Fine-grained evaluation of these models on some
interesting categories still needs more work.

Quantitative evaluation of generative models is mostly concerned with fidelity
and diversity of the entire scenes rather than scene components or individual
objects (with few exceptions [14, 1]). Here, we emphasize on fine-grained
evaluation of models, in particular their ability to generate photorealistic
faces. Notice that our work is different from those studies that attempt to
build models for generating portraits or evaluating such models (_e.g._
StyleGAN [7]). Instead, here we are interested in evaluating the quality of
generated faces in cluttered scenes containing multiple objects. To this end,
we utilize text to image generative models to synthesize scenes. We then use a
face detector to detect faces in these images. Finally, we use the well-
established Fréchet Inception Distance (FID) [5] to evaluate the quality of
the generated faces against a set of real faces.

##  2 Comparison

###  2.1 Models

We consider the following three models:

  1. 1.

Stable Diffusion [11]111https://en.wikipedia.org/wiki/Stable_Diffusion.
Released by StabilityAI in 2022, this model is primarily used to generate
detailed images conditioned on text descriptions. It can also be applied to
other tasks such as inpainting, outpainting, and image translation. This model
is trained on 512 $\times$ 512 images from a subset of the LAION-5B database.
It uses a frozen CLIP ViT-L/14 text encoder to condition the model on text
prompts. With its 860M UNet and 123M text encoder, the model is relatively
lightweight and runs on a GPU with at least 10GB VRAM. We use the Colab
notebook provided by Hugging
Face222https://colab.research.google.com/github/huggingface/notebooks/blob/main/diffusers/stable_diffusion.ipynb
to run Stable Diffusion.

  2. 2.

Midjourney
(https://www.midjourney.com/)333https://en.wikipedia.org/wiki/Midjourney. This
model was created by an independent research lab with the same name. It can
synthesize images from textual descriptions and is currently in open beta.
Midjourney tends to generate surrealistic images and is popular among artists.
We used a collection of images generated by this model available via this
Kaggle link.

  3. 3.

DALL-E 2 [10]444https://en.wikipedia.org/wiki/DALL-E,
https://openai.com/dall-e-2/ is created by OpenAI and is a successor of
DALL-E. It can create more realistic images than DALL-E at higher resolutions
and can combine concepts, attributes, and styles. DALL-E 2 is trained on
approximately 650 million image-text pairs scraped from the Internet. Since
DALL-E 2 code is not available, we were not able to generate images on a large
scale. We accessed the system via their portal by manually entering the
prompts and saving the results.

###  2.2 Data

To evaluate the models, two sets of faces are required: a) generated faces,
and b) real faces.

  * •

Generated faces. We avoided crawling the web for generated faces to reduce the
potential biases. Indeed, people usually tend to post high quality faces in
social media (_a.k.a._ cherry picking). Instead, we use the captions from the
COCO dataset555https://cocodataset.org/ (captions_train2017.json) as prompts
to synthesize images. Images generated by Stable Diffusion and DALL-E 2 have
size of 512 $\times$ 512, while images by Midjourney are of variable size. To
increase the chance of generated images to contain faces, we chose captions
that had any word from the following list of words: [‘person’, ‘man’, ‘woman’,
‘men’, ‘women’, ‘kid’, ‘child’, ‘face’, ‘girl’, ‘boy’]. We then ran the
MediaPipe face
detector666https://google.github.io/mediapipe/solutions/face_detection.html
twice: first on the entire image to detect faces, and a second time on
individual detections to prune false positives. Finally, we manually removed
the remaining false positives. Detected faces were resized to 100 $\times$ 100
pixels. In total, we collected 15,076 generated faces, including 8,050 by
Stable Diffusion, 6,350 by Midjourney, and 676 by DALL-E 2.

  * •

Real faces. We ran the face detector on the COCO training set (train2017.zip)
similar to above to extract faces. In addition, we added 13,233 faces from the
Labeled Faces in the Wild (LFW) 777http://vis-www.cs.umass.edu/lfw/ dataset
[6]. We cropped the central 100 $\times$ 100 pixels area from the 250 $\times$
250 faces of LFW. In total, we collected 30,000 real faces.

We removed faces (both generated and real) that were highly occluded (_e.g._
person eating a large piece of food, or fruit or wearing a mask) as well as
too dark or too blurry faces. We also removed dull faces, animal faces,
drawings, and cartoons. Faces with eyeglasses were kept.

###  2.3 Evaluation Scheme

Several scores have been proposed for evaluating generative models [2, 3]. The
FID score is the most commonly used one. We utilize the implementation from
https://github.com/mseitzer/pytorch-fid. Each time we shuffled the faces
(generated and real) and randomly selected 5,000 faces from each set, and
computed the FID score between the two sets. We repeated this procedure 10
times and computed the mean and standard deviation of FID across the runs.
Since DALL-E 2 faces are less than 5K, we selected faces with replacement for
this model. To compute the FID for real faces, we first shuffle the set of
real faces and then split the first 10K faces into two sets of 5K faces. The
FID between these two sets is then computed.

###  2.4 Results

Sample images and faces generated by three systems are shown in Fig. 2 and
Fig. 3, respectively. Qualitative inspection shows that models are able to
generate high quality images. The generated images have high fidelity, are
diverse, and most of the time are physically plausible (with few exceptions
_e.g._ the right woman in the last row of Fig. 2). The generated faces (Fig.
3) in general look very good, although not as good as the faces generated by
models specifically trained on portraits (_e.g._ StyleGAN [7]).

FID scores are shown in Fig. 1. Stable Diffusion scores a lower FID, hence
generating better faces than the other two models. The quality of the
generated faces, however, is still much worse than the real faces, witnessed
by the large gap in FID of models and real faces. The poor performance of
Midjourney can be partly due to its generated faces being surrealistic and
anime.

There are three possible reasons why DALL-E 2 performs worse than Stable
Diffusion. First, OpenAI introduced some deepfake safeguards during DALL-E 2
training to prevent it from memorizing faces that often appear on the
Internet. Second, DALL-E 2 is optimized for images with a single focus of
attention. Thus, this model does better at generating portraits888DALL-E 2 is
able to generate realistic portraits. Please see this twitter post for some
samples. of imaginary people than faces in complex scenes. Third, lower
performance can be attributed to the smaller set of images for this model
compared to the other two. It is known that FID is sensitive to the number of
images. The higher number of images, the more reliable the score. We address
this issue in the following.

To make a fair comparison of the systems, we limited the number of faces to
676 which is the number of DALL-E 2 faces. Results are shown in the right
panel of Fig. 1. As can be seen, FID values are higher now, but still Stable
Diffusion wins. We believe larger sets of faces (_e.g._ in the order of 20,000
faces) can better distinguish the models. We will leave this to future
investigations.

Some low quality generated faces by models are shown in Figs. 4, 5, and 6.
Models often struggle in generating eyeglasses, eyeballs, occluded faces,
profile faces, and maintaining symmetry.

Figure 1: Left: FID scores of models over random sets of 5000 faces. Right:
Results with samples of size 676 per model. Notice that the lower the FID, the
better. Results are averaged over 10 runs. Figure 2: Samples of real images
(top row) and generated images. Figure 3: Samples of real faces (top row) and
generated faces.

