This is a pre-print.
Aside from formatting differences, it is identical to the version published
in Springer Lecture Notes for Computer Science Vol. 13886
11institutetext:  Lund University LTH 22institutetext: Stanford University
33institutetext: German Aerospace Center DLR 44institutetext: Technical
University of Munich

#  Out-of-Distribution Detection for Adaptive Computer Vision

Simon Kristoffersson Lind –<EMAIL_ADDRESS>11
Rudolph Triebel –<EMAIL_ADDRESS>3344
Luigi Nardi –<EMAIL_ADDRESS>1122
Volker Krueger –<EMAIL_ADDRESS>11

###### Abstract

It is well known that computer vision can be unreliable when faced with
previously unseen imaging conditions. This paper proposes a method to adapt
camera parameters according to a normalizing flow-based out-of-distibution
detector. A small-scale study is conducted which shows that adapting camera
parameters according to this out-of-distibution detector leads to an average
increase of 3 to 4 percentage points in mAP, mAR and F1 performance metrics of
a YOLOv4 object detector. As a secondary result, this paper also shows that it
is possible to train a normalizing flow model for out-of-distribution
detection on the COCO dataset, which is larger and more diverse than most
benchmarks for out-of-distibution detectors.

###### Keywords:

Autonomous Systems Out-of-Distribution Detection Normalizing Flows Object
Detection.

#### Corresponding Author:

Simon Kristoffersson Lind

##  1 Introduction

In the past decade computer vision has become a de facto component in
autonomous systems. However, it is well known that vision can be unreliable
when faced with new, previously unseen situations. Uncertainty is especially
abundant in the field of robotics, where the autonomous agents are expected to
perform tasks in real-world scenarios that often differ significantly from any
training data.

There is an abundance of research to suggest that Out-of-Distribution (OOD)
detection can help improve the reliability of vision algorithms [6]. Relevant
literature reasons that OOD detection allows systems to take action when
uncertainty is encountered [6]. However, there is little research to show that
OOD detection helps the reliability of autonomous systems in practice. Most
literature on OOD detection tends to focus either on synthetic benchmarks, or
controlled real-world scenarios where in- and out-of-distribution examples are
very distinct. Neither of these scenarios are reflective of the diverse
situations that may be encountered by autonomous systems in practice [6].

OOD detectors are commonly used by setting a threshold OOD score [9], and
simply discarding any input that falls beyond this threshold. While this makes
intuitive sense, on the basis that vision systems tend to be unreliable when
faced with OOD data, it only utilises one end of the OOD scores. We
hypothesize that a machine learning model will on average perform more
reliably when an input has a lower OOD score. Therefore we propose to use an
OOD detector as a quality metric, or un-normalized confidence; the lower the
OOD score, the more certain we can be about our vision system’s output.

###  1.1 Our Contributions

  * •

We expand the current knowledge on OOD detection by showing that it is
possible to train normalizing flow models for OOD detection on large, diverse
datasets while getting sensible results.

  * •

We introduce a novel method for utilising OOD detection as a quality metric in
autonomous systems and show that it can lead to reliability improvements for
vision tasks.

##  2 Related Work

There has been some previous work investigating the uses of OOD detection in
autonomous systems. For example Wellhausen et al. [17] perform anomaly
detection on image data collected from a robot in real-world terrain. However,
their main purprose is the evaluation of different OOD detection algorithms,
and not an application of OOD detection to real-world operation.

Yuhas et al. [20] evaluate OOD detection as an emergency breaking system for
an autonomous car, though not in real-world operation, but on a custom test
track.

McAllister et al. [12] perform real-world crash avoidance experiments with an
autonomous car. However, their method does not perform OOD detection directly.
Instead they use a variational autoencoder to generate in-distribution samples
from an out-of-distribution image, which they use as a measure of uncertainty.

Additionally, there is an abundance of work on OOD detection where the authors
have taken care to ensure that in- and out-distributions are disjoint, for
example [4, 7, 10, 18, 15, 13]. These works either use entirely different
datasets as in- and out-distributions, or separate classes as in- and out-
distributions. It is our belief that neither of these scenarios are reflective
of real-world operation.

##  3 Background

###  3.1 Out-of-Distribution Detection

Conceptually, out-of-distibution detection is simple. First an in-distribution
is defined, for example the training data for a machine learning-based vision
system. Then, everything that deviates from said in-distribution is said to be
out-of-distribution [19].

More formally, we assume that our vision system operates on data that is
sampled I.I.D from some distribution $P_{train}(\mathbf{x})$. In practice,
however, the system may encounter data that comes from a different
distribution, in which case the system will produce unreliable results.
Therefore, it is desirable to detect when data lies outside of the original
distribution $P_{train}(\mathbf{x})$.

###  3.2 Normalizing Flows

This section aims to provide a basic introduction to normalizing flows. For
more details we refer to Papamakarios et al. [14].

Normalizing flows have emerged in the last decade as a way to model complex
probability distributions [14]. Let us assume that we want to evaluate or
sample from a distribution $p_{x}(\mathbf{x})$, $\mathbf{x}\in\mathbb{R}^{D}$,
and that $p_{x}$ is intractable to evaluate or sample from. A normalizing flow
can allow us to evaluate or sample from $p_{x}$ by transforming
$p_{x}(\mathbf{x})$ into a tractable distribution $p_{u}(\mathbf{u})$,
$\mathbf{u}\in\mathbb{R}^{D}$.

In order to model $p_{x}(\mathbf{x})$, an invertible transformation $T$ is
constructed such that

$\mathbf{x}=T(\mathbf{u}),\ \ \mathbf{u}=T^{-1}(\mathbf{x})$

where $\mathbf{u}\sim p_{u}(\mathbf{u})$. Here, we are free to define
$p_{u}(\mathbf{u})$, for example
$\mathbf{u}\sim\mathcal{N}(\mathbf{0},\mathbf{1})$.

By additionally requiring both $T$ and $T^{-1}$ to be differentiable, it is
possible to recover $p_{x}(\mathbf{x})$ from $p_{u}(\mathbf{u})$ [16, 3, 14]:

$p_{x}(\mathbf{x})=p_{u}(\mathbf{u})|\det
J_{T}(\mathbf{u})|^{-1}=p_{u}(T^{-1}(\mathbf{x}))|\det
J_{T^{-}1}(\mathbf{x})|$ (1)

where $J_{T}$ and $J_{T^{-1}}$ are the Jacobians of $T$ and $T^{-1}$
respectively.

This type of transformation is called a diffeomorphism. It has been shown that
it is always possible to construct a diffeomorphism $T$ under reasonable
assumptions about $p_{x}(\mathbf{x})$ and $p_{u}(\mathbf{u})$ [14]. However,
analytically constructing $T$ is often infeasible in practice. Therefore, it
is desirable to learn an approximation of $T$. Henceforth we will denote
$T(\mathbf{x};\mathbf{\Theta})$ as an approximation of $T$ with learnable
parameters $\mathbf{\Theta}$.

An important property is that the composition of two diffeomorphisms
$T_{2}\circ T_{1}$ is also a diffeomorphism [14]:

$(T_{2}\circ T_{1})^{-1}=T_{1}^{-1}\circ T_{2}^{-1}$ $\det J_{T_{2}\circ
T_{1}}(\mathbf{u})=\det J_{T_{2}}(T_{1}(\mathbf{u}))\cdot\det
J_{T_{1}}(\mathbf{u})$

This property opens up for many ways to construct powerful approximations
$T(\mathbf{x};\mathbf{\Theta})$. In this work, we focus on a particular model
of so-called affine coupling layers [14].

Affine coupling layers first split the input $\mathbf{x}$ into two parts [14]:

$\mathbf{x}=[x_{i}\ |\ i\in\mathcal{I}],\ \ \mathcal{I}=1,2,\ldots,D$
$\mathbf{x}_{1}=[x_{i}\ |\ i\in\mathcal{A}],\ \ \mathbf{x}_{2}=[x_{i}\ |\
i\in\mathcal{B}],\ \ \mathcal{A}\cup\mathcal{B}=\mathcal{I},\ \
\mathcal{A}\cap\mathcal{B}=\emptyset.$

While $\mathcal{A}$ and $\mathcal{B}$ can be chosen arbitrarily, the most
common choice is to simply split $\mathbf{x}$ into two halves:

$\mathcal{A}=1,2,\ldots,\frac{D}{2},\ \
\mathcal{B}=\frac{D}{2}+1,\frac{D}{2}+2,\ldots,D.$

Then, $\mathbf{x}_{1}$ is transformed as a function of $\mathbf{x}_{2}$, while
$\mathbf{x}_{2}$ is left as-is:

$\displaystyle\mathbf{x}_{1}^{\prime}=[x_{i}^{\prime}|i\in\mathcal{A}]=\boldsymbol{\alpha}\cdot\mathbf{x}_{1}+\boldsymbol{\beta},$
$\displaystyle\text{where}\ \
\boldsymbol{\alpha}=\exp(F_{1}(\mathbf{x}_{2};\mathbf{\Theta}_{1})),\ \
\boldsymbol{\beta}=F_{2}(\mathbf{x}_{2},\mathbf{\Theta}_{2}).$

Here, $\cdot$ denotes element-wise multiplication, and $F_{1},F_{2}$ are
arbitrary functions, for example neural networks. The output of the coupling
layer is then simply the reconstruction of $\mathbf{x}$ from
$\mathbf{x}_{1}^{\prime}$ and $\mathbf{x}_{2}$:

$\displaystyle
T_{coupling}(\mathbf{x};\mathbf{\Theta}_{1},\mathbf{\Theta}_{2})=\mathbf{x}^{\prime}=[x_{i}^{\prime}|i\in\mathcal{I}],\
\ \text{where}\ \ x_{i}^{\prime}=x_{i}\forall i\in\mathcal{B}$

It is trivial to see that this construction results in an invertible and
differential transformation, since $\mathbf{x}_{1}$ is subject to an affine
transformation and $\mathbf{x}_{2}$ remains the same. Coupling layers also
offer the benefit of a simple Jacobian [14]:

$J_{T_{coupling}}=\mathbf{1}F_{1}(\mathbf{x}_{2};\mathbf{\Theta}_{1})$

where $\mathbf{1}$ denotes a row vector of ones – in other words, summing the
elements of $F_{1}(\mathbf{x}_{2};\mathbf{\Theta}_{1})$.

A common way to construct powerful learnable transformations is to simply
compose several of these coupling layers, while changing which parts of
$\mathbf{x}$ are modified [14]. This approach is used in for example RealNVP
[5], and it is what we use in our experiments.

###  3.3 Normalizing Flows for Image Out-of-Distribution Detection

Formulating OOD detection using normalizing flows is simple since we can
directly compute the likelihood using (1), which represents the belief that an
example $\mathbf{x}$ is in-distribution [9]. Note that a high value for
$p_{x}(\mathbf{x})$ equals a low likelihood of being OOD. It is otherwise
common for OOD detection systems to use the opposite formulation, where a high
value equals high likelihood of being OOD.

