#  torchgfn: A PyTorch GFlowNet library

Salem Lahlou<EMAIL_ADDRESS>
Mila, Université de Montréal Joseph D. Viviano<EMAIL_ADDRESS>
Mila Victor Schmidt<EMAIL_ADDRESS>
Mila, Université de Montréal

###### Abstract

The increasing popularity of generative flow networks (GFlowNets or GFNs) is
accompanied with a proliferation of code sources. This hinders the
implementation of new features, such as training losses, that can readily be
compared to existing ones, on a set of common environments. In addition to
slowing down research in the field of GFlowNets, different code bases use
different conventions, that might be confusing for newcomers. torchgfn is a
library built on top of PyTorch, that aims at addressing both problems. It
provides user with a simple API for environments, and useful abstractions for
samplers and losses. Multiple examples are provided, replicating published
results. The code is available in https://github.com/saleml/torchgfn.

##  1 Introduction

Generative Flow Networks (GFlowNets; GFNs; Bengio et al., 2021a, b) are
probabilistic models over discrete sample spaces with a compositional
structure. They are also stochastic sequential samplers that generate objects
from a target distribution, which is given by its unnormalized probability
mass function $R$, referred to as the reward function.

torchgfn serves the purpose of fast prototyping GFlowNet related algorithms.
It decouples the environment definition, the sampling process, and the
parametrization used for the GFN loss.

##  2 Details about the code base

The code base structure is illustrated in Figure 1. In this section, we
describe its components with more details.

Figure 1: Hierarchy of the codebase. States and Actions are abstractions used
to interface between the stateless Environments and Containers, which are
generic objects used by the remainder of the codebase. Containers are utilized
by both Samplers and Losses to implement the core functionality required to
train a GFN, other than the standard nerual network tooling provided by
Pytorch.

###  2.1 Defining an environment

A pointed DAG environment (or GFN environment, or environment for short) is a
representation for the pointed DAG. The abstract class Env specifies the
requirements for a valid environment definition. To obtain such a
representation, the environment needs to specify the following attributes,
properties, or methods:

  * •

The action_space. Which should be a gymnasium.spaces.Discrete object for
discrete environments. The last action should correspond to the exit action.

  * •

The initial state $s_{0}$, as a torch.Tensor of arbitrary dimension.

  * •

(Optional) The sink state $s_{f}$, as a torch.Tensor of the same shape as
$s_{0}$, used to represent complete trajectories only (within a batch of
trajectories of different lengths), and never processed by any model. If not
specified, it is set to torch.full_like(s_0, -float(’inf’)).

  * •

The method make_States_class that creates a subclass of States. The instances
of the resulting class should represent a batch of states of arbitrary shape,
which is useful to define a trajectory, or a batch of trajectories. $s_{0}$
and $s_{f}$, along with a tuple called state_shape should be defined as class
variables, and the subclass (of States) should implement masking methods, that
specify which actions are possible, in a discrete environment.

  * •

The methods maskless_step and maskless_backward_step that specify how an
action changes a state (going forward and backward). These functions do not
need to handle masking, checking whether actions are allowed, checking whether
a state is the sink state, etc… These checks are handled in Env.step and
Env.backward_step

  * •

The log_reward function that assigns a nonnegative reward to every terminating
state (i.e. state with only $s_{f}$ as a child in the DAG). If log_reward is
not implemented, reward needs to be.

If the states (as represented in the States class) need to be transformed to
another format before being processed (by neural networks for example), then
the environment should define a preprocessor attribute, which should be an
instance of the base preprocessor class. If no preprocessor is defined, the
states are used as is (actually transformed using IdentityPreprocessor, which
transforms the state tensors to FloatTensors). Implementing your own
preprocessor requires defining the preprocess function, and the output_shape
attribute, which is a tuple representing the shape of one preprocessed state.

Optionally, you can define a static get_states_indices method that assigns a
unique integer number to each state if the environment allows it, and a
n_states property that returns an integer representing the number of states
(excluding ) in the environment. get_terminating_states_indices can also be
implemented and serves the purpose of uniquely identifying terminating states
of the environment.

For more details, take a look at HyperGrid, an environment where all states
are terminating states, or at DiscreteEBM, where all trajectories are of the
same length but only some states are terminating.

###  2.2 Estimators and Modules

Training GFlowNets requires one or multiple estimators. As of now, only
discrete environments are handled. All estimators are subclasses of
FunctionEstimator, implementing a __call__ function that takes as input a
batch of States.

  * •

LogEdgeFlowEstimator. It outputs a (*batch_shape, n_actions) tensor
representing $\log F(s\rightarrow s^{\prime})$, including when
$s^{\prime}=s_{f}$.

  * •

LogStateFlowEstimator. It outputs a (*batch_shape, 1) tensor representing
$\log F(s)$. When used with forward_looking=True, $\log F(s)$ is parametrized
as the sum of a function approximator and $\log R(s)$ \- which is only
possible for environments where all states are terminating.

  * •

LogitPFEstimator. It outputs a (*batch_shape, n_actions) tensor representing
$logit(s^{\prime}\mid s)$, such that $P_{F}(s^{\prime}\mid
s)=softmax_{s^{\prime}}\ logit(s^{\prime}\mid s)$, including when
$s^{\prime}=s_{f}$.

  * •

LogitPBEstimator. It outputs a (*batch_shape, n_actions - 1) tensor
representing $logit(s^{\prime}\mid s)$, such that $P_{B}(s^{\prime}\mid
s)=softmax_{s^{\prime}}\ logit(s^{\prime}\mid s)$.

Defining an estimator requires the environment, and a module instance. Modules
inherit from the GFNModule class, which can be seen as an extension of
torch.nn.Module. Alternatively, a module is created by providing which module
type to use (e.g. ”NeuralNet” or ”Uniform” or ”Zero”). A basic MLP is provided
as the NeuralNet class, but any function approximator should be possible.

Said differently, a States object is first transformed via the environment’s
preprocessor to a (*batch_shape, *output_shape) float tensor. The
preprocessor’s output shape should match the module input shape (if any). The
preprocessed states are then passed as inputs to the module, returning the
desired output (either flows or probabilities over children in the DAG).

Each module has a named_parameters functions that returns a dictionary of the
learnable parameters. This attribute is transferred to the corresponding
estimator.

Additionally, a LogZEstimator is provided, which is a container for a scalar
tensor representing $\log Z$, the log-partition function, useful for the
Trajectory Balance loss for example. This estimator also has a
named_parameters function.

###  2.3 Samplers

An ActionsSampler object defines how actions are sampled at each state of the
DAG. As of now, only DiscreteActionsSamplers are implemented. The require an
estimator (of $P_{F}$, $P_{B}$, or edge flows) defining the action
probabilities. These estimators can contain any type of modules (including
random action sampling for example). A BackwardDiscreteActionsSampler class is
provided to sample parents of a state, which is helpful to sample trajectories
starting from their last states.

They are at the core of TrajectoriesSamplers, which implements the
sample_trajectories method, that sample a batch of trajectories starting from
a given set of initial states or starting from $s_{0}$.

###  2.4 Losses

GFlowNets can be trained with different losses, each of which requires a
different parametrization. A parametrization is a dataclass, which can be seen
as a container of different estimators. Each parametrization defines a
distribution over trajectories, via the parametrization.Pi method, and a
distribution over terminating states, via the parametrization.P_T method.

