# Data-Efficient Learning for Sim-to-Real Robotic Grasping
using Deep Point Cloud Prediction Networks

Xinchen Yan
University of Michigan Work done during internship with Google Brain and X.
Mohi Khansari
X Inc  Jasmine Hsu
Google Brain  Yuanzheng Gong
X Inc  Yunfei Bai
X Inc  Sören Pirk
Google Brain  Honglak Lee
Google Brain

###### Abstract

Training a deep network policy for robot manipulation is notoriously costly
and time consuming as it depends on collecting a significant amount of real
world data. To work well in the real world, the policy needs to see many
instances of the task, including various object arrangements in the scene as
well as variations in object geometry, texture, material, and environmental
illumination. In this paper, we propose a method that learns to perform table-
top instance grasping of a wide variety of objects while using no real world
grasping data, outperforming the baseline using 2.5D shape by 10%. Our method
learns 3D point cloud of object, and use that to train a domain-invariant
grasping policy. We formulate the learning process as a two-step procedure: 1)
Learning a domain-invariant 3D shape representation of objects from $\sim$76K
episodes in simulation and $\sim$530 episodes in the real world, where each
episode lasts less than a minute and 2) Learning a critic grasping policy in
simulation only based on the 3D shape representation from step 1. Our real
world data collection in step 1 is both cheaper and faster compared to
existing approaches as it only requires taking multiple snapshots of the scene
using a RGBD camera. Finally, the learned 3D representation is not specific to
grasping, and can potentially be used in other interaction tasks. Video demos
can be accessed via https://sites.google.com/site/shapeawaresimtoreal/.

##  1 Introduction

Learning a domain-invariant representation for object manipulation in real-
world environments with minimum supervision is a fundamental challenge in
vision and robotics. State-of-the-art learning systems rely on collecting
large-scale datasets with interaction labels (e.g., success or failure) using
multiple robots (e.g., KUKA robotic arms) to update the model parameters in
parallel via end-to-end training [32]. The cost of deploying these learning
systems to a new setting (e.g. a novel task under different environment) is
very high and the efficiency is limited by the number of robots that can be
deployed in parallel. Therefore, existing learning based methods also use
simulation data to alleviate the data collection problem. However, the
existence of the sim-to-real gap imposes additional difficulty in transferring
from simulation to the real world. Recent work mainly focuses on domain
randomization by generating diverse configurations in simulation [10, 49, 23].
Other approaches use pixel-level or feature-level domain adaptation [3, 14,
24]. At best these methods still require a significant amount of unlabelled
real world data, which is still costly and time-consuming to collect and may
not transfer to a new task.

Figure 1: Architecture overview: (a) we use an object detection network to
obtain object-centric color, depth and mask images; (b) Our point cloud
prediction network allows us to generate a 3D point cloud of the detected
object; (c) Finally, we use a grasping critic network to predict a grasp.

In contrast to end-to-end training frameworks, we consider learning visual
structures as an intermediate representation for sim-to-real transfer. More
specifically, we embrace the recent advances in using deep neural networks to
predict 3D structures (e.g., 3D voxel grids, point clouds, and triangle
meshes) from single-view observation [13, 22, 51]. Compared to 2D sensory
input, 3D structure is known to be very useful for shape-based object
manipulation such as grasping [18, 31, 2]. For example, the geometric center
of an object, is a shape feature useful for its localization and manipulation,
which can be inferred directly from 3D structures [57].

Although shape prediction modules can be useful, it remains a non-trivial task
to apply the existing work to robotic platform in the real world. First,
inferring a depth image solely from a single-view RGB image based on
traditional computer vision techniques introduces ambiguities. Recent
learning-based work on image-to-depth prediction have demonstrated good
performance on the depth prediction from a single RGB camera [11, 60, 17],
however, applications of these methods to robotics tasks are not yet well
explored. Second, even with additional depth channel as input, sim-to-real
transfer is not straightforward. For example, depth sensors often inject
arbitrary noise (e.g. when the object is dark or transparent, or when there is
misalignment between depth and RGB) in the real world.

In this work, we design a novel shape prediction model that generates full 3D
point clouds of an object from a single-view RGBD image sampled from a
sequence containing multiple snapshots of the scene. We further explore cross-
view consistency as the self-supervision signal for training, as multiple
input views share the same intermediate representation given camera
transformations (e.g., rotating the 3D shape from one view to another).
Compared to other 3D representations such as voxels [56, 59] and triangle
meshes [27, 54], 3D point clouds are lightweight (i.e. low-dimensional) and
free from aliasing artifacts under camera transformations.

In summary, our contributions are:

  * •

we present a self-supervised shape prediction framework that reconstructs full
3D point clouds as representation for robotic applications;

  * •

we show data-efficient and robust sim-to-real transfer using this self-
supervised framework;

  * •

we demonstrate the application of the predicted point cloud on the robotic
task of table-top instance grasping using zero real-world grasping data.

##  2 Related Work

Learning to interact with objects is a wide and actively studied field of
vision and robotics research. Many approaches are based on using visual
features obtained from RGB or RGBD images to identify objects and grasping
points [47, 38, 30, 19, 29, 42, 44]. While early approaches focus on studying
the problem of grasping based on traditional techniques, such as logistic
regression or learning probabilities of grasp success [47, 38], more recent
approaches often rely on deep neural networks to extract more nuanced features
from images. This ranges from detecting objects [30] and their pose [19], to
learning grasp types from kinesthetic demonstrations [29], under gripper pose
uncertainty [26], or following an unsupervised learning scheme [9]. The
effectiveness of deep neural networks is unparalleled, however, training the
networks requires large-scale labeled datasets to generalize to unseen objects
[44]. Other approaches for robotic grasping focus on identifying the grasp
affordance of objects [8, 28] or by categorizing them according to their
function [41].

Another line of research focuses on reconstructing objects and scenes as 3D
triangle meshes, which in turn can be used to enable more informed robotic
behavior [33, 50]. Reconstructing objects from incomplete scans is
challenging. Various approaches exist to reconstruct the geometry of objects,
while considering the many facets of the problem [39, 40, 16, 7, 34, 48]. Very
recently, Henderson and Ferrari [22] introduced a network architecture to
generate 3D meshes, while only providing single image supervision. At a higher
level, our method is related to recent work on deep 3D representation learning
for robot-object interaction [12, 58, 55]. Moreover, to facilitate the
learning of 3D shape representations and the grasping of objects, recent
efforts concentrate on curating large shape and grasp repositories [4, 37,
36].

It has been recognized that point sets can serve as an effective
representation to obtain additional information of objects. Several recent
approaches learn representations to generate point clouds of shapes [13, 1,
15]. More closely targeted to robotics, Xu et al. [57] and Wang et al. [53]
propose fusion networks to extract pixel-wise dense feature embeddings for
estimating 3D bounding boxes and 6D object pose respectively. These approaches
focus on robotic vision, and unlike them we do not rely on the 6D object pose
as a label for training.

Reconstructing the geometry of an object allows identifying grasping points
more precisely and thereby to control the grasp [18, 31, 2]. To this end,
Varley et al. [51] and Yan et al. [58] recently proposed to reconstruct shapes
by learning geometry-aware representations based on 3D occupancy grids. Grids
serve as an efficient representation but they obscure the natural invariance
of 3D shapes under geometric transformations and only support to represent
shapes at low resolution. Similar to our method these approaches also aim at
reconstructing objects or object parts to enable more robust grasping.
However, unlike them we focus on the self-supervised reconstruction of full 3D
point clouds of objects. Point clouds enable a more robust sim-to-real
transfer and thereby facilitate efficient training.

Methods on sim-to-real transfer aim at training neural networks on simulated
data with the goal to operate on real data at inference time. This is also
known as domain adaptation, where a model is trained with data points of a
source domain to then generalize to a target domain [43, 6]. As simulated data
can be generated efficiently with ground-truth labels, a number of methods
focus on sim-to-real transfer by enhancing or synthesizing images [47, 52,
20]. More recently, Bousmalis et al. [3] introduced a generative adversarial
network pipeline to enhance simulated data to significantly reduce the number
of real-world data samples for a grasping task. Fang et al. [14] propose a
multi-task domain adaptation framework composed of three grasp prediction
towers for instance grasping in cluttered scenes. Finally, James et al. [24]
introduce a two-stage generator pipeline to translate simulated and real
images into a canonical representation to realize sim-to-real transfer. While
existing work mostly focuses on introducing architectures to operate on
images, we argue that point clouds of objects serve as an effective
representation to facilitate sim-to-real transfer.

Figure 2: Overview of our object detection and point cloud prediction
networks: we detect an object and obtain its cropped color and depth images
along with a binary mask based on an object detection network. We then use the
5 channels (RGBD-M) to train a point cloud prediction network that allows us
to predict a 3D point cloud (PC) of the detected object. Using the depth image
of the object and its mask, we compute a ground truth label for object depth.
The network is trained against an image-based loss of the projected 3D point
cloud and the ground truth mask of the object.