Training a normalizing flow for OOD detection is done by minimizing the KL-
divergence between $p_{x}(\mathbf{x})$ and $p_{u}(\mathbf{u})$, which results
in the following loss function [14]:

$\mathcal{L}=-\frac{1}{N}\sum_{n=1}^{N}\log
p_{u}(T^{-1}(\mathbf{x}_{n}))+\log|\det J_{T^{-1}}(\mathbf{x}_{n})|+C$

where $C$ is a constant.

Since we only need the inverse transformation $T^{-1}$ for OOD detection, we
simply let $T^{-1}$ denote the forward pass of our affine coupling layers.

Despite seeming like perfect candidates for OOD detecion, since equation (1)
allows for the direct computation of the approximate probability distribution,
normalizing flows often fail in the OOD detection setting [9]. Kirichenko et
al. [9] suggest that common image flow models, such as RealNVP[5], simply
learn pixel correspondences, and as such fail to grasp any semantic
information that makes up the distribution. Instead of training a flow model
on images directly, Kirichenko et al. suggest training a flow model on
features extracted from another pretrained image network.

##  4 Our Method

Based on our hypothesis that a lower OOD score would lead to better
reliability, we propose to let a vision system adapt, instead of simply
discarding OOD data. In order to bring this into practice, we create a
framework for adaption around an existing vision task, namely object detection
with a YOLOv4 [2] network.

Most commercially available cameras expose a number of parameters that can
alter the visual appearance of a captured image. Common examples are:
saturation, contrast, and exposure. These camera parameters are a natural
candidate for adapting to various imaging conditions.

Our proposed method is to simply adapt camera parameters in order to minimize
the OOD score. Intuitively, our method can be deployed in two ways:

  * •

either: if an image is marked as OOD, adapt camera parameters to minimize OOD
score,

  * •

or: continuously adapt camera parameters to minimize OOD score.

While most cameras have some built-in measures to adapt, for example auto-
exposure, these are often not sufficient. Figure 1 shows two images of the
same scene to illustrate this. In one picture, the camera’s default settings
fail to produce an image suitable for object detection. With hand-tuned
parameters however, a more suitable image is acquired. Note also the yellow
bounding box indicates that YOLOv4 can detect the dog in the hand-tuned image,
which is not possible with the camera’s default settings.

Figure 1:  Left: camera’s default parameters, auto-exposure and auto-white-
balance enabled. Right: hand-tuned camera parameters.

##  5 Experiments and Results

In this section we outline our two main experiments. Section 5.1 outlines the
normalizing flow model used in our experiments. For our first experiment, in
section 5.3, we train and validate a normalizing flow model as an OOD detector
using the COCO dataset [11] as our in-distribution. Then, in section 5.4, we
take the model from section 5.3 and apply it to a real-world experiment
testing our proposed method from section 4.

###  5.1 Normalizing Flow Model

For our experiments we construct a normalizing flow model from affine coupling
layers. We use a model very similar to the one used by Kirichenko et al. [9].
Our model consists of a series of 10 affine coupling layers, alternating
between transforming the first and second half of the input vector. Each
coupling layer consists of two shared linear layers with 512 hidden units
followed by two parallel linear layers, one for $\boldsymbol{\alpha}$ and one
for $\boldsymbol{\beta}$. This is illustrated in Figure 2.

Linear | tanhLinearxLinear | ReLULinear | ReLU$\boldsymbol{\alpha}$$\boldsymbol{\beta}$ Figure 2: Illustration of coupling layer layout.

We train our model on features extracted from a YOLOv4 [2] object detection
network. Training is done for 200 epochs with a batch size of 128, using the
Adam [8] optimizer with a learning rate of $10^{-4}$.

###  5.2 Intel RealSense D435

Our experiments use an Intel RealSense D435[1], which is a RGB+Depth camera.
In our experiments, however, we only utilise the RGB sensor. Like most
commercially available cameras, the D435 exposes a number of parameters that
affect the resulting image in various ways. For our experiments we manipulate
a selection of these parameters, specifically: backlight compensation,
brightness, contrast, exposure, gain, saturation, and sharpness.

###  5.3 Training Normalizing Flows on a Large Diverse Dataset

In this section, we train a normalizing flow model using the entire COCO
training dataset as our in-distribution. We then test the model by comparing
the log-likelihood outputs from different input image distributions.
Specifically, we divide this section into two smaller experiments:

  * •

One where we compare log-likelihoods of COCO images to randomly generated
images.

  * •

One where we capture many images of a static scene with randomized camera
parameters, and compare their log-likelihood to COCO images.

####  5.3.1 For our first experiment

we run the trained model on all images in the COCO training, validation and
testing datasets, and record the resulting log-likelihood scores. In order to
show that the model can distinguish images that are definitely not part of the
in-distribution, we also record log-likelihood scores from randomly generated
images. These random images were generated according to:

$\displaystyle\sigma$ $\displaystyle\sim\operatorname{Uniform}(1,256)$
$\displaystyle v_{i,j}$ $\displaystyle\sim\mathcal{N}(0,\sigma)$
$\displaystyle pixel_{i,j}$ $\displaystyle=\min(255,|v_{i,j}|),\ \ \forall\
i,j.$

Our primary motivation behind using this formula is to generate images with
varying pixel ranges, while also generating some mostly-black and mostly-white
images. By varying $\sigma$ per image, we vary the pixel range in different
images. When $\sigma$ is small, we get mostly-black images, and when $\sigma$
is large we get mostly-white images due to the $\min$ operation. Figure 3
shows histograms of randomly generated images, along with the COCO training,
validation, and testing datasets.

Figure 3:  Normalized histogram plot of log-likelihoods from COCO training,
validation and test datasets, as well as random images. The histogram has been
cut at -1000 log-likelihood for improved readability.

####  5.3.2 For our second experiment

we explore whether our normalizing flow can distinguish between different real
images. We place a stationary D435 camera in our lab, and capture a large
number of images. For each image, all camera parameters are randomized
according to a uniform distribution. In order to be certain that all variance
in log-likelihood scores are caused by changing camera parameters, we take
precautions to minimize the amount of natural light entering our lab. Then, we
capture 10000 images with the ceiling lamp on, and 10000 with the ceiling lamp
off. The number 10000 was chosen arbitrarily. Figure 4 shows the corresponding
histograms of log-likelihood scores, along with the COCO training data for
comparison.

Figure 4:  Normalized histogram plot of log-likelihood values. Blue: COCO
training images. Orange: 10000 images in our lab with randomized camera
parameters, without any lights on. Green: 10000 images in our lab with
randomized camera parameters, with ceiling lights on.

###  5.4 Parameter Optimization

Here, we present results from a small-scale experiment conducted in one of our
robot labs. In order to make the experiment as realistic as possible, the
experiment was set up using one of our robots. Our robot has an industrial arm
with an Intel Realsense D435 camera attached at the gripper.

For this experiment, we set up an optimization procedure using camera
parameters as our input, and the log-likelihood from our normalizing flow
model as output, as illustrated in Figure 5. More formally, let $\theta$
denote the camera parameters, and $C(\theta)$ denote the camera (a function
that takes camera parameters and produces an image $\mathbf{x}$), then we
solve:

$\operatorname*{arg\,max}_{\theta}\ \log p_{x}(C(\theta))$
$=\operatorname*{arg\,max}_{\theta}\ \log p_{u}(T^{-1}(C(\theta)))+\log|\det
J_{T^{-1}}(C(\theta))|.$

backlight compensationbrightnesscontrastexposuregainsaturationssharpnessD435
CameraYOLOv4Normalizing flow$\log p_{x}(\mathbf{x})$ImageFeatures Figure 5:
Parameter optimization setup.

For the optimization procedure, we used a very simple evolutionary
optimization with elitism selection, a population of 50, and a mutation rate
of 20%. No crossover was used. Optimization was carried out for a total of 200
evaluations. Population size, mutation rate, and number of evaluations were
chosen based on prior experience with evolutionary optimization. A small
number of trial runs confirmed that these values work well.

In our lab, we set up 13 different scenarios with varying COCO objects,
viewing angles, and lighting conditions. We took care to make the scenarios as
challenging as possible by introducing difficult lighting conditions,
reflective surfaces, and background clutter. With the robot stationary,
optimization was performed as described above. During optimization, the images
with the highest and lowest log-likelihood scores were saved along with their
corresponding parameters, and compared to images captured with the camera’s
default parameter settings. Henceforth, these will be referred to as best,
worst and default parameters, respectively.

After the optimization procedure, all images were annotated with bounding
boxes for the COCO objects present. They were then fed through YOLOv4. Mean-
Average-Precision (mAP), mean-Average-Recall (mAR), and F1 scores were
computed and compared between the worst, default, and best camera parameters.

Table 1 shows average mAP, mAR, and F1 scores for best, default and worst
camera parameters. In table 2 we show statistics for the improvement in mAP,
mAR, and F1 scores of the best parameters compared to the default parameters.
Similarly, Table 3 shows statistics for the improvement of the best compared
to the worst parameters.

Table 1: Average mAP, mAR, and F1 score for best, default, and worst parameters. | mAP | mAR | F1
---|---|---|---
Best | 0.7977 | 0.3283 | 0.4605
Default | 0.7591 | 0.2977 | 0.4219
Worst | 0.1731 | 0.0431 | 0.0687
Table 2: mAP, mAR, and F1 improvement. Best compared to default parameters. | Min | Max | Average
---|---|---|---
mAP | -0.2889 | +0.5778 | +0.0386
mAR | -0.1083 | +0.25 | +0.0306
F1 | -0.1056 | +0.349 | +0.0386
Table 3: mAP, mAR, and F1 improvement. Best compared to worst parameters. | Min | Max | Average
---|---|---|---
mAP | -0.104 | +0.95 | +0.6247
mAR | +0.1375 | +0.4278 | +0.2852
F1 | +0.1794 | +0.5703 | +0.3918

Figure 6 shows box plots of the improvements in mAP, mAR, and F1 scores for
the best parameters compared with the default and worst parameters
respectively.

Figure 6:  Boxplot of mAP, mAR, and F1 improvement. Left: Best compared to
default parameters. Right: Best compared to worst parameters.

In Figures 7, 8, and 9 we show example scenarios where optimization led to
varying degrees of success. Each figure includes images along with the
respective F1 scores from YOLOv4. Figure 7 shows examples where parameter
optimization resulted in better object detection than the default parameters.
Figure 8 shows examples where optimiztion resulted in no significant
difference in object detection. Finally, Figure 9 shows the only scenario
where optimization resulted in significantly worse object detection.

Worst | Default | Best
---|---|---
|  |
F1: 0.0 | F1: 0.4464 | F1: 0.5595
|  |
F1: 0.0 | F1: 0.0 | F1: 0.3490
Figure 7:  Examples where parameter optimization resulted in an improvement in object detection. Left: lowest log-likelihood. Center: default parameters. Right: highest log-likelihood.  Worst | Default | Best
---|---|---
|  |
F1: 0.3864 | F1: 0.6127 | F1: 0.6289
|  |
F1: 0.0 | F1: 0.4793 | F1: 0.4865
Figure 8:  Examples where parameter optimization resulted in no significant difference in object detection. Left: lowest log-likelihood. Center: default parameters. Right: highest log-likelihood.  Worst | Default | Best
---|---|---
|  |
F1: 0.0 | F1: 0.6172 | F1: 0.5115
Figure 9:  Example where parameter optimization resulted in worse performance
in object detection. Left: lowest log-likelihood. Center: default parameters.
Right: highest log-likelihood.

