# Persistent-Transient Duality: A Multi-mechanism Approach for Modeling Human-
Object Interaction

Hung Tran1, Vuong Le2, Svetha Venkatesh1, Truyen Tran1
1Applied AI Institute, Deakin University, 2Amazon
{tduy, svetha.venkatesh<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Humans are highly adaptable, swiftly switching between different modes to
progressively handle different tasks, situations and contexts. In Human-object
interaction (HOI) activities, these modes can be attributed to two mechanisms:
(1) the large-scale consistent plan for the whole activity and (2) the small-
scale children interactive actions that start and end along the timeline.
While neuroscience and cognitive science have confirmed this multi-mechanism
nature of human behavior, machine modeling approaches for human motion are
trailing behind. While attempted to use gradually morphing structures (e.g.,
graph attention networks) to model the dynamic HOI patterns, they miss the
expeditious and discrete mode-switching nature of the human motion. To bridge
that gap, this work proposes to model two concurrent mechanisms that jointly
control human motion: the Persistent process that runs continually on the
global scale, and the Transient sub-processes that operate intermittently on
the local context of the human while interacting with objects. These two
mechanisms form an interactive Persistent-Transient Duality that
synergistically governs the activity sequences. We model this conceptual
duality by a parent-child neural network of Persistent and Transient channels
with a dedicated neural module for dynamic mechanism switching. The framework
is trialed on HOI motion forecasting. On two rich datasets and a wide variety
of settings, the model consistently delivers superior performances, proving
its suitability for the challenge.

##  1 Introduction

Figure 1: We model the human motions by a duality of two types of processes: A
single _Persistent process_ (red box) considers the whole activity plan and
trajectory (red curve) and use dense relational structure (red graph). Its
children _Transient processes_ (green boxes) work on the small and local
spatiotemporal scope of a human and the interacted objects using egocentric
structure (green graph). They get turn on and off on-demand by the Persistent
process.

In a planned activity that involves interaction with objects, the patterns of
human motion vary greatly, contextualized to the situation and stages of the
plan. The variety of patterns comes from the fact that the subject needs to
switch between multiple modes of operations: navigating the global scene
according to the plan, and occasionally concentrate on a small set of objects
to act on them. This mode-switching nature requires a machine model to adapt
quickly in structure, representation, and inference mechanisms to follow the
patterns of the behavior. This observation is confirmed by neuroscience
findings that human brain activity contains transient networks that deals with
particular situation [4]. In cognitive science, human activities are also
proved to follow parent-child planning patterns [2].

Recent advancements in graph neural networks [43] allow motion models to
dynamically adjust their relational structure to adapt to changing situations
[8]. However, with a singular inference mechanism, they can only have _gradual
adaptation_ : smoothly adjusting parameters of the same model. They cannot
account for quick and abrupt changes __ regarding the discrete switching
between distinctive mechanisms and as a result fail to keep up with the
movement patterns. This limitation is inherent in human-object interaction
motion (HOI-M) prediction. Fig. 1 visualizes an example activity of cooking a
dinner meal. Here the subject navigates around the kitchen following a recipe
and consider the whole kitchen floor plan (red trajectory in Fig. 1).
Occasionally, they deviate to perform a particular action by interacting with
an object (green sections in Fig. 1) such as moving a bowl, where only the
bowl needs their attention. When this happens, these models will continue to
consider the interacted object as an equal member of the scene and miss its
importance as the direct recipient of the action.

To address this limitation, we explore a new modeling paradigm that factorizes
the human-object interaction into two internal types of processes: a
_Persistent process_ (red box in Fig. 1) which maintains a continuous large-
scale activity progress; and _Transient sub-processes_ (green boxes in Fig. 1)
which has an adaptive life cycle and a personalized structure to reflects the
small-scale local interaction with objects. The Transient act as Persistent’s
sub-processes, and they can be turned on or off by a switch (on/off arrows in
Fig. 1) operating on signal from the main Persistent process. The _parent-
child relationship_ connects the two types of processes and constitute the
_Persistent-Transient Duality_.

This modeling is related to similar transient concepts in other fields such as
control theory [22], electrical [19], and chemical engineering [41, 14]. In
computer engineering, the parent-child relationship also resembles the
operating system and the children task-specific processes.