##  3 Methods

Given a set of RGBD observations, we aim at learning an intermediate
representation that is compact, domain-invariant, semantically interpretable,
and directly applicable for object manipulation. In particular, we use a point
cloud of a target object of interest as our intermediate representation for
learning due to the following reasons: (1) Point clouds are more lightweight
and flexible compared to other 3D representations such as voxel grids and
triangle meshes. (2) Point clouds describe the full 3D shape of a target
object, which is inherently invariant to surface textures or environmental
conditions. (3) A point cloud representation can directly be used to localize
objects in the scene, hence simplifying the tasks when training a policy. Our
approach includes two steps: 1) Learning a domain-invariant representation
using visual observations from simulation and real world, which will be
described in Section 3.1, and 2) Learning an object manipulation policy such
as grasping using the representation from step 1, which will be described in
Section 3.2.

###  3.1 Learning domain-invariant representation

Given a set of RGBD observations
$\left\\{\mathcal{O}_{1},\mathcal{O}_{2},\cdots,\mathcal{O}_{N}\right\\}$,
where $\mathcal{O}_{n}\in R^{h\times w\times 4}$ is an individual observation
which could come from simulation or real world, our goal is to learn a domain-
invariant point cloud representation $\mathcal{P}$ reflecting the 3D geometry
of the target object. These $N$ observations can be easily obtained by using a
mobile manipulator moving around and taking snapshots of the workspace from
different angles. We assume each object may be present in more than one
snapshot as it allows to reconstruct the 3D geometry of the object better.
However, we do not impose any explicit constraint on the number of snapshots
that the object should be present in. Please note that the depth values from
RGBD observations forms a 2.5D representation of the objects (e.g., visible
part subject to noise) and thus do not provide the full 3D geometry.
Furthermore, there is a reality gap between the depth values in simulation and
real world, hence making a policy that is solely trained in simulation quite
ineffective in the real world.

#### Self-supervised labeling.

While target point clouds for supervised learning of a deep network can be
easily obtained in simulation, this task becomes notoriously costly and time-
consuming for real data. Furthermore, the presence of noise and unmodeled
nonlinear characteristics in a depth sensor make the learning harder,
especially in the context of transfer learning. To address this challenge, we
base our framework on the recent work of learning to reconstruct 3D object
geometry using view-based supervision with differential re-projection
operators [25, 35].

We represent a point cloud $\mathcal{P}$ of an object $\mathcal{O}$ as a set
of $K$ points $\mathcal{P}=\\{\mathbf{p}_{k}=(x_{k},y_{k},z_{k})|1\leq k\leq
K\\}$, where $x_{k}$, $y_{k}$, $z_{k}$ are coordinates regarding $k$-th point
$\mathbf{p}_{k}$ along xyz axis, respectively. Without loss of generality, we
assume the point cloud coordinates are defined in the camera frame. We assume
the ground-truth point cloud annotation is not directly available in the real
world data and thus use multi-view projections as the supervision signal. More
specifically, we use the camera intrinsic matrix $E$ to obtain the 2D
projection $\mathcal{M}$ in the image space from the point cloud $\mathcal{P}$
(e.g., homogeneous coordinate $(u_{k},v_{k},1)$ is projected from
$(x_{k},y_{k},z_{k})$):

$\displaystyle(u_{k},v_{k},1)^{\top}$ $\displaystyle\sim
E\;(x_{k},y_{k},z_{k})^{\top}$ (1a) $\displaystyle\mathcal{M}$
$\displaystyle=\left\\{(u_{k},v_{k})\right\\}_{k=1}^{K}$ (1b)

For localization, the corresponding tight bounding box can be derived from the
2D projection: $\mathcal{B}=(u^{\text{mid}},v^{\text{mid}},w,h)$, where
$u^{\text{mid}}$, $v^{\text{mid}}$, $w$, $h$ represents the bounding box
center and size, respectively.

We collect $N$ RGBD snapshots from various scenes in simulation and real world
by moving a mobile manipulator around the workspace. For the real world
dataset we use Mask-RCNN [21] to detect object bounding boxs $\mathcal{B}^{n}$
and their associated mask at each frame. For the simulation dataset bounding
boxes can directly be obtained. Note that it is quite common that multiple
objects may be present in each snapshot. We denote the data associated to the
$m$-th object in the $n$-th frame by $(.)^{m,n}$. We also denote the number of
objects in the $n$-th frame by $C_{n}$. Next, we use the mask for each object
to extract its associated depth values from the depth channel in each
observation and then use the camera intrinsic matrix $E^{n}$ to obtain
$\mathcal{M}^{m,n}$ from the depth values. At the end of this step, we obtain
$\mathcal{M}^{m,n}$ and $\mathcal{B}^{m,n}$ for all $1\leq n\leq N$ and $1\leq
m\leq C_{n}$.

Our deep net provides an estimate of the point cloud $\hat{\mathcal{P}}$ as
the output which can be used to determine $\hat{\mathcal{M}}^{m,n}$ and
$\hat{\mathcal{B}}^{m,n}$ using Eq. 1. Then we define the loss function for
training the domain invariant point cloud representation as follows:

$\displaystyle\mathcal{L}_{\theta}=\sum_{n=1}^{N}\sum_{m=1}^{C_{n}}\Big{(}\lambda^{\mathcal{B}}\mathcal{L}_{\theta}^{\mathcal{B}}(\hat{\mathcal{B}}^{m,n},\mathcal{B}^{m,n})$
$\displaystyle+$
$\displaystyle+\lambda^{\mathcal{M}}\mathcal{L}_{\theta}^{\mathcal{M}}(\hat{\mathcal{M}}^{m,n},\mathcal{M}^{m,n})\Big{)}+\lambda^{\theta}\|\theta\|$
(2)

where $\lambda^{\mathcal{B}}$, $\lambda^{\mathcal{M}}$, and $\lambda^{\theta}$
are weighting coefficients, $\mathcal{L}_{\theta}^{\mathcal{B}}$ is the Huber
loss between the estimated and labeled bounding box, and
$\mathcal{L}_{\theta}^{\mathcal{M}}$ is the projected point-cloud prediction
loss based on [25]. We extend their method in the following way: we do not
rely on obtaining a full 3D point cloud as this is more difficult to
accomplish in real-world environments. Moreover, while [25] uses synthetic 3D
point clouds for training to reconstruct normalized shape centered at the
origin, our goal is to reconstruct shapes at real-world scale.

Network architecture. Figure 2 illustrates the network architecture we use for
shape prediction. Similarly to [25], we use a network that is composed of
several encoder-decoder modules [13] and a fully-connected layer to predict
the point clouds. However our proposed architecture is different from [25] in
three ways in order to make it applicable for real world robotics setting: (1)
we use object masks as an additional input channel to handle situations when
multiple objects are present in the scene (a very common situation in robot
settings). Therefore, the number of input channels to our network is five
(RGBD-Mask), (2) we introduce a dynamic image cropping step on the RGBD-M
channels. This allows to get a more focused view of the target instance, and
(3) to account for the dynamic cropping, we add an additional input right
after the encoder to provide the network with the adapted camera intrinsic
characteristic resulted from cropping.

###  3.2 Learning Point cloud-based Grasping Policy

In this section we describe how the learned point cloud representation can be
used to perform table-top instance object grasping. In this work, we use a
critic network $\mathcal{G}(\mathcal{P},\mathbf{s},\mathcal{T}_{R}^{C})$ to
predict the probability of the success for a sample table-top grasp
$\mathbf{s}\in R^{4}$ based on the predicted point cloud $\mathcal{P}$ for the
target object, and the transformation from the robot base to the camera frame.
In our setup the sample $\mathbf{s}=(\mathbf{p},\psi)$ is composed of the 3D
gripper position $\mathbf{p}$ with respect to the robot base and the gripper
yaw rotation $\psi$.

#### Network architecture.

Figure 3 shows the architecture of our grasp prediction network. For the
preprocessing, we first transform the point cloud $\mathcal{P}$ to the
proposed grasp frame using:

$\mathcal{P}_{s}=\mathcal{T}_{s}^{R}\mathcal{T}_{R}^{C}\mathcal{P}$ (3)

where $\mathcal{T}_{s}^{R}$ can be directly calculated based on the sample
grasp pose $\mathbf{s}$. We then shuffle the order of points in
$\mathcal{T}_{s}^{R}$ to allow the network to adapt to variations in the order
of point clouds.

Our grasping critic network is derived from PointNet [45] architecture, which
is composed of 4 fully-connected layers each followed by a ReLU activation
function with BatchNorm. The last layer is linear and reduces the output size
to 1. It is followed by a sigmoid activation function to provide the grasp
success.

#### Generating grasping data for training.

We collected data for training the grasp policy only in simulation. We use a
heuristic grasping policy for the data collection as follows: (1) Compute the
center of volume of the object $\mathbf{p}_{\text{obj}}$ based on the
predicted point cloud, (2) Set the translation part of the grasp pose to
$\mathbf{p}_{\text{obj}}$ plus some random noise $\mathbf{p}_{\epsilon}\in
R^{3}$, i.e.
$\mathbf{p}_{\text{grasp}}=\mathbf{p}_{\text{obj}}+\mathbf{p}_{\epsilon}$, (3)
Randomly draw a yaw angle from a uniform distribution in the range
$[-\pi/2,\pi/2]$.