Currently, the implemented losses are:

  * •

Flow Matching (Bengio et al., 2021a, b)

  * •

Detailed Balance (Bengio et al., 2021b)

  * •

Modified Detailed Balance (Deleu et al., 2022)

  * •

Trajectory Balance (Malkin et al., 2022)

  * •

Sub-trajectory Balance (Madan et al., 2023)

  * •

Log Partition Variance loss (Zhang et al., 2023)

###  2.5 Solving for the flows using Dynamic programming

A simple script that propagates trajectories rewards through the DAG to define
edge flows in a deterministic way (by visiting each edge once only) is
provided in dynamic_programming.py.

##  3 Examples

###  3.1 The train.py script

A sample script is provided in order to replicate existing published results
on some environments. Here are a few command line examples to run the script:

    $ python train.py --env HyperGrid --env.ndim 4 --env.height 8
      --n_iterations 100000 --loss TB
    $ python train.py --env DiscreteEBM --env.ndim 4 --env.alpha 0.5
      --n_iterations 10000 --batch_size 64 --temperature 2.
    $ python train.py --env HyperGrid --env.ndim 2 --env.height 64
      --n_iterations 100000 --loss DB --replay_buffer_size 1000
      --logit_PB.module_name Uniform --optim sgd --optim.lr 5e-3
    $ python train.py --env HyperGrid --env.ndim 4 --env.height 8
      --env.R0 0.01 --loss FM --optim adam --optim.lr 1e-4

###  3.2 Complete example

⬇

1import torch

2from tqdm import tqdm

3

4from gfn import LogitPBEstimator, LogitPFEstimator, LogZEstimator

5from gfn.envs import HyperGrid

6from gfn.losses import TBParametrization, TrajectoryBalance

7from gfn.samplers import DiscreteActionsSampler, TrajectoriesSampler

8

9if __name__ == "__main__":

10

11 env = HyperGrid(ndim=4, height=8, R0=0.01) # Grid of size 8x8x8x8

12

13 logit_PF = LogitPFEstimator(env=env, module_name="NeuralNet")

14 logit_PB = LogitPBEstimator(

15 env=env,

16 module_name="NeuralNet",

17 torso=logit_PF.module.torso, # To share parameters between PF and PB

18 )

19 logZ = LogZEstimator(torch.tensor(0.0))

20

21 parametrization = TBParametrization(logit_PF, logit_PB, logZ)

22

23 actions_sampler = DiscreteActionsSampler(estimator=logit_PF)

24 trajectories_sampler = TrajectoriesSampler(env=env,
actions_sampler=actions_sampler)

25

26 loss_fn = TrajectoryBalance(parametrization=parametrization)

27

28 params = [

29 {

30 "params": [

31 val for key, val in parametrization.parameters.items() if "logZ" not in key

32 ],

33 "lr": 0.001,

34 },

35 {"params": [val for key, val in parametrization.parameters.items() if
"logZ" in key], "lr": 0.1},

36 ]

37 optimizer = torch.optim.Adam(params)

38

39 for i in (pbar := tqdm(range(1000))):

40 trajectories = trajectories_sampler.sample(n_trajectories=16)

41 optimizer.zero_grad()

42 loss = loss_fn(trajectories)

43 loss.backward()

44 optimizer.step()

45 if i % 25 == 0:

46 pbar.set_postfix({"loss": loss.item()})

##  4 Conclusion and Future work

The library will continuously be developed and improved to incorporate new
features, including the support of continuous environments (Lahlou et al.,
2023). With time, we hope the community will use it to benchmark new features
against a suite of agreed upon environments.

Acknowledgments and Disclosure of Funding

The authors would like to thank Mo Tiwari, Edward Hu, Tristan Deleu, Daniel
Jenson, Eric Elmoznino for useful discussions and feedback.

## References

  * Bengio et al. (2021a) Emmanuel Bengio, Moksh Jain, Maksym Korablyov, Doina Precup, and Yoshua Bengio.  Flow network based generative models for non-iterative diverse candidate generation.  In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, _Advances in Neural Information Processing Systems_ , 2021a.  URL https://openreview.net/forum?id=Arn2E4IRjEB.
  * Bengio et al. (2021b) Yoshua Bengio, Tristan Deleu, Edward J. Hu, Salem Lahlou, Mo Tiwari, and Emmanuel Bengio.  Gflownet foundations, 2021b.
  * Deleu et al. (2022) Tristan Deleu, António Góis, Chris Emezue, Mansi Rankawat, Simon Lacoste-Julien, Stefan Bauer, and Yoshua Bengio.  Bayesian structure learning with generative flow networks.  In _Uncertainty in Artificial Intelligence_ , pages 518–528. PMLR, 2022.
  * Lahlou et al. (2023) Salem Lahlou, T. Deleu, Pablo Lemos, Dinghuai Zhang, Alexandra Volokhova, Alex Hernández-García, L’ena N’ehale Ezzine, Y. Bengio, and Nikolay Malkin.  A theory of continuous generative flow networks.  _International Conference on Machine Learning (ICML)_ , 2023.
  * Madan et al. (2023) Kanika Madan, Jarrid Rector-Brooks, Maksym Korablyov, Emmanuel Bengio, Moksh Jain, Andrei Nica, Tom Bosc, Yoshua Bengio, and Nikolay Malkin.  Learning gflownets from partial episodes for improved convergence and stability.  _International Conference on Machine Learning (ICML)_ , 2023.
  * Malkin et al. (2022) Nikolay Malkin, Moksh Jain, Emmanuel Bengio, Chen Sun, and Yoshua Bengio.  Trajectory balance: Improved credit assignment in gflownets.  _Neural Information Processing Systems (NeurIPS)_ , 2022.
  * Zhang et al. (2023) David W Zhang, Corrado Rainone, Markus Peschl, and Roberto Bondesan.  Robust scheduling with gflownets.  _arXiv preprint arXiv:2302.05446_ , 2023.


# Atypical antiferromagnetic ordering in single crystalline quasi-2D honeycomb
magnet YbI3

Nashra Pistawala  Department of Physics, Indian Institute of Science Education
and Research, Pune 411008, Maharashtra, India  Luminita Harnagea  I-HUB
Quantum Technology Foundation, Indian Institute of Science Education and
Research, Pune 411008, India  Sitaram Ramakrishnan  Department of Physics,
Indian Institute of Science Education and Research, Pune 411008, Maharashtra,
India  I-HUB Quantum Technology Foundation, Indian Institute of Science
Education and Research, Pune 411008, India  Priyanshi Tiwari  UGC-DAE
Consortium for Scientific Research, University Campus, Khandwa Road, Indore
452001, Madhya Pradesh, India  M. P. Saravanan  UGC-DAE Consortium for
Scientific Research, University Campus, Khandwa Road, Indore 452001, Madhya
Pradesh, India  Rajeev Rawat  UGC-DAE Consortium for Scientific Research,
University Campus, Khandwa Road, Indore 452001, Madhya Pradesh, India  Surjeet
Singh<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Science Education and Research, Pune 411008, Maharashtra, India

###### Abstract

Here, we study YbI3, a quasi-2D layered material with Yb atoms arranged on an
ideal honeycomb network of edge-sharing YbI6 octahedra, analogous to the low-
temperature phase of $\alpha-$RuCl3. High quality single crystals of YbI3 are
grown from Yb and I as starting precursors, using the vapor transport
technique. The grown crystals are characterized by single crystal x-ray
diffraction, Raman spectroscopy, magnetization, and heat capacity probes. The
crystal-field split ground state of Yb3+ in YbI3 is a well-isolated Kramers
doublet with an effective moment $\rm J_{eff}=1/2$. Upon cooling, the low-
temperature heat capacity of YbI3 reveals a broad peak at $\rm T_{1}=0.95$ K
due to short-range ordering of the Yb moments, followed by a sharp peak at
$\rm T_{2}=T_{N}=0.6$ K due to long-range ordering. The magnetic behavior is
found to be weakly anisotropic with $\chi^{\parallel}>\chi^{\perp}$, where
$\chi^{\parallel}$ and $\chi^{\perp}$ refers to the in-plane ($H\parallel ab$)
and out-of-plane ($H\perp ab$) susceptibilities. The 2 K isothermal
magnetization saturates at $\rm\approx~{}1.5~{}\mu_{B}/Yb^{3+}$ (in-plane) and
$\rm\approx~{}1~{}\mu_{B}/Yb^{3+}$ (out-of-plane), suggesting the anisotropy
to be easy-plane type. The low-temperature heat capacity, well below TN, is
found to vary as Tα with $\alpha~{}\approx~{}2.5$, indicating a possible
unconventional magnetic ground state for YbI3.

