# Object Propagation via Inter-Frame Attentions for Temporally Stable Video
Instance Segmentation

Anirudh S Chakravarthy1,2 Won-Dong Jang2 Zudi Lin2 Donglai Wei2 Song Bai3
Hanspeter Pfister2
1Birla Institute of Technology and Science  Pilani  India
2School of Engineering and Applied Sciences  Harvard University  Cambridge
USA
3University of Oxford  UK

###### Abstract

Video instance segmentation aims to detect, segment, and track objects in a
video. Current approaches extend image-level segmentation algorithms to the
temporal domain. However, this results in temporally inconsistent masks. In
this work, we identify the mask quality due to temporal stability as a
performance bottleneck. Motivated by this, we propose a video instance
segmentation method that alleviates the problem due to missing detections.
Since this cannot be solved simply using spatial information, we leverage
temporal context using inter-frame attentions. This allows our network to
refocus on missing objects using box predictions from the neighbouring frame,
thereby overcoming missing detections. Our method significantly outperforms
previous state-of-the-art algorithms using the Mask R-CNN backbone, by
achieving 36.0% mAP on the YouTube-VIS benchmark. Additionally, our method is
completely online and requires no future frames. Our code is publicly
available at https://github.com/anirudh-chakravarthy/ObjProp.

##  1 Introduction

Figure 1: We demonstrate the problem of temporal stability in video instance
segmentation. Due to object displacements, mask and class predictions are
temporally inconsistent in MaskTrack R-CNN [6] (second row). Our method
alleviates the issue of temporal stability (third row).

In this work, we propose a video instance segmentation algorithm based on the
Mask R-CNN pipeline [4]. We focus on the problem of temporal instability in
video instance segmentation (Figure 1). There are many reasons for temporal
instability: missing proposals from a region proposal network,
misclassification of the object’s class, or aliasing from small visual
displacements. We address temporal instability by propagating masks using
object boxes to neighbouring frames to complement missing detections. Mask
propagation through bounding boxes enables tracking of objects even when the
detector misses the object’s bounding box in the current frame. To this end,
we use the attention mechanism. Our propagation network predicts an attention
map propagated from previous frames to the current frame. We apply the
attention map to current frame features, which allows us to fill in absent
instance masks and overcome temporal instability.

We have three main contributions in this work. First, we identify the temporal
instability for video instance segmentation. Second, by propagating object
masks through an inter-frame attention mechanism, we generate temporally
coherent and spatially accurate mask tracks. Third, our method outperforms the
conventional Mask R-CNN methods on the on the YouTube-VIS dataset [6].

Figure 2: We add the mask propagation branch to MaskTrack R-CNN [6]. Given two
frames from a video, we propagate an attention map from frame $t-\delta$
(randomly sampled) to frame $t$ to predict segmentation masks corresponding to
missing instances.

##  2 Method

Table 1: Oracle study using MaskTrack R-CNN [6] on our mini-validation set from the YouTube-VIS dataset. The highest gain achieved by correcting masks is highligted.  Box | Class | Mask | Track | mAP | Gain
---|---|---|---|---|---
– | – | – | – | 36.8 | –
✓ | – | – | – | 41.9 | +5.1
✓ | ✓ | – | – | 53.1 | +11.2
✓ | ✓ | ✓ | – | 85.3 | +32.2
✓ | ✓ | ✓ | ✓ | 100.0 | +14.7

###  2.1 Oracle study

In order to examine the problem of temporal stability in greater detail, we
perform an oracle testing in Table 1 using MaskTrack R-CNN [6]. MaskTrack
R-CNN achieves 36.8 mAP on the mini-validation set. Replacing the detected
boxes with the closest ground-truth boxes based on intersection over union
(IoU) achieves a performance of 41.9 mAP. Completely replacing the box head
with ground truth boxes and category labels leads to an improvement of 11.2
mAP. When we replace the mask head with ground truth masks, a significant
performance improvement is observed from 53.1 mAP to 85.3 mAP. Furthermore,
the oracle tracking improves the performance by 14.7 points.

Motivated by this study, we found that the temporal instability of masks is
the critical bottleneck, which means there is considerable room for
improvement in mask generation and tracking. Thus, we focus on improving the
quality of masks, especially alleviating the temporal stability problem in
video instance segmentation.

###  2.2 Mask Propagation with Inter-frame Attentions

Figure 3: An illustration of our attention-guided mask propagation system
which propagates instance masks from frame $t-\delta$ to frame $t$. Our
framework can be summarized in three steps. 1) An affinity matrix is computed
between the two frames. 2) Next, the instance-specific box-level mask and
affinity matrix are used to propagate an attention map. 3) The attention map
is applied on frame $t$ and the segmentation mask is predicted using a mask
prediction head.

Due to object deformation or aliasing, per-frame object instance segmentation
models struggle to segment objects consistently throughout the video. This
leads to several missing detections throughout the video, which drastically
affects the segmentation results. Using 3D convolutional neural networks
(CNNs) poses a couple of challenges. First, 3D convolutions are
computationally expensive. Second, since the large memory footprint constrains
the number of images in memory, the learned temporal interaction is limited.

Instead, we propagate masks from previous frames $t-\delta$ to the current
frame $t$ to compensate the missing detections due to the lack of temporal
context. As illustrated in Figure 2, we add our propagation module upon the
MaskTrack R-CNN pipeline. Our propagation module enables learning of the
temporal context without 3D convolutions. In this work, we improve the
transition-based propagation method using attention mechanism [5]. Our inter-
frame attentions can robustly propagate masks between frames considering the
temporal context.

Our propagation module with inter-frame attentions is illustrated in Figure 3.
For object propagation, we set the backbone features of frames $t-\delta$ and
$t$ and a binary map of an object to propagate at frame $t-\delta$ as input.
Our propagation module will output a mask of the target object at frame $t$.

Inter-frame affinity. We first measure inter-frame affinities between two
frames $t-\delta$ and $t$. The input backbone features from frames $t-\delta$
and $t$ are resized into the stride of 16 of the image resolution and then
concatenated across each level of the feature pyramid. We represent these
processed features as $\mathbf{F}_{t}$ and $\mathbf{F}_{t-\delta}$ for frames
$t$ and $t-\delta$, respectively. By using these features, we compute the
transition matrix to measure inter-frame feature affinity between each spatial
location. The inter-frame affinity matrix $\mathbf{W}_{t-\delta\rightarrow
t}\in\mathbb{R}^{HW\times HW}$ is computed as

$\mathbf{W}_{t-\delta\rightarrow t}=\mathbf{F}_{t}\circ\mathbf{F}_{t-\delta},$
(1)

where $\circ$ is a matrix multiplication operator. Each element of the inter-
frame affinity matrix represents the affinity between corresponding two
locations in $t$ and $t-\delta$ frames. We normalize the affinity matrix to
make the sum of each row to be 1.

Attention estimation via object propagation. Similar to TVOS [7], we aim to
propagate masks using the transition matrix. However, instead of propagating
pixel-level segmentation masks, we use a binary object map, which serves as a
loose estimate for the pixel-level mask. The input binary object map for frame
$t-\delta$ is generated by marking all pixels within the instance-specific
bounding box. In addition to the object map, we also generate a binary map for
the background by inverting the object’s binary map to take into account the
background information when computing inter-frame attentions. We vectorize
both binary object and background maps for matrix multiplication. Using the
transition matrix, the binary object map, $\mathbf{b}_{t-\delta}$, is
propagated to the current frame $t$ by

$\mathbf{a}_{t}=\mathbf{W}_{t-\delta\rightarrow t}\circ\mathbf{b}_{t-\delta}.$
(2)

Since we propagate both object and background maps, we apply softmax to find
the regions of the object. We represent the softmax output of the propagated
object map as $\hat{\mathbf{a}}_{t}$ and use it as an attention map. Unlike
existing attention-based algorithms which allow the machine to prioritize
important features, we explicitly supervise the attention module to learn
where to focus by propagating temporal information. This allows us to maximize
the temporal context.

Attention-based mask prediction. Our next aim is to predict a mask using the
attention map, $\hat{\mathbf{a}}_{t}$, and the features from frame $t$,
$\mathbf{F}_{t}$. We first apply the attention map to the features by
conducting element-wise multiplication to each spatial location. We feed the
attention-guided features to a mask prediction module, which consists of four
convolution modules (a combination of a convolution layer and ReLU), one
deconvolution module, and prediction module (a combination of a convolution
layer and a sigmoid layer).