Figure 3:  Overview of our point cloud-based grasping network. The point
clouds are first transformed to the gripper frame, followed by a
classification network adapted from PointNet [45] architecture with 5 fully-
connected layers in total.

We then evaluate the grasp success by moving the arm first to a pre-grasp pose
$\mathbf{s}^{*}$, where $\mathbf{s}^{*}$ is a pose exactly above $s$ with some
height difference constant, i.e.
$\mathbf{s}^{*}-\mathbf{s}=(0,0,\Delta{h},0)$. This allows to properly pose
the robot end-effector with respect to the object before attempting the grasp.
Then the robot is moved to pose $\mathbf{s}$, and finally we command the robot
to close its parallel-jaw gripper. The robot is then commanded to lift the
object by moving back to $\mathbf{s}^{*}$. Then the grasp success is evaluated
by checking whether the object is moved above the table. This evaluation can
be done easily since we have access to the ground truth object pose in
simulation. The training data is collected by running simulation robots in
parallel and stored for training the off-policy grasping network.

#### Runtime evaluation.

One of the main advantages of using point clouds is that we directly have
access to the 3D location of an object in the scene. Given this knowledge, the
simplest approach to infer a grasp pose is to randomly sample several grasp
pose candidates around the object pose (similar to our data collection
procedure), and then take the command with the highest grasp success
probability. However, to get better results, we use the cross-entropy method
(CEM) [46] to find the most suited grasp pose by using a simple derivative-
free optimization technique. Using CEM, at each iteration we sample a batch of
$N_{\text{CEM}}$ grasp candidates. Then evaluate these samples and pick
$N_{\text{elite}}$ best ones ($N_{\text{elite}}\ll N_{\text{CEM}}$). Next, we
fit a Gaussian distribution to $N_{\text{elite}}$ and then sample a new batch
of size $N_{\text{CEM}}$. We repeat this process a few times until either a
grasp candidate above the desired success probability threshold $\alpha$ is
achieved or when the CEM reaches the maximum number of allowed iterations. In
our implementation we use $N_{\text{CEM}}=100$, $N_{\text{elite}}=10$,
$\alpha=0.9$ and max number of iterations 3.

Note that since we are directly sampling the grasp pose, we can directly
impose additional constraints to our sampling strategy (e.g., limiting samples
to the desired workspace as well as removing kinematically infeasible
samples).

##  4 Experiments

In the following we provide details on our real and simulated datasets and
discuss results on shape prediction and grasping performance with a real
robot.

###  4.1 Datasets.

Table 1: Statistics of shape prediction datasets.

Dataset (split) | # objects | # episodes | # object-centric seqs
---|---|---|---
ShapeNet | 1,345 | 38,653 | 189,424
Kitchenware (train) | 69 | 37,568 | 142,293
Kitchenware (test) | 20 | 3,092 | 11,956
Real (train) | 110 | 534 | 38,301
Real (val) | 34 | 49 | 2,720
Real (test) | 43 | 291 | 18,462

Table 2: Statistics of simulated grasping datasets. Dataset (split) | # objects | # episodes
---|---|---
Kitchenware (train) | 69 | 358,286
Kitchenware (test) | 20 | 2,111
Procedural | 1,000 | 633,069

#### Shape prediction datasets.

We use both real-world data and data generated from simulation for learning
domain-invariant 3D shapes. As shown in Table 1, our real-world dataset
contains 187 distinct object instances in total, covering 8 object categories:
balls, bottles cans, bowls, cups, mugs, wine glasses, and plates. To obtain
sufficient objects for training our shape prediction model, we use additional
CAD models from ShapeNet [4] and Kitchenware [58]. We use an off-the-shelf
simulator PyBullet [5] to create virtual scenes in the simulation. For
ShapeNet, we select $1,345$ objects from 6 categories: bags, bottles, bowls,
cans, jars, and mugs, which are similar to the objects of our real-world
dataset. Moreover, we use Procedural dataset [3] containing $1,000$
procedurally generated object shapes for evaluation.

#### Simulated grasping datasets.

Assuming our shape representation is domain-invariant, we only generate
grasping data in simulation. As shown in Table 2, we use the Kitchenware and
Procedual dataset [3] for training. Additionally, we use a subset of the
Kitchenware dataset for grasping evaluation in simulation. These are held-out
objects which have never been used for learning the domain-invariant 3D
shapes.

Figure 4:  Overview of the dataset used for learning the domain-invariant
shape prediction model. We visualize the object instances used for training
the point cloud prediction model in the real world and simulation (e.g.,
Kitchenware dataset and ShapeNet subset) from top to bottom.  Figure 5:
Examples of data collection for shape prediction in the simulated (a) and real
environment (b) and for instance grasping in the simulated (c) and real (d)
setup.

#### Scene construction and pre-processing.

As illustrated in Figure 5, in both real and simulation environment, we put a
table in front of the mobile manipulator and randomly set the table height
between 30 and 50 centimeters. In the simulation, we perform additional
randomization over the table location and object textures (e.g., texture
pattern of the background, color of the table). We place 4 to 5 objects on the
table with arbitrary location and orientation. Given the scene arrangement, we
place the mobile manipulator at 5 different angles looking at the table from
one side. For each viewpoint, the robot takes one snapshot containing RGB and
depth images with a resolution of $512\times 640$.

For object detection and segmentation we used Mask R-CNN [21]. Given an image
of a scene, Mask-RCNN detects and segments objects above a threshold, which
generates bounding boxes and segmentation masks for each instance of an
object. For our setup we are interested in detecting four types of object
categories: bottles, wine glasses, cups, and bowls. If bounding boxes overlap
we use an IOU threshold of 0.5 to remove duplicate objects as part of the non-
maximum suppression step. For each detected object, we generated a cropped
image and re-scale it to size $192\times 256$. For multiple snapshots taken at
the same scene, we associate objects across snapshots based on the additional
depth information. For the simulated environment, we introduce $4$ virtual
viewpoints to obtain a full 360 degree capture of the scene. The robot looks
at the table from different viewpoints. We then extract object point clouds
based on the object detection inference results.

Figure 6: Visualizations of point clouds generated with our point prediction
network. From left to right: the input image, the generated 3D point cloud,
and two different views with the projected point cloud as an overlay on the
detected object. Our method allows us to produce meaningful point clouds from
real and simulated test data (Kitchenware, Procedural).

###  4.2 Experimental Results

#### Evaluation metrics.

We consider both shape prediction performance and grasping performance for
evaluation. For shape prediction, we use bounding boxes from the simulation or
Mask-RCNN as ground-truth. We project the predicted 3D point cloud to 2D and
evaluate the overlap of the prediction and the ground truth. To measure the
performance, we run the instance grasping and compute the grasping success
rate. Please note that our method relies on object detection based on Mask-
RCNN, which may fail to reliably detected objects. While this can consequently
also result in grasping failure, we consider this as an orthogonal problem
that is outside the scope of this work.

#### Evaluation on shape prediction.

To decide whether our shape prediction model is able to generate meaningful
point clouds from a single RGBD observation, we conducted qualitative and
quantitative evaluations on predicted 3D point clouds. In Figure 6 we
illustrate that our shape prediction model is able to successfully predict 3D
point clouds. The point clouds (second column) are shown from the view of the
input image. One advantage of point clouds as representation is that they can
be easily transformed from one view to another. The right-most column in
Figure 6 shows the projected 2D points still align with the object when looked
at from another view. This also addresses the concern that our model does not
learn the trivial mapping from input object mask. It is worth noting that our
shape prediction model also generalizes to unseen categories from Procedural
dataset (see the last two rows of Figure 6).

For a quantitative evaluation, we use more than 10K image patch sequences
containing the object instances of the test set of both Kitchenware and our
real data. We compute the averaged IOU of the 2D projections with ground-truth
masks (from the simulation or Mask-RCNN) and summarize the results in Table 3.
We also conducted an ablation study on the number of viewpoints used during
training. The results are reported in Table 3. The model performs poorly when
only one view is used for supervision, however, the performance increases when
multiple views are provided.

Table 3:  Shape Prediction IOU on unseen objects.

Dataset $/$ # views | 1 | 2 | 4 | full
---|---|---|---|---
Kitchenware | 0.188 | 0.659 | 0.803 | 0.803
Real | 0.186 | 0.492 | 0.625 | 0.626

Table 4:  Grasping success rate on unseen objects.

Dataset | 2.5D shape | Reconstructed 3D shape
---|---|---
Kitchenware | 68% | 64%
Real | 51% | 61%

Figure 7: Grasping sequence evaluation: we visualize the real world grasping
sequences for the baseline model (left) and our model (right). For both, the
first two columns show the depth and RGB image before we run grasping trials.
The third column illustrates the pre-grasp state where the robot arm has been
moved on top of the target instance. The last column shows the final state
after the grasping has been executed.

#### Evaluation on instance grasping.

Finally, we evaluated whether the learned critic model is able to guide the
CEM policy for grasping on both simulated and real data (the dataset contains
$20$ and $43$ unseen objects respectively). Specifically, we execute 100
grasping trials in the real world and in the simulation. We report the average
grasping success rate in Table 4. During execution, we follow the same
protocol used for generating the simulation dataset for training (see Figure
5). The robot is expected to pick up one specific object from the table and
drops it into a bin. If the grasp was not successful, we manually remove the
object from the table. This prevents the model from repeatedly grasping the
same object.