We model this concept into a multi-channel neural network called _Persistent-
Transient Duality_(PTD) networks for Human-object interaction motion (HOI-M)
prediction. The _Persistent channel_ contains a recurrent relational network
operating on the global scene spatially and throughout the session temporally.
The _Transient channels_ instead have contextualized structures constructed on
the spot whenever the human subjects shift the priority toward interacting
with a subset of objects. The life cycles of these spontaneous channels are
managed by a neural _Transient Switch_ , which anticipates the initialization
and termination of many Transient processes along a single Persistent process.

The benefit of PTD is demonstrated via motion forecasting experiments on WBHM
and Bimanual Actions datasets where it sets new SOTA performances and
generalizability tests. Being a generic framework for human behavior modeling,
PTD is readily applicable to other problems such as pedestrian trajectory
prediction of which preliminary adaptation and experiment are shown in the
supplementary.

The key contributions of this work are:

1\. The exploration of _the new Persistent-transient duality concept_ to model
the multi-mechanism nature of human behavior reflected in the large-small
temporal scale and global-local spatial scope of HOI motions.

2\. A _parent-children neural framework_ with egocentric design that applies
the PTD concept in HOI-M prediction.

3\. The extensive analysis demonstrating that PTD _sets new SoTA in HOI-M
prediction_ across multiple datasets and settings, and generalizes better to
new scenarios.

##  2 Related Work

Human-object interaction in videos is traditionally formulated for predictions
of human action and object affordance labels [30, 37, 26, 15]. Approaches
include modeling the activity continuity with CNNs [30] and RNNs [37], or
modeling the interactions using GNNs [25, 18], which can be improved with
dynamic topology [36, 39]. Recent works also leverage the power of
transformers [42] to model the spatio-temporal relationship between humans and
objects in the videos [26]. In this paper, we consider human-object
interaction motion (HOI-M) prediction, the task of forecasting the concrete
future locations of humans and objects in an activity which is more well-
defined and easier to evaluate than the vague affordance labels traditionally
used in HOI. Also, this task is more complex and requires more precise
modeling structures, such as the duality proposed in this work.

Human body motion predictions are done with MLPs [6, 21], RNNs [16, 17, 38, 3,
45] or GNNs [25, 9, 29, 11, 12, 34, 40, 32, 31] and can be embedded in
generative models [20, 23, 10, 27]. More recent works started to consider
relations with surroundings in the form of intention toward destinations [7]
or interaction with other entities [8, 1]. We advance this line by considering
the directional relations between human and objects beyond the simplistic
homogenous graphs such as in [8] through a new multi-mechanism adaptive
structural of the persistent-transient duality.

##  3 Method

Figure 2: The architecture of _Persistent-transient Duality_ Networks (PTD).
The _persistent channel_ is a heterogeneous graph network connecting all
entities at the same time step. In complement, the _Transient channel_ zooms
into the local context of each human (circles) when they interact with
surrounding objects (triangles). This channel works with the egocentric
structure, under the viewpoint of the human subject. The transient channels
are initialized and terminated on-demand, controlled by the _Transient
Switches_ (diamonds) which turn on (filled) or off (empty) the corresponding
Transient processes.

###  3.1 Preliminaries

We consider the problem of modeling the sequential motions of $N$ entities of
humans and objects, where each entity $i$ is represented by a class label
$c_{i}$ and sequential visual features
$X_{i}=\left\\{x_{i}^{t}\right\\}_{t=1}^{T}$. Given the class labels and the
sequential features, we want to predict the future motions in the next $L$
steps, $Y_{i}=\left\\{y_{i}^{t}\right\\}_{t=T+1}^{T+L}$. Each time instance of
features $x_{i}^{t}$ of humans contains skeleton positions; while those of
objects include their object types and bounding boxes.

This problem is traditionally approached under the single-mechanism assumption
that the inference dynamics stay the same during the whole session [8]. In
this section, we present our new multi-mechanism paradigm, modeled with the
persistent-transient duality, to capture how humans maintain their global
operation mode while swiftly adapt to emerging local interaction situations.