Loss functions. The propagation loss $L_{prop}$ consists of two terms, the
mask propagation loss $L_{prop}^{m}$ and the attention loss $L_{prop}^{a}$.
$L_{prop}^{m}$ is computed identically to the mask head in Mask R-CNN [4]. The
attention loss, $L_{prop}^{a}$, is computed as follows:

$L_{prop}^{a}=-\sum_{i=1}^{H}\sum_{j=1}^{W}y_{ij}y_{ij}\log\tilde{y}_{ij}+(1-y_{ij})\log(1-\tilde{y}_{ij}),$
(3)

where $y_{ij}$ is the value in the ground-truth attention map at location
$(i,j)$, and $\tilde{y}_{ij}$ is the predicted attention value.

Training. We use Mask R-CNN [4] with ResNet-50 backbone pre-trained on COCO
dataset. Our model is trained on 4 Tesla V100 GPUs. We use a batch size of 24,
with 6 images on each GPU. We train our model for 12 epochs using SGD
optimizer with a learning rate of 0.005, which is decayed by a factor of 10 at
8 and 11 epochs.

Inference. During inference, our framework is completely online and does not
require any future frames. We store the output predictions for previous frames
in memory and propagate instance masks to the current frames. Using mask
propagation, we are able to alleviate the temporal instability problem. When a
bounding box is missing from the current frame, we propagate an instance mask
from the frame history to the current frame to segment the missing object
instance. Therefore, we use mask propagation as an empty instance filling
mechanism.

##  3 Experiments

Video clip |  |  |  |
---|---|---|---|---
MaskTrack R-CNN |  |  |  |
Ours |  |  |  |
| Frame 6 | Frame 8 | Frame 9 | Frame 11
Figure 4: We compare results obtained using our approach (third row) with
MaskTrack R-CNN predictions (second row).  Table 2: Comparison of video
instance segmentation methods on the YouTube-VIS validation dataset [6]. The
best results are boldfaced.

Method | Backbone | mAP | AP@50 | AP@75 | AR@1 | AR@10
---|---|---|---|---|---|---
MaskTrack R-CNN | ResNet-50 | 30.3 | 51.1 | 32.6 | 31.0 | 35.5
STEm-Seg | ResNet-50 | 30.6 | 50.7 | 33.5 | 31.6 | 37.1
SipMask | ResNet-50 | 32.5 | 53.0 | 33.3 | 33.5 | 38.9
Ours | ResNet-50 | 36.0 | 59.4 | 39.2 | 39.1 | 47.7

We use the YouTube-VIS [6] validation set to compare video instance
segmentation methods. For evaluation, we follow 3 metrics. We use 1). mean
Average Precision over the video sequence (mAP), 2). Average Precision over
the video sequence at 50% and 75% IOU thresholds, and 3). Average Recall for
the highest 1 and 10 ranked instances per video.

We compare our method with three state-of-the-art methods; MaskTrack R-CNN
[6], STEm-Seg [1], and SipMask [3]. Note that direct comparison with MaskProp
[2] is not fair due to the complex architecture, hence, we compare our method
with Mask R-CNN based approaches.

It is observable that our method comprehensively outperforms all four
conventional methods on all evaluation metrics. We achieve nearly 3.5% greater
mAP than the closest method on the benchmark, which demonstrates the
effectiveness of our approach. In Figure 4, we illustrate the results of our
method (third row) compared to MaskTrack R-CNN (second row). We observe that
our inter-frame attention propagation head leads to temporally consistent
segmentation tracks throughout the video.

##  4 Conclusions

In this work, we introduce an inter-frame attention propagation network for
video instance segmentation. Using box-level instance masks from the frame
history, we propagate an attention map onto the current frame which is used to
generate an instance-specific segmentation mask. Our method is online and
requires limited computational overhead. Using inter-frame attentions, we
achieve state-of-the-art results on the YouTube-VIS benchmark using the Mask
R-CNN pipeline. Qualitative results demonstrate the effectiveness of our
approach in alleviating missing detections due to temporal stability problem.

## References

  * [1] Ali Athar, Sabarinath Mahadevan, Aljoša Ošep, Laura Leal-Taixé, and Bastian Leibe.  STEm-Seg: Spatio-temporal Embeddings for Instance Segmentation in Videos.  In ECCV, 2020.
  * [2] Gedas Bertasius and Lorenzo Torresani.  Classifying, Segmenting, and Tracking Object Instances in Video with Mask Propagation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9739–9748, 2020.
  * [3] Jiale Cao, Rao Muhammad Anwer, Hisham Cholakkal, Fahad Shahbaz Khan, Yanwei Pang, and Ling Shao.  SipMask: Spatial Information Preservation for Fast Image and Video Instance Segmentation.  In ECCV, 2020.
  * [4] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick.  Mask r-cnn.  In Proceedings of the IEEE international conference on computer vision, pages 2961–2969, 2017.
  * [5] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention is All you Need.  In NIPS, 2017.
  * [6] Linjie Yang, Yuchen Fan, and Ning Xu.  Video instance segmentation.  In Proceedings of the IEEE International Conference on Computer Vision, pages 5188–5197, 2019.
  * [7] Yizhuo Zhang, Zhirong Wu, Houwen Peng, and Stephen Lin.  A Transductive Approach for Video Object Segmentation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6949–6958, 2020.


11institutetext: Human Information Engineering Lab, Department of
Bioinformatic Engineering, Graduate School of Information Science and
Technology, Osaka University, Osaka 5650871, Japan
11email<EMAIL_ADDRESS>22institutetext: Center for Information
and Neural Networks, 1-4 Yamadaoka, Suita, Osaka 5650871, Japan

# Modulation of 2-Point Discrimination Threshold Under Cheminally Induced
Global Stimulation

Junjie Hua 11 0000-0001-8918-6291 Masahiro Furukawa 1122 0000-0003-0421-8106
Taro Maeda 1122 0000-0002-9762-1086

###### Abstract

The two-point discrimination threshold (2PDT) serves as a critical indicator
in the study of tactile acuity, representing the minimal distance at which an
individual can differentiate two distinct points of contact on the skin. This
measurement is instrumental in exploring the neural mechanisms underlying
tactile perception. On the other hand, tactile acuity can be modulated by
global stimulation. Prior research indicates that experimental inflammation
induced by an application of capsaicin cream increases 2PDT. In our study, we
applied chemicals (oregano, menthol, and Sichuan pepper) to selectively
activate receptors that usually respond to mild physical stimuli to
investigate their influence on 2PDT without inducing inflammation. The results
unveiled a pronounced augmentation of 2PDT following any form of global
stimulation. Intriguingly, the cumulative effect of the chemical mix on 2PDT
appeared to be additive. These observations suggest that Wide Dynamic Range
(WDR) neurons, functioning as relay nuclei with projections for touch, warmth,
and cold sensations, play a pivotal role in this process. In lateral
connection structures parallel to afferent nerve transmission pathways with
WDR neurons as relay nuclei, global stimulation amplifies excitatory
connections over inhibitory ones, thereby elevating the two-point
discrimination threshold.

###### Keywords:

Two-point discrimination threshold Cheminally Global stimulation
Psychophysics.

##  1 Introduction

Tactile acuity refers to the sharpness or sensitivity of the sense of touch.
It is usually measured by the two-point discrimination threshold (2PDT), which
is the smallest distance between two points that can be perceived as separate
stimuli by the skin [3]. The fingertips, palms, and forehead have very high
resolution and therefore are the most sensitive areas of the body for two-
point discrimination [12]. Other areas like the forearms, calves, and shoulder
are much lesser sensitive [12]. This metric not only reflects the tactile
acuity of the sensory system but also provides a window into the neural
mechanisms underlying spatial discrimination and somatosensory processing.

On the other hand, it has been found that tactile acuity can be modulated by
global stimulation. Kauppila reported that experimental inflammation induced
by an application of capsaicin cream increases 2PDT [8]. The authors concluded
that this is a result of an increase in the receptive fields of neurons in the
nervous system based on the previous finding in animal experiments that
capsaicin induces an increase in the size of receptive fields of the dorsal
horn pain-related neurons. However, the use of just a single type of
stimulation that can induce inflammation left open the possibility that
combinations of stimulation that do not induce inflammation might modulate
tactile acuity.