As reported in Table 4, CEM policy guided by our critic model achieves 64%
instance grasping success rate in simulation and 61% in the real world
execution with zero real data for training. This is a significant improvement
over previous work that aims at sim-to-real transfer at image-level (e.g., [3]
achieved 23.53% indiscriminative grasping success trained only with simulated
data).

To evaluate the performance of grasping success with our reconstructed 3D
point clouds as representation for sim-to-real transfer, we learn a baseline
critic model that only uses raw 2.5D sensor depth as the input. As reported in
Table 4, the CEM policy guided by the baseline critic model achieves slightly
higher grasping success rate in simulation (e.g., 68% vs. 64%), as it is
trained on the clean ground-truth point cloud. However, policy guided by the
baseline model suffers severely from domain shift, as the success rate drops
from 68% to 51% when applied in the real-world environment. As we used the
same architecture and training set for both our model and the baseline, we
believe the performance gap is the result of domain shift (e.g., the unmodeled
noise from the depth input, the noise introduced by our Mask-RCNN model). Due
to our domain-invariant 3D representation, the policy based on our critic
model achieved 10% higher success rate compared to the policy based on raw
depth from the sensor. These results are shown in Table 4 and Figure 7. To
summarize, the improvement illustrates clear advantages of our compact and
geometry-aware domain-invariant representation.

##  5 Conclusions and Future Work

In this work we presented a novel self-supervised approach that learns to
perform table-top instance grasping of objects using no real world grasping
data. The proposed framework consists of a shape prediction model that learns
a domain-invariant 3D point cloud representation of objects by operating on
single RGBD images containing the same object from different viewpoints.
Moreover, the 3D point cloud is further utilized to perform instance grasps
based on a critic model learned with simulated grasping data only.

Experimental results have demonstrated that (1) our shape prediction model is
able to learn domain-invariant 3D shapes in real world settings from single
RGBD image observations with self-supervision; (2) the policy guided by our
representation generalizes significantly better in the real world compared to
previous state-of-the-art based on end-to-end training and a policy based on a
2.5D shape representation.

In a broader context, our shape-aware representations provide a better
understanding of objects and thereby have the potential to enable more robust
robotic behavior towards grasping and other manipulation tasks. As avenues for
future work, it would be interesting to explore the potential of such
representations with respect to more diverse object categories or end-effector
configurations as well as a larger number of tasks.

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# Quantum criticality in Ce1-xSmxCoIn5

D. L. Kunwar  Department of Physics, Kent State University, Kent, Ohio, 44242,
USA  R. B. Adhikari  Department of Physics, Kent State University, Kent, Ohio,
44242, USA  N. Pouse  Center for Advanced Nanoscience, University of
California, San Diego, La Jolla, California 92093, USA  Department of Physics,
University of California at San Diego, La Jolla, CA 92903, USA  M. B. Maple
Center for Advanced Nanoscience, University of California, San Diego, La
Jolla, California 92093, USA  Department of Physics, University of California
at San Diego, La Jolla, CA 92903, USA  M. Dzero  Department of Physics, Kent
State University, Kent, Ohio, 44242, USA  C. C. Almasan  Department of
Physics, Kent State University, Kent, Ohio, 44242, USA

###### Abstract

Motivated by the possibility of observing the co-existence between magnetism
and unconventional superconductivity in heavy-fermion Ce1-xSmxCoIn5 alloys, we
studied how the samarium substitution on the cerium site affects the magnetic
field-tuned-quantum criticality of stoicheometric CeCoIn5 by performing
specific heat and resistivity measurements. By applying an external magnetic
field, we have observed Fermi-liquid to non-Fermi-liquid crossovers in the
temperature dependence of the electronic specific heat normalized by
temperature and of the resistivity. We obtained the magnetic-field-induced
quantum critical point (QCP) by extrapolating to zero temperature the
temperature - magnetic field dependence at which the crossovers take place.
Furthermore, a scaling analysis of the electronic specific heat is used to
confirm the existence of the QCP. We have found that the magnitude of the
magnetic-field-induced QCP decreases with increasing samarium concentration.
Our analysis of heat capacity and resistivity data reveals a zero-field QCP
for $x_{\textrm{cr}}\approx 0.15$, which falls inside the region where Sm ions
antiferromagnetism and superconductivity co-exist.

###### pacs:

71.10.Hf, 71.27.+a, 74.70.Tx

###### pacs:

71.10.Ay, 74.25.F-, 74.62.Bf, 75.20.Hr

##  I Introduction

CeCoIn5 is an unconventional superconductor in the family of the ‘115’ heavy
fermion materials with a fairly high transition critical temperature of Tc at
2.3 K. A consensus exists by now that the unconventional superconductivity in
the ‘115’ system is likely governed by its proximity to an antiferromagnetic
critical point at zero temperature [1, 2]. Generally, these materials can be
tuned to a quantum phase transition at a quantum critical point (QCP) by
either chemical substitutions [3, 4], pressure [5, 6], or by applying an
external magnetic field [7, 8]. Consequently, interactions between conduction
electrons and critical fluctuations associated with the underlying QCP lead to
a manifestation of quite unusual physical properties in both normal and
superconducting phases of the ‘115’ materials [9, 10]. The fact that a QCP
does exist in these materials is usually elucidated by performing a scaling
analysis of the thermodynamic response functions, such as specific heat and
magnetic susceptibility.

A significant amount of experimental data, as well as theoretical results,
strongly suggest that the Cooper pairing in CeCoIn5, albeit unconventional, is
mediated by the interaction between conduction electrons and localized
magnetic moments of partially filled $f$-shells of cerium ions. A tendency
towards an antiferromagnetic ordering itself is driven by the exchange
interaction between cerium magnetic moments. The exchange interaction - a
driver for an antiferromagnetic transition - is also thought to produce soft
long-range bosonic modes - a pairing glue for the conduction electrons - to
ultimately induce superconductivity with $d$-wave order parameter [11].

Naturally, the chemical substitution of magnetic (and nonmagnetic) rare earth
ions for magnetic Ce3+ not only allows one to study the effect of inter-site
interactions on the QCP, but also separate the single-ion Kondo effect from
effects associated with the magnetic exchange interaction. Indeed, the
underlying quantum phase transition may shift under the effect of an external
magnetic field or pressure. As a result, one may expect that it would affect
the superconducting transition temperature, as well as the thermodynamic
response functions in the normal state. Thus, if we were to believe the
hypothesis that QCP is governing the unusual transport and thermodynamic
properties observed in superconducting and normal states [12, 2, 13],
systematic studies of the ‘115’-based alloyed compounds could offer an
opportunity to get a deeper insight as well as to unveil differences and/or
similarities pertaining to both normal and superconducting states. An
additional important aspect of the problem consists in the fact that chemical
substitutions inevitably bring disorder into a system, so that the putative
QCP may or may not be smeared by the effects associated with the induced
spacial inhomogeneities.

Recently, there have been several reports on alloys of Ce1-x _M_ xCoIn5, where
$M$ is a member of lanthanide family with unfilled $f$-orbital shells. The
general motivation for studying such alloys lies in an attempt to stabilize
the phase of co-existence between superconductivity of the host ’115’ system
and magnetism governed by the impurity magnetic moments. Indeed, one can
envision a scenario in which the suppression of superconductivity would be
slow enough so that both Néel ($T_{N}$) and superconducting ($T_{c}$) critical
temperatures are finite in some region of the phase diagram. In fact, similar
effects have been observed in iron-based superconductors [14], although
magnetism in those materials is itinerant and the co-existence between
$s^{\pm}$-wave superconductivity and spin-density-wave order is possible due
to the Anderson theorem [15, 16].

Ce1-xYbxCoIn5 is one example of such an alloy: although Yb is supposed to be
in the magnetic Yb3+ valence configuration, with an increase in ytterbium
concentration, its valence configuration changes from magnetic to a non-
magnetic Yb2+. Nevertheless, one of the intriguing results is that the
magnetic-field-induced QCP ($H_{\textrm{QCP}}$) of the stoichiometric compound
is fully suppressed in the alloy Ce1-xYbxCoIn5 for $x>0.1$ without
substantially affecting the unconventional superconductivity and non-Fermi-
liquid (NFL) behavior [17]. Specifically, a zero-field QCP separating
paramagnetic and antiferromagnetic phases is observed in Ce0.91Yb0.09CoIn5 and
its presence has been confirmed by the scaling analysis of the specific heat
data [18]. This is a surprising result given the fact that the dependence of
the superconducting critical temperature on ytterbium concentration does not
display any correlation with the suppression of the $H_{\textrm{QCP}}$.