##  I Introduction

Quantum magnetism is one of the frontier areas in condensed matter physics,
showcasing unusual ground states, such as, spin liquids with emergent
quasiparticles, and exotic spin textures driven by competing interactions [1,
2]. The magnetic interactions in these systems are generally frustrated, which
prevents simple, ordered spin arrangements, leading to complex or
unconventional ground states. The spin systems based on triangular, Kagome, or
pyrochlore lattices are model geometrically frustrated systems, where the
competing interactions result from the lattice geometry which does not allow
all the nearest-neighbor couplings to be simultaneously satisfied.

The Kitaev model, introduced by Alexei Kitaev in the year 2006 [3], opened-up
a new frontier for exploring magnetic frustration. This groundbreaking
theoretical framework describes a spin-1/2 system on a two-dimensional
honeycomb lattice, where the bond-dependent nearest-neighbor couplings give
rise to a strong frustration. The ground state of Kitaev model is a novel
quantum spin liquid with Majorana fermions as quasiparticle excitations.

The quest for experimental realization of Kitaev model gained momentum
following the seminal work by Jackeli and Khaliullin [4], who laid out a
roadmap for uncovering potential Kitaev candidates among the spin-orbit-
assisted Mott insulators, specifically the $4d^{5}$ ruthenates and $5d^{5}$
iridates. In these compounds, the complex interplay between the strong spin-
orbit coupling, octahedral crystal electric field splitting, and electronic
correlations give rise to a $\rm J_{eff}=1/2$ Kramers doublet ground state
with highly anisotropic bond-dependent interactions, which serves as a
prerequisite for realizing the Kitaev model.

The electronic and magnetic properties of candidate materials, including,
Na2IrO3 [5, 6, 7], $\alpha$\- Li2IrO3 [8, 9], $\beta$-Li2IrO3 [10, 11],
$\gamma$-Li2IrO3 [12], H3LiIr2O6 [13], Cu2IrO3 [14], Cu3LiIr2O6 [15],
Ag3LiIr2O6 [16],$\alpha$-RuCl3 [17, 18, 19, 20, 21, 22, 23] has been
extensively studied in recent years to explore the manifestations of Kitaev
physics.

The $f$-electron systems, based on the rare-earths with $\rm J_{eff}=1/2$
Kramer’s doublet ground state and edge-shared octahedra have also been
theoretically predicted to host Kitaev physics [24, 25, 26, 27]. However,
suitable materials embodying this concept remain sparse. Xing et al. recently
investigated the crystal growth and magnetic properties of YbCl3, aiming to
explore its potential for Kitaev physics [28]. YbCl3 is a two-dimensional van
der Waals material, where the Yb atoms form a slightly distorted honeycomb
net, where each Yb atom has nearest-neighbors at 3.886 Å and 3.864 Å [29]. The
previous studies reveal short-range ordering of Yb3+ moments ($\rm
J_{eff}=1/2$) around 1.2 K and a Néel-type antiferromagnetic long-range order
at 0.6 K, with Yb spins antiferromagnetically aligned within each honeycomb
plane, pointing primarily along the a-axis with a small tilt of
$\sim~{}5^{\circ}$ along the c-axis [28, 30, 31].

A recent study by Matsumoto et al. provides evidence for Bose-Einstein
condensation in YbCl3. When the magnetic field is aligned parallel to the
a-axis, YbCl3 has been argued to undergo a transition from Heisenberg to XY-
like state, ultimately achieving a field-polarized state at $\rm H_{s}=5.93$
T, marked as the quantum critical point. Upon reducing the field slightly
below $H_{s}$, YbCl3 allegedly behaves as a two-dimensional gas of bosons,
showing signatures of Bose-Einstein condensation possibly stabilized by a
small but non zero interlayer interaction ($\Lambda_{\perp}\approx$ 0.1 mK and
$\Lambda_{\parallel}\approx$ 5 K, where $\Lambda_{\perp}$ and
$\Lambda_{\parallel}$ are interlayer and intralayer nearest neighbor
interactions) [32]. Hence, YbCl3 is posited as an exemplary two-dimensional
system for studying quantum magnetism on a honeycomb lattice.

Conversely, YbBr3, also comprising $\rm J_{eff}=1/2$ Heisenberg spins on a
honeycomb lattice, demonstrates no long-range ordering down to 100 mK. Based
on their inelastic neutron scattering data, Wessler et al. argued that despite
lacking Kitaev-type anisotropic interactions, YbBr3 exhibits pronounced
quantum fluctuations, perhaps signature of a quantum spin liquid ground state
[33]. Considering the novel magnetic behavior associated with the YbCl3 and
YbBr3, we set out to investigate the previously unexplored member YbI3 of this
family.

Here, we report the crystal growth of a new two-dimensional van der Waals
material, YbI3, using the vapor transport technique. YbI3 crystallizes in
$R\bar{3}$ symmetry, determined using the single crystal x-ray diffraction. In
this structure, the Yb3+ ions form an ideal honeycomb network, with
equidistant nearest neighbors with the honeycomb plane. This structure is
analogous to the low-temperature (T $<$ 150 K) phase of $\alpha$-RuCl3,
suggesting that YbI3 is a possible candidate material for studying Kitaev
physics.

The ground state of Yb3+ in YbI3 is shown to be a well-isolated Kramers
doublet with an effective moment, $\rm J_{eff}=1/2$. The magnetic behavior is
found to be slightly anisotropic. While the in-plane saturation moment at 2 K
is $\rm\approx 1.5~{}\mu_{B}$/Yb3+, it is $\rm\approx 1~{}\mu_{B}$/Yb3+ for
the out-of-plane orientation, suggesting an easy-plane behavior analogous to
YbCl3 (here in-plane and out-of-plane refers to the orientation of the applied
field with respect to the ab-plane). The low-temperature heat capacity shows a
broad peak at T1 = 0.95 K due to short-range ordering of the Yb moments, and a
sharp peak at T2 = 0.6 K that marks the onset of long-range antiferromagnetic
ordering. The value of critical exponent $\alpha$ obtained from fitting the
low-temperature heat capacity using the power-law is 2.5, different from
$\alpha=3$ for a 3D Heisenberg antiferromagnets or $\alpha=2$ for the
Heisenberg spins on a 2D lattices, suggesting a possible unconventional
quantum ground state for this system.