##  6 Discussion

###  6.1 Training on the COCO Dataset

When training on the full COCO dataset we see two important indicators that
the model is learning properly. First, as seen in Figure 3, the output log-
likelihoods span a wide range of values. It would be a red flag if all log-
likelihoods were too similar, indicating that the learned probability
distribution might be flat. Second, we observe that the validation and test
datasets are distributed the same way as the training set, while the random
images are given distinctly lower log-likelihood scores.

In figure 4, we see that the model is able to distinguish between real images
from different distributions. As such, we have found no evidence to suggest
that the size and diversity of the COCO dataset should pose any problems for
OOD detection. This in turn suggests that normalizing flow models can be
readily applied to diverse real-world scenarios.

Additionally, figure 4 shows that varying parameters in the D435 over the same
scene can result in a wide range of log-likelihood values. This gives merit to
the idea that camera parameters form a good basis for adapting to various
imaging conditions.

###  6.2 Parameter Optimization Experiment

Table 1 tells us at a glance that our optimized camera parameters acheived the
best average scores across the board. Looking at Table 2, we see an average
improvement of 3 to 4 percentage points in terms of all metric scores. From
the box plot in Figure 6, we see that most scenarios yield only a small
difference in object detection compared to the default parameters. However,
most importantly we see that the trend is a positive improvement. We also
observe that the positive outliers are significantly larger than the negative
outliers, again suggesting a positive trend.

Perhaps unsurprisingly, Table 3 shows a massive improvement (28 to 62
percentage points on average) between the best and worst parameters found. The
box plot in Figure 6 further highlights this improvement.

In Figures 7, 8, and 9 we note that some of the parameter-optimized pictures
look better to the human eye, while others look worse despite resulting in
better object detection. This highlights the important fact that there is no
equivalence between images that look good to the human eye, and images that
are good for object detection.

Perhaps the most important result of this experiment is not the fact that
parameter-optimization can lead to better object detection, but instead that
it seems feasible to adapt and recover from a very poor visual scenario.
Extrapolating from the fact that different camera parameters could result in
images where YOLO performs well, and where it fails almost completely on the
same scenario, it seems feasible that these parameters have the necessary
capacity to adapt to a wide range of difficult visual conditions. This is
given further merit by the fact that the varying camera parameters resulted in
widely varying log-likelihood values in figure 4. As such, when the
normalizing flow model detects something with extremely low log-likelihood
value, it is most likely possible to adapt camera parameters to get an image
more suitable for object detection. While these experiments have only been
concerned with object detection, we expect that this behaviour generalises to
other vision tasks as well.

##  7 Conclusion

In this paper we have shown that it is possible to train a normalizing flow
model for OOD detection on a large and diverse dataset such as COCO. We have
also conducted a real-world experiment which shows that a normalizing flow OOD
detector can be used as a quality metric for a vision task. Optimizing camera
parameters with respect to the output from our normalizing flow OOD model
yielded, on average, a 3 to 4 percent unit improvement in the reliability of
YOLOv4 in terms of object detection performance.

##  8 Future Work

This paper lays the groundwork for a larger framework of quality metrics for
vision task. In order to further expand this framework, a larger benchmark
dataset is beneficial. It is also an interesting research direction to explore
whether image improvements can be learned based on this OOD metric, rather
than going through the somewhat slow optimization process used in our
experiments.

####  8.0.1 Acknowledgements

This research is funded by the Excellence Center at Linköping-Lund in
Information Technology (ELLIIT), and the Wallenberg AI, Autonomous Systems and
Software Program (WASP). Computations for this publication were enabled by the
supercomputing resource Berzelius provided by the National Supercomputer
Centre at Linköping University and the Knut and Alice Wallenberg foundation.
Additionally, Luigi Nardi was supported in part by affiliate members and other
supporters of the Stanford DAWN project – Ant Financial, Facebook, Google,
Intel, Microsoft, NEC, SAP, Teradata, and VMware.

## References

  * [1] Intel realsense depth camera d435, https://www.intelrealsense.com/depth-camera-d435/
  * [2] Bochkovskiy, A., Wang, C.Y., Liao, H.Y.M.: Yolov4: Optimal speed and accuracy of object detection (2020). https://doi.org/10.48550/ARXIV.2004.10934, https://arxiv.org/abs/2004.10934
  * [3] Bogachev, V.I., Ruas, M.A.S.: Measure theory, vol. 1. Springer (2007)
  * [4] Deecke, L., Vandermeulen, R., Ruff, L., Mandt, S., Kloft, M.: Image anomaly detection with generative adversarial networks. In: Berlingerio, M., Bonchi, F., Gärtner, T., Hurley, N., Ifrim, G. (eds.) Machine Learning and Knowledge Discovery in Databases. pp. 3–17. Springer International Publishing, Cham (2019)
  * [5] Dinh, L., Sohl-Dickstein, J., Bengio, S.: Density estimation using real nvp. (2016), https://arxiv.org/abs/1605.08803
  * [6] Gawlikowski, J., Tassi, C.R.N., Ali, M., Lee, J., Humt, M., Feng, J., Kruspe, A., Triebel, R., Jung, P., Roscher, R., Shahzad, M., Yang, W., Bamler, R., Zhu, X.X.: A survey of uncertainty in deep neural networks (2021). https://doi.org/10.48550/ARXIV.2107.03342, https://arxiv.org/abs/2107.03342
  * [7] Hsu, Y.C., Shen, Y., Jin, H., Kira, Z.: Generalized odin: Detecting out-of-distribution image without learning from out-of-distribution data. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (June 2020)
  * [8] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization (2014), https://arxiv.org/abs/1412.6980
  * [9] Kirichenko, P., Izmailov, P., Wilson, A.G.: Why normalizing flows fail to detect out-of-distribution data. Advances in neural information processing systems 33, 20578–20589 (2020)
  * [10] Liang, S., Li, Y., Srikant, R.: Principled detection of out-of-distribution examples in neural networks. CoRR abs/1706.02690 (2017), http://arxiv.org/abs/1706.02690
  * [11] Lin, T.Y., Maire, M., Belongie, S., Bourdev, L., Girshick, R., Hays, J., Perona, P., Ramanan, D., Zitnick, C.L., Dollár, P.: Microsoft coco: Common objects in context (2014). https://doi.org/10.48550/ARXIV.1405.0312, https://arxiv.org/abs/1405.0312
  * [12] McAllister, R., Kahn, G., Clune, J., Levine, S.: Robustness to out-of-distribution inputs via task-aware generative uncertainty. In: 2019 International Conference on Robotics and Automation (ICRA). pp. 2083–2089 (2019). https://doi.org/10.1109/ICRA.2019.8793552
  * [13] Mohseni, S., Pitale, M., Yadawa, J., Wang, Z.: Self-supervised learning for generalizable out-of-distribution detection. Proceedings of the AAAI Conference on Artificial Intelligence 34(04), 5216–5223 (Apr 2020). https://doi.org/10.1609/aaai.v34i04.5966, https://ojs.aaai.org/index.php/AAAI/article/view/5966
  * [14] Papamakarios, G., Nalisnick, E.T., Rezende, D.J., Mohamed, S., Lakshminarayanan, B.: Normalizing flows for probabilistic modeling and inference. J. Mach. Learn. Res. 22(57), 1–64 (2021)
  * [15] Ren, J., Liu, P.J., Fertig, E., Snoek, J., Poplin, R., Depristo, M., Dillon, J., Lakshminarayanan, B.: Likelihood ratios for out-of-distribution detection. In: Wallach, H., Larochelle, H., Beygelzimer, A., d'Alché-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems. vol. 32. Curran Associates, Inc. (2019), https://proceedings.neurips.cc/paper/2019/file/1e79596878b2320cac26dd792a6c51c9-Paper.pdf
  * [16] Rudin, W.: Real and Complex Analysis. Tata McGraw-Hill Education (2006)
  * [17] Wellhausen, L., Ranftl, R., Hutter, M.: Safe robot navigation via multi-modal anomaly detection. IEEE Robotics and Automation Letters 5(2), 1326–1333 (2020). https://doi.org/10.1109/LRA.2020.2967706
  * [18] Winkens, J., Bunel, R., Roy, A.G., Stanforth, R., Natarajan, V., Ledsam, J.R., MacWilliams, P., Kohli, P., Karthikesalingam, A., Kohl, S., Cemgil, A.T., Eslami, S.M.A., Ronneberger, O.: Contrastive training for improved out-of-distribution detection. CoRR abs/2007.05566 (2020), https://arxiv.org/abs/2007.05566
  * [19] Yang, J., Zhou, K., Li, Y., Liu, Z.: Generalized out-of-distribution detection: A survey (2021). https://doi.org/10.48550/ARXIV.2110.11334, https://arxiv.org/abs/2110.11334
  * [20] Yuhas, M., Feng, Y., Ng, D.J.X., Rahiminasab, Z., Easwaran, A.: Embedded out-of-distribution detection on an autonomous robot platform. In: Proceedings of the Workshop on Design Automation for CPS and IoT. p. 13–18. Destion ’21, Association for Computing Machinery, New York, NY, USA (2021). https://doi.org/10.1145/3445034.3460509, https://doi-org.ludwig.lub.lu.se/10.1145/3445034.3460509


# Disordered ground state in the spin-orbit coupled $J_{\rm eff}=1/2$ cobalt-
based metal-organic framework magnet with orthogonal spin dimers

Sebin J. Sebastian  School of Physics, Indian Institute of Science Education
and Research Thiruvananthapuram-695551, India  S. Mohanty  School of Physics,
Indian Institute of Science Education and Research Thiruvananthapuram-695551,
India  A. Nath  School of Chemistry, Indian Institute of Science Education and
Research Thiruvananthapuram-695551, India  M. P. Saravanan  UGC-DAE Consortium
for Scientific Research, University Campus, Khandwa Road, Indore 452001, India
S. Mandal  School of Chemistry, Indian Institute of Science Education and
Research Thiruvananthapuram-695551, India  A. A. Tsirlin  Felix Bloch
Institute for Solid-State Physics, Leipzig University, 04103 Leipzig, Germany
R. Nath<EMAIL_ADDRESS>School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India

###### Abstract

We present the magnetic properties of a strongly spin-orbit coupled quantum
dimer magnet based on Co2+. The metal-organic framework compound
Co2(BDC)2(DPTTZ)2$\cdot$DMF features Co2+ dimers arranged nearly orthogonal to
each other, similar to the Shastry-Sutherland lattice. Our assessment based on
the magnetization and heat capacity experiments reveals that the magnetic
properties at low temperatures can be described by an effective $J_{\rm
eff}=1/2$ Kramers doublet and the ground state is a singlet with a tiny spin
gap. Although the magnetic susceptibility could be analyzed in terms of the
interacting dimer model with an isotropic intra-dimer coupling $J_{0}/k_{\rm
B}\simeq 7.6$ K, this model fails to reproduce the shape of magnetization
isotherm and heat capacity data. A model of isolated spin dimers with the
anisotropic exchange couplings $J_{xy}\simeq 3.5$ K and $J_{z}\simeq 11$ K
provides an adequate description to the magnetic susceptibility, magnetization
isotherm, and heat capacity data at low temperatures. Interestingly, no field-
induced quantum phase phase is detected down to 100 mK around the critical
field of gap closing, suggesting the absence of Bose-Einstein condensation of
triplons and establishing isolated dimers with a negligible interdimer
coupling.