Another example is provided by Ce1-xSmxCoIn5. Samarium Sm3+ replaces Ce3+ in
CeCoIn5 and, unlike Yb, it remains in the same integer valence configuration.
One needs to keep in mind that existing ‘electron-hole symmetry’ between
4$f^{1}$ (Ce${}^{3+})$ and 4$f^{5}($Sm${}^{3+})$ valence configurations
implies that larger magnetic moments are introduced into the system through
this substitution without changing the carrier density. The phase diagram in
the temperature-doping $(T-x)$ plane can be generated based on the $T_{N}$ and
$T_{c}$ measured from resistivity $\rho(x)$ and specific heat $C/T$ of
Ce1-xSmxCoIn5 for $0\leq x\leq 0.3$ in zero field, Fig. 1. Therefore, in
contrast to Ce1-xYbxCoIn5, superconductivity in this alloy is completely
suppressed at $x\approx 0.18$ and long-range antiferromagnetic (AFM) order
emerges in the sub-lattice of Sm moments in Ce1-xSmxCoIn5 for $x\approx 0.10$.
While fairly fast (in comparison with Ce1-xYbxCoIn5 alloys), the suppression
of superconductivity is not surprising given the sizable magnetic moment of
samarium. The fact that the AFM order emerges before superconductivity has
been fully suppressed does provide long thought playground to investigate
possible co-existence between unconventional superconductivity and magnetism.
Furthermore, at the critical concentration
$x_{\textrm{cr}}^{\textrm{Sm}}\approx 0.1$ where the Néel temperature for the
samarium sublattice vanishes, an antiferromagnetic QCP may be also present.

Figure 1: (Color online) Temperature - Sm concentration $T-x$ phase diagram of
Ce1-xSmxCoIn5. The $T_{c}(x)$ and $T_{N}(x)$ data shown here are in part take
from Ref. [19] and from present work. The solid lines are theoretical fits to
the experimental data points. We fitted the superconducting critical
temperature using Abrikosov-Gor’kov expression for $T_{c}(x)$ for
superconductors with $d$-wave symmetry of the order parameter. The Néel
temperature $T_{N}$ depends linearly on $x$ at small enough values of $x$ as
can be shown by Quantum Monte Carlo simulations and can also be seen by
employing a perturbation theory [20, 21]. Inset: Dependence of the magnetic-
field-induced quantum critical point $H_{QCP}$, which separates magnetic and
non-magnetic states at zero temperature, is shown as a function of samarium
concentration. When $x>0.16$ $H_{QCP}$ of stoicheometric CeCoIn5 becomes fully
suppressed implying the absence of the anti-ferromagnetic transition in the
sub-lattice of Ce moments even at absolute zero temperatures.

Samarium substitutions must also affect the magnitude of $H_{\textrm{QCP}}$
separating the anti-ferromagnetic and paramagnetic phases of cerium sub-
lattice at zero temperature. Specifically, as in the case of Yb substitutions,
one may expect that the value of $H_{\textrm{QCP}}$ will be suppressed to zero
for some critical concentration $x_{\textrm{cr}}$ of samarium ions,
$H_{\textrm{QCP}}(x_{\textrm{cr}})=0$. In the inset of Fig. 1 we see that the
magnetic field induced quantum phase transition of the cerium lattice is
clearly affected by the samarium substitutions. In the light of our discussion
above, one may then wonder whether $x_{\textrm{cr}}$ satisfies
$x_{\textrm{cr}}^{\textrm{Sm}}\leq x_{\textrm{cr}}\leq x^{*}$ (with
$T_{c}(x^{*})=0$) falls into the region where antiferromagnetism induced by
the ordering of Sm moments and superconductivity may co-exist or not. These
considerations have lead us to select the samples $x=0.10$ and $x=0.15$ to
investigate the changes in the magnitude of $H_{\textrm{QCP}}$ through
specific heat and resistivity measurements done in the presence of an applied
magnetic field ($H$). In what follows, we present the results of specific heat
and resistivity measurements on Ce0.9Sm0.10CoIn5 and Ce0.85Sm0.15CoIn5 to
elucidate the role played by quantum critical fluctuations and their effect on
unconventional superconductivity. Our investigation reveals that quantum
critical fluctuations strongly correlate with unconventional
superconductivity. The field driven QCP is suppressed with increasing Sm
concentration. The observed pronounced crossovers from FL to NFL behavior, as
well as the scaling analysis of the $C_{e}/T$ on both alloys show that the
normal state of Ce0.85Sm0.15CoIn5 is quantum critical.

##  II Experimental Details

Single crystals of Ce1-xSmxCoIn5 ($x=0.10$ and $0.15$) were grown using the
molten In flux method in alumina crucibles, as described in Ref. [22]. The
composition and crystal structure were determined from X-ray powder
diffraction (XRD) and energy dispersive X-ray (EDX) techniques. The crystals’
actual chemical composition is the same as that of the nominal doping, as
confirmed by the EDX using FFI Quanta 600 scanning electron microscope
equipped with an INCA EDX detector from Oxford Instruments [19].

The crystals were cut into a typical size of 2.0x0.5x0.17 mm3, with the
$c$-axis along the shortest dimension of the crystals. These single crystals
were first polished with sandpaper and then etched in a 5% HCl solution for
three hours to remove the indium left on the surface during the growth
process. Then they were washed thoroughly with ethanol to remove any acid and
impurities.

Heat capacity measurement was performed in an external magnetic field in the
range $0\leq H\leq 14$ T applied perpendicular to the $ab$-plane of each
crystal and in the temperature range 0.42 K $\leq T\leq 8$ K. The data were
obtained using a relaxation technique in the He-3 option of the Quantum Design
Physical Properties Measurement System.

Four gold leads were attached to each crystal using silver-based epoxy, with
the current $I\parallel a$-axis of the single crystal. We performed
temperature ($T$) dependent electrical resistivity $(\rho(T))$ measurements
between 0.5 K and 300 K in zero magnetic field to extract the superconducting
transition temperature and the Kondo lattice coherence peaks. The resistivity
in field $(\rho(H))$ was measured by scanning temperature from 0.6 K to 10 K
for selected field values from 6 T to 14 T. Then in-plane transverse
magnetoresistivity ($\Delta\rho_{a}^{\perp}/\rho_{a}$) was measured by
scanning the field from -14 T to 14 T for selected temperatures between 2 and
50 K.

##  III Results

###  III.1 Ce0.90Sm0.10CoIn5

Figure 2: (Color online) Semi-Log plot of the electronic specific heat
normalized by temperature $T$, $C_{e}/T$ vs $T$ of Ce0.90Sm0.10CoIn5 measured
in applied magnetic field $H\parallel c$ axis over the temperature range 0.42
K $\leq T\leq 10$ K. $T^{C}_{FL}$ represents the temperature at which the
behavior changes from Fermi-liquid to non-Fermi liquid. Figure 3:  (Color
online) Low temperature $T$ resistivity $\rho$ plotted as a function of T2 in
the field range 6 T$\leq H\leq$ 14 T for Ce0.90Sm0.10CoIn5. The solid lines
represent linear fits of the data with $\rho(H)=\rho_{0}+AT{{}^{2}}$. Inset:
Plot of the slope $A$ as a function of magnetic field $H$. The red solid line
is a fit of the data.

#### Specific heat data.

Figure 2 shows the temperature dependence of the electronic specific heat
normalized by temperature $C_{e}(T)/T$ of Ce0.90Sm0.10CoIn5 measured over the
field range 6 T $\leq H\leq$ 14 T and temperature range 0.4 K $\leq T\leq$ 8
K. We have subtracted the specific heat [19] of the non-magnetic reference
compound LaCoIn5 from the measured specific heat to get the electronic and
magnetic contributions to the specific heat. Then, to obtain the electronic
contribution to the specific heat, we subtracted the high temperature tail of
the Schottky anomaly due to quadrupolar and magnetic spin splitting of Co and
In nuclei [12]. All the data shown in Fig. 2 are normal state results since
superconductivity is suppressed in this temperature range measured at fields
above 4 T (see purple data taken at 4 T in the inset to Fig. 6).

Applying an external magnetic field, we observed definite crossovers from
constant $C_{e}/T$ vs $T$ at low temperatures, i.e. Fermi-liquid behavior, to
logarithmic temperature-dependent $C_{e}/T$ with $C_{e}/T\propto$ ln T-0.4 at
high temperatures, i.e., non-Fermi liquid behavior. For the measured
temperature range, the clearly visible FL regime ($C_{e}/T$ = constant) at low
temperatures is observed for $H\geq 6$ T. The data taken in all these applied
magnetic fields exhibit the FL to NFL crossovers at a temperature
$T^{\mathrm{C}}_{\mathrm{FL}}$ that shifts towards higher temperatures with
increasing $H$, as shown by the vertical dashed lines of Fig. 2. The
$T^{\mathrm{C}}_{\mathrm{FL}}$ from each specific heat data measured at
different fields are extracted and plotted as shown in the $H-T$ phase diagram
of Fig. 5.