##  3 Conclusion

To the best of our knowledge, we are the first to evaluate the quality of
generated faces in the wild. We found that Stable Diffusion generates more
realistic faces. In the majority of cases, however, it is possible for a human
to tell whether a face is real or generated, indicating that there is still a
large gap to close. To this end, we suggest the following directions for
future work in this area:

  * •

Due to limited access to DALL-E 2, we were not able to include a lot of faces
from this model in our evaluation. We encourage researchers to consider larger
sets of generated faces, especially from DALL-E 2 to compare and evaluate
models.

  * •

We found that some faces look very realistic, triggering the thought that
maybe they have been copied from the training set, in whole or in part. We
encourage the creators of these systems to either conduct memorization tests,
or release their training set for independent researchers to investigate this.
We also found that some generated faces contain watermarks.

  * •

Our data and methodology allows probing the generative models for their
potential use in generating deepfakes, as well as the societal biases inherent
in them. For example, annotating the faces in our dataset in terms of race,
gender, and age can inform the community about the algorithmic biases in the
generators.

  * •

FID score uses embeddings from models trained on ImageNet. This can be
problematic for evaluating the quality of faces. Classification models
specifically trained on face datasets might be better options. In addition to
FID, other scores such as SSIM [12], LPIPS (learned perceptual image patch
similarity) [13], and human judgments can be used to evaluate the generators.
Further, it might be a good idea to align the faces before computing the
scores.

  * •

Our preliminary probing reveals that Stable Diffusion also generates better
images than the other two models in terms of FID (results not shown). We also
found that scaling the images (50, 100, 200, 300, 500 squares) does not change
the FID score much.

  * •

Future work can also investigate finer facial details such as expressions
(sad, happy, angry), age (young _v_ s. old), and viewpoint specifications
(profile _v_ s. frontal). Lastly, models can be compared over other
interesting categories such as humans, cats, dogs, cars, and bedrooms.

Figure 4: Low quality faces generated by Stable Diffusion.

Figure 5: Low quality faces generated by Midjourney.

Figure 6: Low quality faces generated by DALL-E 2.

## References

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# Magnetic properties of a spin-orbit entangled $J_{\rm eff}$ = 1/2 honeycomb
lattice

J. Khatua  Department of Physics, Indian Institute of Technology Madras,
Chennai 600036, India  Q. P. Ding  Ames National Laboratory, U.S. DOE, and
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  M. S. Ramachandra Rao  Department of Physics, Nano Functional Materials
Technology Centre and Materials Science Research Centre, Indian Institute of
Technology Madras, Chennai-600036, India  Quantum Centre of Excellence for
Diamond and Emergent Materials, Indian Institute of Technology Madras, Chennai
600036, India.  K. Y. Choi  Department of Physics, Sungkyunkwan University,
Suwon 16419, Republic of Korea  A. Zorko  Jožef Stefan Institute, Jamova c.
39, SI-1000 Ljubljana, Slovenia  Faculty of Mathematics and Physics,
University of Ljubljana, Jadranska u. 19, SI-1000 Ljubljana, Slovenia  Y.
Furukawa  Ames National Laboratory, U.S. DOE, and Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50011, USA  P.Khuntia
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Madras, Chennai 600036, India  Quantum Centre of Excellence for Diamond and
Emergent Materials, Indian Institute of Technology Madras, Chennai 600036,
India.

###### Abstract

The interplay between spin-orbit coupling, anisotropic magnetic interaction,
frustration-induced quantum fluctuations, and spin correlations can lead to
novel quantum states with exotic excitations in rare-earth-based quantum
magnets. Herein, we present the crystal structure, magnetization, electron
spin resonance (ESR), specific heat, and nuclear magnetic resonance (NMR)
experiments on the polycrystalline samples of Ba9Yb2Si6O24, in which Yb3+ ions
form a perfect honeycomb lattice without detectable anti-site disorder. The
magnetization data reveal antiferromagnetically coupled spin-orbit entangled
$J_{\rm eff}$ = 1/2 degrees of freedom of Yb3+ ions in the Kramers doublet
state. The ESR measurements reveal that the first excited Kramers doublet is
32.3(7) meV above the ground state. The specific heat results suggest the
absence of any long-range magnetic order in the measured temperature range.
Furthermore, the 29Si NMR results do not indicate any signature of magnetic
ordering down to 1.6 K, and the spin-lattice relaxation rate reveals the
presence of a field-induced gap that is attributed to the Zeeman splitting of
Kramers doublet state in this quantum material. Our experiments detect neither
spin freezing nor long-range magnetic ordering down to 1.6 K. The current
results suggest the presence of short-range spin correlations in this spin-
orbit entangled $J_{\rm eff}$ =1/2 rare-earth magnet on a honeycomb lattice.