##  II Experimental Methods

Single crystals of YbI3 are grown using the vapor transport technique in a
two-zone furnace. The details of the crystal growth are given in sec. III.1.
The x-ray diffraction patterns are recorded using a Bruker D8 advance
diffractometer in the Bragg Brentano geometry employing Cu Kα
($\lambda=1.5406$ Å) radiation. The $2\Theta$ values ranged from 5∘ to 90∘
with a step size 0.02∘. For this purpose, a thin plate-like single crystal was
placed over a oven-dried glass-slide and completely covered by a layer of
Kapton tape to avoid degradation due to moisture.

Single crystal x-ray diffraction (SCXRD) was carried out using a Bruker Smart
Apex Duo diffractometer employing Mo-Kα radiation ($\lambda=0.70173$ Å) at 250
K and 150 K using an open-flow nitrogen cryostat. A tiny crystal of
approximately 1.630 $\mu$m lateral dimensions was covered in paratone oil to
avoid exposure to the atmosphere and mounted on a goniometer loop. The
intensities were integrated using the Crystalis Pro [34]. The refinement was
performed using Jana 2006 [35].

The layered morphology of the single crystals was confirmed using a scanning
electron microscope (SEM) (FEI Make), and the Chemical composition was
verified using an Energy Dispersive x-ray (EDX) (Quanta 2003D) detector
attached to the SEM instrument in the Secondary Electron (SE) mode. The EDX
spectra was collected at $15-20$ different points, spread across the platelet-
shaped single crystal specimen (see the Supplementary Material [36]), to
obtain the averaged elemental composition. The elemental mapping was also
performed over a micron size area of the single crystal to check the
compositional homogeneity of the grown crystal.

Raman spectra were collected at room temperature in the back scattering
configuration using a Horiba Jobin-Yvon LabRAM HR spectrometer equipped with
liquid nitrogen cooled Charge-Coupled Detector (CCD) and a laser of 532 nm as
the source of excitation. The excitation was maintained at 75% of the maximum
power, and the accumulation time for each spectrum was 10 s with 30 iterations
each time to improve the resolution and intensity of the Raman modes. For the
purpose of obtaining Raman spectrum, the YbI3 crystal was coated with a thin-
layer of Apiezon N grease to prevent degradation due the measurement.

Magnetic measurement between $2-300$ K are performed at the UGC-DAE CSR,
Indore using the VSM probe in a Physics Property Measurement System (PPMS),
Quantum design, USA. The heat capacity was measured $0.11$ K to $4$ K using a
dilution insert in the PPMS at UGC-DAE-CSR, Indore. Due to highly hygroscopic
nature of YbI3, the crystal was covered with an unknown amount of Apiezon N
grease, hence the addenda could not be measured accurately. However, in the
temperature range of our measurement, the magnetic specific heat of YbI3 is
$3-4$ orders of magnitude higher than that of the Apiezon N grease, and can
therefore be safely ignored while analyzing the measured data.

##  III Results and Discussion

###  III.1 Crystal growth of YbI3

Figure 1: (a) Schematic of vapor transport reaction for YbI3 in a sealed
quartz ampoule kept in a temperature gradient furnace. (b) Actual growth
ampoule (left), containing several small crystals, and a large cm-sized single
crystal (middle). A scanning electron microscope image of a small crystal
specimen used for chemical analysis (right). (c) x-ray diffraction was
conducted in Bragg Brentano geometry, displaying the (0 0 l) reflections which
confirms the ab - plane of the single crystal specimen shown in the inset.

High-quality single crystals of YbI3 are grown using the physical vapor
transport (self-transport) technique. Ytterbium powder (99.9% REO, Alfa Aesar)
and Iodine flakes (ACS reagent, $\geq$ 99.8%, Sigma Aldrich) are used in their
elemental form as starting precursors. The stoichiometric quantities of the
starting materials are loaded in a quartz ampoule, measuring approximately 15
cm in length, with an inner diameter of about 2 cm and wall-thickness of 2 mm,
resulting in a volume of around $40-50$ cm3. To compensate for iodine
sublimation losses during the evacuation process, an additional 10% excess of
iodine is included. The entire procedure of handling and storing the
precursors is carried in an argon-filled glove box with oxygen and moisture
content $<$ 0.1 ppm. Before loading the precursors into the ampule, thorough
cleaning steps are undertaken. The ampule undergoes sonication in acetone,
followed by rinsing with deionized water. Subsequently, it is preheated to
1000∘C to eliminate residual moisture and organic impurities sticking to its
inner walls. The filled ampule is evacuated down to $10^{-5}$ torr, and
thereafter flame-sealed under dynamic vacuum. Given iodine’s high vapor
pressure at room temperature (approximately 100 Pa at 309 K), the ampule was
kept under an ice bath during evacuation and sealing to minimize iodine
losses. The sealed ampoule is then kept in a two-zone furnace under a
temperature gradient $\Delta$T = 100∘C, with the source temperature set at
750∘C ($\rm T_{source}$) and the sink temperature($\rm T_{sink}$) at 650∘C.
The schematic of vapor transporter reaction is shown in Fig. 1(a). Optimal
growth conditions for producing the largest and highest quality single
crystals involve simultaneous cooling of both the zones at a rate of 0.3∘C h-1
down to $\rm T_{source}=600^{\circ}C$ and $\rm T_{sink}=400^{\circ}C$.
Subsequently, both zones of the furnace are cooled simultaneously to room
temperature.

The experiments performed using different growth conditions are summarized in
Table S2. (See Supplemental Material) [36]. Shiny, plate-like single crystals,
with typical lateral dimensions of 1 cm $\times$ 1 cm, and dark-
brown/transparent appearance, as shown in Fig. 1(b), are obtained. The plate-
like morphology of the grown crystals is a reflection of the underlying
layered or quasi-two-dimensional structure of YbI3. A scanning electron
microscope image of a small crystal specimen is shown in Fig. 1(b). The
chemical composition, obtained using EDX, confirms the 1:3 ratio of Yb:I (see
Fig. S1 and Table S2. of the Supplemental Material) [36].

The x-ray diffraction pattern collected from a plate-like single crystal
specimen is shown in Fig. 1(c). The specimen used is shown in the inset. The
pattern is collected in the Bragg Brentano geometry, revealed diffraction
peaks with miller indices $(00l)$, where $l=3,6,...$, suggesting that the
large lateral surface of the measured specimen is oriented perpendicular to
the c-axis.

The crystals are found to be extremely hygroscopic, decomposing into white
powder and eventually into a liquid droplet when exposed to the ambient
atmosphere. Therefore, due care is required during sample preparation and
while loading the sample for measurements in a cryostat as described in Sec.
II.

###  III.2 Crystal Structure

Figure 2: Crystal structure of YbI3. (a) Projection of the ab-plane showing
the arrangement of Yb ions on a honeycomb lattice. The hexagonal rings formed
by the Yb atoms, exemplified using green solid lines, is regular with each
side 4.226(1) Å. (b) Analogous image for YbCl3 is shown for comparison. In
this case the hexagonal ring is slightly distorted with Yb-Yb distances 3.886
Å and 3.864 Å [29]. (c) Stacking of the honeycomb layers of YbI3 perpendicular
to the c-axis. The nearest neighbor Yb-Yb distance along the c-axis is
6.803(4) Å. (d) Analogous image for YbCl3 is shown for comparison, where the
nearest neighbor Yb-Yb separation is 6.313 Å [29]. (e) and (f) comparison of
Yb-I-Yb bond angles along the three nearest neighbors in the ab-plane. These
angles are uniformly 91.76(3)∘ in YbI3, and 96.12∘ and 96.73∘ in YbCl3 [29].
(g) A regular octahedron of I atoms around the Yb atom. Each Yb-I bond
distance is close to 2.942(3) Å.