#### Resistivity data.

Figure 3 shows the $T^{2}$ dependence of resistivity of Ce0.90Sm0.10CoIn5
measured in different $H$. The resistivity is linear in $T^{2}$ at low
temperatures: $\rho(H,T)=\rho_{0}+AT^{2}$, where $\rho_{0}$ is the residual
resistivity, and $A(H)$ is a constant that measures the strength of electron-
electron interactions. The linear in $T^{2}$ behavior of resistivity at low
$T$ and high $H$ reveals the recovery of the FL behavior. Its deviation from
linearity with increasing temperature is the signature of the NFL behavior
[23]. The crossovers from linear (FL) to non-linear (NFL) $T^{2}$ dependence
of $\rho(H,T)$ at low temperatures represented by
$T^{\mathrm{R}}_{\mathrm{FL}}$, shift towards higher temperatures with
increasing magnetic field. The $T^{\mathrm{R}}_{\mathrm{FL}}$ from each
resistivity data measured at different fields are extracted and plotted as
shown in the $H-T$ phase diagram of Fig. 5.

The slope $A$ of the linear fit (solid lines in Fig. 3) increases with
decreasing field (see inset to Fig. 3). Its field dependence follows
$A(H)\propto(H-H_{\textrm{QCP}})^{-a}$ with $H_{\textrm{QCP}}$ = 2.5 T and $a$
=1.05. Thus, this fitting reveals the quantum critical behavior for this Sm
doping, with a field-induced quantum critical point of 2.5 T. We note that an
$H_{\textrm{QCP}}\approx 5$ T has been extracted through a similar type of
analysis of resistivity data in the parent compound CeCoIn5 [23, 8]. As we
will show below, an extrapolation to zero temperature of the linear fit of
$T^{\mathrm{C}}_{\mathrm{FL}}(H,T)$ and $T^{\mathrm{R}}_{\mathrm{FL}}(H,T)$ vs
$T$ also gives $H_{QCP}=2.5$ T (see Fig. 5).

#### Magnetoresistivity data.

Figures 4(a) and 4(b) show the in-plane transverse magnetoresistivity (MR)
$\Delta\rho_{a}^{\perp}/\rho_{a}$ vs $H$ of Ce0.90Sm0.10CoIn5 measured over
the temperature range 2 K $\leq T\leq$ 50 K. These figures reveal pronounced
crossovers from positive to negative MR that become flatter with increasing
temperature. The peaks in MR take place at temperatures smaller than the
coherence temperature $T_{\textrm{coh}}=41$ K.

We extracted the field $H_{\textrm{max}}$ at which the magnetoresistivity is
maximum [see main panel of Fig. 4(a) and 4(b)] and plotted $H_{\textrm{max}}$
vs $T$ in the inset to Fig. 4(a). This plot shows a non-monotonic dependence
of $H_{\textrm{max}}$ vs $T$, with a maximum at around 19 K. For $T>19$ K, the
value of $H_{\textrm{max}}$ decreases with increasing temperature, revealing
the coherent Kondo lattice behavior in this $T$ range. However, the positive
MR that decreases with decreasing $T$ in the low temperature range ($T<$ 19 K)
indicates unconventional Kondo lattice behavior and is attributed to field
quenching of the AFM spin fluctuations [24] responsible for the NFL behavior,
previously observed in CeCoIn5 [24] and Ce1-xYbxCoIn5 [17]. Therefore, the
positive MR measured for the $x=0.10$ single crystals at $T<$ 19 K is due to
the AFM spin fluctuations. The extrapolation of the low $T$ linear fit of
$H_{\textrm{max}}$ vs $T$ to zero temperature give $H_{QCP}=2.5$ T for this Sm
doping.

#### Magnetic-field-induced quantum critical point.

Based on features extracted from $C_{e}/T$, $\rho$, and
$\Delta\rho_{a}^{\perp}/\rho_{a}$ data, we generated the $H-T$ phase diagram
shown in Fig. 5. Specifically, the FL to NFL crossover temperatures for
different $H$ values are extracted from the $C_{e}/T$ vs $T$ data of Fig. 2 as
$T^{\mathrm{C}}_{\mathrm{FL}}$ (blue triangles) and $\rho$ vs $T^{2}$ data of
Fig. 3 as $T^{\mathrm{R}}_{\mathrm{FL}}$ (red circles). The linear fit of
$T^{\mathrm{C}}_{\mathrm{FL}}$ and $T^{\mathrm{R}}_{\mathrm{FL}}$ from both
measurements extrapolates to zero temperature at $H_{\textrm{QCP}}=2.5$ T. In
addition, the linear extrapolation of the peak
$H^{\mathrm{MR}}_{\mathrm{max}}$ (black squares) vs $T$ (extracted from
$\Delta\rho_{a}^{\perp}/\rho_{a}$ of Fig. 4) to zero temperature gives the
same $H_{\textrm{QCP}}=2.5$ T.

Figure 4: (Color online) (a) In-plane transverse magnetoresistivity
$\Delta\rho_{a}^{\perp}/\rho_{a}$ vs magnetic field $H$ of Ce0.90Sm0.10CoIn5
measured over a temperature $T$ range 2 K$\leq T\leq$ 19 K. Inset: $T$
dependence of the characteristic field $H_{max}$ corresponding to the maximum
of the transverse MR. The solid line in the inset is the linear fit of the low
$T$ data. (b) $\Delta\rho_{a}^{\perp}/\rho_{a}$ vs $H$ measured over a $T$
range 21 K$\leq T\leq$ 50 K Figure 5: (Color online) Magnetic field-
temperature $(H-T)$ phase diagram of Ce0.90Sm0.10CoIn5 with $H\perp ab$ plane.
The solid and dash lines are linear fits of the the data extrapolated to zero
temperature. Figure 6: (Color online) Scaling of the Sommerfeld coefficient
$\gamma$ according to
$\gamma(H)-\gamma(2.5$T$)\propto(H-2.5$T$)^{0.71}f[(H-H(2.5$T$)/T^{2.5}]$. We
obtained the best scaling shown on the main panel with a logarithmic
dependence of $\gamma(2.5$T) vs $T$ at temperatures $T\leq 6$ K. Inset: Semi-
log plot of $\gamma\equiv C_{e}/T$ vs temperature $T$ measured at 2.5 T and 4
T and their normal state fit with $C_{e}/T$ = 0.65 - 0.4ln$T$.

#### Quantum critical scaling of heat capacity.

When the system is tuned to a quantum critical point by a magnetic field
$H=H_{\textrm{QCP}}$, the time-scale of the quantum critical fluctuations is
governed by temperature only, i.e., $\tau=\hbar/k_{B}T$. As a consequence, the
relevant dynamical response functions exhibit the $\omega/T$ scaling, where
$\omega$ is the characteristic frequency on which the system is probed [25,
26]. When $H$ serves as a tuning parameter, the effect of quantum critical
fluctuations in the thermodynamic or transport quantities is manifested in
their dependence on the ratio of $H-H_{\textrm{QCP}}$ and $T$, as well as a
typical energy scale describing the source of quantum fluctuations.

In order to further confirm that the anomalous temperature dependence of
$C_{e}/T$ is governed by quantum critical fluctuations and that $H_{QCP}=2.5$
T for the $x=0.10$ samples, we show that $\gamma(H,T)\equiv C_{e}/T$ is
governed by the critical free energy density
$f_{\textrm{cr}}=a_{0}r^{\nu(d+z)}f_{0}(T/r^{\nu
z})=a_{0}T^{(d+z)/z}\tilde{f}_{0}(r/T^{1/\nu z})$, where $a_{0}$ is a
constant, $f_{0}$ and $\tilde{f}_{0}$ are scaling functions,
$r\propto(H-H_{\textrm{QCP}})$, $d$ is the dimensionality of the system, $z$
is the dynamical critical exponent, and $\nu$ is the critical exponent
describing the dynamical correlations between Ce moments. Therefore, based on
the arguments of Ref. [27] and the scaling analysis performed for CeCoIn5 [8],
we performed the scaling analysis using the function
($\gamma(H)-\gamma(H_{\textrm{QCP}}))\propto(H-H_{\textrm{QCP}})^{\alpha}f[(H-H_{\textrm{QCP}})/T^{\beta}]$,
where $\alpha\equiv\nu(d+z)$ and $\beta$ represents the scaling dimension of
$H$. The best scaling we obtained (Fig. 6) confirms that
$H_{\textrm{QCP}}=2.5$ T for Ce0.90Sm0.10CoIn5 and gives $\alpha$ = 0.71, and
$\beta$ = 2.5. The scaling of $\gamma(H,T)$ spans both the FL regime at low
temperatures and the NFL regime at high temperatures, with all five data sets
for different $H$ values overlapping over the $T$ range 0.42 K $\leq T\leq$ 8
K. A very similar scaling has been observed in the stocheometric compound
CeCoIn5 [8] and YbRh2Si2 [28]

It is instructive to compare these results for the values of $\alpha$ and
$\beta$ of Ce0.90Sm0.10CoIn5 with those obtained for Ce1-xYbxCoIn5 [18]. For
the latter, we found $\alpha_{\textrm{Yb}}=0.71$ and
$\beta_{\textrm{Yb}}=1.2$, while for the former $\alpha_{\textrm{Sm}}=0.71$
and $\beta_{\textrm{Sm}}=2.5\approx 2\beta_{\textrm{Yb}}$. On the other hand,
both $\alpha_{\textrm{Sm}}$ and $\beta_{\textrm{Sm}}$ match the corresponding
values found for the stocheometric compound. Given the fact that magnetic
field, on one hand, serves as a tuning parameter to the QCP, while, on the
other hand, suppresses the magnetic fluctuations by direct coupling to the
magnetic moments of Ce ions, the relatively high value of $\beta$ signals that
the region of quantum critical fluctuations is broader for Ce0.90Sm0.10CoIn5
compared to Ce1-xYbxCoIn5. This is also consistent with the stronger
suppression of superconductivity in the samarium alloys.