††preprint: APS

##  I Introduction

Quantum fluctuations induced by frustration, spin correlations, and spin-orbit
entanglement can stabilize exotic states in quantum materials Witczak-Krempa
_et al._ (2014); Takagi  _et al._ (2019); Balents (2010). Understanding
emergent quantum phenomena and associated elementary excitations is one of the
attractive pursuit in quantum condensed matter Witczak-Krempa  _et al._
(2014); Takagi  _et al._ (2019); Balents (2010); Khuntia (2019). The two-
dimensional geometrically frustrated triangular and kagomé lattices have been
studied extensively to realize exotic many-body quantum phenomena such as
quantum spin liquid (QSL) in condensed matter Khuntia  _et al._ (2020);
Balents (2010); Broholm  _et al._ (2020); Wen  _et al._ (2019). A QSL is a
highly entangled state of matter wherein spins do not exhibit long-range
magnetic order even at $T\rightarrow 0$ owing to strong quantum fluctuations
and intertwining of competing degrees of freedom Balents (2010). This QSL
state is often characterized by exotic quasi-particle excitations such as
spinons or Majorana fermions with fractional spin quantum number Anderson
(1973); Kitaev (2006), which are different from the conventional magnon
excitations with integer spin quantum numbers, as usually observed in
magnetically ordered systems Matan  _et al._ (2006).
The quest for such unconventional states of matter was triggered by two
theoretical scenarios. The first is the resonating valance bond state proposed
by P. W. Anderson in 1973 Anderson (1973). The second one is the Kitaev QSL on
the honeycomb lattice wherein S = 1/2 spin is predicted to fractionalize into
emergent Majorana fermions and localized $Z_{2}$ fluxes Kitaev (2006);
Baskaran  _et al._ (2007); Motome and Nasu (2020).
The Kitaev model on the honeycomb lattice with $J_{\rm eff}$ = 1/2 degrees of
freedom demonstrated that the bond dependent Ising interactions provide an
alternative route to realize a frustration-driven Kitaev spin-liquid state
with Majorana fermions Kitaev (2006); Kitagawa  _et al._ (2018). This has
sparked significant research interest in strong spin-orbit coupled 4$d$ and
5$d$ honeycomb magnets, including iridates A2IrO3 (A = Li+, Na+) Choi  _et
al._ (2012); Chaloupka  _et al._ (2010) and ruthenate $\alpha$-RuCl3 Plumb
_et al._ (2014); Sears  _et al._ (2020); Banerjee  _et al._ (2016); Takagi
_et al._ (2019). Beyond 4$d$/5$d$ ion based honeycomb magnets, the search for
Kitaev materials has been extended to 3$d$ transition metal Co2+ ion based
honeycomb compounds such as Na2Co2TeO6 and Na3Co2SbO6, in which oxygen ligands
of 3$d$ ions form nearly regular octahedra with small trigonal distortions Liu
and Khaliullin (2018); Liu  _et al._ (2020). Recent reports demonstrate that
the combination of spin-orbit coupling and strong Hund’s couplings can host
pseudospin-1/2 degrees of freedom with Kitaev interactions in the
aforementioned cobaltates Kim  _et al._ (2021); Songvilay  _et al._ (2020);
Motome  _et al._ (2020); Liu and Khaliullin (2018); Stavropoulos  _et al._
(2019). However, most of the 3$d$, 4$d$ and 5$d$ based honeycomb magnets show
magnetic ordering at low temperatures due to the presence of inevitable
defects, site disorder, and additional exchange interactions in real materials
that destabilize the Kitaev QSL ground state Bahrami  _et al._ (2022).
The essential ingredients to realize Kitaev spin liquid are spin–orbit
entangled $J_{\rm eff}$ = 1/2 moments, electron correlations with bond-
directional Ising exchange interactions, and low dimensionality Trebst and
Hickey (2022). It is interesting in this respect that, many recent studies
have found signatures of collective quantum phenomena in low dimensional rare-
earth magnets with anisotropic interaction between pseudospin-1/2 moments Wu
_et al._ (2019); Li  _et al._ (2015, 2016); Arh  _et al._ (2022); Dun  _et
al._ (2021); Hu  _et al._ (2020); Hester  _et al._ (2019).
Theoretically, it has been proposed that Yb-based honeycomb magnets may offer
a more faithful realization of Kitaev physics due to strong localization and
spin-orbit coupling of 4$f$ electrons compared to their 4$d$ or 5$d$
counterparts Jang  _et al._ (2020); Li  _et al._ (2017a); Luo and Chen (2020).
In sharp contrast to the long-range Néel order state in an isotropic nearest-
neighbor exchange model on the honeycomb lattice Fouet  _et al._ (2001); Xing
_et al._ (2020); Sala  _et al._ (2019); Hao  _et al._ (2020); Xing  _et al._
(2020); Sala  _et al._ (2019, 2021); Matsumoto  _et al._ (2022), the strong
quantum fluctuations induced by further nearest-neighbor frustrated exchange
interaction can destabilize the Néel order even in bipartite honeycomb
lattices Merino and Ralko (2018); Ferrari and Becca (2020); Ferrari  _et al._
(2017); Wessler  _et al._ (2020). The bipartite spin-lattice of rare-earth-
based honeycomb magnets offers a promising venue to host spiral spin-liquid
state with fracton quadrupole excitations Niggemann  _et al._ (2019); Gao  _et
al._ (2022), multiple-$q$ states in the presence of magnetic field Shimokawa
_et al._ (2019), lattice nematic phase Zhang and Lamas (2013), and
Berezinskii-Kosterlitz-Thouless phase Caci  _et al._ (2021); Waibel  _et al._
(2015). In this context, structurally perfect novel rare-earth 4$f$ based
honeycomb magnets wherein the combination of spin-orbit coupling and sizable
crystal electric field allows for the realization of an effective spin 1/2
system with large anisotropy and strong spin correlation. These spin-orbit
driven materials offer an alternate route for the experimental search for
exotic quantum many-body phenomena including Kitaev QSL Luo and Chen (2020);
Clark  _et al._ (2019); Jang  _et al._ (2020); Li (2019); Bu  _et al._ (2022);
Bordelon  _et al._ (2021); Wu  _et al._ (2021); Li  _et al._ (2021); Tian  _et
al._ (2021); Zhu  _et al._ (2018); Arh  _et al._ (2022); Hester  _et al._
(2019); Zhang  _et al._ (2022); Paiva  _et al._ (2005); Li  _et al._ (2010);
Huang  _et al._ (2016); Moessner and Sondhi (2001); Coletta  _et al._ (2011).
Herein, we present our results on a promising rare-earth-based quantum magnet
Ba9Yb2Si6O24, where the magnetic Yb3+ ions form a perfect honeycomb lattice in
the $ab$-plane Brgoch  _et al._ (2013). Magnetic susceptibility data suggest
the realization of a spin-orbit entangled $J_{\rm eff}$ = 1/2 moments of Yb3+
ions that is consistent with a low-energy Kramers doublet state at low
temperature. As revealed by ESR, the ground state is well isolated from the
first excited Kramers doublet, with an energy separation of 32.3(7) meV. The
Curie-Weiss fit of low-temperature magnetic susceptibility data reveals the
presence of antiferromagnetic interactions in the ground state and detects
neither spin glass nor long range magnetic ordering down to 1.9 K. Specific
heat measurements further support the absence of long range magnetic order
down to 1.9 K. 29Si NMR measurements in weak magnetic fields confirm the
absence of long range magnetic order in the measured temperature range. The
NMR spin-lattice relaxation rate in high-magnetic fields suggests the presence
of a field-induced gap due to the Zeeman splitting of the lowest Kramers
doublet state. Our investigation also reveals the presence of short-range spin
correlations in this rare-earth honeycomb magnet.

##  II EXPERIMENTAL DETAILS

Polycrystalline samples of Ba9Yb2Si6O24 (henceforth; BYSO) were prepared by a
conventional solid state reaction of appropriate stoichiometry amounts of
BaCO3 (99.997 %, Alfa Aesar), SiO2 (99.999 %, Alfa Aesar), and Yb2O3 (99.998
%, Alfa Aesar). Prior to use, we preheated BaCO3, and Yb2O3 at 100∘C and
800∘C, respectively, to remove moisture and carbonates. All the reagents were
thoroughly ground to obtain homogeneous mixtures. The powder mixtures were
pelletized and sintered at 900∘C for 24 hours in air to decompose carbon
dioxide. In order to obtain the desired phase, the pelletized sample was fired
at 1350∘C for 72 hours with several intermittent grindings. The powder x-ray
diffraction (XRD) patterns were measured by employing a smartLAB Rigaku x-ray
diffractometer with Cu Kα radiation ($\lambda$ = 1.54 Å).

Figure 1: (a) Rietveld refinement profile of the room-temperature x-ray
diffraction data of Ba9Yb2Si6O24. The black circle represents the
experimentally observed data points and the orange solid line is the
calculated data. The rows of vertical bars are the Bragg reflection positions
for Ba9Yb2Si6O24 (olive bars), Ba2SiO4 (violet bars) and Yb2O3 (pink bars).
The blue line is the difference between observed and calculated intensities.
(b) A single unit cell of the trigonal crystal structure of Ba9Yb2Si6O24. The
nearest-neighbor oxygen atom of Yb3+ ions form a YbO6 octahedra. The possible
in-plane nearest-neighbor and inter-plane exchange interactions through the
bridges Yb-O-Si-O-Yb, and Yb-O-Si-O-Si-O-Yb are shown, respectively. (c)
Structure depicting nearest-neighbor (5.79 Å) Yb3+ ions which form honeycomb
planes. There are three such consecutive honeycomb planes in the unit cell of
Ba9Yb2Si6O24.