##  I Introduction

Quantum phase transition (QPT) in a spin system is a transition between two
phases at absolute zero temperature driven by quantum fluctuations [1,
*Vojta2069]. Such a phase can be accessed upon variation of an external
control parameter that has direct bearing on the system properties. To study
QPT, antiferromagnetic (AFM) spin-$1/2$ dimers are well suited systems, which
are characterized by a singlet ($\lvert S,S_{z}\rangle=\lvert 0,0\rangle$)
ground state with entangled spins and an excitation gap in the energy spectrum
[3]. Application of external magnetic field leads to a closing of the
excitation gap and breaking of SU(2) symmetry that in turn stabilizes the
magnetic long-range order (LRO). This non-trivial field-induced LRO can be
described well as the Bose-Einstein condensation (BEC) of spin-1 triplons [4,
3, 5]. Experimentally, BEC physics is demonstrated in compounds featuring
coupled spin dimers, even-leg ladders, and bond-alternating spin chains [6, 7,
8, 9, 10, 11]. Another fascinating field-induced quantum effect is the
appearance of magnetization plateaus, typically observed in spin systems with
orthogonal dimers, such as in the Shastry-Sutherland lattice SrCu2(BO3)2 [12,
13].

In the past, oxide materials with several transition-metal and few rare-earth
ions (e.g. Yb3+) are studied in the context of field-induced transitions [14,
15, 16, 17, 18, 9, 19]. Among these systems, Co2+-based compounds are quite
attractive as they evince peculiar features, such as the $J_{\rm eff}=1/2$
Kramers-doublet ground state, similar to Yb3+ ($4f$) and Ce3+ ($4f$) magnets
[19, 20, 21], single-ion anisotropy [22] etc. Furthermore, strong spin-orbit
coupling (SOC) and crystal electric field (CEF) in Co2+ systems introduce
bond-dependent anisotropy of the exchange couplings, thus offering a promising
playground to realize exotic phases, including field-induced quantum spin
liquid (QSL) in Kitaev materials [23, 24], BEC of triplons [25], and
Berezinskii-Kosterlitz-Thoueless (BKT) transition [26]. Recently, BEC of
triplons is observed in a $4f$-based dimer magnet Yb2Si2O7 but with an
unusually asymmetric BEC dome [19]. This asymmetry is sometimes attributed to
the anisotropy in the microscopic exchanges in the Hamiltonian [27, 28].
Similar physics is also envisaged for Co2+-based compounds because of the
exchange anisotropy [29]. In this regard, Co2+-based metal-organic compounds
are ideal contenders and provide a convenient testing ground for field-induced
studies due to reduced exchange couplings compared to their inorganic
counterparts. This leads to lower critical fields, enabling the experimental
techniques to have access to the complete phase diagram using continuous-field
laboratory magnets [16, 30, 17].

In this paper, we report the structural and magnetic properties of a
Co2+-based metal-organic framework Co2(BDC)2(DPTTZ)2$\cdot$DMF (abbreviated as
Co-MOF), investigated via magnetization and heat capacity measurements down to
100 mK, followed by exact diagonalization calculations. Co-MOF crystallizes in
an orthorhombic space group $Iba2$ (No. 45) with the lattice parameters
$a=34.7269(1)$ Å, $b=17.1760(3)$ Å, $c=17.0489(5)$ Å, and
$\alpha=\beta=\gamma=90^{\circ}$ [31, 32]. It contains two
crystallographically in-equivalent cobalt atoms [Co(1) and Co(2)]. Each Co(II)
center is coordinated with O and N atoms in a distorted heteroleptic
octahedron (CoO4N2) geometry with Co-O and Co-N distances in the ranges of
$1.988(4)-2.232(4)$ Å and $2.134(3)-2.160(3)$ Å, respectively. Two in-
equivalent cobalt octahedra are connected via two C atoms and form a Co(1) –
Co(2) dimer. These dimer units are further connected through phenyl ring of
benzene dicarboxylate (BDC) ligands to form a corrugated two-dimensional layer
[Fig. 1(a)]. Each layer is pillared by DPTTZ
($N,N^{\prime}$-di(4-pyridyl)thiazolo-[5,4-d] thiazole) units to form an
overall three-dimensional framework. Interestingly, these Co-Co dimers are
arranged in an orthogonal fashion, reminiscent of the famous Shastry-
Sutherland lattice [see Fig. 1(b)]. The possible intradimer interaction $J$
($d_{\rm Co-Co}\simeq 4.04$ Å) and interdimer interaction $J^{\prime}$
($d_{\rm Co-Co}^{\rm avg}\simeq 10.597$ Å) are also depicted in Fig. 1(b).

Figure 1:  (a) Structure of Co-MOF in one unit cell showing the formation of
spin dimers by the CoO4N2 units and the interdimer connectivity. (b) Schematic
diagram showing the orthogonal arrangement of dimers and the possible exchange
couplings.

##  II Methods

The single-crystal synthesis of Co-MOF involves two steps. In the first step,
the precursor DPTTZ was prepared by mixing 200 mg of dithiooxamide and 0.4 mL
of 4-pyridinecarboxaldehyde in 10 mL of high-performance liquid chromatography
$N,N^{\prime}$-dimethylformamide (DMF), and refluxed at 153∘C for 5 h [33].
The yellow-colored needle-shaped crystals of DPTTZ were formed after slow
cooling. The crystals were washed in water, dried under vacuum, and stored
under ambient conditions. In the second step, 0.027 mmol (7.85 mg)
Co(NO3)2.6H2O, 0.03 mmol (4.98 mg) terephthalic acid, and 0.025 mmol (7.4 mg)
DPTTZ were mixed in 2:0.5 DMF : H2O ratio and sonicated for 1 h. The resulting
solution was transferred to a teflon lined vessel and kept in an autoclave at
100∘C for two days. Co-MOF crystals were obtained after slow cooling of the
autoclave. The crystals were washed several times with DMF and dried under
vacuum. The crystals were maintained under ambient conditions prior to further
experiments.

Figure 2:  Powder XRD pattern of Co-MOF measured at room temperature. The open
circles represent the experimental data and the solid red line represents the
Le-Bail fit. The Bragg positions are indicated by blue vertical bars and the
bottom solid green line indicates the difference between the experimental and
calculated intensities.

A large number of small single crystals were crushed into fine powder and the
phase purity of the product was confirmed via powder x-ray diffraction (XRD)
at room temperature using a PANalytical x-ray diffractometer (CuKα radiation,
$\lambda_{\rm avg}\simeq 1.5418$ Å). Le-Bail analysis of the powder XRD
pattern was performed using the MAG2POL software package [34]. Figure 2 shows
the powder XRD data along with the fit at $T=300$ K. The structural parameters
given in Ref. [31] were used as initial parameters. The obtained lattice
parameters $a=34.9769(1)$ Å, $b=17.1990(1)$ Å, $c=17.1076(1)$ Å, and
$\alpha=\beta=\gamma=90^{\circ}$, are in close agreement with the previous
report [31].

DC magnetization ($M$) of the powder sample was measured using a
superconducting quantum interference device (SQUID) magnetometer (MPMS-3,
Quantum Design). The measurements were performed in the temperature range of
1.8 K $\leq T\leq 380$ K and in the magnetic field range of 0 T $\leq H\leq 7$
T. The SQUID enabled us to measure the isothermal magnetization at $T=0.4$ K
with the help of a 3He attachment. Heat capacity ($C_{\rm p}$) was measured on
a small piece of sintered pellet using the thermal relaxation technique over a
large temperature range in two physical property measurement systems (PPMS,
Quantum Design). For $T>2$ K, measurements were done in a 9 T PPMS while for
0.1 K $\leq T\leq 4$ K we have used a dilution insert in a 14 T PPMS.

The magnetization and heat capacity of the anisotropic spin dimer are
simulated via exact diagonalization using the fulldiag utility of the ALPS
package [35], where the number of sites is taken to be $Z=2$.

##  III Results

###  III.1 Magnetization

Temperature-dependent magnetic susceptibility $\chi~{}(\equiv M/H)$ measured
in different magnetic fields is shown in Fig. 3(a). At the lowest measured
field ($\mu_{0}H=0.01$ T), $\chi(T)$ passes through a broad maximum at $T^{\rm
max}_{\chi}\simeq 3$ K, which is the signature of an antiferromagnetic (AFM)
short-range order, typical for a low-dimensional spin system. With increasing
field, the position of the broad maximum shifts towards low temperature, as
expected. Below 3 K, $\chi(T)$ exhibits a rapid decrease, suggesting a singlet
ground state or the opening of a spin gap, which necessitates additional
investigation [36, 37]. No anomaly associated with the magnetic LRO is
observed down to 1.8 K.

Figure 3: (a) $\chi(T)$ of Co-MOF measured in different applied fields. The
dash-dotted line represents the fit of the $\mu_{0}H=0.01$ T data by the
interacting dimer model [Eq. (2)] while the solid line is the simulation of
isolated spin dimers with anisotropic interactions ($J_{z}=11$ K and
$J_{xy}=3.52$ K). Inset: CW fit to the low-$T$ $1/\chi$ data (after
subtracting the van-Vleck contribution). (b) $M$ vs $H$ and $dM/dH$ vs $H$ on
the left and right $y$-axes, respectively, measured at $T=0.4$ K. The
horizontal dash-dotted line marks the van-Vleck contribution. The dashed line
shows the Brillouin fit [Eq. (4)] to magnetization in the gaped regime. The
dash-dotted and solid lines are the simulations for isolated spin dimers with
isotropic and anisotropic interactions, respectively.

To extract key magnetic parameters, we conducted the $\chi(T)$ analysis
employing the modified Curie-Weiss (CW) law:

$\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm CW}},$ (1)

where $\chi_{0}$ is the temperature-independent contribution consisting of the
core diamagnetic susceptibility ($\chi_{\rm core}$) and van Vleck
paramagnetism ($\chi_{\rm VV}$) of the open shell of the Co2+ ions present in
the sample. The second term in Eq. (1) represents the CW law with the Curie
constant $C=N_{\rm A}\mu_{\rm eff}^{2}/3k_{\rm B}$, where $N_{\rm A}$ is the
Avogadro number, $k_{\rm B}$ is the Boltzmann constant, $\mu_{\rm
eff}=g\sqrt{S(S+1)}\mu_{\rm B}$ is the effective magnetic moment, $g$ is the
Landé $g$-factor, $\mu_{\rm B}$ is the Bohr magneton, $S$ is the spin quantum
number, and $\theta_{\rm CW}$ is the characteristic CW temperature. Above 150
K, $\chi(T)$ for $\mu_{0}H=0.01$ T was fitted by Eq. (1) that yields
$\chi_{0}\simeq 7.3\times 10^{-4}$ cm3/mol, the high-$T$ (HT) effective moment
$\mu_{\rm eff}^{\rm HT}\simeq 4.53~{}\mu_{\rm B}$/Co2+, and the high-$T$ CW
temperature $\theta_{\rm CW}^{\rm HT}\simeq-4~{}$K. The value of $\mu_{\rm
eff}^{\rm HT}$ corresponds to $S=3/2$ with $g\simeq 2.33$ for the $3d^{7}$
configuration. A slightly larger $\mu_{\rm eff}^{\rm HT}$ value compared with
the theoretical spin-only value ($\sim 3.87~{}\mu_{B}$/Co2+ for $S=3/2$ with
$g=2.0$) indicates anisotropy in $g$-factor stemming from the pseudo-trigonal
crystal field as well as additional orbital contribution, as observed in
several Co2+-based systems [38, 39].