Figure 7: (Color online) Semi-log plot of the electronic specific heat
normalized by temperature $C_{e}/T$ vs temperature $T$ of Ce0.85Sm0.15CoIn5
measured with applied magnetic fields $H\parallel c$-axis over the temperature
range 0.42 K $\leq T\leq$ 10 K. $T^{C}_{FL}$ represents the temperature at
which the behavior changes from Fermi liquid to non-Fermi liquid.

A major obstacle in performing the scaling analysis was to determine the
normal state $\gamma(T,2.5$T$)$ at low temperatures because the $x$ = 0.10
sample exhibits superconductivity below 1.1 K in the presence of a magnetic
field of 2.5 T. We were able to overcome this problem by determining the $T$
dependence of the $\gamma(T,4$T) down to 0.42 K and taking advantage of the
fact that $\gamma(T,4$T) and $\gamma(T,2.5$T) completely overlap in the normal
state, i.e., for $T>1.2$ K (see inset to Fig 6). We found that the
$C_{e}(T,H)/T$ follows a logarithmic $T$-dependence with $C_{e}(T,H)/T$ = 0.64
+ ln$T^{-0.4}$. This result implies that proximity to the underlying quantum
critical point affects the thermodynamic properties significantly.

###  III.2 Ce0.85Sm0.15CoIn5

#### Heat capacity.

In order to determine the effect of doping on the value of $H_{\textrm{QCP}}$
and to search for a critical Sm concentration for which $H_{QCP}=0$, we
performed similar specific heat and resistivity measurements as a function of
temperature and magnetic field on single crystals with a slightly higher Sm
concentration, i.e. Ce0.85Sm0.15CoIn5. Figure 7 shows the temperature
dependence of $C_{e}/T$ measured in fields 8 T $\leq H\leq$ 14 T and at low
temperatures, i.e., 0.4 K $\leq T\leq$ 8 K. The normal-state results shown in
Fig. 7 reveal sharp crossovers from constant $C_{e}/T$ (FL at low $T$) to
logarithmic $T$-dependent $C_{e}/T$, with $C_{e}/T=1+lnT^{-0.64}$ ( NFL at
high $T$). The crossovers temperature $T^{\mathrm{C}}_{\mathrm{FL}}$ also
shifts towards higher temperatures with increasing $H$ for $H\geq$ 6 T.

Figure 8: (Color online) Low temperature resistivity $\rho$ vs temperature
$T^{2}$ of Ce0.85Sm0.15CoIn5 measured in a magnetic field $H$ with 6 T $\leq
H\leq$ 14 T. The solid lines are linear fits to the data with
$\rho(T,H)=\rho_{0}+AT^{2}$. Inset: $H$ dependent coefficient $A$ of
$\rho(T,H)$.

#### Resistivity.

Figure 8 shows the $T^{2}$ dependence of $\rho(H,T)$ of Ce0.85Sm0.15CoIn5
measured from 0.6 K to 5 K and for 6 T $\leq H\leq$ 14 T. As in the case of
the single crystals with $x$ = 0.10, the resistivity for the $x=0.15$ samples
is linear in $T^{2}$ at low temperature, signaling the presence of the Fermi-
liquid behavior. The $H$ dependence of the coefficient $A$ (see inset to Fig.
8) is best fitted with the function $A(H)\propto(H-H_{\textrm{QCP}})^{-a}$,
with $H_{QCP}$ = 0 and $a$ = 1.1. Since $A(H)$ vs $H$ curve diverges for
$H_{QCP}$ = 0, Ce0.85Sm0.15CoIn5 is quantum critical.

Figure 9: (Color online) Magnetic field-temperature $(H-T)$ phase diagram of
Ce0.85Sm0.15CoIn5 with $H\perp ab$ plane. Figure 10: (Color online) Scaling of
the Sommerfeld coefficient $\gamma$ according to
$\gamma(H)-\gamma(0$T$)\propto H^{0.8}f(H)/T^{2.5}$ with $\gamma$ = $C_{e}/T$.
We obtained the best scaling shown on the main panel with a logarithmic
dependence of $\gamma(0$T) vs $T$ for temperatures $T\leq$ 6 K. Inset: Semi-
log plot of $\gamma\equiv C_{e}/T$ vs temperature $T$ measured at 0 T and its
normal state fit with $C_{e}(T,0)/T$ = 1 - 0.65 ln$T$.

To check that the samples with $x=0.15$ are, indeed, quantum critical, we
plotted in Fig. 9 the $H-T$ phase diagram for this Sm-doped single crystals.
The FL to NFL crossovers extracted from $C_{e}/T$ are represented by
$T^{\mathrm{C}}_{\mathrm{FL}}$ (solid blue triangles), and these crossovers
extracted from $\rho(T,H)$ are represented by
$T^{\mathrm{R}}_{\mathrm{FL}}$(solid red circles). The FL to NFL crossovers
obtained from both measurements are in excellent agreement. The linear
extrapolation of the fit of these crossovers to zero temperature reveals that
$H_{QCP}$ = 0, indicating that the sample Ce0.85Sm0.15CoIn5 is, indeed, at the
critical doping.

#### Quantum critical scaling of heat capacity.

We further checked whether Ce0.85Sm0.15CoIn5 is quantum critical by performing
the scaling analysis as discussed above. We obtained the best scaling, shown
in Fig. 10, with $H_{\textrm{QCP}}=0$, $\alpha=0.8$ and $\beta=2.5$. The
scaling of $\gamma\equiv C_{e}/T$ spans both the FL at low temperatures and
the NFL regime at high temperatures, with all four data sets measured at
different $H$ values overlapping over the temperature range 0.42 K $\leq T\leq
5$ K. This scaling further indicates that Ce0.85Sm0.15CoIn5 is quantum
critical and exhibits NFL behavior.

In order to determine $\gamma(0$T) required for the scaling of Fig. 10, we
fitted the $C_{e}(T$,0)$/T$ data for $T>1.91$ K of the $x$ = 0.15 sample with
a logarithmic $T$-dependence; i.e., $C_{e}(T$,0)$/T$ = 1 + ln$T^{-0.64}$. The
data for $T<1.5$ K display a stronger than logarithmic increase with
decreasing $T$, most likely due to the presence of long range AFM fluctuations
and superconductivity. This is consistent with the phase diagram of Fig. 1,
where $T_{N}=0.8$ K. Nevertheless, we used the logarithmic $T$ dependence of
$\gamma(0$T) obtained by fitting the data at $T>1.91$ K over the whole $T$
range down to 0.42 K

Combining the experimental results shown above, we have generated an
$H_{\textrm{QCP}}$ vs Sm concentration phase diagram of Ce1-xSmxCoIn5, for the
doping $x$ = 0.00, 0.10, 0.15, as shown in the inset to Fig. 1. This phase
diagram shows that $H_{\textrm{QCP}}$ is suppressed with increasing Sm3+
doping and becomes zero for $x\geq$ 0.15. Therefore, at zero temperature in
the region for $x<0.1$ (main panel of Fig. 1) superconductivity and long-range
AFM order of Ce moments coexist. For $0.1\leq x\leq$ 0.15, superconductivity,
long-range AFM order of Ce moments, and long-range AFM of Sm ions coexist at
zero temperature. In region $0.15\leq x\leq$ 0.17 superconductivity co-exists
with long-range AFM order of Sm moments, while there is only long range AFM
order due to Sm ions when $x\geq 0.17$.

##  IV Discussion and outlook

Anomalous thermodynamic and transport low-temperature properties of complex
materials have long been associated with underlying QCPs. In this regard,
CeCoIn5 as well as the other members of the ’115’ family of compounds are not
exception. The existence of the $H_{\textrm{QCP}}$ in CeCoIn5 has been already
established independently by several groups. Our present study further
justifies the validity of using the conceptual framework of quantum
criticality to account for the observed anomalies in Ce1-xSmxCoIn5.

We can also estimate the fluctuation correction to the heat capacity in
external $H$. The details of the calculation are given in the Appendix A, so
here we only present the results in Fig. 11. We found that due to presumably
strong coupling between itinerant carriers and localized moments of Ce ions,
the fluctuation correction to the heat capacity in the close vicinity to QCP
does have a power-law temperature dependence $\delta C_{\textrm{sf}}\propto
T^{\alpha}$ with $\alpha\approx 0.45$. When the system is de-tuned from the
proximity to the QCP, the exponent $\alpha$ increases and becomes
$\alpha\approx 1.5$ at very low temperatures.