###  3.2 The Persistent-Transient Duality

We consider the motions of the human bodies and the interacted objects to be
caused by two mechanisms: 1. the navigation according to the global activity
progress; and 2. the individual interactions with objects, which happens in
the local contexts. This multi-mechanism factorization can be viewed as a
persistent-transient process duality: The _persistent process_ operates in
large activity temporal scale and with a global spatial perspective. Its
counterpart, the _transient process,_ is local in time and egocentric to each
human subject’s spatial viewpoint.

We model this duality by a hierarchical neural network called _Persistent-
Transient Duality Networks (PTD)_ whose architecture is drawn in Fig. 2.
Within PTD, the _Persistent process_ is modeled as a single _Persistent
channel_ , which operates recurrently along the whole activity sequence. Its
children, the _Transient channel_ instances, are initiated with personalized
structures and representations whenever a human has a new interaction with the
surrounding. When the interaction is over, the outdated transient channel is
terminated, and the control is returned to the persistent channel waiting in
the background. These transient life cycles are managed by the neural modules
called _Transient Switches_.

###  3.3 Persistent Channel

The Persistent channel models the common global view of all humans and objects
in the scene. It takes the form of a heterogeneous graph attention network
[44] with two node types corresponding to human and object entities and _dense
spatial edges_ connecting entities at the same time step. We extend this model
by adding _recurrent temporal edges_ along the temporal dimension for each
entity:

$z_{i}^{t}=\left[x_{i}^{t},m_{i}^{t},m_{i,\mathcal{T}\rightarrow\mathcal{P}}^{t}\right],\hskip
8.53581pth_{i}^{t}=\text{RNN}_{c_{i}}\left(z_{i}^{t},h_{i}^{t-1}\right),$ (1)

where $\text{RNN}_{c_{i}}$ is a recurrent unit (e.g., GRU) that corresponds to
the class of the $i^{\text{th}}$ entity. The input $z_{i}^{t}$ of
$\text{RNN}_{c_{i}}$ is formed using the the entity’s intrinsic features
$x_{i}^{t}$ (skeleton for humans and bounding boxes for objects), the spatial
messages $m_{i}^{t}$ gathered from spatial edges, and a message from the
active transient channel $m_{i,\mathcal{T}\rightarrow\mathcal{P}}^{t}$ (Sec.
3.4).

The spatial message $m_{i}^{t}$ in Eq. 1 is aggregated from the _spatial
edges_ of the graph via attention:
$m_{i}^{t}=\text{Attn}\left(u_{i}^{t},\left\\{u_{j}^{t}\right\\}_{j\neq
i}\right)$ with $u\\_^{t}=\left[x\\_^{t};h\\_^{t-1}\right]$. Here and
throughout this work, we use the GAT-based attention function [43],
$\text{Attn}\left(q,V\right)$, defined over the query $q$ and the identical
key/value pairs $V=\\{v_{j}\\}_{j=1}^{N}$ by:

$\text{Attn}\left(q,V\right)\coloneqq\sum_{j=1}^{N}\text{softmax}_{j}\left(\sigma\left(\text{MLP}\left(\left[\boldsymbol{W_{q}}q;\boldsymbol{W}_{v}v_{j}\right]\right)\right)\right)\boldsymbol{W}_{v}v_{j},$
(2)

where $\boldsymbol{W_{q}}$ and $\boldsymbol{W}_{v}$ are the learnable
embedding weights for the query and key/value, $\left[\cdot;\cdot\right]$ is
the concatenation, $\sigma\left(\cdot\right)$ is a non-linear activation.

The Persistent channel generates two outputs from its hidden state: the
message to the transient channel of each human
$m_{i,\mathcal{P}\rightarrow\mathcal{T}}^{t}$ and the future position of each
entity $\hat{y}_{i,\mathcal{P}}^{t}$:

$m_{i,\mathcal{P}\rightarrow\mathcal{T}}^{t}=\text{MLP}\left(h_{i}^{t}\right),\qquad\hat{y}_{i,\mathcal{P}}^{t}=\text{MLP}\left(h_{i}^{t}\right).$
(3)

These prediction outputs are later combined with those from the Transient
channel as detailed in Sec. 3.6.

###  3.4 Transient Channel

Operating with a consistent graph, the Persistent process alone cannot adapt
quickly enough to emerging events that require different perspectives,
particularly when the human interacts with an object. When these cases are
detected, PTD initiates a _Transient process_ that (1) zooms in to the
relevant context and (2) take the viewpoint of the subject.