At temperatures below 40 K, $1/\chi$ exhibits a subtle slope change, deviating
from the high-$T$ CW behavior. Spin-orbit coupling splits the Co2+ multiplet
into several Kramers doublets, and the physics at low temperatures is fully
determined by the lowest Kramers doublet that can be treated as
pseudospin-$\frac{1}{2}$. This type of crossover from the high-$T$ $S=3/2$ to
the low-$T$ $J_{\rm eff}=1/2$ behavior is a common character of Co2+-based
compounds [40, 41, 42]. To understand the low-temperature $\chi(T)$ data
precisely, one requires a proper estimation of $\chi_{\rm VV}$ that arises
from excitations between the ground-state Kramers doublet and excited states.
After subtracting $\chi_{\rm VV}$ (obtained from the magnetization isotherm at
$T=0.4$ K) from $\chi(T)$, $1/(\chi-\chi_{\rm VV}$) shows a clear slope change
and a linear regime below 40 K [inset of Fig. 3(a)]. A CW fit [Eq. (1)] in the
low-$T$ (LT) regime (8 K - 30 K) returns $\mu_{\rm eff}^{\rm LT}\simeq
3.40~{}\mu_{\rm B}$/Co2+ and $\theta_{\rm CW}^{\rm LT}\simeq-1.9$ K. The
$\mu_{\rm eff}^{\rm LT}$ value corresponds to a $J_{\rm eff}=1/2$ moment with
an average $g\simeq 3.93$. Such a large value of $g$ compared to the free
electron value $g=2$ reflects the effect of spin-orbit coupling [43, 44, 42,
41]. The negative value of $\theta_{\rm CW}^{\rm LT}$ confirms the dominant
AFM interactions at low temperatures.

From the crystal structure, while considering the spin-lattice of Co atoms, we
anticipate the formation of magnetic dimers of Co2+ ions in an orthogonal
fashion, reminiscent of Shastry-Sutherland lattice. As a first step, to
determine the potential exchange couplings, we fitted the $\chi(T)$ data using
the following expression:

$\chi(T)=\chi_{0}+\frac{C_{\rm imp}}{T}+\chi_{\rm d}(T).$ (2)

Here, the second term is the Curie law and $\chi_{\rm d}(T)$ is the expression
for susceptibility of AFM spin-1/2 coupled dimer model which has the form [45]

$\chi_{\rm d}(T)=\frac{N_{\rm A}g^{2}\mu_{\rm B}^{2}}{k_{\rm
B}T[{3+\exp({J/k_{\rm B}T})+zJ{{}^{\prime}}/k_{\rm B}T}]}.$ (3)

Here, $J/k_{\rm B}$ is the intra-dimer exchange coupling, $J^{\prime}/k_{\rm
B}$ represents the inter-dimer exchange coupling, and $z$ is the number of
nearest-neighbor spins. The fit using Eq. (2) in the low temperature regime
(1.80 K $\leq~{}T\leq$ 40 K) [see Fig. 3(a)] returns $\chi_{0}\simeq 0.0132$
cm3/mol, $C_{\rm imp}\simeq 0.0678$ cm3-K/mol, $g\simeq 4.15$, $z=2$,
$J/k_{\rm B}\simeq 7.6$ K, and $J{{}^{\prime}}/k_{\rm B}\simeq 0.07$ K. The
obtained value of $C_{\rm imp}$ corresponds to $\sim 4.1$% of spin-1/2
impurities. The value of $J/k_{\rm B}\simeq 7.6$ K with $g\simeq 4.15$ sets
the critical field of gap closing $H_{\rm C1}=\Delta_{0}/g\mu_{\rm B}\simeq
2.7$ T where $\Delta_{0}\simeq J/k_{B}$ is the spin gap between the singlet
and triplet states of the dimer [10].

To track the magnetization response with respect to the applied field, we
measured an isothermal magnetization [$M(H)$] at $T=0.4$ K up to 7 T. For a
gapped spin system, magnetization ($M$) typically remains zero below $H_{\rm
C1}$, while above this field it is expected to increase up to the saturation
field $H_{\rm C2}$ that marks the onset of the fully polarized state [46, 47].
As shown in Fig. 3(b), in the low-field regime, $M$ exhibits a wide plateau up
to $H\sim 2$ T, clearly suggesting a singlet ground state and the critical
field required to overcome the spin gap is about $H_{\rm C1}\simeq 2$ T. This
plateau with non-zero magnetization can be attributed to the paramagnetic
impurities/defects expected in a powder sample, which saturate above 1 T.
Above $H_{\rm C1}\simeq 2$ T, $M$ increases sharply and saturates in high
fields. It is also observed that $M$ increases weakly with increasing field
beyond the saturation field $H_{\rm C2}$. This weak increase in high fields is
primarily due to the van Vleck contribution ($\chi_{\rm VV}$), expected for
Co2+ systems in an octahedral environment [48]. The slope and $y$-intercept of
a linear fit to the data above $\sim 4$ T yields $\chi_{\rm VV}\simeq
2.46\times 10^{-2}$ cm3/mol and the saturation magnetization of $M_{\rm
S}\simeq 2.05~{}\mu_{\rm B}$/f.u., respectively. This value of $M_{\rm S}$ is
in good agreement with $M_{sat}=gJ_{\rm eff}\mu_{\rm B}\simeq 2.0~{}\mu_{\rm
B}$ expected for $J_{\rm eff}=1/2$ with powder averaged $g\simeq 4.0$. Thus,
the magnetization isotherm analysis also validates the $J_{\rm eff}=1/2$
ground state of Co2+.

In order to quantitatively estimate the impurity contribution and to visualize
zero magnetization below $H_{\rm C1}$, we fitted the data below 2 T [see Fig.
3(b)] by the following equation [10]

$M=\chi H+N_{\rm A}\mu_{\rm B}f_{\rm imp}S_{\rm imp}g_{\rm imp}\times B(x),$
(4)

where $\chi$ is the intrinsic susceptibility of the sample, $f_{\rm imp}$ is
the molar fraction of the impurities, $g_{\rm imp}$ is the impurity
$g$-factor, $S_{\rm imp}$ represents the impurity spin, $B(x)$ is the
Brillouin function taken for the spin value $S_{\rm imp}(x)$, and the argument
($x$) can be defined by $x=g_{\rm imp}\mu_{\rm B}S_{\rm imp}H/[k_{\rm
B}(T+\theta_{\rm imp})]$. The fit yields the following parameters: $f_{\rm
imp}\simeq 0.039$, $S_{\rm imp}=0.5$, $g_{\rm imp}\simeq 4.15$, and
$\theta_{\rm imp}\simeq 0.48$ K. This amount of impurity matches the one found
from the susceptibility fit. Further, after subtraction of the impurity
contribution, the corrected magnetization stays at zero value up to $H_{\rm
C1}$, as expected in the gapped state.

###  III.2 Heat Capacity

Figure 4:  (a) $C_{\rm p}$ vs $T$ measured in zero field along with the
calculated phonon contribution $C_{\rm ph}$. Inset: Zero-field $C_{\rm p}(T)$
in the low-temperature regime with the solid line representing the nuclear
contribution $C_{\rm n}$. (b) $C_{\rm mag}/T$ and $\Delta S_{\rm mag}$ vs $T$
on the left and right $y$-axes, respectively. Inset: $C_{\rm p}(H)$ measured
at $T=0.1$ K.

Temperature-dependent heat capacity [$C_{\rm p}(T)$] measured in zero field
and down to 100 mK is shown in Fig. 4(a). It depicts a well-defined broad
maximum at around $\sim 3$ K, a hallmark of short-range AFM order. No
signature of magnetic LRO is observed down to the lowest temperature. At very
low-$T$s, a slight upturn is observed, which we attribute to the nuclear
contribution appearing due to a small hyperfine field in this temperature
regime, produced by Co or other atoms present. We estimated the nuclear
contribution by fitting the zero-field $C_{\rm p}(T)$ data below 0.2 K with
$C_{\rm n}(T)=\alpha_{\rm Q}/T^{2}$ [inset of Fig. 4(a)], where the
coefficient $\alpha_{\rm Q}$ is related to the nuclear-level splitting, both
quadrupolar and Zeeman. The fit yields $\alpha_{\rm Q}\simeq 1.6\times
10^{-3}$J K/mol, which aligns well with the values reported for other Co-based
systems [42]. The data after subtraction of the nuclear contribution is also
presented in the inset of Fig. 4(a) where $C_{\rm p}$ decreases towards zero
on cooling, thus reflecting the singlet ground state.

In general, for a magnetic insulator, $C_{\rm p}(T)$ in zero field has two
major contributions: phononic $C_{\rm ph}(T)$ and magnetic $C_{\rm mag}(T)$
parts. Above 10 K, $C_{\rm p}(T)$ is dominated by $C_{\rm ph}$, while at low
temperatures, the dominant contribution is due to $C_{\rm mag}$. To estimate
$C_{\rm ph}(T)$, experimental heat capacity data above 10 K were fitted
empirically using a polynomial of the form [49]

$C_{\rm ph}(T)=aT^{3}+bT^{5}+cT^{7}+dT^{9}.$ (5)

From the fit [Fig. 4(a)], the obtained coefficients are $a\simeq 3.26\times
10^{-2}$ J mol-1 K-4, $b\simeq 1.06\times 10^{-4}$ J mol-1 K-6, $c\simeq
2.04\times 10^{-7}$ J mol-1 K-8, and $d\simeq-2.2\times 10^{-10}$ J mol-1
K-10. $C_{\rm mag}$ was obtained by subtracting $C_{\rm ph}$ and $C_{\rm n}$
from the total $C_{\rm p}$. Furthermore, $C_{\rm mag}(T)$ was used to estimate
the magnetic entropy [$\Delta S_{\rm mag}(T)$] by integrating $C_{\rm
mag}(T)/T$ in the measured $T$-range. As shown in the right $y$-axis in Fig.
4(b), $\Delta S_{\rm mag}(T)$ $[=\int_{\rm 0.1\,K}^{T}\frac{C_{\rm
mag}(T^{\prime})}{T^{\prime}}dT^{\prime}$] reaches a well-defined plateau
close to 90% of $R\ln 2$, expected for a pseudospin-$\frac{1}{2}$ spin system.
This again indicates that the low-temperature properties of Co-MOF are
governed by the $J_{\rm eff}=1/2$ Kramers doublet [42].