In this paper, we aim to explore the influence of global stimulation that does
not induce inflammation on the two-point discrimination threshold. Evidence of
nonpainful stimulation would question the involvement of pain-related neurons.
Traditional tactile devices provide temperature sensations through heat
exchange with the skin [14] or force feedback through pressure and vibration
[2]. These methods require contact with the skin, which means that in addition
to generating the desired temperature and mechanical stimuli, there is
additional pressure caused by skin contact. This additional pressure from skin
contact is something we want to avoid in our experiments. Therefore, in this
study, we use chemicals applied to the skin surface to activate receptors
corresponding to thermal and mechanical stimuli. This method of using
chemicals can avoid the additional pressure caused by skin contact.

##  2 Related Work

The process of how physical stimuli transform into perception starts with the
detection of various stimuli. Thermoreceptors and mechanoreceptors are two
types of sensory receptors in the body that detect changes in temperature and
mechanical forces, respectively. Thermoreceptors are specialized nerve cells
that are sensitive to temperature changes. The functioning of thermoreceptors
is closely linked to specific ion channels, primarily transient receptor
potential (TRP) channels. Different TRP channels respond to different
temperature ranges. In the range of warmth, TRPV1 is sensitive to chemical
irritants (e.g., capsaicin) as well as to heat. Its heat threshold is in the
vicinity of 41– 43∘C. Thus, activation of TRPV1 is usually accompanied by
inflammation [8]. Similar to TRPV1, TRPV3 is also involved in the sensation of
temperature. However, it responds to lower temperatures, with a heat threshold
below 39∘C [4]. Thus, activation of TRPV3 hardly induces pain. Natural
monoterpenoid carvacrol from plant oregano is a known chemical irritant that
specifically activates the TRPV3 channel [13]. Additionally, in the range of
cold, TRPM8 was reported to have a threshold of 25 to 27∘C [4]. Menthol was
found to sensitize the response of TRPM8 to cold.

Mechanoreceptors are sensory receptors that respond to mechanical pressure or
distortion. They are involved in sensations such as touch, pressure, and
vibration. Rapidly Adapting type 1 (RA1) mechanoreceptors, also known as
Meissner’s corpuscles, are one of the mechanoreceptors found in the skin. They
are responsible for detecting light touch and vibrations at low frequencies
(about 10-50 Hz). It has been found that bioactive component of Sichuan pepper
(also known as Szechuan pepper), hydroxy-$\alpha$-sanshool, activates
RA1/Meissner fibers [5]. The tingling sensation caused by Sichuan pepper on
the lip does not result from pain. Instead, it results from vibration. Thus,
Sichuan pepper can be used to activate the mechanoreceptor while not inducing
pain.

When these receptors are activated, they send signals through peripheral
nerves to the dorsal horn of the spinal cord. In the dorsal horn, primary
afferent neurons synapse with second order neurons. These neurons can be
unimodal, processing one type of signal, or Wide Dynamic Range (WDR) neurons,
which handle multiple types of signals. The sensory signals are then relayed
by these second order neurons through various ascending pathways to the
thalamus and cerebral cortex. It is in the cerebral cortex, particularly the
somatosensory cortex located in the parietal lobe, where this sensory
information is processed and turned into conscious perception. Given the
physiological findings described above, it is feasible to utilize chemicals to
selectively activate receptors for perception. In fact, there has been some
recent work on perception studies using chemicals. A study used capsaicin and
menthol to selectively activate thermoreceptors and an illusion called Thermal
Grill Illusion which is usually induced by physical warm and cold stimuli was
induced [6]. Therefore, in this study, we used chemicals applied to the skin
surface to activate receptors corresponding to thermal and mechanical stimuli
to investigate their influence on tactile acuity.

##  3 Material and Methods

Fifteen naive participants, aged 21 – 31 (all males), participated in all the
three experiments.

In this experiment, participants placed their right forearm on a table while
receiving various global stimulations on their right volar forearm, varying in
modality and intensity. The study focused on measuring the 2PDT. The sequence
of modalities and intensities was randomized. For 2PDT assessment, stimuli
with point-to-point distances ranging from 0 to 100 mm, increasing in 10 mm
increments, were applied. A 0 mm distance indicated a single-point stimulus,
applied along the forearm’s proximal-distal axis. Each distance, including 0
mm, was tested twice in a randomized order. The stimulation site near the
elbow was consistently positioned at the midpoint of the global stimulus’s
edge closest to the elbow. Participants, with eyes closed, answered the number
of perceived points (1 or 2 points). Each condition involved 22 stimuli,
administered at minimum 10-second intervals. To avoid influence from previous
conditions, as suggested by a pilot study indicating a return to baseline
after one hour, a minimum two-hour interval was maintained between different
stimulus conditions.

To assess the 2PDT, a Vernier caliper was employed (refer to Fig. 1). This
caliper was modified to include two probes attached to its jaws, each probe
consisting of a resin cylinder with a 3 mm diameter. Care was taken to ensure
simultaneous contact of both cylinders with the skin during stimulation. The
duration of each stimulation was set to 2 sec. Single-point stimuli were
defined as the condition where the distance between the two cylinders was
zero.

Figure 1: Mechanical stimulator used in this study. One Vernier caliber was
used. The caliber had two probes attached to the jaws of the caliber. The
probes were made of two resin cylinders (3 mm diameter). It was ensured that,
during the stimulation, both cylinders impacted the skin simultaneously.

The 2PDT was determined for each condition individually. Prior to analysis,
responses indicating one or two points were recoded from 1 or 2 to 0 or 1,
respectively. The threshold was then estimated through logistic regression,
fitting the data to a sigmoidal curve represented by:

$y=\frac{1}{1+e^{(a(b-x))}}$ (1)

In this equation, $x$ represents the distance between points (in millimeters),
$b$ denotes the point at which $y$ equals 0.5, and $a$ is the slope of the
curve at $y=0.5$. Here, $y=0.5$ corresponds to the separation distance at
which participants perceived 50% of the stimuli as two distinct points. During
logistic regression, $x$ and $y$ values were derived from experimental data,
while the fitting process estimated parameters $a$ and $b$.

Table 1: Experimental conditions in this study

Condition | Exp1 | Exp2 | Exp3 | Ingredients ratio (Total volume 0.5mL) | Threshold (mm)
---|---|---|---|---|---
Placebo |  $\surd$1 | $\surd$ | $\surd$ | Salad oil | 28.7
Warm | $\surd$ | $\surd$ |  | Oregano oil | 39.1
Warm$\\_$d |  | $\surd$ |  | Oregano : Salad = 1 : 1 | 36.7
Cold | $\surd$ | $\surd$ |  | Menthol oil | 38.9
Cold$\\_$d |  | $\surd$ |  | Menthol : Salad = 1 : 1 | 36.6
Mechanical | $\surd$ |  | $\surd$ | Sichuan pepper oil | 39.7
Mechanical$\\_$d |  |  | $\surd$ | Sichuan pepper : Salad = 1 : 1 | 38.5
Mix1 |  | $\surd$ | $\surd$ | Oregano : Menthol = 1 : 1 | 47.6
Mix1$\\_$d |  | $\surd$ | $\surd$ | Oregano : Menthol : Salad = 1 : 1 : 2 | 39.4
Mix2 |  |  | $\surd$ | Oregano : Menthol : Sichuan pepper = 1 : 1 : 1 | 51.3
Mix2$\\_$d |  |  | $\surd$ | Oregano : Menthol : Sichuan pepper : Salad = 1 : 1 : 1 : 3 | 45.3

  * 1

The $\surd$ mark means the condition was used in the analysis of the
corresponding experiment.

##  4 Experiment 1

###  4.1 Procedure

We initially verified that the application of global chemically-induced warm,
cold, and mechanical stimuli leads to an increase in the 2PDT. Salad oil was
employed as a control substance. To stimulate warmth receptors, pure oregano
oil (R V Essential), known for its specific activation of the TRPV3 channel
[13], was used. For cold receptors, menthol oil (Komamori, 30% menthol
concentration) was applied. Additionally, Sichuan pepper oil, with a
hydroxy-$\alpha$-sanshool concentration of 4.36g/kg, was utilized to activate
mechanoreceptors. Using eyedropper pipettes and cotton swabs, 0.5 mL of these
oils were uniformly applied to a 40 mm x 40 mm area on the proximal forearm.
The 2PDT test was performed 30 minutes following the application.