In our framework, the Transient process is implemented by a neural _Transient
channel_ , available one for each _human_ entity _._ We propose an _egocentric
recurrent graph networks_ for the Transient channel. The egocentric property
of this model is the most important aspect that separates it from the global
view of its parent persistent channel. This egocentric design reflects in
three aspects of _computational structure, feature representation_ , and
_inference logic_.

##### Egocentric computational structure

In switching from the global to personalized view, we start by forming an
_egocentric Transient graph_
$\mathcal{G}_{i}^{t}=\left(\mathcal{V}_{i}^{t},\mathcal{E}_{i}^{t}\right)$ at
time $t$ of the human $i$ and their relations with the objects interacted
with. For a particular human, subscript $i$ will be omitted for conciseness.
The egocentric characteristic of $\mathcal{G}^{t}$ reflects in its star-like
structure: the nodes $\mathcal{V}^{t}$ includes a single _center node $r$_ for
the considering human, and _leaf nodes_ of indices $\left\\{l\right\\}_{l\neq
r}$ for objects. The dynamic edges $\mathcal{E}^{t}$ connect the center with
the leaves in two directions: _inward edges_ $e_{l\rightarrow r}^{t}$ reflect
which objects the human may consider to interact with, and the _outward edges_
$e_{r\rightarrow l}^{t}$ represents the objects are being manipulated by the
human. In our implementation, the existences of the edges are determined by
thresholding the center-leaf distances $d_{lr}^{t}$:

$\begin{cases}e^{t}\\_\in\mathcal{E}^{t}&\text{if }d_{lr}^{t}\leq\beta\\_\\\
e^{t}\\_\notin\mathcal{E}^{t}&\text{otherwise}\end{cases},$ (4)

where $\\_$ indicates either the inward or onward, $\beta\\_$ is a pair of
threshold hyper-parameters. This thresholding effectively thins out the
neighbors, making the graph more efficient and localized around the center
node. Inward threshold is commonly smaller than outward one, because humans
subject’s attention, hence motion, are affected by objects before and after
objects get directly manipulated. The edges are estimated dynamically for each
time step, allowing the graph’s topology to evolve within one single Transient
session.

Figure 3: The Transient Channel features an egocentric structure of the center
node (circle) and leaves (triangles) with sparse edges (colored arrows). When
being active, it uses the egocentric features $\bar{x}_{r}^{t}$ and the
persistent message $m_{\mathcal{P}\rightarrow\mathcal{T}}^{t}$ to update its
state $s_{r}^{t}$. The life-cycle of this channel is determined by the
Transient Switch (blue diamond) visualized in Fig. 4.

##### Egocentric representation.

Figure 4: The Transient Switch predicts the transient score $p_{r}^{t}$ from
the hidden states $h_{r}^{t}$ and the input $z_{r}^{t}$ of the persistent RNN.
This score is modulated by a discount factor $\gamma_{r}^{t}$ computed from
the minimum distance $d_{r,min}^{t}$ via the discount gate (trapezoid).

Paired with the egocentric graph structure, the geometrical features of the
entities are also transformed into the egocentric coordinate system
corresponding to the viewpoint of the human center node $r$:

$\displaystyle\bar{x}\\_^{t}$
$\displaystyle=f_{\text{ego}}\left(x\\_^{t},x_{r}^{t}\right)=x\\_^{t}-\text{centroid}(x_{r}^{t}),$
(5)

where $\\_$ indicate both center or leaf nodes. Conceptually, this change of
system puts various patterns of the human’s motion into the same aligned
space. In the global view, features of two similar actions can be vastly
different if they are far away. After this transformation, they are aligned
into the common egocentric view of the center human subject. This alignment
filters out the irrelevant global information, and facilitates efficient
inference of the egocentric model.

##### Egocentric inference.

Operating on the transient graph structure, the RNN hidden state of each node
$s_{-}^{t}$ is updated with the input aggregated through attention-based
message passing along the edges in $\mathcal{E}^{t}$. See Fig. 3 for an
illustration.