Magnetization measurements were carried out using the VSM option of Quantum
Design, Physical Properties Measurement System (QD, PPMS) in the temperature
range 1.9 K $\leq\textit{T}\leq$ 300 K in magnetic fields 0 T
$\leq\mu_{0}H\leq$ 7 T. Specific heat measurements were performed using QD,
PPMS by thermal relaxation method, in the temperature range 1.9 K
$\leq\textit{T}\leq$ 250 K in magnetic fields 0 T $\leq\mu_{0}H\leq$ 7 T. The
electron spin resonance (ESR) spectrum was measured at 9.40 GHz K on a
commercial X-band Bruker E500 spectrometer working at 9.40 GHz in the
temperature range 4 K $\leq$ $T$ $\leq$ 140 K. At low temperatures, the
microwave power was varied between 0.01 mW to prevent signal saturation, while
at high temperatures, it was set to 1 mW to avoid signal diminishment. To
enhance the signal-to-noise ratio, the magnetic field was modulated with a
frequency of 100 kHz and an amplitude of 0.5 mT, resulting in derivative ESR
spectra. Field-swept 29Si ($I$ = 1/2, and gyromagnetic ratio 8.4577 MHz/T) NMR
measurements down to 1.6 K at several frequencies were carried out on a home-
made phase-coherent spin-echo pulse spectrometer equipped with a 9 T Oxford
magnet. NMR spectra measurements were carried out using a standard Hahn-echo
sequence while the 29Si NMR spin-lattice relaxation time was extracted from
the recovery of longitudinal nuclear magnetization $M(t)$ after a time delay
$t$ following a saturation pulse sequence.

##  III RESULTS

Table 1: The Rietveld refinement parameters obtained from the analysis of the XRD data taken at room temperature. The Rietveld refinements were carried out with space group R$\bar{3}$ and yields unit cell parameters $a$ = b = 10.002 Å,  c = 22.127 Å and $\alpha$ = 90∘, $\beta$ = 90∘, $\gamma$ = 120∘. The goodness of Rietveld refinement was confirmed by the following factors: $\chi^{2}$ = 3.823; and Rexp= 5.25.  Atom | Wyckoff position | x | y | z | Occ.
---|---|---|---|---|---
Ba1 | 3a | 0 | 0 | 0 | 1
Ba2 | 18f | 0.333 | 0.666 | 0.004 | 1
Ba3 | 18f | 0.029 | 0.668 | 0.109 | 1
Yb | 6c | 0 | 0 | 0.164 | 1
Si | 18f | 0.336 | 0.012 | 0.073 | 1
O1 | 18f | 0.347 | 0.065 | 0.006 | 1
O2 | 18f | 0.480 | 0.158 | 0.102 | 1
O3 | 18f | -0.002 | 0.173 | 0.107 | 1
O4 | 18f | 0.137 | 0.468 | 0.094 | 1

###  III.1 XRD and structural details

In order to confirm the phase purity and obtain structural atomic parameters,
Rietveld refinement of the X-ray diffraction data was performed using the
FULLPROF suite Rodríguez-Carvajal (1993). The XRD results reveal that the
polycrystalline samples of BYSO contain a few percentage of non-magnetic
Ba2SiO4 and magnetic Yb2O3 secondary phases, which have not much effect on the
overall magnetic properties of the material studied here. In the literature,
it is observed that Ba2SiO4 (BSO) and Yb2O3 impurities are unavoidable in the
polycrystalline samples of barium and silicon-based magnetic materials Brgoch
_et al._ (2013); Jang  _et al._ (2021) as well as for some Yb-based magnets,
respectively Grußler  _et al._ (2023); Tokiwa  _et al._ (2021); Xie  _et al._
(2022). To quantify the percentage of the dominant and secondary phases, we
performed a three-phase Rietveld refinement. The initial atomic coordinates
were taken from refs. Brgoch  _et al._ (2013) and Banjare  _et al._ (2020) for
the dominant BYSO phase and the secondary BSO phase, respectively.
Figure 1 (a) depicts the Rietveld refinement of the XRD data, suggesting that
our polycrystalline samples contain $\approx$ 94 % of the dominant BYSO phase
and $\approx$ 4 % non-magnetic BSO and $\approx$ 2 % magnetic Yb2O3 secondary
phases. The Rietveld refinement reveals that BYSO crystallizes in a trigonal
crystal structure with space group R$\bar{3}$. Table 1 lists the atomic
parameters obtained from the Rietveld refinement. The Yb atoms occupy only one
Wyckoff atomic site 6$c$ and form a six-coordinated YbO6 octahedron with
neighboring O atoms. The possible in-plane exchange interactions via Yb-O-Si-
O-Yb super-exchange pathways is shown in Figure 1 (b). More interestingly,
first nearest Yb3+ neighbors (5.79 Å) constitutes two-dimensional honeycomb
layers perpendicular to the crystallographic $c$-axis and there are three such
well-separated honeycomb layers in one unit cell of BYSO (Figure 1 (c)).
From the structural point of view, BYSO is a bit different from the honeycomb
lattice YbCl3, which crystallizes in the monoclinic crystal structure (space
group $C2/m$ with lattice parameters $a$ = 6.732 Å, $b$ = 11.620 Å and $c$ =
6.328 Å, $\alpha$ = $\gamma$ = 90.00∘ and $\beta$ = 110.551∘) Sala  _et al._
(2019). However, BYSO is structurally similar to the honeycomb lattice of
YbBr3, which crystallizes in the trigonal crystal structure (space group
R$\bar{3}$) with the lattice parameters $a=b$ = 6.971 Å, $c$ = 19.103 Å,
$\alpha$ = $\beta$ = 90∘ and $\gamma$ = 120∘ Wessler  _et al._ (2020).
Although both systems belong to the same crystal class, the ground state
properties of BYSO are expected to be different due to significant differences
in the ratio of $c/a$ and the exchange paths compared to YbBr3 Wessler  _et
al._ (2020).

Figure 2:  (a) Temperature dependence of magnetic susceptibility in several
magnetic fields. The inset (left bottom corner) shows the comparison of the
zero-field cooled (ZFC) and field cooled (FC) magnetic susceptibility as a
function of temperature in a magnetic field $\mu_{0}H$ = 0.01 T. (b)
Temperature dependence of inverse magnetic susceptibility in a magnetic field
$\mu_{0}H$= 1 T with Curie-Weiss fit in the high-temperature (red line) and
the low temperature (orange line) regions. The top inset shows the Curie-Weiss
fit of inverse magnetic susceptibility in the low temperature region. The
bottom inset shows the estimated Curie-Weiss temperature as a function of
upper limit of temperature range in the Curie-Weiss fit where the constant
value of Curie-Weiss temperature at low temperatures is shown by dotted orange
line. (c) Isotherm magnetization as a function of magnetic field at several
temperatures where the solid line represents the Brillouin function fit for
paramagnetic Yb3+ spins with $J_{\rm eff}$ = 1/2 moment.