###  4.2 Results of Experiment 1

Each subplot in the graph represents a comparison between the control dataset
and one of the other datasets (warm, cold, or mechanical) using a sigmoid
function fit (see Fig. 2). The x-axis of each subplot stands for the distance
between the two points. The y-axis of each subplot stands for the perceived
number of points. The gray curve in each subplot stands for the results under
the placebo condition. Compared to the placebo, chemically-induced warm, cold,
and mechanical stimulation caused a rightward shift in the curve representing
perceived numbers of points, indicating a reduction in sensitivity.
Specifically, a greater point distance was required for participants to
perceive them at an equivalent level to the placebo condition. Under warm,
cold, and mechanical stimulation, the Two-Point Discrimination Threshold
(2PDT) increased by 10.4 mm, 10.2 mm, and 11 mm, respectively. Additionally,
this figure includes a 95$\%$ confidence interval for each condition.
Throughout the experiment, no instances of pain were reported by any of the
participants.

Figure 2: Results of Experiment 1. Fitted psychometric functions to the
perceived number of points when globally applied with chemical-induced
stimuli. The gray curve in each subplot stands for the results under the
placebo condition (salad oil). The red, blue, and orange curve stands for the
results under chemical-induced warm (oregano), cold (menthol), and mechanical
(Sichuan pepper) stimulation.

##  5 Experiment 2

###  5.1 Procedure

In experiment 1, we confirmed that the application of chemically induced
stimulation (warm/cold/mechanical) enlarges 2PDT. In experiment 2, we
investigated the mixture effect of warm and cold stimulation. Specifically, we
tested whether the warm and cold stimulation is additive. Undiluted and 50$\%$
concentration conditions tested are listed in Tab. 1. Prior to the experiment,
we mixed according to the proportions shown and stored the mixture in plastic
bottles. The experimental methods were the same as in Experiment 1.

Figure 3: Results of Experiment 2. We investigated the mixture effect of
chemical-induced warm and cold stimulation. The gray bar stands for the result
under the placebo condition. Other bars stand for the changes due to
corresponding conditions (Tab. 1). (a) Undiluted conditions. (b) 50$\%$
concentration conditions.

###  5.2 Results of Experiment 2

The point of subjective equality (PSE) of each corresponding condition in
Experiment 2 is shown in Fig. 3 and Tab. 1. Results across participants were
used to calculate the PSE. Fig. 3.(a) illustrates the results under undiluted
conditions. The gray bar stands for the result under the placebo condition.
The red, blue, and green bars stand for the change due to warm, cold, and mix1
(Oregano:Menthol = 1:1) stimuli, respectively. The effect of mix1 was greater
than the effect under both warm and cold conditions. A similar relationship
was observed when the stimuli were diluted (Fig. 3.(b)). The effect of
mix1$\\_$d was greater than the effect under warm$\\_$d and cold$\\_$d
conditions. This result suggests that the combined effect of chemical-induced
warm and cold stimuli on 2PDT appeared additive when compared to the influence
of each stimulus individually. Additionally, compared with the results under
undiluted conditions, the 2PDT under the corresponding diluted conditions was
smaller (Tab. 1), indicating that the 2PDT is influenced by concentration.
Throughout the experiment, no instances of pain were reported by any of the
participants.

##  6 Experiment 3

###  6.1 Procedure

In experiment 2, we investigated the mixture effect of warm and cold
stimulation. This can be seen as an interaction in the category of thermal
sensation. In experiment 3, we investigated the mixture effect of thermal
stimulation and mechanical stimulation. We tested whether the mixture of warm
and cold and mechanical stimulation was additive. Undiluted and 50$\%$
concentration conditions tested are listed in Tab. 1. The experimental methods
were the same as in Experiments 1 and 2.

Figure 4: Results of Experiment 3. We investigated the mixture effect of
chemical-induced thermal and mechanical stimulation. The gray bar stands for
the result under the placebo condition. Other bars stand for the changes due
to corresponding stimuli. (a) Undiluted conditions. (b) 50$\%$ concentration
conditions.

###  6.2 Results of Experiment 3

PSE of each corresponding condition in Experiment 3 is shown in Fig. 4 and
Tab. 1. Results across participants were used to calculate the PSE. Fig. 4.(a)
illustrates the results under undiluted conditions. The purple bar was higher
than the green and orange bars, indicating that the effect of mix2 (Oregano :
Menthol : Sichuan pepper = 1:1:1) was greater than the effect under mix1
(Oregano:Menthol = 1:1) and mechanical conditions. A similar relationship was
observed when the stimuli were diluted (Fig. 4.(b)). The effect of mix2$\\_$d
was greater than the effect under mix1$\\_$d and mechanical$\\_$d conditions.
These results suggest that the combined effect of chemical-induced thermal and
mechanical stimuli on 2PDT appeared additive when compared to the influence of
each stimulus individually. Additionally, the influence of concentration was
also observed. Throughout the experiment, no instances of pain were reported
by any of the participants.

##  7 Discussion

In this study, we demonstrated a modulation of the Two-Point Discrimination
Threshold (2PDT) in response to global warm, cold, and mechanical stimuli,
without triggering pain. To investigate the effects of these stimuli on 2PDT,
while avoiding skin deformation and inadvertent mechanical stimuli that could
distort the results, we applied chemicals (oregano, menthol, and Sichuan
pepper) to selectively activate receptors that usually respond to mild warm,
cold, and mechanical stimuli. Our results indicated a notable increase in 2PDT
across all conditions—warm, cold, or mechanical stimulation—suggesting a
consistent modulation of tactile acuity in the absence of inflammation.
Furthermore, we observed that the combined effect of mixed chemical stimuli on
2PDT seemed to be cumulative.

It has been previously reported that capsaicin-induced inflammation globally
increases the 2PDT [8]. In our study, we demonstrate that global application
of chemically induced thermal and mechanical stimuli similarly elevates 2PDT
but does so without inducing inflammation (Experiment 1). This finding
indicates that tactile acuity can be modulated by multimodal stimuli
independently of inflammation, suggesting that the regulatory mechanism is not
mediated by nociceptive specific neurons in the nervous system. Our results
imply the existence of a relay nucleus capable of processing both thermal and
mechanical stimuli.

Previous research has established that Wide Dynamic Range (WDR) neurons in the
spinal cord are responsive to various sensory inputs, including mechanical,
warm, and cold stimuli [10, 4]. Furthermore, these neurons are known to be
activated by both innocuous and noxious thermal stimuli [9, 7]. Our findings
indicate that WDR neurons may serve as integral relay nuclei in the spinal
cord, with projections accommodating touch, warmth, and cold stimuli. This
suggests a critical role for WDR neurons in the modulation of tactile acuity.

The variation in the 2PDT was not constant but influenced by the concentration
of the stimuli. Further, our data from Experiments 2 and 3 demonstrate that a
combination of multiple stimuli produces a more pronounced effect than any
single stimulus. Specifically, the aggregate impact of the chemical mixture on
2PDT seemed to be additive in nature compared to the individual effects of
each stimulus. These findings suggest that signals related to warm, cold, and
mechanical stimuli converge on Wide Dynamic Range (WDR) neurons, and the
integration of these signals might amplify the stimulation to levels typically
elicited by more intense stimuli. This potent activation of WDR neurons
results in a significant increase in 2PDT. Additionally, the cross-modal
additivity of these signals indicates the presence of an “intensity channel”
in the human body, based on WDR neurons [1]. Within this channel, the specific
nature of the stimulus is secondary.

Here, we discuss the mechanism of two-point discrimination sensation. Mechan-
oreceptors transduce mechanical deformation of the skin, induced by a
mechanical stimulator, into neural signals (refer to Fig. 5.(a)). When two
stimuli are applied within the same receptive field, they are perceived as a
single point. However, if one stimulus falls within an adjacent receptive
field, two-point discrimination becomes unfeasible, resulting in a single-
point perception [11]. Thus, for effective two-point discrimination, an
unstimulated receptive field separating the two stimuli is essential. The
receptor responses are relayed to the spinal cord, where lateral interneurons
modulate the activity, inhibiting adjacent second order neurons (A′, B′, and
C′) [11]. The processed signals are then conveyed to the brain, culminating in
the recognition of two distinct points.

Figure 5: Mechanism of two-point discrimination sensation and mechanism of
enlargement of two-point discrimination threshold by globally warm, cold, and
mechanical stimulation.