Figure 11: (Color online) Fluctuation correction to the heat capacity due to
system’s proximity to magnetic QCP as a function of temperature for various
values of the parameter $\eta\propto(H-H_{\textrm{QCP}})$. The temperature is
given in the units of the characteristic energy scale for the magnetic
subsystem. Although $\delta C_{\textrm{sf}}(T)\propto T^{\alpha}$ has a power-
law (and not logarithmic) temperature dependence, the exponent $\alpha$ shows
strong dependence on the parameter $\eta$, a feature which is also observed
experimentally.

Samarium substitution on cerium sites brings about a novel feature into the
phase diagram: the possible co-existence between superconductivity and
antiferromagnetic ordering of Sm local moments in the narrow region of
$0.1\leq x\leq 0.17$, Fig 1. The Néel temperature vanishes at $x\approx 0.1$
giving rise most likely to an antiferromagnetic QCP. In this regard we would
like to note that quantum critical fluctuations may already manifest
themselves in the superconducting state. Indeed, recent theoretical works have
shown the the effect of quantum critical fluctuations can be also probed in
the superconducting state by studying the temperature dependence of
thermodynamic functions such as heat capacity [29, 30] and London penetration
depth [30]. In particular, it has been shown that in a fully gapped
superconductor, quantum fluctuations produce power-law dependences in both of
these quantities on the background of the (mean-field) exponential temperature
dependence. In the context of Ce1-xSmxCoIn5, it would be intriguing to probe
the variation in the temperature dependence of the heat capacity of the
$x=0.1$ samples in the superconducting state. From the point of view of the
theoretical analysis, this problem poses an additional challenge since the
fermionic spectrum is not fully gaped.

##  V Summary

Through the measurements of specific heat, resistivity, and magnetoresistivity
on both Ce0.90Sm0.10CoIn5 and Ce0.85Sm0.15CoIn5 samples, we observe that the
FL regime recovery is established with increasing $H$ in both $\rho(H,T)$ and
$C_{e}(H,T)/T$ along with the low T evolution of positive magnetoresistivity.
We conclude that $H_{\textrm{QCP}}$ decreases with increasing Sm
concentration. The single crystal with $x$ = 0.15 exhibits zero temperature
quantum criticality associated with the antiferromagentic ground state of Ce
ions. However, the lower doped crystal Ce0.90Sm0.10CoIn5 reveals
$H_{\textrm{QCP}}$ at 2.5 T. In addition, crossovers ($T\approx 19$ K) from
positive to negative MR extending to higher temperatures are the evidence that
the change of spin fluctuations with $H$ may be strongly associated to the
critical nature of the slope $A(H)$ of $\rho(H,T)$, suggesting that the QCP is
magnetic in nature. As compared to the parent compound and low doped sample
($x$ = 0.1), the normal-state transport and thermodynamic properties of
Ce0.85Sm0.15CoIn5 are controlled by the presence of QCP alone. Moreover, the
scaling analysis of the $C_{e}/T$ data provides strong evidence for the
existence of $H_{\textrm{QCP}}$. Similarly, excellent fits of $C_{e}(H,T)/T$
and $\rho(T,H)$ data measured at several $H$ also suggest that the QCP is
antiferromagnetic in nature as supported by the spin fluctuation theory.

##  VI Acknowledgments

The work was supported by the National Science Foundation under grants Nos.
DMR-1904315 and NSF-DMR-BSF-2002795 at Kent State University, and by the US
Department of Energy, Office of Basic Energy Sciences, Division of Materials
Sciences and Engineering, under Grant No. DE-FG02-04ER46105 at UCSD.

##  Appendix A Temperature correction to the heat capacity from quantum
critical spin-fluctuations

The contribution of the quantum critical spin-fluctuations to the free energy
is [29, 30]:

$\delta
F_{\textrm{sf}}=3T\sum\limits_{m=-\infty}^{\infty}\int\frac{d^{2}{\mathbf{q}}}{(2\pi)^{2}}\log\left[\chi^{-1}({\mathbf{q}},\Omega_{m})\right].$
(1)

Here
$\chi^{-1}({\mathbf{q}},\Omega_{m})=\nu\left(E_{\mathbf{q}}^{2}-i|\Omega_{m}|/\omega_{\textrm{sf}}+\Omega_{m}^{2}/\omega_{0}^{2}\right)$,
$E_{\mathbf{q}}=\sqrt{\eta+({\mathbf{q}}/Q_{0})^{2}}$, $\nu$ is inverse
proportional to the static spin-susceptibility at $T=0$, parameter $\eta$
controls the proximity to the magnetic QCP, $\Omega_{m}=2\pi Tm$ is the
bosonic Matsubara frequency, $\omega_{\textrm{sf}}$ is an energy scale which
describes the interaction between the localized moments and conduction
conductions, $\omega_{0}$ is a typical energy scale of the magnetic system
[31, 32] and the numerical pre-factor takes into account three fluctuating
modes: two transverse and one longitudinal.

Since the spin-fluctuation propagator is a non-analytic function of
$\Omega_{m}$, in order to evaluate the free energy we will need to keep the
real part of the expression under the integrals only. In the Matsubara
summation (1) we can single out the $m=0$ term and reduce the remaining
summation over $m\geq 1$:

$\delta F_{\textrm{sf}}=3T\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\log(\nu
E_{\mathbf{q}}^{2})+6T\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\log\left[\prod\limits_{n=1}^{\infty}\left(1+\frac{\omega_{0}^{2}E_{\mathbf{q}}^{2}}{\Omega_{n}^{2}}\right)\prod\limits_{m=1}^{\infty}\left(\frac{\nu\Omega_{m}^{2}}{\omega_{0}^{2}}\right)\prod\limits_{l=1}^{\infty}\left(1-\frac{i\Omega_{l}/\omega_{\textrm{sf}}}{E_{\mathbf{q}}^{2}+\frac{\Omega_{l}^{2}}{\omega_{0}^{2}}}\right)\right].$
(2)

The first term in this expression represents a ”zero-point-motion” correction
and, therefore, does not produce the temperature dependent contribution to the
heat capacity.

In the second term, there are three products under the logarithm which we will
discuss them separately below. The first product evaluates to

$\begin{split}&\prod\limits_{m=1}^{\infty}\left(1+\frac{\omega_{0}^{2}E_{\mathbf{q}}^{2}}{\Omega_{m}^{2}}\right)=\frac{\sinh(\pi
z)}{\pi z},\quad z=\frac{\omega_{0}E_{\mathbf{q}}}{2\pi T}.\end{split}$ (3)

To evaluate the second product we need to use regularization scheme to assign
it a finite value. We use the zeta-function regularization scheme [30] to find

$\sum\limits_{m=1}^{\infty}\log\left(\frac{\nu\Omega_{m}^{2}}{\omega_{0}^{2}}\right)=\log\left(\frac{\omega_{0}}{\sqrt{\nu}T}\right).$
(4)

Finally, for the third product we clearly need to evaluate the real part only.
It obtains:

$\begin{split}\prod\limits_{l=1}^{\infty}\sqrt{1+\frac{(\Omega_{l}/\omega_{\textrm{sf}})^{2}}{\left(E_{\mathbf{q}}^{2}+\frac{\Omega_{l}^{2}}{\omega_{0}^{2}}\right)^{2}}}=\frac{\sinh(r_{\mathbf{q}}^{+})\sinh(r_{\mathbf{q}}^{-})}{2\sinh^{2}(\pi
z)},\end{split}$ (5)

where we introduced auxiliary variables

$r_{\mathbf{q}}^{\pm}=\frac{\omega_{0}^{2}}{2\sqrt{2}\omega_{\textrm{sf}}T}\sqrt{1+\frac{2\omega_{\textrm{sf}}^{2}}{\omega_{0}^{2}}E_{\mathbf{q}}^{2}\pm\sqrt{1+\frac{4\omega_{\textrm{sf}}^{2}}{\omega_{0}^{2}}E_{\mathbf{q}}^{2}}}.$
(6)

Thus, the expression for the fluctuation correction to the free energy is

$\delta
F_{\textrm{sf}}=6T\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\log\left[\frac{\sinh(r_{\mathbf{q}}^{+})\sinh(r_{\mathbf{q}}^{-})}{\sinh(\omega_{0}E_{\mathbf{q}}/2T)}\right].$
(7)

The fluctuation correction to the heat capacity is directly obtained from (7),
$\delta C_{\textrm{sf}}/T=-\partial^{2}(\delta F_{\textrm{sf}})/\partial
T^{2}$. We find

$\begin{split}\delta
C_{\textrm{sf}}(T)=6\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}&\left[\frac{(r_{\mathbf{q}}^{+})^{2}}{\sinh^{2}(r_{\mathbf{q}}^{+})}+\frac{(r_{\mathbf{q}}^{-})^{2}}{\sinh^{2}(r_{\mathbf{q}}^{-})}\right.\\\
&\left.-\frac{\omega_{0}^{2}E_{\mathbf{q}}^{2}}{4T^{2}\sinh^{2}(\omega_{0}E_{\mathbf{q}}/2T)}\right].\end{split}$
(8)

Our numerical analysis of this expression shows that at low temperatures,
$T\ll\omega_{0}$ and $\omega_{\textrm{sf}}\approx\omega_{0}$, $\delta
C_{\textrm{sf}}(T)\propto T^{\alpha}$ with $\alpha\approx 0.45$ at the QCP.

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