###  III.2 Magnetic susceptibility

The temperature dependence of the magnetic susceptibility of BYSO in several
magnetic fields is shown in Figure 2 (a). The absence of any anomaly indicates
that the Yb3+ moment does not undergo any long-range magnetic order in the
measured temperature range of investigation. The bottom inset of Figure 2 (a)
depicts the zero-field cooled (ZFC) and field-cooled (FC) susceptibility and
reveals no bifurcation, which suggests the absence of spin-freezing at least
above 1.9 K.
Above 110 K, the inverse magnetic susceptibility data (see Figure 2 (b)) were
fitted with the Curie-Weiss law, $\chi=C/(T-\theta_{\rm CW})$, where C is the
Curie constant and $\theta_{\rm CW}$ is the Curie-Weiss temperature Bain and
Berry (2008). The high-temperature Curie-Weiss fit (red line in Figure 2 (b))
yields $\theta_{\rm CW}$ = $-$ 111 K and effective moment $\mu_{\rm eff}$ =
4.53 $\mu_{B}$. The estimated large negative $\theta_{\rm CW}$ is attributed
to the energy scale of crystal field excitations. The obtained effective
moment $\mu_{\rm eff}=4.53$ $\mu_{\rm B}$ is close to that expected for free
Yb3+ ions (4$f^{13}$; $L$ = 3, $S$= 1/2). As Yb3+ is a Kramers ion, generally
the strong crystal electric field splits the eight-fold degenerate $J$ = 7/2
multiplet into four Kramers doublets. At low temperatures, Yb3+ ions acquire
spin-orbit entangled $J_{\rm eff}$ = 1/2 moment. In such a scenario, the low-
temperature magnetic properties are mainly governed by the exchange
interactions between the $J_{\rm eff}$ = 1/2 moment of Yb3+ ions in the lowest
Kramers doublet state while the higher doublet states are important to
understand the magnetic properties at high temperatures and high magnetic
fields Pocs  _et al._ (2021).
In order to obtain information concerning the Kramers doublet ground state and
the nature of magnetic interactions, the Curie-Weiss fit to the low-
temperature inverse susceptibility data is required. However, accurately
estimating the actual Curie-Weiss temperature for rare-earth magnets is very
crucial due to the high sensitivity of the fit parameters to the fitting range
Arh  _et al._ (2022). To obtain a rough idea of the Kramers doublet state and
dominant magnetic interactions between Yb3+ moments in BYSO, the low-
temperature magnetic susceptibility data were fitted with different upper
temperature limit while the lower temperature limit was fixed to 4 K as
described in ref. Arh  _et al._ (2022) (see the bottom inset of Figure 2 (b)).
Following this method, the temperature-independent fit parameters were found
to be $\theta_{\rm CW}$ = $-$ 0.94 $\pm$ 0.01 K and $\mu_{\rm eff}=2.46$
$\mu_{\rm B}$. The obtained value of effective moment, $\mu_{\rm eff}=2.46$
$\mu_{\rm B}$ is considerably smaller than $\mu_{\rm eff}$ = 4.53 $\mu_{B}$
expected for free Yb3+ ions. It implies the presence of a Kramer doublet state
with $J_{\rm eff}$ = 1/2 low-energy state of Yb3+ ions Xing  _et al._ (2020).
The obtained negative Curie-Weiss temperature, $\theta_{\rm CW}$ = $-$ 0.94 K,
suggests the presence of a weak antiferromagnetic interaction between Yb3+
spins.

Figure 3: (a) The ESR spectra of Ba9Yb2Si6O24 at various temperatures
(circles), with corresponding best fits with isotropic Lorentzian line shape
(solid lines) above 70 K. The spectra are shifted vertically for clarity. (b)
The temperature dependence of the ESR line width (circles), with the solid
line representing the fit to the Orbach model, yielding the energy gap of
$\Delta=32.3(7)$ meV (see text for details).

Figure 2 (c) depicts the isothermal magnetization as a function of the
magnetic field up to 7 T at several temperatures. For temperatures ($T\geq$ 2
K) well above the interaction energy scale, one can model the observed
isothermal magnetization following the Brillouin function, $M$/$M_{s}$ =
$B_{1/2}$ ($y$), where
$B_{J}(y)=[\frac{2J+1}{2J}coth[\frac{y(2J+1)}{2J}]-\frac{1}{2J}coth\frac{y}{2J}]$
is the Brillouin function, $M_{s}$ (= g$J\mu_{B}$) is the saturation
magnetization and $y=g\mu_{B}J\mu_{0}H/k_{B}T$ is the ratio of the Zeeman
energy of magnetic moment to the thermal energy, $\mu_{B}$ is the Bohr
magneton, and $g$ is the Lande’s g-factor. The solid lines in Figure 2 (c) for
2 K, 5 K, 10 K and 15 K are the Brillouin function fit which yields an average
Lande’s g-factor, $g$ = 2.45, while $J$ was fixed to 1/2, consistent with the
lowest Kramers doublet state of Yb3+ ions in this temperature regime.

###  III.3 Electron spin resonance

Electron spin resonance (ESR) measurements were performed on the powder sample
of BYSO at $T\geq$ 4 K. The ESR spectra exhibit pronounced broadening with
increasing temperature, making the signal too broad for reliable analysis
above 140 K (Figure 3 (a)). Below 50 K, the spectra become structured,
however, their line shape does not correspond to a single powder pattern due
to anisotropic $g$ factors. Above 70 K, the spectra can be nicely fitted with
a single Lorentzian curve, revealing motional narrowing effects on the line
shape Zorko (2018).
The temperature dependence of the ESR line width (Figure 3 (b)) is due to
crystal-electric-field (CEF) fluctuations, as often encountered in rare-earth
magnets Arh  _et al._ (2022), where excited CEF levels are relatively low in
energy. The broadening is due to the Orbach process, involving two-phonon
scattering via excited CEF levels Abragam and Bleaney (2012). Indeed, the
experimental line width $\Delta B$ is well described with the expression
$\Delta B(T)$ = $\Delta B_{0}$ $+$ [$f$/(exp($\Delta/k_{B}T$) $-$1)], where
the constant term $\Delta B_{0}$ arises from the magnetic anisotropy in the
CEF ground state, while the exponential term describes the Orbach relaxation
with $f$ being the scaling factor between the ESR line width and the spin
fluctuation frequency, and $\Delta$ being the energy gap between the CEF
ground state and the lowest excited state. The fit of the model to the
experimental data yields $\Delta B_{0}=51(3)$ mT, $f=13(1)$ T, and
$\Delta=32.3(7)$ meV. This gap value is very similar to the gap of 39.4 meV
observed in YbMgGaO4 Li  _et al._ (2017b) or 34.8 meV observed in NaYbO2 Ding
_et al._ (2019). Both materials possess YbO6 octahedra with frustrated two-
dimensional arrangements, similar to BYSO.

###  III.4 Specific heat

In order to gain further insights into ground state properties, we performed
the temperature dependence of specific heat ($C_{\rm p}(T)$) of BYSO down to
1.9 K in several magnetic fields up to 7 T. The temperature dependence of
zero-field $C_{\rm p}/T$ data in the temperature range 1.9 K $\leq$ T $\leq$
250 K is shown in Figure 4 (a). It is noted that in zero magnetic field an
anomaly appears at $T_{\rm N}$ = 2.26 K that is attributed to the transition
temperature of the unreacted minor impurity phase of Yb2O3 in BYSO (Figure 4
(b)) Moon  _et al._ (1968, 2004); Grußler  _et al._ (2023); Tokiwa  _et al._
(2021). However, the Yb3+ moments decorating the two dimensional honeycomb
lattice do not undergo long-range magnetic order down to 1.9 K.