We propose a mechanism for the enlargement of the two-point discrimination
threshold in response to global stimuli, encompassing warmth, cold, and
mechanical factors. Upon exposure to these stimuli, the corresponding
receptors are activated, transmitting signals to the spinal cord. At this
juncture, Wide Dynamic Range (WDR) neurons serve as key relay nuclei [10, 4].
These neurons integrate signals of various modalities, leading to an additive
effect that results in increased firing frequency of WDR neurons, compared to
when a single stimulus type is received. Concurrently, excitatory interneurons
amplify the activity of A′, B′, and C′ neurons. The processed stimuli are then
relayed to the brain, where they stimulate A′′, B′′, and C′′ neurons.
Consequently, stimuli originally perceived as two distinct points are now
interpreted as a single point, thereby expanding the two-point discrimination
threshold (refer to Fig. 5.(b)).

In an investigation of the two-point fused sensory area and the influence of
acupuncture stimulation, previous authors observed an expansion of the fused
sensory area following acupuncture [11]. They suggested that acupuncture-
induced afferent nerve impulses interfere with the lateral inhibitory
mechanism through inhibitory connections. In this model, acupuncture
administration leads to an expansion of the fused sensory area. Because of the
inhibitory connections, second order neurons at greater distances cannot be
activated. This implys that variations in acupuncture intensity (or the output
of the relay nucleus) do not alter the extent of expansion. This
interpretation, however, contrasts with the findings of our current study. To
explain our results, we propose a model based on excitatory connections. In
this framework, excitatory connections facilitate the activation of second
order neurons at greater distances, resulting in a progressive enlargement of
the 2PDT.

Moreover, considering that the mechanical stimulator generates mechanical
stimuli, it is possible that the second order neurons (A′, B′, and C′) in Fig.
5 are WDR neurons. When these neurons receive excitatory inputs from far away
due to the global application of stimulation, it may be manifested as an
increase in the receptive field. This hypothesis aligns with the observed
increase in the receptive fields of WDR neurons following capsaicin
application, as reported in a prior study [8].

##  8 Conclusion

It was shown that chemical-induced warm, cold, and mechanical stimuli lead to
an increase in the two-point discrimination threshold (2PDT) without inducing
inflammation. Moreover, the combined effect of chemical-induced stimuli on
2PDT appeared to be additive. These findings suggest that Wide Dynamic Range
(WDR) neurons, which act as relay nuclei with projections for touch, warmth,
and cold sensations, play a crucial role in this process. In lateral
connection structures parallel to afferent nerve transmission pathways with
WDR neurons as relay nuclei, global stimulation amplifies excitatory
connections over inhibitory ones, thereby elevating the two-point
discrimination threshold.

####  8.0.1 Acknowledgements

This research was supported by JSPS KAKENHI Grant-in-Aid for Scientific
Research (A) Grant Number 19H01121.

## References

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# Magnetic field tuning of the valley population in the Weyl phase of Nd2Ir2O7

Itzik Kapon<EMAIL_ADDRESS>Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland  Carl
Willem Rischau  Department of Quantum Matter Physics, University of Geneva, 24
Quai Ernest-Ansermet, 1211 Geneva, Switzerland  Bastien Michon  Department of
Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211
Geneva, Switzerland  Kai Wang  Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland  Bing
Xu  Department of Physics and Fribourg Center for Nanomaterials, University of
Fribourg, Fribourg, Switzerland  Qiu Yang  Department of Physics, University
of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Satoru Nakatsuji
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan  Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba
277-8581, Japan  Institute for Quantum Matter and Department of Physics and
Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA  Trans-
scale Quantum Science Institute, University of Tokyo, Hongo, Bunkyo-ku, Tokyo
113-8654, Japan  Dirk van der Marel  Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland

(August 30, 2024)

###### Abstract

The frustrated magnet Nd2Ir2O7, where strong correlations together with spin-
orbit coupling play a crucial role, is predicted to be a Weyl semimetal and to
host topological pairs of bulk Dirac-like valleys. Here we use an external
magnetic field to manipulate the localized rare earth 4f moments coupled to
the 5d electronic bands. Low energy optical spectroscopy reveals that a field
of only a few teslas suffices to create charge compensating pockets of holes
and electrons in different regions of momentum space, thus introducing a
valley population shift that can be tuned with the field.

##  I Introduction

Dirac materials are understood today in terms of common topological
properties, although they encompass vast range of compounds with different
underlying physical mechanisms which bring upon the low energy Dirac
dispersion Vafek and Vishwanath (2014); Armitage  _et al._ (2018). These
materials, be that two (2D) or three (3D) dimensional, host Dirac cones in
their electronic band structure, in the bulk or on the surface, where the
conduction and valence bands are degenerate at a discrete set of points in the
Brillouin zone, and disperse linearly around these points. Graphene is the
most famous 2D one, however many more materials belong to this family: Weyl
semimetals like TaAs Lv  _et al._ (2015), Mn3Sn Nakatsuji  _et al._ (2015) and
CoSi Rao  _et al._ (2019), the topological insulator Bi2Se3, for the spin-
density wave state of BaFe2As2 Richard  _et al._ (2010) and it has been
predicted for the pseudogap phase of the cuprate high-$T_{c}$ superconductors
Borne  _et al._ (2010), to only name a few. Recently, it has been proposed to
use these materials for a new type of electronics utilizing the Dirac valleys,
namely, “valleytronics" Vitale  _et al._ (2018). The main challenge towards
achieving this is valley polarization - the ability to selectively control the
different Dirac cones with external forces, like circularly polarized light or
magnetic field.

Here we show that the pyrochlore iridate Nd2Ir2O7 is a new candidate for
future valleytronics technologies. It has been theoretically predicted to be a
Weyl semimetal Wan  _et al._ (2011), thus hosting distinguishable Dirac nodes.
The sister compound Pr2Ir2O7 was experimentally reported to be a correlated
Weyl semimetal under external strain Li  _et al._ (2021); Ohtsuki  _et al._
(2019). In our compound, the localized 4f electronic magnetic moments on the
Nd sites interact with the itinerant 5d electrons from the Ir. The Ir
electrons carry an effective pseudospin $j=1/2$ due to spin-orbit interaction
and crystal field splitting. They order antiferromagnetically (AFM) at
$T_{N}=37$ K in the non-collinear all-in-all-out (AIAO) state with the Ir
moments pointing inside or outside the tetrahedra, and create an exchange
field of $\Delta_{0}=6.5$ K on the Nd sites. The Nd moments become detectable
at $T\approx 13$ K and are also arranged in the AFM AIAO configuration. An
open question concerns the influence of the Nd moments on the Ir ones.
Moreover, the effect of a magnetic field could be amplified at the Ir sites by
the strong Nd moments and the f-d interaction Tian  _et al._ (2016).

We follow this approach, and use magneto-optical spectroscopy and magneto-
calorimetry combined with mean-field calculations to show that magnetic field
of seven tesla creates charge compensated pockets at the different Weyl nodes
in the Brillouin zone. This is due to the f-d interaction, where the Nd
magnetic moments are re-oriented with the field and serve as field boosters at
the Ir sites, thus altering the band structure. Therefore, it is possible to
tune the valley populations with modest fields. It has also been suggested
lately Wang  _et al._ (2020) that this material demonstrates unconventional
free charge, based on the quadratic in temperature dependence of the Drude
spectral weight without magnetic field. We comment on this interpretation and
explain this phenomenon in the framework of charge compensated Weyl points.

Figure 1: Optical conductivity data. a. Real part of the optical conductivity
$\sigma_{1}$ as function of energy for different magnetic fields at $T=6$ K.
The main field effects are red shift of the spectra and the increase of the
Drude component. b. Same as (a) but for low energy. DC conductivity data are
presented as solid circles. Vertical dashed lines (a and b) mark the energy
below which the fits are extrapolated outside our measured data range. c.
Temperature evolution of the relative reflectivity at H=7T. $R_{0}$ is the
zero-field reflectivity and $\delta R=R(H)-R_{0}$. Above $T\approx 13$ K we
are not able to detect any change with field. d. Fits to the reflectivity data
at 0T (black) and 7T (red) with Drude part taken as constant with field
(blue). Figure 2: Drude spectral Weight at $T=6$ K. The free charge carrier
density doubles from 0 to 7 tesla, supporting the creation of charge
compensated pockets. Antiferromagnetic domain walls exist until 2-3 tesla, as
known from Ma  _et al._ (2015). Error bars correspond to $1\%$ error of the
reflectivity. Figure 3: Simulation data. a) Ir average magnetic moments as
function of field. Black squares represent Ir pseudospin along the $(1,1,1)$
field direction, whereas red circles represent the other three Ir’s. b)
Illustration of the 4-in-0-out and 3-in-1-out configurations of the Nd
moments. The Nd atoms reside on the tetrahedron’s corners. c) The first
Brillouin zone with high symmetry points marked. d) Illustration of two pairs
of Weyl points for $H=7$ T in the $\Gamma L_{1}L_{2}$ plane. Orange plane
represents the chemical potential. Small holes and electrons pockets are seen
in the two nonequivalent points $L_{1}$ and $L_{2}$. e-h) Band structure along
$\Gamma L_{1}$ line for various fields, where a transition from Weyl semimetal
to Weyl metal is clearly observed.