The symmetry of YbI3 was found to be trigonal $R\bar{3}$ (# 148) from single
crystal x-ray diffraction, which is in agreement with the L Asprey et al.[37].
Temperature-dependent single crystal x-ray diffraction measurements at 250 K
and 150 K show an absence of structural transition and that the symmetry
remains trigonal as shown in Fig. S2. (see Supplemental Material) [36], unlike
$\rm\alpha-RuCl_{3}$, which exhibits a change of symmetry from room-
temperature monoclinic ($C2/m$) to low-temperature trigonal ($R\bar{3}$)
symmetry below a temperature of 150 K [38] The symmetry of the lattice is
further substantiated by the reciprocal space cuts shown

Crystal structure of YbI3 in the ab-plane is shown in Fig. 2(a), illustrating
the honeycomb network of Yb atoms similar to $\rm\alpha$-RuCl3 [39] and YbBr3
[33]. Each Yb atom is coordinated to six neighboring I atoms in an almost
ideal edge-sharing octahedral arrangement. In the honeycomb layer, the Yb-Yb
bond distance is 4.226(1) Å, while the distance along the c-axis is 6.803(4) Å
as shown in Fig. 2(c), where the stacking of the honeycomb layers
perpendicular to the c-axis can be seen. The Yb-I-Yb bond angle in the
honeycomb layer is 91.76(3)∘ (see panel (e) in Fig. 2). The octaheron of
ligands I atoms around the central Yb atom is shown in Fig. 2(g).

The details of the data collection strategy, crystallographic information and
refinement parameters are summarized in Table 1. The atomic positions with
their corresponding isotropic displacement parameters (ADPs) are listed in
Table 2.

The crystal structure of YbI3 is different from YbCl3, where the later
crystallizes with a monoclinic $C2/m$ crystal structure [28, 29], analogous to
the room-temperature phase of $\alpha-$RuCl3. However, unlike $\alpha-$RuCl3,
which undergoes a structural phase transition to $R\bar{3}$ symmetry below 150
K, YbCl3 remains in the $C2/m$ symmetry over the whole temperature range. In
$C2/m$ symmetry, the hexagonal ring of the Yb atoms is not perfect with two
different Yb-Yb bond distances of 3.886 Å and 3.864 Å. The Yb-Cl-Yb bond
angles also take two different values, namely 96.12∘ and 96.73∘, thus lacking
the $C_{3}$ symmetry [29]. The effect of symmetry breaking has been discussed
in the recent study by Kim et. al [40]. The $C_{3}$ symmetry with 90∘ bond
angles and larger inter-layer separation along the c-axis are ideal criteria
for realizing the Kitaev model [41], which minimizes the Heisenberg
interaction, allowing Kitaev interactions to dominate, as shown in the case of
$\alpha-$ RuCl3 [42, 39]. As mentioned above, YbCl3 has a slightly distorted
honeycomb net with different bond angles and with more number of nearest
neighbors along the c-axis [29]. YbBr3, on the other hand, has a similar
structure as that of YbI3 and low-temperature phase of $\alpha-$ RuCl3 but
with small site-disorder on the Yb site [33]. To summarize, between the three
Yb based halides compared here, YbI3 appears to have the closest resemblance
to the low-temperature phase of $\alpha-$ RuCl3.

Table 1: Crystallographic data and refinement parameters for SCXRD of YbI3 single crystal at 250 K . Temperature (K) | 250
---|---
Crystal system | Trigonal
Lattice system/Setting | Hexagonal
Space group (Point group) |  $R\bar{3}$ ( $C_{3i}^{2}$)
No. | 148
$a$ (Å) | 7.3201(8)
$c$ (Å) | 20.4545(27)
Volume (Å3) | 949.2(3)
$Z$ | 6
Wavelength (Å) | 0.71073
Detector distance (mm) | 40
Scan type | Full sphere ($\phi$, $\omega$ scans)
Rotation per image (deg) | 0.8
Exposure time (sec) | 10
$(\sin(\Theta)/\lambda)_{max}$ (Å-1) | 0.743374
Absorption, $\mu$ (mm-1) | 29.261
Tmin, Tmax | 0.84, 1.08
Criterion of observability | $I>3\sigma(I)$
No. of reflections measured | 10072
Unique reflections (obs/all) | 561/728
$R_{int}$ (obs/all) | 0.1022/0.1054
No. of parameters | 14
$R_{F}$ (obs) | 0.0556
$wR_{F}$ (all) | 0.0720
GoF (obs/all) | 3.98/3.50
$\Delta\rho_{min}$, $\Delta\rho_{max}$(e Å-3) | -3.59, 3.77
Table 2: Atomic Coordinates x, y, z and atomic displacement parameters(ADPs) of YbI3 single crystal at 250 K in Å2 . Unique reflections (obs/all) = 561/728, criterion of observability: $I>3\sigma(I)$, $R_{F}=0.0556$, No. of refined paramateres = 14, Refinement method used: least-squares on F, Space group: $R\bar{3}$. The complete data including the anisotropic atomic displacement parameters, are shown in Table S3.(see Supplemental Material) [36]. Atoms | Wyck | x | y | z | Occ. | $U_{iso}^{eqi}$
---|---|---|---|---|---|---
Yb | 6c | 0 | 0 | 0.3337(4) | 1 | 0.0182(3)
I | 18f | 0.3229(1) | 0.3341(1) | 0.4165(1) | 1 | 0.0228(4)

###  III.3 Raman Spectroscopy

As discussed above, YbI3 has a Trigonal symmetry with the point group symmetry
$C_{3i}^{2}$. According to the group theory, the irreducible representation of
the zone-center Raman active phonon modes in YbI3 are
$\Gamma(C_{3i}^{2})=4A_{g}+4^{1}E_{g}+4^{2}E_{g}$. However, if the inter-layer
coupling is treated as a small correction, the single-layer of YbI3 has the
$D_{3d}$ point group symmetry, rendering six Raman active modes,
$\Gamma(D_{3d})=4E_{g}+2A_{1g}$ Fig. III.3 shows the observed Raman spectrum
for YbI3 single crystal. The whole spectrum could be nicely fitted using a
total of six Lorentzian lineshapes corresponding the six expected Raman active
phonon modes. The modes are labeled from $\rm P_{1}$ to $\rm P_{6}$ in the
increasing order of frequency as
$\rm\Delta\omega=38.4~{}(P_{1}),56.4~{}(P_{2}),69.4~{}(P_{3}),94.0~{}(P_{4}),118.5~{}(P_{5})$,
and $\rm 142.5~{}(P_{6})$ cm-1. The modes P1, P3, P4, and P6 have the $\rm
E_{g}$ symmetry, whereas symmetry of P2 and P5 is $\rm A_{1g}$. Additionally,
four weak modes at $\rm 61.6~{}(M_{1}),74.3~{}(M_{2}),114.1~{}(M_{3})$, and
$\rm 137.5~{}(M_{4})$ are also observed. The presence of weak extra modes in
the Raman spectrum are not unusual. These are often assigned as infrared
active modes rendered Raman active due to lowering of the local symmetry at
some sites [43]. In some cases, the extra modes are assigned to second-order
Raman scattering [44]. The observed spectrum (including some of the weak extra
modes) agrees well with that reported for the isostructural compound DyI3
[45]. The observed phonon modes for YbI3 and DyI3, and their corresponding
symmetries in the $D_{3d}$ representation, are listed in Table S4.(see
Supplemental Material) [36].