Figure 4:  Temperature dependence of zero-field specific heat divided by
temperature, $C_{\rm p}/T$, where the red line is the fitting curve by the
Debye model (see text) which accounts for phonon contributions. (b)
Temperature dependence of $C_{\rm p}/T$ in several magnetic fields. The
anomaly at $\sim$ $T$ = 2.26 K in zero-field corresponds to the transition
temperature of minor impurity phase of Yb2O3 in BYSO. (c) Temperature
dependence of magnetic specific heat in several magnetic fields where the
solid line depicts the fit using Eq. (1).

In order to extract the magnetic specific heat associated with Yb3+ spins from
the total specific heat, we model the lattice contribution due to phonons
using the Debye function i.e., $\textit{C}_{\rm lat.}(\textit{T})=9k_{\rm
B}[\sum_{n=1}^{2}C_{n}(\frac{\textit{T}}{\theta_{D_{n}}})^{3}\int_{0}^{\theta_{D_{n}}/\textit{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx]$,
where $\theta_{D}$ is the Debye temperature, while R and kB are the molar gas
constant and Boltzmann constant, respectively. The solid red line in Figure 4
(a) is the fitted lattice contributions that was obtained with $\theta_{D1}=$
234 K and $\theta_{D2}$ = 456 K. In the fits, two coefficients were fixed in
the ratio CD1:CD2 = 17 : 24 that corresponds to the ratio of the number of
heavy and light atoms in BYSO Ortega-San Martin  _et al._ (2006); Khatua  _et
al._ (2022a); Kim  _et al._ (2019). The associated magnetic contribution
$C_{\rm mag}(T)$ was extracted after subtracting lattice contribution and is
shown in Figure 4 (c) as a function of temperature.
The temperature dependence of $C_{\rm p}(T)/T$ in magnetic fields up to 7 T is
shown in Figure 4 (b). In a magnetic field of 3 T, only a broad maximum is
observed around 2.3 K in the specific heat while the anomaly due to Yb2O3 is
fully suppressed. This observation suggests that the low-field specific heat
data most likely show a broad maximum at substantially lower temperatures due
to the weak exchange coupling between $J_{\rm eff}$ = 1/2 moments of Yb3+ ions
Khatua  _et al._ (2022b); Ulaga  _et al._ (2023). Above 3 T, this broad
maximum progressively shifts toward higher temperatures and it behaves like a
Schottky anomaly, suggesting the presence of a field polarized state with a
field-induced gap in high magnetic fields Luo and Chen (2020). A similar
scenario has been suggested in the honeycomb lattice YbCl3 Hao  _et al._
(2020) and other low-dimensional frustrated magnets Wu  _et al._ (2019); Bader
_et al._ (2022). In BYSO, the field-induced gap is attributed to the Zeeman
splitting of the lowest Kramers doublet state, effectively surpassing the
intrinsic exchange interactions between $J_{\rm eff}$ = 1/2 moments in the
ground-state Kramers doublet. To estimate the gap value, the high-field
magnetic specific heat data were fitted a two-level Schottky expression of the
specific heat

$C_{\rm sch.}=fR\left(\frac{\Delta_{\rm s}}{k_{B}T}\right)^{2}\frac{{\rm
exp}(\Delta_{s}/k_{B}T)}{(1+{\rm exp}(\Delta_{\rm s}/k_{B}T))^{2}},$ (1)

where $\Delta_{\rm s}$ is the gap induced by the Zeeman splitting of the
ground state Kramers doublet of Yb3+ ion, and f measures the fraction of Yb3+
spins which contribute to the splitting of the ground-state doublet. The
fitted solid lines in Figure 4 (c) were obtained using Eq.1. Figure 5 (e)
displays the corresponding estimated field-induced gap. The estimated gap
value is consistent with that obtained from nuclear magnetic resonance spin-
lattice relaxation experiments (see Figure 5 (d)). The estimated fraction of
Yb3+ spins was close to 1, suggesting that almost all the Yb3+ spins
contribute to the observed effect Jiang  _et al._ (2022); Khatua  _et al._
(2022a). The spin correlations between 4$f$ moments generally develop at low
temperature in view of the weak exchange interaction between magnetic moments
in these materials. This calls for sub-Kelvin thermodynamic experiments to
gain deeper insights into the ground state of this 4$f$-based honeycomb
antiferromagnet Bag  _et al._ (2021).

Figure 5: The field-swept 29Si-NMR spectra measured at a constant frequency
$\nu$ = 61 MHz at various temperatures. The blue dashed vertical line
corresponds to the zero-shift reference position of 7.2123 T. The blue curve
at the bottom shows the calculated asymmetric NMR spectrum due to hyperfine
anisotropy ($K_{\rm iso}$ and $K_{\rm ax}$) with nearly zero inhomogeneous
magnetic broadening at $T$ = 1.6 K. The red curves are calculated spectra by
taking into consideration the inhomogeneous magnetic broadening introduced by
convoluting the Lorentzian function with the full width at half maximum
defined as a parameter $\Delta\nu$. The slightly asymmetric shape in the
spectra at higher temperatures above $\sim$ 60 K is probably due to the 29Si
NMR signal at zero-shift position from the nonmagnetic impurity of Ba2SiO4 sm
. (b) Temperature dependence of $K_{\rm iso}$ and $K_{\rm ax}$. The inset
shows $K_{\rm iso}$ and $K_{\rm ax}$ versus magnetic susceptibility $\chi$.
The solid and dashed lines are linear fits, $K_{\rm iso}$ = 0.45 $\chi$ \+
0.032 and $K_{\rm ax}$ = 0.95 $\chi$ $-$ 0.012, respectively with units of %
for NMR shifts and cm3/mol for magnetic susceptibility. (c) The temperature
dependence of $\Delta\nu$ (inhomogeneous magnetic broadening) determined by
the simulation of the NMR spectra measured at different frequencies (i.e.,
different magnetic field). (d) The temperature dependence of the 29Si NMR
spin-lattice relaxation rate (1/$T_{1}$) measured at five different fields on
a log-log scale. The inset shows the 1/$T_{1}$ as a function of inverse
temperature (1/$T$) for different magnetic fields on a semilogarithmic scale.
The black lines in the inset represent the fits with a phenomenological model
valid for thermally activated behavior of 1/$T_{1}$ as discussed in the text.
(e) Magnetic field dependence of the magnitude of the field-induced gap
estimated from the 1/$T_{1}$ and heat capacity measurements.