In detail, for the _center node_ , the inward messages from its leaves are
aggregated into:
$m_{r}^{t}=\text{Attn}\left(\begin{array}[]{l}u_{r}^{t},\left\\{u_{l}^{t}\right\\}_{e_{l\rightarrow
r}^{t}\in\mathcal{E}^{t}}\end{array}\right)$, where
$u^{t}\\_=\left[\bar{x}\\_^{t};s\\_^{t-1}\right]$ and
$\text{Attn}\left(q,V\right)$ is defined in Eq. 2. This message is combined
with the egocentric features $\bar{x}_{r}^{t}$ and the persistent channel’s
message $m_{\mathcal{P}\rightarrow\mathcal{T}}^{t}$ (Eq. 3) to update the
recurrent state $s_{r}^{t}$:

$s_{r}^{t}=\text{RNN}_{r}\left(\left[\bar{x}_{r},m_{r}^{t},m_{\mathcal{P}\rightarrow\mathcal{T}}^{t}\right],s_{r}^{t-1}\right).$
(6)

For _leaf nodes_ , when they are interacted by the center node (indicated via
$e_{r\rightarrow l}^{t}$), they receive a center-broadcasted message
$m_{l}^{t}=\text{MLP}\left(\left[\bar{x}_{r}^{t};s_{r}^{t-1}\right]\right)$
and update their state $s_{l}^{t}$:

$s_{l}^{t}=\begin{cases}\text{\text{RNN}}_{l}\left(\left[\bar{x}_{l}^{t};m_{l}^{t}\right],s_{l}^{t-1}\right)&\textrm{if
}e_{r\rightarrow l}^{t}\in\mathcal{E}^{t}\\\
s_{l}^{t-1}&\text{otherwise}\end{cases},$ (7)

The updated hidden states are used to generate the messages sent to the
persistent channel $m_{\mathcal{T}\rightarrow\mathcal{P}}^{t}$ and the
transient predictions about the future position $\hat{y}_{-}^{\mathcal{T},t}$
of the entities:

$\displaystyle
m_{\mathcal{T}\rightarrow\mathcal{P}}^{t}=\text{MLP}\left(s_{r}^{t}\right),\quad\hat{y}_{-,\mathcal{T}}^{t}$
$\displaystyle=f_{\text{ego}}^{-1}\left(\text{MLP}\left(s_{-}^{t}\right)\right),$
(8)

where $f_{\text{ego}}^{-1}$ is the inverse function of Eq. 5 converting the
egocentric back to global coordinates. The Transient predictions
$\hat{y}_{-}^{\mathcal{T},t}$ are then combined with the Persistent
counterparts as described in Sec. 3.6. As a key modeling feature, the
egocentric design plays a crucial goal in the power of the PTD which will be
demonstrated later in the Ablation studies (Sec. 4.2).

###  3.5 Switching Transient Processes

The life cycles of the Transient processes are managed based on the situation
of human’s activity. These decisions are made by a neural _Transient Switch_
(See Fig. 4) that makes switch-on and switch-off decisions of the Transient
process for each human entity. The switch-on probability $p_{r}^{t}$ is
computed by considering the current hidden state $h_{r}^{t}$ and the input
$z_{r}^{t}$ of the persistent RNN in Eq. 1:

$\hat{p}_{r}^{t}=\gamma_{r}^{t}\cdot\text{sigmoid}\left(\text{MLP}\left(\left[h_{r}^{t};z_{r}^{t}\right]\right)\right),$
(9)

where $h_{r}^{t}$ and $z_{r}^{t}$ is the current hidden state and the input of
the persistent RNN. The discount factor $\gamma_{r}^{t}\in[0,1]$ responds to
subject’s distance to the nearest object $d_{r,min}^{t}$:

$\gamma_{r}^{t}=\exp\left(-\eta\cdot d_{r,min}^{t}\right),$ (10)

where $\eta$ is a learnable decay rate. This factor acts as a disruptive
shortcut gate that modulates the switching decision based on the spatial
evidence of the interaction.

Finally, the binary switch decision is made by thresholding the score with a
learnable threshold $\theta$: the switch is $on$ when
$\hat{p}_{r}^{t}>=\theta$, and is _off_ otherwise. When the switch changes
from _off_ to _on_ , a new Transient process is created for the subject. It
will run until the switch turns _off_ , then the persistent process again
becomes the single operator.

###  3.6 Future prediction

In PTD, future motions are predicted autoregressively: After running through
the observed sequence $X^{1:T}$, PTD keeps unrolling to predict the requested
$L$ time steps and feeds its prediction back as input to keep unrolling.