Figure 3: Raman spectrum of YbI3 single crystal. The six Raman active phonon
modes are labelled from $P_{1}$ to $P_{6}$.

###  III.4 Heat Capacity

Figure 4: The temperature variation of heat capacity ($\rm C_{p}$) and
magnetic entropy released ($\rm S_{mag}$): (a) $C_{p}/T$ is shown as a
function of temperature. Inset: solid red line is a fit to the low temperature
experimental data for $\alpha=2.5$. The fit using $\alpha=1,2$ is also shown
for reference (see text for details). (b) $\rm S_{mag}$ is shown as a function
of temperature. The purple shaded region indicates the error bars.

Fig. 4(a) shows the heat capacity $\rm C_{p}$ plotted as $\rm C_{p}/T$ as a
function of temperature from $\rm 0.1~{}K\leq T\leq 1.5~{}K$. The heat
capacity is an additive quantity, comprising various contributions, and can be
expressed as: $\rm C_{p}(T)=C_{mag}+C_{latt}+C_{nuc}$, where $\rm C_{mag}$ is
contribution to heat capacity arising from the 4f electrons of Yb3+, $\rm
C_{nuc}$ is the nuclear Zeeman contribution, and $\rm C_{latt}$ is the lattice
contribution. In the temperature range of measurement, the lattice
contribution due to phonons becomes extremely small and can therefore be
neglected. As shown in Fig. 4(a), the $\rm C_{p}/T$ exhibits three distinct
features at temperatures T1, T2, and T∗. The broad anomaly around 0.95 K,
labeled as $T_{1}$, is attributed to the short-range ordering of the Yb3+
moments, whereas the sharp $\lambda$-type anomaly at 0.6 K, labeled as
$T_{2}$, has been tentatively attributed to their long-range ordering. The
upturn below T∗ is due to nuclear Zeeman splitting caused by the internal
field generated by the ordering of Yb moments [46].

Here, it would be instructive to bring a comparison with much more extensively
studied YbCl3. In case of YbCl3, a similar broad peak attributed to short-
range ordering appears at 1.2 K, followed by weaker anomalies at 0.6 K and 0.4
K upon further cooling [28]. While the 0.6 K anomaly is linked to the long-
range ordering of the Yb moments, the the 0.4 K anomaly is sample-dependent
and has been attributed to the presence of defects/imperfections or
impurities. In contrast, in the YbI3, the peak corresponding to the long-range
ordering at T2 = 0.6 K is sharp and well-defined, while the 0.4 K anomaly
could not be detected in YbI3.

We fitted the heat capacity data of YbI3 below 0.3 K using the expression:
$\rm C_{p}=AT^{-2}+\gamma T^{\alpha}$, where the first term in the equation is
the high-temperature approximation of the two-level Schottky equation [47],
and the second term is written based on the assumption that $\rm C_{mag}$
follows a certain power-law behavior in the ordered state. The data fits well
as shown with a red solid curve in the inset of Fig. 4(a). The values obtained
for the parameters A, $\gamma$ and $\alpha$ from the fitting are: $\rm
A=3.67\times 10^{-4}~{}J~{}mole^{-1}$ K,
$\rm\gamma=2.2~{}J~{}mole^{-1}~{}K^{-3.5}$, and $\alpha=2.5$. For YbCl3, the
exponent $\alpha$ is $1.5$ [32].The fit using the $\alpha=1,2$ is shown in the
inset of Fig. 4(a) using blue and green solid curves, respectively. The
fitting clearly indicates that the value of exponent $\alpha$ is 2.5 in case
of YbI3. In the magnetically ordered state of traditional antiferromagnets,
the exponent $\alpha$ is expected to be 3, whereas in the case of layered
antiferromagnets the expected value of $\alpha$ should be 2. On the other
hand, $\alpha\approx~{}1$, indicating a linear temperature dependence of low-
temperature heat capacity conventionally holds for the spinons in gapless
quantum spin liquids; whereas a more unconventional quadratic temperature
dependence ($\alpha\approx 2$) is expected for a gapless Dirac fermions
[48].The value of $\alpha\approx 2.5$ in the present case of YbI3 is possibly
an indication of an unusual ground state. Further, experiments in the presence
of applied magnetic field, currently not accessible to us, will be
illuminating in understanding the ground state of YbI3 accurately.

Fig. 4(b) shows the magnetic entropy released as a function of temperature
calculated using $\rm S_{mag}=\int_{0}^{T_{0}}C_{mag}(T)/TdT$. We extracted
$\rm C_{mag}$ by subtracting the nuclear contribution from the total heat
capacity. As mentioned in the beginning (Sec. II), the addenda accounting for
the contribution from the Apiezon N grease used for thermal contact and for
covering the same to prevent it from decomposing in the process of loading for
low-temperature measurements has not been subtracted. However, at low-
temperatures, where $\rm C_{mag}$ is overwhelmingly large, we do not expect
the Apiezon N contribution to be of much significance. The temperature
variation of $\rm S_{mag}$ is shown in Fig. 4(b). Above 0.5 K, $\rm S_{mag}$
rises sharply, tapers a bit around 2 K, but do not show any signs of
saturation up to 4 K. At 4 K, the value approaches 82% of Rln2. Since the
measurements were only conducted up to 4 K, the exact temperature at which the
magnetic entropy plateaus at Rln2 is not known. However, previous reports on
YbCl3 [28] suggest that this plateau will possibly occur at a slightly higher
temperature ( close to 10 K). A value of $\rm S_{mag}~{}\approx$ Rln2, shows
that the crystal field split ground state of Yb3+ in YbI3 is a Kramer’s
doublet. To establish this further, we investigated the magnetic behavior of
our YbI3 crystals, disccussed next.

###  III.5 Magnetization

Figure 5: The magnetic susceptibility ($\chi$) and isothermal magnetization
M(H) of YbI3: (a) Temperature variation of $\rm\chi_{\parallel}$ and
$\rm\chi_{\perp}$ measured under an applied field magnetic field of 10 KOe.
Inset: $\rm\chi^{-1}(T)$ plotted as a function of temperature for $H\parallel
ab$. The solid line through the data points in the inset is the modified
Curie-Weiss fit (see text for details). (b) M(H) at 2 K for $H\perp ab$ and
$H\parallel ab$. The dashed lines represent linear extrapolation of the high-
field data to $\rm H=0$. Inset: derivative dM/dH plot.

Magnetic susceptibility ($\chi$) is measured by applying a magnetic field of
10 kOe, in the parallel ($\chi_{\parallel}$) (in-plane) and perpendicular
($\chi_{\perp}(T)$) (out-of-plane) orientations. The temperatures variation of
in-plane and out-of-plane susceptibilities are shown in Fig. 5(a). For the
out-of-plane orientation, the zero-field-cooled (ZFC) and field-cooled (FC)
curves overlap over the whole temperature range, but in the in-plane
susceptibility, a very weak splitting is observed below approximately
150$~{}K$ (not shown).

The high temperature $\rm 1/{\chi}$ data ($150\leq T\leq 300$) are fitted
using a modified Curie-Weiss (CW) expression:
$\rm{\chi}={\chi_{0}+C/(T-\Theta_{CW})}$, where C is the Curie-constant from
which the value of the effective magnetic moment ($\rm\mu_{eff}$) can be
derived using $\rm\mu_{eff}=\sqrt{8C}$; $\rm\Theta_{CW}$ is the Weiss
temperature, and $\rm\chi_{0}$ is a small temperature independent term that
usually arise due to core diamagnetism and/or van Vleck type paramagentism.