###  III.5 Nuclear magnetic resonance

In order to track the intrinsic static magnetic susceptibility and spin
dynamics of BYSO, we also performed nuclear magnetic resonance (NMR)
measurements on 29Si (nuclear spin $I$ = 1/2, gyromagnetic ratio
$\gamma$N/2$\pi$ = 8.4577 MHz/T). Figure 5 (a) shows the field-swept 29Si NMR
spectra at several temperatures at $\nu$ = 61 MHz. The smooth evolution of the
field-swept NMR spectra in the entire temperature range without developing
rectangular shape or splitting of the 29Si line suggests the absence of long-
range magnetic ordering in the compound Yamada and Sakata (1986); Wang  _et
al._ (2021); Chakraborty  _et al._ (2021)
At high temperatures, the NMR spectra are relatively narrow, but as the
temperature drops, they start widening and exhibit anisotropic behavior at low
temperatures. The asymmetric shapes of the spectra were well reproduced by the
calculated spectra where we introduced an anisotropy in NMR shift $K$. The red
curves in Figure 5 (a) show the calculated powder-pattern spectra with
isotropic and axial components of NMR shifts ($K_{\rm iso}$ and $K_{\rm ax}$),
which reproduced the observed spectra well. Here, we calculated the spectra by
taking into consideration the short-range spin correlations that cause
inhomogeneous magnetic broadening, which is introduced by convoluting the
Lorentzian function with the full width at half maximum defined as a parameter
$\Delta\nu$. The blue curve at the bottom shows the calculated asymmetric NMR
spectrum due to hyperfine anisotropy ($K_{\rm iso}$ and $K_{\rm ax}$) with
nearly zero inhomogeneous magnetic broadening at $T$ = 1.6 K. The NMR shift
$K$ is described by $K$ = $K_{\rm iso}$ \+ $K_{\rm ax}$(3cos${}^{2}\theta-1$),
where $\theta$ is the angle between the principal axis of hyperfine tensor at
the Si site and the external magnetic field. At higher temperatures above
$\sim$ 60 K, the relatively narrow spectra also show slightly asymmetric
shapes but with tails on the higher magnetic field side, in contrast to those
observed at low temperatures. From the detailed analysis, it turned out that
the slightly asymmetric shape of the spectra at higher temperatures above
$\sim$ 60 K can be explained by an additional 29Si NMR signal at the zero-
shift position (the red curves in Figure 5 (a) for $T$ $>$ 60 K) which is
probably due to the signal from the nonmagnetic impurity of Ba2SiO4 sm .
Figure 5 (b) shows the temperature dependence of $K_{\rm iso}$ and $K_{\rm
ax}$ determined by fitting the spectra. $K_{\rm iso}$ and $K_{\rm ax}$ are
nearly independent of temperature within our experimental uncertainty above 40
K, however, both show a clear increase below $\sim$ 40 K beyond our
experimental uncertainty. It is interesting to point out that 40 K is close to
the crossover temperature between the $J_{\rm eff}$ = 1/2 and $J$ = 7/2 states
as shown in the magnetic susceptibility measurements (see Figure 2). Below 7
K, both $K_{\rm iso}$ and $K_{\rm ax}$ saturate to finite values, whose
behavior is ascribed to the strong polarization of the Yb3+ moments in high-
magnetic fields. The obtained NMR shift can be expressed as $K(T)$ = $K_{\rm
0}$ \+ $K_{\rm spin}(T)$ = $K_{\rm 0}$ \+ ($A_{\rm hf}/N_{\rm
A}\mu_{B}$)$\chi(T)$, where the first term ($K_{\rm 0}$) represents the sum of
temperature-independent contribution arising from the orbital susceptibility
and chemical shift, the second term is the spin part of $K$ which accounts for
the temperature-dependent intrinsic spin susceptibility of Yb3+ spins. Here
$A_{\rm hf}$ is the hyperfine coupling constant, and $N_{\rm A}$ the Avogadro
number. To extract the hyperfine coupling constants, the NMR shifts versus
magnetic susceptibility (at 1 T) was plotted with temperature as an implicit
parameter in the inset of Figure 5 (b) (also known as Clogston-Jaccarino plot)
Clogston and Jaccarino (1961). From the linear fit in the temperature range of
$T$ = 7 - 150 K, $A_{\rm hf}$ and $K_{\rm 0}$ are estimated to be $A_{\rm
hf,iso}$ = 25 $\pm$ 3 Oe/$\mu_{B}$ and $K_{\rm 0,iso}$ = 0.032 % for the
isotropic part and $A_{\rm hf,ax}$ = 53 $\pm$ 10 Oe/$\mu_{B}$ and $K_{\rm
0,ax}$ = $-$0.012 % for axial part, respectively.
The broadening of NMR spectra with decreasing temperature is due to the
increase of the distributions in $K_{\rm ax}$ and $K_{\rm iso}$. The
temperature and magnetic field dependence of the inhomogeneous broadenings is
shown in Figure 5 (c), where $\Delta\nu$ is plotted as a function of
temperature. $\Delta\nu$ depends on temperature and increases with increasing
magnetic field. This is consistent with the magnetization measurements since
$\Delta\nu$ is related to the magnetization.
The NMR spin-lattice relaxation rate ($T_{1}^{-1}$) probes the low-energy spin
excitations related to the dynamic spin susceptibility governed by
fluctuations of electron spin at the nuclear sites through hyperfine
interactions. Figure 5 (d) depicts the temperature dependence of $T_{1}^{-1}$
with three distinct regions of different spin-lattice relaxation rates. In the
entire temperature range of investigation, the relaxation rates were estimated
from fitting the recovery of longitudinal nuclear magnetization $M$($t$) by a
single exponential function, $M_{z}(t)=(M_{\rm 0}-M(t))/M_{0}=A\ \
\textnormal{exp}(-t/{\textit{T}_{1}})$, where $M_{\rm 0}$ is the equilibrium
magnetization, $M_{z}(t)$ is the magnetization at time $t$ after the
saturation pulse, and $A$ is a constant. It is noted that all measured
relaxation curves were well-fitted with a single component of $T_{1}$,
implying a homogeneous distribution of spin-lattice relaxation rates in this
4$f$ honeycomb lattice system. Upon cooling, the $T_{1}^{-1}$ increases till
the plateau around 50 K in magnetic fields $\mu_{0}H$ = 2.5 and 7.2 T. This
behavior is often observed in rare-earth magnets due to the depopulation of
the crystal electric field and is due to the Orbach mechanism that is
responsible for the ESR line broadening in BYSO Khatua  _et al._ (2022b).
Below 50 K, $T_{1}^{-1}$ first remains nearly temperature independent down to
a field-dependent characteristic temperature, as shown by the dotted curve in
Figure 5 (d). Although $T_{1}^{-1}$ slightly decreases with decreasing
temperature in a high magnetic field of 7.2 T, the nearly temperature-
independent behavior of $T_{1}^{-1}$ in the intermediate temperature range
suggests that the relaxation rate is dominated by the paramagnetic spin
fluctuations of Yb3+ spins in the crystal field ground state. One can expect
the correlated magnetism of Yb3+ spins and a field-polarized phase at low
temperatures owing to the weak exchange interaction between Yb3+ spins typical
for rare-earth-based quantum magnets Khatua  _et al._ (2022a). The relaxation
rate $T_{1}^{-1}$ decreases rapidly below the characteristic temperature
$T^{*}$, suggesting that the applied magnetic field opens a gap which normally
occurs when the Zeeman energy exceeds the interaction energy between Yb3+
ions. To estimate the value of the gap in the presence of a magnetic field, we
present $T_{1}^{-1}$ as a function of inverse temperature in a semi-log plot
for $\mu_{0}H$ $>$ 1.35 T as shown in the inset of Figure 5 (d). The solid
line represents the fit to the experimental data using the phenomenological
model relevant for thermally activated behavior of magnetic moments i.e.,
$T_{1}^{-1}$ $\propto$ exp($-\Delta_{\rm s}/k_{B}T$), where $\Delta_{s}$ is
the gap value to the Zeeman splitting of the ground state Kramers doublet in
the presence of a magnetic field. We find a linear variation of the gap with
the applied magnetic field (see Figure 5 (e)), as expected when the Zeeman
energy overcomes the exchange energy. The estimated $\Delta_{s}$ from the NMR
measurements is in good agreement with the gap values obtained from the
specific heat measurements (Figure 4 (c)).