At each future time step $t$, the predictions from Persistent and Transient
channels $\hat{Y}_{\mathcal{P}}^{t}$, $\hat{Y}_{\mathcal{T}}^{t}$ (Eq. 3, Eq.
8) are combined with the priority on the Transient predictions. For a human
entity, if its Transient channel is activated, the Transient prediction will
be chosen; otherwise, the Persistent prediction will be used. For an object
entity, if it receives an active outward Transient edge, it will take that
channel’s prediction. If it receives multiple outward edges, it uses the
prediction from the channel with the highest transient score
$\hat{p}_{r}^{t}$. Otherwise, it uses the persistent prediction by default.

###  3.7 Model Training

The model is trained end-to-end with three losses: prediction loss of humans
and objects and switch loss:

$\displaystyle\mathcal{L}$
$\displaystyle=\lambda_{h}\mathcal{L}_{pred,h}+\lambda_{o}\mathcal{L}_{pred,o}+\lambda_{switch}\mathcal{L}_{switch}.$
(11)

The Prediction loss $\mathcal{L}_{\text{pred}}$ measures the mismatch between
predicted values $\hat{Y}$ and ground truth $Y$, implemented as their
Euclidean distance:

$\mathcal{L}_{\text{pred}}=\left\|\hat{Y}^{T+1:T+L}-Y^{T+1:T+L}\right\|_{2}.$
(12)

Even though the Transient switch can be implicitly trained with the prediction
loss, the gradient flowing through this binary gate can be weak. To directly
supervise the Transient switch, we further introduce the Switch loss:

$\mathcal{L}_{switch}=\text{BCE}\left(\hat{P}^{1:T+L},P^{1:T+L}\right),$ (13)

where $\hat{P}^{t}$ and $P^{t}$ are the predicted and ground truth switch
scores (Eq. 9) of all human entities at time step $t$.

##### Setting switch ground truth label

$P^{t}$ from data is an interesting modeling topic. The term represents the
true moment where the human $r$ turns on their “Transient mode”. A simple way
to learn it is through self-supervision using a binary label $q^{t}$ on
whether an interaction happens at that time. In particular, it is determined
by the outward edges $e_{r\rightarrow l}^{t}$ in the Transient graph (see Sec.
3.4): $q^{t}=1$ if $\exists l:e_{r\rightarrow l}^{t}\in\mathcal{E}^{t}$, and
$q^{t}=0$ otherwise.

However, is this the true ground truth to human behavior switch? Humans
usually foresee their interaction and change their behavior before the
observable interaction occurs; therefore, $q^{t}$ is actually too late to turn
on the Transient channel. We resolve this mismatch by using the future ground
truth labels of deviations for the current label of Transient switch:

$p^{t}=q^{t}\vee q^{t+1}\vee...\vee q^{t+\omega},$ (14)

where $\vee$ indicates ’bit-wise or’ operator, and $\omega$ is the postdating
window meta-parameter, whose values are examined in the Supp, Sec. 5.

##### Multistage training

We train our model in two stages: we first use teacher-forcing [28] to use the
ground truth position as the input in the prediction stage, preventing the
model from accumulating errors and facilitate faster training. Then, in the
second stage, we fine-tune the model with the unrolling mechanism introduced
in Sec. 3.6.

This way of training can be thought of as a curriculum learning technique [5]
used in previous works [1, 25] where we initially train the model with an easy
problem, then increase the difficulty of the problem in the later epochs.

###  3.8 Modeling scope and limitations

The modeling of the PTD in this paper makes an assumption that the Transient
processes of different persons are independent of each other. Extensions can
be made to break this assumption and support more complex cases, such as
collaborative and competitive multi-agent systems.

The model also assumes that there is a hard temporal border between the
persistent and transient processes, which may be an approximation as humans
sometimes have a mixed-up thinking and acting mechanisms. Future works may
involve allowing the two processes to take over each other more softly, hence
can cover these cases.

The PTD formulation in this section is done particularly for HOI-M
forecasting. Modifications may be necessary for other applications. Example
adaptation for _pedestrian trajectory prediction_ is given in Supp. Sec 9.

##  4 Experiments

We examine the effectiveness of PTD via quantitative evaluations,
generalization trial, visual analysis, and ablation studies on two HOI-M
prediction datasets: WBHM [33] and Bimanual Actions [13].