Figure 5:  (a) $C_{\rm p}$ vs $T$ measured at low temperatures in different
applied fields. (b) $C_{\rm p}/T$ vs $T$ in the same temperature range.

In order to detect possible field-induced phases between $H_{\rm C1}$ and
$H_{\rm C2}$, we also measured $C_{\rm p}(T)$ in different fields at low
temperatures [see Fig. 5(a)]. With the application of field, the broad maximum
shifts towards low temperatures, as expected for an AFM short-range-order. No
feature/peak of field-induced magnetic LRO reminiscent of BEC of triplons is
observed down to 100 mK for $H>H_{\rm C1}$. In order to visualize the absence
of magnetic LRO, in Fig. 5(b), we plotted $C_{\rm p}/T$ vs $T$ measured in
various applied fields that display no clear anomaly. The absence of a field-
induced magnetic LRO suggests almost non-interacting dimers.

###  III.3 Discussion

Figure 6:  The tentative energy diagram that depicts the splitting of the
${{}^{4}}F$ state of Co2+ under an octahedral crystal field and of the
${{}^{4}}T_{\rm 1g}$ state under SOC. $\chi(T)$ ($\mu_{0}H=0.01$ T) is plotted
in the right $y$-axis in order to visualize the temperature regimes where the
high ($S=3/2$) and low-spin ($J_{\rm eff}=1/2$) states are effective. The
maximum in $\chi(T)$ is arising due to the spin dimers.

According to Hund’s rule, the ground state of a free Co2+ ion (3$d^{7}$) is
${{}^{4}}F$ with the orbital angular momentum $L=3$ and total spin $S=3/2$.
This multiplet is split by the crystal field as well as SOC. In Fig. 6, we
show the typical energy-level diagram of a Co2+ ion in the crystal
environment. Cubic crystal field (CF) splits the ${{}^{4}}F$ multiplet into
three multiplets: ${{}^{4}}A_{\rm 2g}$, ${{}^{4}}T_{\rm 2g}$, and
${{}^{4}}T_{\rm 1g}$. For the octahedral geometry, the ${{}^{4}}T_{\rm 1g}$
multiplet is expected to be the ground state. The energy level
(${{}^{4}}T_{\rm 1g}$) is further influenced by the SOC
($\hat{\mathcal{H}}_{\rm SOC}=\lambda L\cdot S$), which splits this level into
a doublet, quartet, and sextet (six Kramers doublets). The doublet is
separated from the quartet by an energy gap of $\Delta_{\rm
SL}=\frac{3}{2}\lambda$. At low temperatures, the corresponding degeneracy of
the doublets is further lifted and the lowest Kramers doublet becomes the
ground state. Thus, when the temperature is much lower than the spin-orbit
coupling $\lambda$ (i.e. $T\ll|\lambda|/k_{\rm B}$), one expects the magnetism
to be determined by the lowest Kramers doublet with the effective spin $J_{\rm
eff}=1/2$ [22, 50]. Indeed, all our thermodynamic measurements confirm the
$J_{\rm eff}=1/2$ ground state at low temperatures for Co-MOF.

Figure 7:  (a) $\chi T$ vs $T$ is plotted for $\mu_{0}H=0.01$ T. The solid and
dashed lines are fits using Eq. (6) for $T_{0}=0$ K and $T_{0}=-4.46$ K,
respectively. (b) $C_{\rm mag}$ vs $T$ measured down to 0.1 K in zero field.
The dashed and solid lines represent the simulated curves for an isolated
spin-$1/2$ dimer with isotropic ($J=6$ K) and anisotropic interactions
($J_{z}=11$ K and $J_{xy}=3.52$ K), respectively. Inset: Zero-field $C_{\rm
mag}(T)$ in the low-$T$ regime; the solid line is an exponential fit with
$\Delta_{0}^{\rm C}/k_{\rm B}\simeq 4.3$ K.

Upon revisiting the $\chi T$ vs $T$ data [Fig. 7(a)], we observed distinct
slope changes that indicate changes in the effective magnetic moment of Co2+.
The temperature dependence can be fitted by a model of two lowest doublets
separated by the gap $\Delta_{\rm SL}/k_{\rm B}$ as [51]

$\displaystyle\chi=\chi_{0}+\frac{N}{k_{B}(T-T_{0})}\left[C_{0}+C_{1}\left(\frac{2k_{\rm
B}T}{\Delta_{\rm SL}}\right)\tanh\left(\frac{\Delta_{\rm SL}}{2k_{\rm
B}T}\right)\right.$ $\displaystyle\left.+~{}C_{2}\tanh\left(\frac{\Delta_{\rm
SL}}{2k_{\rm B}T}\right)\right].$ (6)

The value of $N$ represents the number of paramagnetic entities, the
coefficients $C_{0}$, $C_{1}$, and $C_{2}$ depend on the electronic wave
function of the two Kramers doublets involved, and $T_{0}$ essentially
represents the energy scale of the exchange interactions. The fit above 10 K
as shown in Fig. 7(a) returns $\chi_{0}\simeq 8.746\times 10^{-4}$ cm3/mol,
$C_{0}/k_{\rm B}\simeq 0.972$ cm3 K/mol, $C_{1}/k_{\rm B}\simeq 0.668$ cm3
K/mol, $C_{2}/k_{\rm B}\simeq 0.249$ cm3 K/mol, and $\Delta_{\rm SL}/k_{\rm
B}\simeq 160$ K with $T_{0}\simeq-4.4~{}$K. The large value of $\Delta_{\rm
SL}$ implies that the system should behave as $J_{\rm eff}=1/2$ at low
temperatures ($T<\Delta_{\rm SL}/k_{\rm B}$). Indeed, our analysis below 30 K
confirms this scenario. The negative value of $T_{0}$ suggests AFM
correlations at low temperatures. In the low-$T$ regime when $T\ll\Delta_{\rm
SL}/k_{\rm B}$, the second term within the parenthesis in Eq. (6) is almost
negligible and provides an effective Curie constant of $C_{\rm
eff}=C_{0}/k_{\rm B}+C_{2}/k_{\rm B}\simeq 1.22$ cm3-K/mol. This value
corresponds to $\mu_{\rm eff}\simeq 3.12~{}\mu_{\rm B}$ that matches closely
with $\mu_{\rm eff}^{\rm LT}$ obtained from the CW fit.

Structurally, Co-MOF presents an ideal case of isolated dimers. Though the
isolated dimer model with the isotropic (Heisenberg) intradimer interaction
fits the $\chi(T)$ data reasonably well, it fails to reproduce the shape of
the magnetization isotherm and $C_{\rm mag}(T)$. A more narrow maximum is
expected in the heat capacity of a Heisenberg spin dimer. Even if we choose
$J=6$ K instead of $J=7.6$ K from the $\chi(T)$ fit in order to reproduce the
position of the maximum, its height should be reduced by 25%, which is
inconsistent with the almost full magnetic entropy recovered in our
measurements. Despite all these adjustments, the peak remains more narrow than
in the experiment, thus suggesting that Co-MOF can not be treated as
Heisenberg spin dimer.

The heteroleptic, distorted CoO4N2 octahedron is elongated along the
$z$-direction as compared to the $xy$-plane. This implies that a trigonal
crystal-field splitting is likely to develop. It leads to a Hamiltonian of the
form $\hat{\mathcal{H}}_{\rm trig}=\delta(3L_{\rm z}^{2}-2)$, where $\delta$
denotes the strength of CFS [52]. It is theoretically predicted that under the
influence of trigonal distortion, isotropic Heisenberg exchange gives way to
an anisotropic one described by the XXZ model [24]. Therefore, in Fig. 7(b) we
simulated the $C_{\rm mag}(T)$ using such a model,

$\mathcal{\hat{\mathcal{H}}}=J_{xy}(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y})+J_{z}S_{i}^{z}S_{j}^{z},$
(7)

with the exchange couplings $J_{xy}\simeq 3.5$ K and $J_{z}\simeq 11$ K. The
anisotropy of $J_{xy}/J_{z}\simeq 0.32$ controls the width and height of the
maximum, so it can be determined with relative accuracy even from powder data.
With the XXZ model, we are also able to replicate $\chi(T)$ and $M$ vs $H$
curves using the same exchange parameters assuming isotropic $g=4.15$ (see
Fig. 3) and the same impurity contribution as in Sec. III.1. The average
exchange coupling $J_{\rm avg}/k_{\rm B}=\frac{2J_{xy}+J_{z}}{3k_{\rm
B}}\simeq 6$ K with $g=4.15$ corresponds to $H_{\rm C1}\simeq 2$ T. Similar
physics of isolated spin dimers with anisotropic interactions is also realized
in $4f$ systems, such as BiYbGeO5 and NaLu0.9Yb0.1Se2 [21, 53].

We further quantified the spin gap by fitting $C_{\rm mag}(T)$ by the low-$T$
approximation of the spin-$1/2$ dimer model: $C_{\rm
mag}\propto\left(\Delta_{0}^{\rm C}/k_{\rm B}T\right)^{2}e^{-\Delta_{0}^{\rm
C}/k_{\rm B}T}$ [16]. The fit below 2 K [inset of Fig. 7(b)] returns
$\Delta_{0}^{\rm C}/k_{\rm B}=4.3(1)$ K and this value seems to be reasonable
when compared with $J_{\rm avg}\simeq 6$ K of the anisotropic XXZ model.

The scenario of isolated spin dimers are also further confirmed from the
magnetization isotherm and $C_{\rm p}$ vs $H$ measurements. In a magnetization
isotherm, the spacing between $H_{\rm C1}$ and $H_{\rm C2}$ is normally
controlled by the magnitude of the inter-dimer coupling $J^{\prime}/k_{\rm
B}$, i.e., increasing the value of $J^{\prime}/k_{\rm B}$ widens the spacing
between them. In such a scenario, the derivative of $M$ ($dM/dH$) vs. $H$
should exhibit two peaks corresponding to $H_{\rm C1}$ and $H_{\rm C2}$ [17].
As presented in Fig. 3(b), our $dM/dH$ vs $H$ plot reveals only one peak at
around $2.2$ T, reflecting the fact that the compound is an isolated dimer
system with negligible inter-dimer coupling [54]. $C_{\rm p}$ vs $H$ measured
at the lowest temperature of $T=0.1$ K is shown in the inset of Fig. 4(b).
$C_{\rm p}$ remains almost zero in low fields due to the singlet ground state.
As the increasing field approaches $H_{\rm C1}$, $C_{\rm p}$ starts
increasing, shows a peak at around 2.2 T, and then decreases further to zero
in higher fields [53]. Usually, in a system of coupled dimers with the spin
gap, $C_{\rm p}$ vs $H$ shows double peaks corresponding to $H_{\rm C1}$ and
$H_{\rm C2}$ [16, 55]. However, for an isolated dimer with negligible
interdimer coupling, one may expect only one peak. In our case, we observed a
single broad peak in $C_{\rm p}(H)$ at $T=0.1$ K that clearly indicates the
proximity of our system to the isolated-dimer model where both $H_{\rm C1}$
and $H_{\rm C2}$ coincide when $T\rightarrow{0}$.