##  IV Discussion

In this work, we have investigated crystal structure and ground state
properties of an unexplored rare-earth based honeycomb spin-lattice
Ba9Yb2Si6O24 through the combination of thermodynamic and microscopic
measurements. Ba9Yb2Si6O24, nearly free from anti-site disorder, constitutes a
perfect honeycomb lattice of Yb3+ ions perpendicular to the crystallographic
$c$-axis. Magnetic susceptibility data suggest the presence of spin-orbit
entangled $J_{\rm eff}$ = 1/2 moments of Yb3+ ions with a Kramers doublet
ground state, which are exchange-coupled by weak antiferromagnetic
interactions at low temperatures. The mean field formula $\theta_{\rm CW}$ =
($-$$zJS(S+1)$)/3$k_{\rm B}$ offers an approximate estimation of the exchange
interaction $J/k_{\rm B}$ between $J_{\rm eff}$ = 1/2 moments of Yb3+ ions in
the $ab$-plane taking into account the nearest-neighbor $z$ = 3 and $S$ =
$J_{\rm eff}$ = 1/2 for BYSO Li  _et al._ (2020). The nearest-neighbor
exchange interaction in the $ab$-plane is found to be roughly 1.3 K while the
dipolar interaction energy Edip$\approx$ $\mu_{0}g_{\rm
avg}^{2}\mu_{B}^{2}/4\pi a^{3}$ $\approx$ 0.3 % of the nearest-neighbor
exchange interactions, where $g_{\rm avg}$ is the powder average Landé $g$
factor and $a$ is the nearest-neighbor Yb-Yb bond length in BYSO. This implies
the presence of a significant super-exchange interaction between $J_{\rm eff}$
= 1/2 moments of Yb3+ ions in BYSO in addition to the dipolar interaction,
analogous to the honeycomb lattice YbBr3 Wessler  _et al._ (2020). In contrast
to the corner and side sharing regular YbBr6 octahedra in the two-dimensional
honeycomb lattice YbBr3, YbO6 octahedra in BYSO are isolated Wessler  _et al._
(2020). Thus, in BYSO, the nearest-neighbor intra-plane superexchange
interaction can only occur via the Yb-O-Si-O-Yb virtual path, which may be one
of the reasons for the weak antiferromagnetic interaction strength between
Yb3+ moments in BYSO. Contrarily, in YbBr3, the bromine ion directly mediates
the nearest neighbor superexchange interaction through the Yb-Br-Yb virtual
electron hopping processes, which results in a little higher interaction
strength ($\approx$ 8 K) Wessler  _et al._ (2020). Furthermore, in BYSO, the
nature of exchange interactions between Yb3+ moments is expected to be
Heisenberg-type as the YbO6 octahedra are connected through intermediate Si4+
ions which prevent one of the essential requirements i.e., Yb-O-Yb bond angles
of 90∘ for stabilizing Kitaev interactions (see Figure 1 (b)) Jackeli and
Khaliullin (2009).
Our ESR results indicate the presence of anisotropic magnetic interaction
between $J_{\rm eff}$ = 1/2 moments of Yb3+ ions. Normally, in the
conventional antiferromagnetically ordered state, the crystallographic site of
the probing nucleus in an NMR experiment becomes inequivalent and senses
different magnetic fields which leads to splitting the NMR spectra when
internal fields at the nucleus are parallel or antiparallel to an external
magnetic field. The absence of NMR line splitting or rectangular spectra
confirms the absence of long-range magnetic order in BYSO down to 1.6 K, which
substantiates that the minor Yb2O3 impurity phase does not significantly
affect the underlying magnetic properties of the material under consideration.
The bulk of the material maintains a dynamic ground state at least down to 1.6
K. In high magnetic fields, the exponential decay of the spin-lattice
relaxation rate at low temperature is attributed to a field-induced gap due to
the Zeeman splitting of the low-energy Kramers doublet state, which typically
takes place when the Zeeman energy exceeds the exchange interaction energy.
Theoretically, it has been suggested that the competing magnetic interactions
between nearest-neighbor and next-neighbor contacts can nevertheless lead to
the realization of a quantum spin liquid state in honeycomb lattices even in
the absence of Kitaev-type interactions Fouet  _et al._ (2001); Lefrançois
_et al._ (2016); Gong  _et al._ (2013); Ganesh  _et al._ (2013). The two-
dimensional honeycomb lattice BYSO seems promising in this direction, and its
low-temperature magnetic properties on single crystal could hold significant
potential for future investigations.

##  V SUMMARY

To summarize, we have synthesized and performed magnetization, specific heat,
ESR, and NMR experiments on a 4$f$ electron-based material Ba9Yb2Si6O24, which
crystallizes in a trigonal crystal structure with the space group R$\bar{3}$.
In this material, the Yb3+ ions constitute a perfect honeycomb lattice in the
$ab$-plane. Magnetization data suggest the pseudospin-1/2 degrees of freedom
of Yb3+ ions in the Kramers doublet state, and these $J_{\rm eff}$ = 1/2 spins
interact antiferromagnetically. The lowest CEF excited Kramers doubled is far
above (at 32.3(7) meV) the ground state, as estimated from ESR.
The absence of long-range magnetic ordering between pseudospin-1/2 moments of
Yb3+ ions in specific heat data, which is corroborated by microscopic NMR
results down to 1.6 K, suggests that the two-dimensional honeycomb lattice
Ba9Yb2Si6O24 can host a spin liquid ground state. The spin-lattice relaxation
rate measurements in high-magnetic fields show the presence of a field-induced
gap due to Zeeman splitting of the Kramers doublet, which is consistent with
specific heat results. Further studies on single crystals of Ba9Yb2Si6O24 are
highly desired to shed more insights into the anisotropic magnetic
interactions, and low-energy excitations. The present family of rare-earth
based honeycomb spin-lattice Ba9$R$2Si6O24 ($R$ = rare-earth ions) with
distinct rare-earth elements, spin-orbit-driven anisotropy, and spin
correlations provide an ideal ground to realize exotic quantum phenomena.

##  VI Acknowledgments

P.K. acknowledges funding by the Science and Engineering Research Board and
Department of Science and Technology, India through research grants. This
research was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering. Ames National
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DEAC02-07CH11358. A.Z. acknowledges the
financial support of the Slovenian Research Agency through the Program No.
P1-0125 and Projects No. N1-0148 and J1-2461.

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