The values of the fitting parameters for the in-plane and out-of-plane
orientation are as follows: $\rm\chi_{0}^{\perp}=3.8\times
10^{-4}~{}emu~{}mole^{-1}~{}Oe^{-1}$, $\rm\Theta_{cw}^{\perp}=-55~{}K$, and
$\rm\mu_{eff}^{\perp}=4.53~{}\mu_{B}/Yb^{3+}$; the corresponding values in the
in-plane orientation are: $\rm\chi_{0}^{\parallel}=1.9\times
10^{-4}~{}emu~{}mole^{-1}~{}Oe^{-1}$, $\rm\Theta_{cw}^{\parallel}=-24~{}K$,
and $\rm\mu_{eff}^{\parallel}=4.53~{}\mu_{B}/Yb^{3+}$. In both cases, the
value of $\rm\mu_{eff}$ agrees well with the calculated value of
$\rm\mu_{eff}=g_{J}\sqrt{J(J+1)}=4.53\mu_{B}$ for Yb3+ ($4f^{13}$: $\rm
L=3,S=1/2$ , and $\rm J=7/2$). The large values of $\Theta_{CW}$ for both the
orientations are unrealistic, which may be due to the large crystal field
splitting of the J = 7/2 manifold. In the presence of an octahedral crystal
field environment of surrounding I- ions, the eight-fold degenerate, $\rm
J=7/2$ manifold in a free Yb3+ ion will split into four Kramers doublet. As a
reference, we note that in YbCl3, the four Kramer’s doublets are located at
energies, 0, 243, 371, and 456 K. Clearly, in the range over which the fitting
is performed, the van Vleck contribution from the higher-lying levels will be
small but not insignificant.

,  Orientation Temperature range (K) $\chi_{0}$ C $\rm\mu_{eff}$ g-factor
$\rm\Theta_{CW}$ Ref. $H\perp ab$ $150-300$ $3.82\times 10^{-4}$ $2.57$ $4.53$
$-$ $-55$ $4-12$ $0~{}^{a}$ $0.98$ $2.80$ $3.2$ $-10$ This work $4-12$
$5.5\times 10^{-3}~{}^{b}$ $0.82$ $2.56$ $3.0$ $-8$ c $3-15$ $0$ $1.12$
$3.0(1)$ $3.5(1)$ $-9$ [28] $H\parallel ab$ $150-300$ $1.95\times 10^{-4}$
$2.57$ $4.53$ $-$ $-24$ $4-12$ $\sim 0~{}^{a}$ $1.32$ $3.24$ $3.8$ $-7$ This
work $4-12$ $16\times 10^{-3}$ b $0.93$ $2.73$ $3.2$ $-5$ c $3-15$ $0$ $1.20$
$3.1(1)$ $3.6(1)$ $-6$ [28] a when $\chi_{0}$ is fixed as 0; b When $\chi_{0}$
is fixed to the value obtained from the slope of the linear-section of the
high field isothermal magnetization plots. c YbCl3

Table 3: Summary of fitting parameters $\chi_{0}$ ($\rm
emu~{}mole^{-1}~{}Oe^{-1}$), C ($\rm emu~{}mole^{-1}~{}Oe^{-1}~{}K$),
$\rm\mu_{eff}$ ($\rm\mu_{B}$), anisoptropic Lande g-factor and
$\rm\Theta_{cw}$ (K) obtained by fitting the magnetic susceptibility data for
in-plane ($H\parallel ab$) and out-of-plane ($H\perp ab$) orientations

In such scenarios where the crystal field splitting is large, it is more
instructive to fit the low-temperature data to estimate the magnetic moment in
the lowest Kramers doublet. We fitted the data in the temperature range from 4
K to 12 K using the modified Curie-Weiss expression. The values of the fitting
parameters thus obtained are listed in Table. 3. For both the orientations, if
we fix $\rm\chi_{0}=0$, we get $\rm\mu_{eff}^{\perp}~{}=~{}$2.8
$\rm\mu_{B}/Yb^{3+}$, $\rm\mu_{eff}^{\parallel}=3.24$ $\rm\mu_{B}/Yb^{3+}$;
and, $\rm\Theta_{CW}^{\perp}=-10$ K, $\rm\Theta_{CW}^{\parallel}=-7~{}K$. In
YbCl3, the corresponding values are [28]: $\rm\mu_{eff}^{\perp}~{}=~{}$3.0
$\rm\mu_{B}/Yb^{3+}$, $\rm\mu_{eff}^{\parallel}=3.1~{}\mu_{B}/Yb^{3+}$; and,
$\rm\Theta_{CW}^{\perp}=-9$ K, $\rm\Theta_{CW}^{\parallel}=-6$ K. The two sets
of values are in good agreement with each other, suggesting some degree of
similarity between the two systems. In both cases, the value of $\Theta_{CW}$
is somewhat higher than expected as we do not expect the exchange interaction
to be of the order of 10 K for the reason that the 4f-electrons in rare-earths
are highly sequestered, shielded by 5d and 6s electrons, which prevents them
from making an appreciable overlap with the ligand p orbitals. For example, in
the Yb-based pyrochlore oxides, the exchange interaction is of the order of 1
K. The other problem with these values is that the estimated Weiss temperature
is comparable from the two orientations, which appears improbable given the
layered nature of these materials, which suggests that the interaction within
the plane should be much stronger than out-of-plane. In YbCl3, the in-plane
nearest-neighbor coupling strength is $\rm\Lambda_{\parallel}~{}\approx~{}5$ K
and $\rm\Lambda_{\perp}/\Lambda_{\parallel}\approx 10^{-5}$ [31]. The exact
reason for the high Weiss temperature in the low-temperature Curie-Weiss fits
is not yet clear. Fixing $\chi_{0}$ to some high value
($\rm\sim~{}10^{-4}-~{}10^{-3}~{}emu~{}mole^{-1}~{}Oe^{-1}$) did not decrease
the Weiss temperature significantly either. We believe that the short-range
magnetic correlations, that persists up to very high temperatures, may have
contributed to the high value of $\rm\Theta_{CW}$. Strong spin fluctuations or
short-range correlations are intrinsic to the layered (or quasi-2D) magnets,
which is reflected in the broad peak in the heat capacity and a nearly
linearly increasing $\rm S_{mag}$ up to temperatures as high as $\sim$ 7TN
(here TN = T2 corresponds to the sharp anomaly in $\rm C_{p}$). The value of
anisotropic g-factor in YbI3 is estimated to be $\rm g_{J}^{\perp}=3.2$ and
$\rm g_{J}^{\parallel}=3.8$, assuming $\rm J_{eff}=1/2$. In comparison, from a
similar analysis the anisotropic g-factor in YbCl3 is $\rm g_{J}^{\perp}=3.5$
and $\rm g_{J}^{\parallel}=3.6$ [28]. Slightly smaller values for the
g-factors are obtained by taking $\chi_{0}$ values from the isothermal
magnetization, as discussed below.