###  III.4 Conclusion

Co-MOF features $J_{\rm eff}=1/2$ Co2+ dimers arranged in an orthogonal
fashion, similar to the famous Shastry–Sutherland lattice. In zero field, a
quantum disordered ground state with the spin gap and no magnetic LRO is
observed. The absence of magnetic LRO in applied fields down to 100 mK
excludes BEC of triplons and underpins the absence of any significant
interdimer coupling. We observed that the ground-state Kramers doublet is
influenced by both trigonal CFS and SOC, causing the spin system to deviate
from the Heisenberg dimer limit. Indeed, the low-temperature properties are
well described by an isolated spin-1/2 anisotropic XXZ dimer model
($J_{xy}\neq J_{z}$). No magnetization plateau is observed despite the
presence of orthogonal dimers. However, this compound can serve as a model
system to study magnetization plateaus if one can introduce a finite inter-
dimer coupling by an appropriate choice of organic ligands.

###### Acknowledgements.

We would like to acknowledge SERB, India for financial support bearing
sanction Grant No. CRG/2022/000997. SJS is supported by the Prime Minister’s
Research Fellowship (PMRF) scheme, Government of India. Computations for this
work were done using resources of the Leipzig University Computing Center.

## References

  * Sachdev [1999] S. Sachdev, Quantum phase transitions, Phys. World 12, 33 (1999).
  * Vojta [2003] M. Vojta, Quantum phase transitions, Rep. Prog. Phys. 66, 2069 (2003).
  * Zapf  _et al._ [2014] V. Zapf, M. Jaime, and C. D. Batista, Bose-Einstein condensation in quantum magnets, Rev. Mod. Phys. 86, 563 (2014).
  * Rice [2002] T. M. Rice, To Condense or Not to Condense, Science 298, 760 (2002).
  * Giamarchi  _et al._ [2008] T. Giamarchi, C. Rüegg, and O. Tchernyshyov, Bose-Einstein condensation in magnetic insulators, Nat. Phys. 4, 198 (2008).
  * Nikuni  _et al._ [2000] T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka, Bose-Einstein Condensation of Dilute Magnons in ${\mathrm{TlCuCl}}_{3}$, Phys. Rev. Lett. 84, 5868 (2000).
  * Jaime  _et al._ [2004] M. Jaime, V. F. Correa, N. Harrison, C. D. Batista, N. Kawashima, Y. Kazuma, G. A. Jorge, R. Stern, I. Heinmaa, S. A. Zvyagin, Y. Sasago, and K. Uchinokura, Magnetic-Field-Induced Condensation of Triplons in Han Purple Pigment ${\mathrm{B}\mathrm{a}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{i}}_{2}{\mathrm{O}}_{6}$, Phys. Rev. Lett. 93, 087203 (2004).
  * Thielemann  _et al._ [2009] B. Thielemann, C. Rüegg, K. Kiefer, H. M. Rønnow, B. Normand, P. Bouillot, C. Kollath, E. Orignac, R. Citro, T. Giamarchi, A. M. Läuchli, D. Biner, K. W. Krämer, F. Wolff-Fabris, V. S. Zapf, M. Jaime, J. Stahn, N. B. Christensen, B. Grenier, D. F. McMorrow, and J. Mesot, Field-controlled magnetic order in the quantum spin-ladder system ${(\text{Hpip})}_{2}{\text{CuBr}}_{4}$, Phys. Rev. B 79, 020408 (2009).
  * Garlea  _et al._ [2007] V. O. Garlea, A. Zheludev, T. Masuda, H. Manaka, L.-P. Regnault, E. Ressouche, B. Grenier, J.-H. Chung, Y. Qiu, K. Habicht, K. Kiefer, and M. Boehm, Excitations from a Bose-Einstein Condensate of Magnons in Coupled Spin Ladders, Phys. Rev. Lett. 98, 167202 (2007).
  * Mukharjee  _et al._ [2019] P. K. Mukharjee, K. M. Ranjith, B. Koo, J. Sichelschmidt, M. Baenitz, Y. Skourski, Y. Inagaki, Y. Furukawa, A. A. Tsirlin, and R. Nath, Bose-Einstein condensation of triplons close to the quantum critical point in the quasi-one-dimensional spin-$\frac{1}{2}$ antiferromagnet ${\mathrm{NaVOPO}}_{4}$, Phys. Rev. B 100, 144433 (2019).
  * Mukharjee  _et al._ [2020] P. K. Mukharjee, K. M. Ranjith, M. Baenitz, Y. Skourski, A. A. Tsirlin, and R. Nath, Two types of alternating spin-$\frac{1}{2}$ chains and their field-induced transitions in $\varepsilon\text{$-$}{\mathrm{LiVOPO}}_{4}$, Phys. Rev. B 101, 224403 (2020).
  * Kodama  _et al._ [2002] K. Kodama, M. Takigawa, M. Horvatić, C. Berthier, H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, and F. Mila, Magnetic Superstructure in the Two-Dimensional Quantum Antiferromagnet ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}){}_{2}$, Science 298, 395 (2002).
  * Kageyama  _et al._ [1999] H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto, and Y. Ueda, Exact Dimer Ground State and Quantized Magnetization Plateaus in the Two-Dimensional Spin System ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}){}_{2}$, Phys. Rev. Lett. 82, 3168 (1999).
  * Laflorencie and Mila [2009] N. Laflorencie and F. Mila, Theory of the Field-Induced BEC in the Frustrated Spin-$\frac{1}{2}$ Dimer Compound ${\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}$, Phys. Rev. Lett. 102, 060602 (2009).
  * Suh  _et al._ [2011] S. Suh, K. A. Al-Hassanieh, E. C. Samulon, I. R. Fisher, S. E. Brown, and C. D. Batista, Nonuniversal magnetization at the BEC critical field: Application to the spin dimer compound Ba3Mn2O8, Phys. Rev. B 84, 054413 (2011).
  * Freitas  _et al._ [2017] R. S. Freitas, W. A. Alves, and A. Paduan-Filho, Magnetic-field-induced ordered phase in the chloro-bridged copper(ii) dimer system $[\mathrm{C}{\mathrm{u}}_{2}{(\mathrm{apyhist})}_{2}\mathrm{C}{\mathrm{l}}_{2}]{(\mathrm{Cl}{\mathrm{O}}_{4})}_{2}$, Phys. Rev. B 95, 184426 (2017).
  * Lancaster  _et al._ [2014] T. Lancaster, P. A. Goddard, S. J. Blundell, F. R. Foronda, S. Ghannadzadeh, J. S. Möller, P. J. Baker, F. L. Pratt, C. Baines, L. Huang, J. Wosnitza, R. D. McDonald, K. A. Modic, J. Singleton, C. V. Topping, T. A. W. Beale, F. Xiao, J. A. Schlueter, A. M. Barton, R. D. Cabrera, K. E. Carreiro, H. E. Tran, and J. L. Manson, Controlling Magnetic Order and Quantum Disorder in Molecule-Based Magnets, Phys. Rev. Lett. 112, 207201 (2014).
  * Zapf  _et al._ [2006] V. S. Zapf, D. Zocco, B. R. Hansen, M. Jaime, N. Harrison, C. D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, and A. Paduan-Filho, Bose-Einstein Condensation of $S=1$ Nickel Spin Degrees of Freedom in ${\mathrm{NiCl}}_{2}\mathrm{\text{$-$}}4\mathrm{SC}({\mathrm{NH}}_{2}{)}_{2}$, Phys. Rev. Lett. 96, 077204 (2006).
  * Hester  _et al._ [2019] G. Hester, H. S. Nair, T. Reeder, D. R. Yahne, T. N. DeLazzer, L. Berges, D. Ziat, J. R. Neilson, A. A. Aczel, G. Sala, J. A. Quilliam, and K. A. Ross, Novel Strongly Spin-Orbit Coupled Quantum Dimer Magnet: ${\mathrm{Yb}}_{2}{\mathrm{Si}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 123, 027201 (2019).
  * Somesh  _et al._ [2023] K. Somesh, S. S. Islam, S. Mohanty, G. Simutis, Z. Guguchia, C. Wang, J. Sichelschmidt, M. Baenitz, and R. Nath, Absence of magnetic order and emergence of unconventional fluctuations in the ${J}_{\mathrm{eff}}=\frac{1}{2}$ triangular-lattice antiferromagnet ${\mathrm{YbBO}}_{3}$, Phys. Rev. B 107, 064421 (2023).
  * Mohanty  _et al._ [2023] S. Mohanty, S. S. Islam, N. Winterhalter-Stocker, A. Jesche, G. Simutis, C. Wang, Z. Guguchia, J. Sichelschmidt, M. Baenitz, A. A. Tsirlin, P. Gegenwart, and R. Nath, Disordered ground state in the spin-orbit coupled ${J}_{\mathrm{eff}}$ = $\frac{1}{2}$ distorted honeycomb magnet ${\mathrm{BiYbGeO}}_{5}$, Phys. Rev. B 108, 134408 (2023).
  * Abragam and Pryce [1951] A. Abragam and M. H. L. Pryce, The theory of paramagnetic resonance in hydrated cobalt salts, Proc. R. Soc. A 206, 173 (1951).
  * Lin  _et al._ [2021] G. Lin, J. Jeong, C. Kim, Y. Wang, Q. Huang, T. Masuda, S. Asai, S. Itoh, G. Günther, M. Russina, Z. Lu, J. Sheng, L. Wang, J. Wang, G. Wang, Q. Ren, C. Xi, W. Tong, L. Ling, Z. Liu, L. Wu, J. Mei, Z. Qu, H. Zhou, X. Wang, J.-G. Park, Y. Wan, and J. Ma, Field-induced quantum spin disordered state in spin-1/2 honeycomb magnet Na2Co2TeO6, Nat. Commun. 12, 5559 (2021).
  * Liu and Kee [2023] X. Liu and H.-Y. Kee, Non-Kitaev versus Kitaev honeycomb cobaltates, Phys. Rev. B 107, 054420 (2023).
  * Sheng  _et al._ [2022] J. Sheng, L. Wang, A. Candini, W. Jiang, L. Huang, B. Xi, J. Zhao, H. Ge, N. Zhao, Y. Fu, J. Ren, J. Yang, P. Miao, X. Tong, D. Yu, S. Wang, Q. Liu, M. Kofu, R. Mole, G. Biasiol, D. Yu, I. A. Zaliznyak, J.-W. Mei, and L. Wu, Two-dimensional quantum universality in the spin-1/2 triangular-lattice quantum antiferromagnet Na2BaCo(PO4)2, Proc. Natl. Acad. Sci. U.S.A. 119, e2211193119 (2022).