##  II Experimental results

For the present experiments we used the single crystal of Ref. Wang  _et al._
, 2020. We measured reflectivity at temperatures between 4 to 300 K at
magnetic fields from 0 to 7 T along the $(1,1,1)$ direction. Optical
conductivity spectra $\sigma_{1}(\omega)$ obtained from the reflectivity data
(see Appendix A) are shown for $T=6$ K for different magnetic fields in Fig. 1
(a) and (b). As the field is increased, the conductivity is rising and red-
shifting throughout the low energy range from 0 to 100 meV. Specifically, the
Drude part is increasing. The strong spectral weight increase of the inter
band transitions from 15-45 meV (see Fig. 1b) brings up the question whether
the increase in the dc conductivity can partly be attributed to the tail of
the enhanced interband optical conductivity. When decomposing using the Drude-
Lorentz model, by construction there is no interband contribution at zero
frequency because of the mathematical properties of the Lorentz component.
That said, in a microscopic calculation it is possible to have a non-zero
interband conductivity at zero frequency, provided that two or more bands are
overlapping and these bands cross right at the Fermi surface. However, we will
see below that such a scenario is not supported by the electronic structure
calculations. Some peaks corresponding to phonons also exhibit an asymmetric
change with the field. To follow the temperature evolution at constant field,
Fig. 1(c) presents the relative change in the measured reflectivity $\delta
R\left(H\right)/R\left(0\right)=\left(R\left(H\right)-R\left(0\right)\right)/R\left(0\right)$
at $H=7$ T for temperatures from 4 to 30 $\leavevmode\nobreak\ K$. Field
response is only observed below $T\approx 13$ K. This is expected, as the Ir
5d electrons carry a small magnetic moment of approximately $0.2\mu_{B}$
relative to the Nd 4f electrons with $2.4\mu_{B}$ as measured by neutron
scattering Tomiyasu  _et al._ (2012), where $\mu_{B}$ is the Bohr magneton.
Nonetheless, one must bear in mind that optical conductivity is mostly
sensitive to the itinerant electrons, i.e., the Ir electronic bands. Thus, it
must be deduced that the external magnetic field indirectly influences the 5d
bands via the 4f moments, whose coupling to the former comprises the biggest
effect on their electronic phase. See the computation and discussion sections
for more details.

Fig. 2 presents the evolution of the Drude spectral weight with field,
$\omega_{p}^{2}\left(H\right)$. Going from zero to seven tesla the spectral
weight doubles. Thus, the free charge carrier density increases with field, as
also observed in transport Ueda  _et al._ (2015); Nakayama  _et al._ (2016).
DC conductivity of our sample is presented as solid circles in Fig. 1(b). To
further evaluate the free charge response to the field, we compare in Fig.
1(d) the fits to the reflectivity data at $T=6$ K for $H=0$ T (green dashed
line) and $H=7$ T (red dotted line). For the latter, we also present the fit
while constraining the Drude part to its value at 0T (blue solid line). The
relative change in reflectivity due to the Drude part comprises a major
contribution in the low energy regime. We will later argue that this
originates from valley population shift.

Next we describe our results from specific heat measurements under magnetic
field. The data (see Appendix B) confirm that the most probable scenario is
that above 2-3 tesla there is only one AF domain in our crystal. It consists
of Nd moments pointing towards the center of the tetrahedron, with three of
them pointing along the favorable field direction, i.e., having positive
projection along $(1,1,1)$. That configuration is named “4-in-0-out" (4-0) and
illustrated in Fig. 3(b) on the left panel. This result is supported by a
previous study that followed the evolution of domain walls in Nd2Ir2O7 Tardif
_et al._ (2015). We extract the exchange field at the Nd sites exerted by the
Ir moments as a function of magnetic field $\Delta_{0}(H)$. This effective
potential will be used in our model for the f-d interaction (see Appendix C).

##  III Model calculations

To interpret the observed behavior, we conducted a Hartree mean field type
calculation, following the tight binding Hamiltonian from Ref. Witczak-Krempa
_et al._ , 2013, to which we added an effective f-d interaction. To this end,
we follow Chen and Hermele Chen and Hermele (2012), starting with a general
term describing coupling between Ir 5d electrons and Ising like Nd moments
pointing in their local $z$ axis:

$H_{fd}=\sum_{\left\langle i,j\right\rangle}\sum_{j\in Nd}\sum_{i\in
Ir}\hat{\tau}_{j}^{z}\left[\left(d_{i\alpha}^{\dagger}\frac{\boldsymbol{\sigma}_{\alpha\beta}}{2}{d}_{i\beta}\right)\cdot\mathbf{v}_{ji}\right],$
(1)

where $\hat{\tau}_{j}^{z}$ is a Pauli matrix representing the j-th Nd Ising
spin, $d_{i\alpha}^{\dagger}$ is a creation operator of 5d electron on the
i-th Ir site with pseudo-spin $\alpha$, $\boldsymbol{\sigma}$ is a vector of
Pauli matrices, and $\mathbf{v}_{ji}$ are vectors describing the symmetry
allowed interaction, and contain two parameters $c_{1}$ and $c_{2}$. Next, we
substitute $\hat{\tau}_{j}^{z}$ with an average magnetic moment $M_{j}^{z}$,
as described hereafter.

Using our specific heat data, we construct an expression for the effective
potential acting on the 4f moments. Taking into account the Zeeman term, it is
given by $\Delta_{j}^{{}^{\prime}}(T,H)=\Delta_{0}(H)+\alpha g\mu_{B}H$, where
$\alpha$ is -1 for the Nd moment pointing anti-parallel to the magnetic field,
and $\alpha=1/3$ for the other three. The effective Hamiltonian for each Nd is
$H_{Nd}^{eff}=-\frac{1}{2}\Delta_{j}^{{}^{\prime}}(T,H)\tau_{j}^{z}$, from
which it is easy to calculate the average magnetic moment
$\left\langle{{M}_{j}^{z}}\right\rangle=\tanh\left(\frac{\beta\Delta_{i}^{{}^{\prime}}\left(\beta,H\right)}{2}\right)$.
Now, we can substitute ${{\hat{\tau}}^{z}_{j}}$ in $H_{fd}$ with
$\left\langle{{M}_{j}^{z}}\right\rangle$. All the parameters in the model,
unless stated otherwise, are given in units of the oxygen mediated nearest
neighbor Ir-Ir hopping, $t_{0}$, as commonly used in theories for pyrochlore
iridates Pesin and Balents (2010); Chen and Hermele (2012); Witczak-Krempa
_et al._ (2013).

Figure 3(a) presents the evolution of Ir magnetic moments with field. While
the Nd moments flip from the “4-0" to the “3-1" configuration, the Ir moments
do not flip with fields up to 7 T. However, their mean field average values
are getting smaller, indicating possible canting of the moments from the local
z easy-axis. Their initial configuration depends on the f-d interaction type -
ferromagnetic or antiferromagnetic, and determined by the values of $c_{1}$
and $c_{2}$. Nonetheless, both types of interaction yield similar band
structures qualitatively. We present a cut along high symmetry line from the
Brillouin zone center $\Gamma=(0,0,0)$ to the edge $L_{1}=\frac{1}{2}(1,1,1)$
in units of reciprocal lattice vectors. Figures 3(d-g) present the band
structure evolution with magnetic field, starting from Weyl semimetal at $H=0$
to Weyl metallic phase at $H>0$ with compensating charge pockets. While
electron pockets are created in the Weyl pair along the $L_{1}$ direction,
hole pockets are created in the other three pairs along the high symmetry
directions $L_{2}=\frac{1}{2}(1,-1,1)$ and its cyclic variations.