Isothermal magnetization, M(H) for out-of-plane ($H\perp ab$) and in-plane
($H\parallel ab$) orientations is shown in Fig. 5 (b). In the out-of-plane
orientation, the magnetization nearly saturates above a field of $H\approx
120$ kOe. The small slope that still persists can be attributed to the van
Vleck term ($\chi_{0}$), which can be calculated by fitting the high-field
data to an expression of the $\rm M=M_{s}^{\perp}$ \+ $\chi_{0}^{\perp}H$,
where M${}_{s}^{\perp}$ is the saturation magnetization in the out-plane
orientation. The fit, which is shown in the Fig. 5(b) using a dashed line
(shown extrapolated to H = 0), gives $\rm M_{s}^{\perp}\approx
1~{}\mu_{B}/Yb^{3+}$ and $\chi_{0}^{\perp}=5.6\times 10^{-3}$ emu mol-1 Oe-1.
Using these values of $\chi_{0}$ for the orientations (instead of fixing them
to zero, as done in the previous paragraph), we get $\rm g_{J}^{\perp}=3.2$
and $\rm g_{J}^{\parallel}=3.0$ (see Table. 3). Combining the two results, we
can conclude that the approximate $\rm g_{J}$ values for the two orientations
are: $\rm g_{J}^{\perp}\approx 3.0\pm 0.2$ (out-of-plane), and $\rm
g_{J}^{\parallel}\approx 3.4\pm 0.3$ (in-plane).

For the in-plane M(H) a more intriguing field variation is observed, as
depicted using the derivative ($d$M/$dH$) plot in the inset of Fig. 5(b). With
increasing field, $d$M/$dH$ exhibits a plateau in the field ranging form 20
kOe to 40 kOe, suggesting a field induced transition. In the high-field region
$d$M/$dH$ becomes almost flat and very small. Analyzing the high-field region
as before yields: $\rm M_{s}^{\parallel}\approx 1.5~{}\mu_{B}/Yb^{3+}$ and
$\chi_{0}^{\parallel}=16\times 10^{-3}$ emu mol-1 Oe-1. Using, the g-factor
obtained for the two orientations from the susceptibility data, a back
calculation of the saturation moment suggests $\rm J_{eff}\leq 1/2$.

###  III.6 YbI3 versus YbCl3

To begin with, the crystal structures of YbI3 and YbCl3 are distinct from one
another. YbI3 exhibits trigonal symmetry, while YbCl3 displays monoclinic
symmetry [29]. This primary difference is due to the variation in the stacking
sequence of YbX3 layers along the c-axis. In YbI3, three layers form a
repeating unit, resulting in nearest neighbor Yb-Yb distance of 6.803 Å along
the c-axis, as shown in Fig. 2(c). In contrast, YbCl3 consists of a single
layer repeated along the c-axis, leading to nearest neighbors at 6.313 Å in
the layers above and below the middle layer, as shown in Fig. 2(d). Beyond the
global symmetry, the local symmetry within a single YbX3 layer in the ab-plane
also differs slightly for the two compound, which may be crucial for
understanding the magnetic behavior of these materials. In YbCl3, the Yb-Yb
honeycomb bond distances are 3.886 Å and 3.864 Å; accordingly, the bond angles
are 96.12∘ and 96.73∘, respectively [29]. Conversely, YbI3 features a uniform
bond distance of 4.226(1) Å and a bond angle of 91.76(3)∘, resulting in an
isotropic honeycomb lattice with equal bond distances and angles. Although the
distortion in YbCl3 is very small, it nevertheless causes the breaking of C3
symmetry, which is preserved in the case of YbI3.

Heat capacity measurements reveal two distinct transitions in both compounds.
In YbI3, short-range ordering occurs at 0.95 K, followed by long-range
ordering at 0.6 K. Similarly, in YbCl3, short-range ordering occurs at 1.2 K,
followed by long-range ordering at 0.6 K. The entropy release for both
compounds approaches Rln2, indicating a $\rm J_{eff}$ 1/2 Kramers doublet
ground state. Regarding differences, the transition at 0.6 K attributed to the
long-range ordering in YbI3 is rather strong, while it is significantly weaker
in YbCl3. On the other hand, the broad hump due to short-range ordering is
much more pronounced in YbCl3 compared to YbI3. If this broad hump is due to
2D spin correlations, as has been argued in the literature [31], we expect it
to be more pronounced in YbI3, where the interlayer separation is larger. This
aspects should be looked into more carefully in the future studies. Another
notable difference in the heat capacity of two compounds is that in YbCl3, a
weak sample dependent anomaly is reported at 0.4 K, which is absent in YbI3.

Magnetic behavior of the two compounds above 2 K is qualitatively similar.
Further low temperature measurements on YbI3 might prove useful in this
context. The saturation moments $\rm M_{s}^{\parallel}\approx
1.5~{}\mu_{B}/Yb^{3+}$ and $\rm M_{s}^{\perp}\approx 1~{}\mu_{B}/Yb^{3+}$ is
nearly the same for both the compounds. The differences may crop-up at lower
temperatures, hence, magnetic measurements below 2 K would be interesting.

In conclusion, YbI3 possesses a more symmetric structure compared to YbCl3,
with C3 symmetry preserved, similar to $\alpha$-RuCl3, whereas YbCl3 exhibits
broken C3 symmetry. The nearest neighbor Yb$-$Yb separation along the c-axis
is found to be larger in YbI3 compared to YbCl3. Heat capacity measurements
show a similar two-step ordering for both compounds: initial short-range
ordering followed by long-range antiferromagnetic ordering. Any further
additional anomaly due to poor quality of the sample is absent in YbI3. In the
paramagnetic state (T $>$ 2 K), the magnetic behavior of YbI3 resembles that
of YbCl3.

##  IV Summary & Conclusions

To conclude, we have successfully grown high-quality single crystals of YbI3
using a vapor transport reaction, taking elemental Yb and I as the starting
precursors. The crystal field split ground state of Yb3+ in YbI3 is a well-
isolated Kramer’s doublet with $\rm g_{J}^{\perp}\approx 3.0\pm 0.2$ (out-of-
plane), and $\rm g_{J}^{\parallel}\approx 3.4\pm 0.3$ (in-plane). The in-plane
isothermal magnetization shows weak a field induced transition, the indicating
presence of slight anisotropy even within the ab-plane.

Heat capacity show a two-step magnetic transition with a broad peak T1 = 0.95
K due to short-range correlations between the Yb moments, followed by a sharp
peak at 0.6 K, signaling transition into a long-range magnetically ordered
state. No additional anomaly, except an uptrun below about 0.25 K due to
nuclear Schottky effect, could be seen.The magnetic entropy, reaches almost
80% of Rln2 near 4 K ( highest temperature in our experiment), consistent with
the Kramer’s doublet ground state. The value of critical exponent $\alpha$ of
2.5 from the heat capacity data suggests a possible unconventional ground
state for YbI3. To conclude, YbI3 appears to harbor very interesting physics
associated with $\rm J_{eff}$ 1/2 on an ideal 2D honeycomb lattice. We believe
that our preliminary study will trigger further experimental investigation to
explore the spin Hamiltonian and exact ground state of this compound.

## Acknowledgments

The authors acknoweldge UGC-DAE-CRS user facility for magnetization and low-
temperature specific heat measurements. S.R. acknowledges I-HUB, National
Mission on Interdisciplinary CyberPhysical Systems (NM-ICPS) of the Department
of Science and Technology, Government of India, for the postdoctoral
fellowship. L.H. and S.S. thank I-HUB (NM-ICPS) for partial financial support.
S.S. thanks financial assistance received under the Scheme for
Transformational and Advanced Research in Sciences (STARS) by the Ministry of
Education, Government of India (sanction order no. STARS/APR2019/PS/358/FS).
N.P. thanks Council for Scientific and Industrial Research (CSIR) for Ph.D.
fellowship. P.T. thanks DST INSPIRE for the Ph.D. fellowship.

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