  * Gao  _et al._ [2022] Y. Gao, Y.-C. Fan, H. Li, F. Yang, X.-T. Zeng, X.-L. Sheng, R. Zhong, Y. Qi, Y. Wan, and W. Li, Spin supersolidity in nearly ideal easy-axis triangular quantum antiferromagnet Na2BaCo(PO4)2, npj Quantum Mater. 7, 89 (2022).
  * Flynn  _et al._ [2021] M. O. Flynn, T. E. Baker, S. Jindal, and R. R. P. Singh, Two Phases Inside the Bose Condensation Dome of ${\mathrm{Yb}}_{2}{\mathrm{Si}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 126, 067201 (2021).
  * Feng  _et al._ [2023] C. Feng, E. M. Stoudenmire, and A. Wietek, Bose-Einstein condensation in honeycomb dimer magnets and ${\mathrm{Yb}}_{2}{\mathrm{Si}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. B 107, 205150 (2023).
  * Zhang  _et al._ [2019] L.-C. Zhang, L. Zhang, G. Qin, Q.-R. Zheng, M. Hu, Q.-B. Yan, and G. Su, Two-dimensional magnetic metal–organic frameworks with the Shastry-Sutherland lattice, Chem. Sci. 10, 10381 (2019).
  * Coak  _et al._ [2023] M. J. Coak, S. P. M. Curley, Z. Hawkhead, J. P. Tidey, D. Graf, S. J. Clark, P. Sengupta, Z. E. Manson, T. Lancaster, P. A. Goddard, and J. L. Manson, Asymmetric phase diagram and dimensional crossover in a system of spin-1/2 dimers under applied hydrostatic pressure, (2023), arXiv:2308.12671 .
  * Zhai  _et al._ [2018] Z.-W. Zhai, S.-H. Yang, M. Cao, L.-K. Li, C.-X. Du, and S.-Q. Zang, Rational Design of Three Two-Fold Interpenetrated Metal–Organic Frameworks: Luminescent Zn/Cd-Metal–Organic Frameworks for Detection of 2,4,6-Trinitrophenol and Nitrofurazone in the Aqueous Phase, Cryst. Growth Des. 18, 7173 (2018).
  * Nath  _et al._ [2023] A. Nath, V. Kumar, A. Shukla, H. N. Ghosh, and S. Mandal, Influence of Molecular Separation on Through-Space Intervalence Transient Charge Transfer in Metal–Organic Frameworks with Cofacially arranged Redox Pairs, Angew. Chem. Int. Ed. 62, e202308034 (2023).
  * Nath  _et al._ [2022] A. Nath, G. M. Thomas, S. Hans, S. R. Vennapusa, and S. Mandal, Crystal Packing-Driven Selective Hg(II) Ion Sensing Using Thiazolothiazole-Based Water-Stable Zinc Metal–Organic Framework, Inorg. Chem. 61, 2227 (2022).
  * Qureshi [2019] N. Qureshi, Mag2Pol: a program for the analysis of spherical neutron polarimetry, flipping ratio and integrated intensity data, J. Appl. Crystallogr. 52, 175 (2019).
  * Bauer  _et al._ [2011] B. Bauer, L. D. Carr, H. G. Evertz, A. Feiguin, J. Freire, S. Fuchs, L. Gamper, J. Gukelberger, E. Gull, S. Guertler, A. Hehn, R. Igarashi, S. V. Isakov, D. Koop, P. N. Ma, P. Mates, H. Matsuo, O. Parcollet, G. Pawłowski, J. D. Picon, L. Pollet, E. Santos, V. W. Scarola, U. Schollwöck, C. Silva, B. Surer, S. Todo, S. Trebst, M. Troyer, M. L. Wall, P. Werner, and S. Wessel, The ALPS project release 2.0: open source software for strongly correlated systems, J. Stat. Mech. 2011, P05001 (2011).
  * Arjun  _et al._ [2019] U. Arjun, K. M. Ranjith, B. Koo, J. Sichelschmidt, Y. Skourski, M. Baenitz, A. A. Tsirlin, and R. Nath, Singlet ground state in the alternating spin-$\frac{1}{2}$ chain compound ${\mathrm{NaVOAsO}}_{4}$, Phys. Rev. B 99, 014421 (2019).
  * Ahmed  _et al._ [2017] N. Ahmed, P. Khuntia, K. M. Ranjith, H. Rosner, M. Baenitz, A. A. Tsirlin, and R. Nath, Alternating spin chain compound ${\mathrm{AgVOAsO}}_{4}$ probed by ${}^{75}\mathrm{As}$ NMR, Phys. Rev. B 96, 224423 (2017).
  * Eichhöfer  _et al._ [2014] A. Eichhöfer, Y. Lan, V. Mereacre, T. Bodenstein, and F. Weigend, Slow Magnetic Relaxation in Trigonal-Planar Mononuclear Fe(II) and Co(II) Bis(trimethylsilyl)amido Complexes—A Comparative Study, Inorg. Chem. 53, 1962 (2014).
  * Iakovleva  _et al._ [2022] M. Iakovleva, T. Petersen, A. Alfonsov, Y. Skourski, H.-J. Grafe, E. Vavilova, R. Nath, L. Hozoi, and V. Kataev, Static magnetic and ESR spectroscopic properties of the dimer-chain antiferromagnet ${\mathrm{BiCoPO}}_{5}$, Phys. Rev. Mater. 6, 094413 (2022).
  * Lines [1963] M. E. Lines, Magnetic Properties of Co${\mathrm{Cl}}_{2}$ and Ni${\mathrm{Cl}}_{2}$, Phys. Rev. 131, 546 (1963).
  * Susuki  _et al._ [2013] T. Susuki, N. Kurita, T. Tanaka, H. Nojiri, A. Matsuo, K. Kindo, and H. Tanaka, Magnetization Process and Collective Excitations in the $S\mathbf{=}1/2$ Triangular-Lattice Heisenberg Antiferromagnet ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathbf{O}}_{9}$, Phys. Rev. Lett. 110, 267201 (2013).
  * Lal  _et al._ [2023] S. Lal, S. J. Sebastian, S. S. Islam, M. P. Saravanan, M. Uhlarz, Y. Skourski, and R. Nath, Double magnetic transitions and exotic field-induced phase in the triangular lattice antiferromagnets ${\mathrm{Sr}}_{3}\mathrm{Co}{(\mathrm{Nb},\mathrm{Ta})}_{2}{\mathrm{O}}_{9}$, Phys. Rev. B 108, 014429 (2023).
  * Li  _et al._ [2020] N. Li, Q. Huang, X. Y. Yue, W. J. Chu, Q. Chen, E. S. Choi, X. Zhao, H. D. Zhou, and X. F. Sun, Possible itinerant excitations and quantum spin state transitions in the effective spin-1/2 triangular-lattice antiferromagnet Na2BaCo(PO4)2, Nat. Commun. 11, 4216 (2020).
  * Ranjith  _et al._ [2017] K. M. Ranjith, K. Brinda, U. Arjun, N. G. Hegde, and R. Nath, Double phase transition in the triangular antiferromagnet Ba3CoTa2O9, J. Phys.: Condens. Matter 29, 115804 (2017).
  * Arjun  _et al._ [2017] U. Arjun, V. Kumar, P. K. Anjana, A. Thirumurugan, J. Sichelschmidt, A. V. Mahajan, and R. Nath, Singlet ground state in the spin-$\frac{1}{2}$ weakly coupled dimer compound ${\mathrm{NH}}_{4}[{({\mathrm{V}}_{2}{\mathrm{O}}_{3})}_{2}{(4,{4}^{{}^{\prime}}-bpy)}_{2}({\mathrm{H}}_{2}{\mathrm{PO}}_{4}){({\mathrm{PO}}_{4})}_{2}]\cdot{}0.5{\mathrm{H}}_{2}\mathrm{O}$, Phys. Rev. B 95, 174421 (2017).
  * Tsirlin  _et al._ [2011] A. A. Tsirlin, R. Nath, J. Sichelschmidt, Y. Skourski, C. Geibel, and H. Rosner, Frustrated couplings between alternating spin-$\frac{1}{2}$ chains in AgVOAsO4, Phys. Rev. B 83, 144412 (2011).
  * Nohadani  _et al._ [2005] O. Nohadani, S. Wessel, and S. Haas, Quantum phase transitions in coupled dimer compounds, Phys. Rev. B 72, 024440 (2005).
  * Shiba  _et al._ [2003] H. Shiba, Y. Ueda, K. Okunishi, S. Kimura, and K. Kindo, Exchange Interaction via Crystal-Field Excited States and Its Importance in CsCoCl3, J. Phys. Soc. Jpn. 72, 2326 (2003).
  * Nath  _et al._ [2015] R. Nath, M. Padmanabhan, S. Baby, A. Thirumurugan, D. Ehlers, M. Hemmida, H.-A. Krug von Nidda, and A. A. Tsirlin, Quasi-two-dimensional $S=\frac{1}{2}$ magnetism of $\mathrm{Cu[}{\mathrm{C}}_{6}{\mathrm{H}}_{2}(\text{COO}{)}_{4}\mathrm{][}{\mathrm{C}}_{2}{\mathrm{H}}_{5}{\mathrm{NH}}_{3}\mathrm{]}{}_{2}$, Phys. Rev. B 91, 054409 (2015).
  * Wellm  _et al._ [2021] C. Wellm, W. Roscher, J. Zeisner, A. Alfonsov, R. Zhong, R. J. Cava, A. Savoyant, R. Hayn, J. van den Brink, B. Büchner, O. Janson, and V. Kataev, Frustration enhanced by Kitaev exchange in a ${\stackrel{{\scriptstyle{}}}{{j}}}_{\text{eff}}=\frac{1}{2}$ triangular antiferromagnet, Phys. Rev. B 104, L100420 (2021).
  * Bartolome  _et al._ [2012] E. Bartolome, P. J. Alonso, A. Arauzo, J. Luzón, J. Bartolome, C. Racles, and C. Turta, Magnetic properties of the seven-coordinated nanoporous framework material Co($bpy$)1.5(NO3)2 ($bpy$ = 4,4’-bipyridine), Dalton Trans. 41, 10382 (2012).
  * Liu  _et al._ [2020] H. Liu, J. c. v. Chaloupka, and G. Khaliullin, Kitaev Spin Liquid in $3d$ Transition Metal Compounds, Phys. Rev. Lett. 125, 047201 (2020).
  * P. Cairns  _et al._ [2022] L. P. Cairns, R. Day, S. Haley, N. Maksimovic, J. Rodriguez, H. Taghinejad, J. Singleton, and J. Analytis, Tracking the evolution from isolated dimers to many-body entanglement in ${\mathrm{NaLu}}_{x}{\mathrm{Yb}}_{1\text{$-$}x}{\mathrm{Se}}_{2}$, Phys. Rev. B 106, 024404 (2022).
  * Tsirlin  _et al._ [2008] A. A. Tsirlin, R. Nath, C. Geibel, and H. Rosner, Magnetic properties of ${\mathrm{Ag}}_{2}\mathrm{V}\mathrm{O}{\mathrm{P}}_{2}{\mathrm{O}}_{7}$: An unexpected spin dimer system, Phys. Rev. B 77, 104436 (2008).
  * Kohama  _et al._ [2011] Y. Kohama, A. V. Sologubenko, N. R. Dilley, V. S. Zapf, M. Jaime, J. A. Mydosh, A. Paduan-Filho, K. A. Al-Hassanieh, P. Sengupta, S. Gangadharaiah, A. L. Chernyshev, and C. D. Batista, Thermal Transport and Strong Mass Renormalization in ${\mathrm{NiCl}}_{2}\mathrm{\text{$-$}}4\mathrm{SC}({\mathrm{NH}}_{2}{)}_{2}$, Phys. Rev. Lett. 106, 037203 (2011).