##  IV Discussion

The magnetic field breaks the symmetry between the otherwise equivalent
positions in reciprocal space along the line connecting $\Gamma$ and $L$; the
field serves as a knob for tuning the valley population. Figure 3(d)
demonstrates two inequivalent pairs of Weyl nodes in the $\Gamma L_{1}L_{2}$
plane. One pair, hosting electron pockets, is along the $\Gamma L_{1}$ line
and the other shifted slightly from the $\Gamma L_{2}$ line hosting hole
pockets. The chemical potential is presented as an orange plane. Previous
study has reported field-induced polarization of Dirac valleys in bismuth Zhu
_et al._ (2012), based on the cyclotron motion of the conduction electrons. In
contrast, the valley polarization of Nd2Ir2O7 capitalizes on the magnetic
structure of the Ir sublattice in this compound and the amplifying effect of
the Nd moments.

Figure 4: Simulated optical conductivity for $\mathbf{U=1.365}$ eV,
$\mathbf{c_{1}=0.026}$ eV and $\mathbf{c_{2}=-0.039}$ eV. The appearance of
Drude spectral weight and an overall red shift of the spectra qualitatively
match the experimental data. In order to match quantitatively, one needs
$t_{0}=0.26$ eV, suggesting a factor of four mass renormalization of the 5d
bands.

The corresponding optical conductivity for various fields is given in Fig. 4.
Qualitatively, the theory and the experimental data agree: the simulation
predicts the appearance of Drude spectral weight and red shift of the
interband transitions upon application of magnetic field. To match the
experimental data one needs $t_{0}=0.26$ eV. From the dispersion shown in Ref.
Zhang  _et al._ , 2017 one can estimate $t_{0}=1.3$ eV, suggesting a factor of
four mass enhancement due to strong correlation effects not captured by the
single-particle tight binding Hamiltonian and the Mean Field approximation.
Energy renormalization of this order due to strong correlations is common in
transition metal oxides Basov  _et al._ (2011); Stricker  _et al._ (2014); van
Mechelen  _et al._ (2008), and observed in SrIrO3 Moon  _et al._ (2008) and in
Sr2IrO4 Wang  _et al._ (2018). Our result is in agreement with angle-resolved
photo-emission spectroscopy measurements, that showed bandwidth narrowing with
respect to LDA+U calculations. Nakayama et al. Nakayama  _et al._ (2016)
reported 40 meV width of the occupied band next to the Fermi energy versus
about 0.5 eV of the calculated bands Shinaoka  _et al._ (2019). Moreover,
coupling of the electronic degrees of freedom to the phononic ones would also
strongly affect the band structure. This kind of coupling is clearly evident
from our optical conductivity data in the form of asymmetric phonons peaks
that also respond asymmetrically to external field. We mention that similar
behavior of band narrowing and electron-boson coupling was observed in
Pr2Ir2O7 Kondo  _et al._ (2015).

One of the signatures of the valley imbalance at finite field (Fig. 3d and 3h)
is a Pauli-blocking of the optical conductivity at low frequencies. The fact
that the tilted Dirac cones are 3-dimensional makes that the Pauli blocking
shows up as a gradual rise starting at $0.05t_{0}=13$ meV and ending
$0.1t_{0}=26$ meV (and an additional one at about half that energy). While
such a gradual rise of the optical conductivity could in principle be revealed
by the reflectivity, the accuracy obtained in the present study (especially
below 30 meV, see Fig. 1c) does not allow to identify this feature in the
experimental data.

Recently some of us reported a $T^{2}$ free carrier spectral weight as
expected for massless Dirac electrons, however the entropy counterpart did not
show up in the specific heat Wang  _et al._ (2020). A scenario of temperature
dependent charge compensated electron and hole pockets that for $T=0$ touch at
$E_{F}$, and overlap for finite temperature was dismissed because there are no
indications for this in the literatureWan  _et al._ (2011); Witczak-Krempa
_et al._ (2013); Wang  _et al._ (2017). However, our present magneto-optics
data suggest precisely such a scenario. In the absence of a magnetic field and
at $T=0$ the Ir 5d bands form Weyl points at $E_{F}$ and the four Nd moments
are in the 4-0 configuration. At finite temperature the fraction of excited
moments is $n(T)=1/(e^{\Delta_{0}/T}-1)\approx T/\Delta_{0}$ where the
approximation is valid for $T\gg\Delta_{0}$. This creates a random exchange
potential $V_{x}$ at the Ir sites, which statistically averages out to zero,
but its square has a finite average $V_{x}^{2}=J_{fd}^{2}T^{2}/\Delta_{0}^{2}$
where $J_{fd}$ characterizes the $fd$ exchange interaction. Substituting
$V_{x}$ for the chemical potential $\mu$ in Eq. 10 of Ashby and Carbotte Ashby
and Carbotte (2014), we obtain for the free carrier spectral weight
$\omega_{p}^{2}=2ge^{2}J_{fd}^{2}T^{2}/(3\pi\hbar^{3}\upsilon_{F}\Delta_{0}^{2})$,
where $g$ is the number of Weyl points in the Brillouin zone and
$\upsilon_{F}$ the Fermi velocity. The only entropy involved is that of the Nd
spin degrees of freedom, which is already accounted for by the Schottky
anomaly.

##  V Summary and conclusions

In summary, optical spectroscopy reveals a red shift of the interband
transitions and an increase of the Drude spectral weight in a magnetic field
below 13 K. Mean field calculations, taking into account interactions between
the 4f and 5d electrons using real measured values extracted from magneto-
calorimetry, show that modest magnetic fields are enough to alter the band
structure, showing that the rare earth Nds serve as magnetic field “boosters".
The field creates charge compensated pockets at the four pairs of different
Weyl points and induces valley population shift. While the model of valley
imbalance offers a plausible explanation that respects the overall charge
neutrality of the system, we do not wish to pretend that it is the only
mechanism that could possibly be at play here. The role of rare earth together
with fields of that scale as valley polarizers should be further studied as
one of the possible mechanisms for valleytronic devices. Further experiments
such as time domain THz pump probe, Kerr rotation and non-linear
electromagnetics Ma  _et al._ (2015) are needed to understand and manipulate
the different Weyl points.

###### Acknowledgements.

We thank Nimrod Bachar, Vladimir Kalnizky, Alexey Kuzmenko, Michael Hermele
and Gang Chen for fruitful discussions. This project was supported by the
Swiss National Science Foundation through project 200020-179157. The work at
the Institute for Quantum Matter, an Energy Frontier Research Center was
funded by DOE, Office of Science, Basic Energy Sciences under Award DE-
SC0019331. This work was partially supported by JST-CREST (JPMJCR18T3).

##  Appendix A Magneto-reflectance measurements

Magneto-reflectance measurements were performed in a cryogenic magnet
connected to Fourier transform spectrometer. Hg and Globar lamps were used as
light sources, together with KBr beam splitter. The iris aperture was 2 mm.
The sample chamber inside the magnet is pumped to high vacuum of about
$10^{-8}$ mbar. The sample and reference mirror were mounted to a sample
holder, which in turn was placed on a motorized arm, thus allowing accurate
movement between the sample and the mirror alternatively while in the magnet.
Light reflected from the sample or the mirror was collected and measured with
a cryogenic Bolometer. The sample surface is along the $(1,1,1)$
crystallographic direction with area of approximately 1 mm2. Reflectivity
measurements were done in near-to-normal incidence geometry. The system was
cooled down slowly to 4 K, and then warmed up to the desired temperature.
Magnetic field was applied along the $(1,1,1)$ direction from 0 T to 7 T. In
order to prevent misalignment and mechanical movement errors, the sample was
measured first with different fields at constant temperature while being kept
stable at the same position. Only then was the mirror measured. To compensate
for drift due to changes with time in the light source and the Bolometer, the
reference was measured both before and after the sample had been measured. The
recorded signal, $I(H)$, was used to extract the field dependent reflectivity
from the zero field reflectivity via the relation $R(H)=\frac{I(H)}{I(0)}\cdot
R(0)$. To obtain the dielectric function $\epsilon(\omega)$ and the optical
conductivity $\mbox{Re}\sigma(\omega)=(\omega/4\pi)\mbox{Im}\epsilon(\omega)$
we fitted the Drude-Lorentz expansion

$\epsilon(\omega)=\epsilon_{\infty}+\sum_{j}\frac{\omega_{p,j}^{2}}{\omega_{0,j}^{2}-\omega^{2}-i\gamma_{j}\omega}$

to the magneto-reflectivity (15-90 meV), the zero field reflectivity (90-500
meV), the zero field room temperature ellipsometery (0.5-2.5 eV) and the DC
conductivity. For the Drude term $\omega_{0}=0$.