###  4.1 Preliminaries

This section describes the datasets, the baselines, and metrics in used.
Further details are in Supp, Sec. 1.

####  4.1.1 Datasets

##### Whole-Body Human Motion (WBHM) Database [33]

is a large-scale dataset featuring 3D motion data of both humans and objects,
which is well suited for this paper. From the raw 3D data, the selected visual
features include 3D skeleton poses of 18 joints for human entities
$(x\in\mathbb{R}^{54})$ and 3D bounding boxes for objects
$(x\in\mathbb{R}^{24})$, sampled at 10Hz, consistent with the compared methods
[8].

##### Bimanual Actions Dataset [13]

contains activities of subjects using both hands to interact with different
objects at the same time. Unlike WBHM with 3D geometrical features, all
features here are in 2D coordinates making the dataset more challenging for
motion prediction. Furthermore, the subjects in this dataset always use two
arms to interact concurrently with different objects, requiring a new
capability of modeling the collaboration between the arms. PTD naturally
support this use case by considering each arm to be one human entity, each
with features of 2D locations of the arm key points and hand bounding box.

####  4.1.2 Compared methods and baselines

We compare PTD with the _HOI-M forecasting SOTAs_ : CRNN-OPM, CRNN-OPM-LI [8]
and the _pose forecasting SOTAs_ : STS-GCN [40], Motion-Mixer [6]. Motion-
Mixer and STS-GCN are retrained using the provided
codes111https://github.com/FraLuca/STSGCN
https://github.com/MotionMLP/MotionMixer. The others are re-implemented with
the settings provided in the original papers. We also use several common
baseline methods of Zero Velocity, Running avg. 2, GRU [35].

####  4.1.3 Evaluation metric

The prediction errors at each time step are calculated as the Euclidean
distances with the ground-truth for both human poses and object bounding boxes
(mm for WBHM, pixel for Bimanual Action Dataset). The error for a sequence is
then computed as the average errors across $L$ prediction frames. Prediction
performances are reported as the mean and std of the average errors of 5
independent runs for humans and objects entities.

###  4.2 Motion forecasting on WBHM Dataset

For WBHM dataset, we follow the common evaluation protocol [8]: to observe for
1 second ($T=10$) and predict the next 2 seconds ($L=20$).

##### Quantitative evaluation.

The average errors reported in Tab. 1 clearly indicates that PTD consistently
outperforms the SoTAs in both human pose and object box forecasting. Supp,
Sec.3 further reports detailed errors at each time step.

| Human | Obj
---|---|---
Zero-Velocity | 176.45 | 128.6
Running avg. 2 | 183.95 | 133.3
GRU [35] | 102.86 $\pm$ 1.4 | 119.64 $\pm$ 1.6
STS-GCN [40] | 101.36 $\pm$ 2.4 | -
Motion-Mixer [6] | 87.35 $\pm$ 1.2 | -
CRNN-OPM [8] | 99.01 $\pm$ 1.1 | 87.52 $\pm$ 1.6
CRNN-OPM-LI [8] | 95.96 $\pm$ 1.7 | 74.27 $\pm$ 1.3
PTD (Ours) | 85.53 $\pm$ 0.9 | 70.69 $\pm$ 0.5
Table 1: The average errors (mm) on WBHM after 5 independent runs. PTD
outperforms other SOTAs in both human and object prediction. The errors at
each time step are given in Supp, Sec. 3.

##### Visual analysis.

Figure 5: Visual Comparison in WBHM. When the situation changes from
interaction-free (a) to interaction-involved (b), PTD _(Upper row_) switches
on its Transient channel with egocentric structures and handles the
interaction accurately; In contrast, CRNN-OPM-LI [8] (_Lower row_) uses a
single mechanism, resulting in the sluggish adaptation of the attention map
and inaccurate predictions.  Figure 6: The switching behavior in WBHM. The
Transient Switch anticipates the beginning (i), stays stable during (ii), and
anticipates the end (iii) of the interaction. This early anticipation is
crucial for timely process switching.