##  Appendix B Specific heat

Specific heat was measured using a PPMS©. First, the thermometers were
calibrated under all the desired magnetic fields to correctly read the sample
temperature. Next, the heat capacity of the addenda, $C_{add}$, was measured
for each field, including a small amount of vacuum grease used as glue and
thermal link between the platform and the sample. Finally, the sample
including the addenda were measured yielding $C_{tot}$ from which the heat
capacity of the sample alone is calculated via
$C_{sample}(H)=C_{tot}(H)-C_{add}(H)$. These values were converted to units of
JK-2mol-1 using the sample mass and the compound molar mass.

Figure 5: Specific heat data as function of magnetic field. The lines are the
fits to data with single type 1 domain.

The exchange potential was calculated from the specific heat data. The Nd ions
occupy the corners of a tetrahedron. We use the label $j=0$ for the Nd atom on
the $(1,1,1)$ corner, and $j=1,2,3$ for the other three. The relevant states
for the specific heat form spin-orbital doublet polarized along the local
quantization axis, which for $j=0$ is $(1,1,1)$. Magnetic order of the Ir
sublattice induces an exchange field at the Nd sites oriented along the local
quantization axis. Two types of domains are possible, which for a given
tetrahedron correspond to exchange fields either pointing all in (type 1
domain) or all out (type 2 domain). At low field these domains alternate and
are separated by domain walls. Here we found that a field of 3 tesla suffices
to favor one type of domain and suppress all domain walls.

An external magnetic field $H$ along $(1,1,1)$ has a projection $f_{j}H$ on
the local quantization axis, where $f_{0}=1$, and for $j=1,2,3$ $f_{j}=-1/3$.
Together with the local exchange field this gives the energy splitting of the
$4f$ doublet

$\Delta_{j}=\sigma V_{df}+gf_{j}\mu_{B}H$

where $g=2.4$ is the gyromagnetic ratio of Nd3+, and $\sigma=1(-1)$
corresponds to domains of type 1 (2).

The specific heat is

$\frac{C(T)}{k_{B}T}=\frac{d}{dT}\sum_{j=0}^{3}\sum_{s=\pm
1}\frac{\ln{\left(1+e^{s\Delta_{j}/T}\right)}}{1+e^{s\Delta_{j}/T}}+aT^{2}-bT^{4}$

where the last two terms describe the phonon contribution. We fitted this
expression to the experimental specific heat data by adjusting $a$, $b$, and
$V_{df}$.

In Fig. 5 $C(T)/T$ is presented for different magnetic fields together with
the fits, assuming $\sigma=1$ (type 1 domains). The corresponding $\Delta_{0}$
and $\Delta_{1}$ is shown in Fig. 6. The field dependent exchange coupling is
given by $V_{df}=6.7+0.285H$ in units of kelvin and with $H$ in units of
tesla.

Fitting the data with $\sigma=-1$ (type 2 domains) or an equal mixture of both
types of domains, gives unreasonably high values of $V_{df}$. With these
parameters we calculate the thermally averaged spin values of the Nd atoms

${M}_{j}^{z}=\arctan\left(\frac{\Delta_{j}}{2T}\right)$

which are inserted in Eq. 1 of the main text to calculate the bandstructure in
the presence the Nd exchange potential on the Ir sites.

Figure 6: Exchange potential at the Nd sites.

##  Appendix C The electron band structure

The electron band structure was calculated using the tight binding Hamiltonian
of Ref. 19 of the main text, which describes electrons with pseudospin $j=1/2$
having an onsite Hubbard U interaction, $H_{U}$, and the Zeeman term:

$\displaystyle
H_{0}=\sum\limits_{s,s^{\prime}}\sum\limits_{\mathbf{R}}\sum\limits_{\left\langle
i,j\right\rangle}d_{\mathbf{R}is}^{\dagger}\left(t_{1}\mathbf{I}+it_{2}\mathbf{d}_{ij}\cdot\boldsymbol{\sigma}\right)_{ss^{\prime}}d_{\mathbf{R}js^{\prime}}+$
$\displaystyle+\sum\limits_{\left\langle\left\langle
i,j\right\rangle\right\rangle}d_{\mathbf{R}is}^{\dagger}\left(t_{1}^{{}^{\prime}}\mathbf{I}+i\left[t_{2}^{{}^{\prime}}{{\mathcal{R}}_{ij}}+t_{3}^{{}^{\prime}}{{\mathcal{D}}_{ij}}\right]\cdot\boldsymbol{\sigma}\right)_{ss^{\prime}}d_{\mathbf{R}js^{\prime}}.$
(2)

Here $d_{\mathbf{R}is}^{\dagger}$ creates an electron at unit cell
$\mathbf{R}$, at Ir site $i$ with pseudospin $s$, $t_{1},t_{2}$ and
$t_{1}^{{}^{\prime}},t_{2}^{{}^{\prime}},t_{3}^{{}^{\prime}}$ are the NN and
NNN hopping amplitudes, respectively, $\boldsymbol{\sigma}$ is a vector of the
Pauli matrices and $\mathbf{d}_{ij},{{\mathcal{R}}_{ij}},{{\mathcal{D}}_{ij}}$
are real geometrical vectors. Mean field decoupling of the Hubbard term gives

$\displaystyle{{H}_{U}}=\sum\limits_{\mathbf{R},i}\left[\frac{2U}{3}{{\left\langle{{\mathbf{j}}_{\mathbf{R},i}}\right\rangle}^{2}}-\frac{4U}{3}\left\langle{{\mathbf{j}}_{\mathbf{R},i}}\right\rangle\cdot{{\mathbf{j}}_{\mathbf{R},i}}\right],$

with
${{\mathbf{j}}_{\mathbf{R},i}}=\frac{1}{2}\sum\limits_{\alpha,\beta\in\left\\{\uparrow,\downarrow\right\\}}{d_{\mathbf{R},i,\alpha}^{\dagger}{{\mathbf{\sigma}}_{\alpha\beta}}{{d}_{\mathbf{R},i,\beta}}}$
the pseudospin operator. We add the f-d interaction term, as described in the
main text, and solve with four mean field parameters for the Ir sublattice
pseudospins. The complex optical conductivity is calculated as follows:

$\displaystyle\sigma(\omega)$ $\displaystyle=$
$\displaystyle\frac{q_{e}^{2}}{\hbar\Omega}\sum\limits_{\mathbf{k},j}^{1^{st}BZ}\mathbf{v}_{j,j}(\mathbf{k})\cdot\mathbf{v}_{j,j}(\mathbf{k})\left(-\frac{\partial
f_{\mathbf{k},j}}{\partial\varepsilon_{\mathbf{k},j}}\right)\frac{i}{\omega+i\delta}$
(3) $\displaystyle+$
$\displaystyle\frac{q_{e}^{2}}{\hbar\Omega}\sum\limits_{\mathbf{k},j\neq
m}^{{1^{st}}BZ}\mathbf{v}_{j,m}(\mathbf{k})\cdot\mathbf{v}_{m,j}(\mathbf{k})\left(\frac{f_{\mathbf{k},j}-f_{\mathbf{k},m}}{\varepsilon_{\mathbf{k},m}-\varepsilon_{\mathbf{k},j}}\right)$
$\displaystyle\times$
$\displaystyle\frac{i\omega}{\omega\left(\omega+i\delta\right)-\left(\varepsilon_{\mathbf{k},m}-\varepsilon_{\mathbf{k},j}\right)^{2}}.$

$\displaystyle\mbox{where }\hbar{{\mathbf{v}}_{j,m}}(\mathbf{k})=$
$\displaystyle\sum\limits_{\eta,\mu}{u_{\eta,j}^{*}(\mathbf{h}){{u}_{\mu,m}}(\mathbf{k})\left[\frac{\partial{{H}_{\eta,\mu}}(\mathbf{k})}{\partial\mathbf{k}}+i\left({{\mathbf{d}}_{\mu}}-{{\mathbf{d}}_{\eta}}\right){{H}_{\eta,\mu}}(\mathbf{k})\right]},$

and $\mathbf{d}_{\mu}$ are the positions of the four Ir atoms in a unit cell
and ${u}_{\mu,m}$ are the matrix elements of the unitary diagonalizing matrix
of the Hamiltonian.

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