We verify the benefit of our model by visualizing the internal output
predictions and graph structures of PTD compared to CRNN-OPM-LI [8]. The upper
row of Fig. 5 shows that PTD could learn to switch the mechanism from
Persistent to Transient when the situation changes from interaction-free (a)
to interaction-involved (b). The Transient graph in _(b)_ reflects the
interactions correctly thanks to it being trained on targeted samples.

In contrast, CRNN-OPM-LI [8] (lower row) holds on to a single global mechanism
and does not evolve adequately for the swift change in the true relational
topology, resulting in inaccurate and unrealistic interactions.

The operation of the Transient Switch is visualized in Fig. 6. When the
interaction is about to occur (i), the switch score $\hat{p}_{r}^{t}$ (Eq. 9)
increases to reflect the prospective of the interaction. When the score passes
the threshold, it switches on the Transient channel a few time-steps before
the interaction can be observed, precisely as designed (Eq. 14). After
maintaining high values during interaction (ii), the score falls when it
anticipates the end of the interaction, deactivating the transient channel
(iii).

##### Generalization analysis.

(a) Keep $L=20$, vary $T$ (b) Keep $T=10$, vary $L$

(c) Increase $T=30$, vary $L$

Figure 7: Generalizability test on varying observed lengths ($T$) and
predicted lengths ($L)$ of the sequences in testing from the ones used in
training.

With more accurate modeling, PTD promises a greater generalization in
predicting the patterns unseen in training. To evaluate such potential, we use
a generalization test [24] to set up a series of experiments where PTD and
other models compete on the test sequences of lengths different from the ones
used in training: (1) observation length $T$ varies, (2) prediction length $L$
varies, and (3) both observation and prediction lengths varies.

The results are measured as the total average errors (mm) of humans and
objects and are visualized in Fig. 7. We exclude STS-GCN and MotionMixer due
to their dependence on the sequence length.

1\. Varying observation length $T$ (Fig. 7a): Interestingly, when observing
longer sequences in testing ($T>10$), the baseline models failed to generalize
and perform worse. In contrast, with mechanism switching, PTD is flexible
enough to take advantage of the longer observed data to gain performance
without re-training. Also, with shorter observed sequences ($T<10$), PTD is
more resilient than other models, showing its capacity of modeling the generic
pattern and avoid overfitting.

2\. Predicting longer sequences. With same observation $(T=10)$ and increased
prediction length $L$ of the testing data (Fig. 7b), PTD keeps the superior
performance by extrapolating well to farther future predictions.

3\. Increasing both lengths. Finally, when both length changes (Fig. 7c), the
result is consistent with case 1. and 2\. where PTD shows its superior ability
to take advantage of more data and generalize well to longer predictions.

##### Ablation studies.

|  Ablation |  Human |  Object
---|---|---|---
1 |  w/o Transient channel |  89.49 |  74.91
2 |  w/o Persistent channel |  91.41 |  78.67
3 |  w/o egocentric property |  87.90 |  72.30
4 |  w/o heuristic switch |  86.44 |  74.85
5 |  w/o $\gamma$ |  86.17 |  70.71
6 |  w/ only $\gamma$ |  87.11 |  76.95
7 |  w/o switching transient |  86.00 |  71.40
8 |  w/o switch loss |  89.59 |  73.17
9 |  w/o multistage training |  90.21 |  72.22
|  Full PTD model |  85.53 |  70.69
Table 2: Ablation studies on WBHM (avg. errors in mm.)

We examine the roles of PTD’s core components by making ablations from the
model and report the performances in Tab. 2. They include:

1. Without Transient channel: Being alone, Persistent Channel performs significantly worse than when with its Transient partner in the duality.

2\. Without Persistent channel: Transient Channel also struggles and gives
worse performance operating alone.

3\. Without Egocentric property, including egocentric structure (Eq. 4) and
representation (Eq. 5), Transient channels operate on a fully-connected graph
with global features. This ablated model suffers a significant drop in
performance, showing the role of egocentric design.

4. Heuristic switch: This experiment probe the need for the _Transient Switch_ (Sec. 3.5) by replacing it with a heuristic rule $\hat{p}_{r}^{t}=d_{r,min}^{t}\leq\beta$. This heuristic switch can still provide benefit compared to single channels (row 1,2). However, being too stiff, it cannot represent the switching patterns and fails to reach the duality’s full potential.

5\. Switch without spatial discount factor: Without $\gamma_{r}^{t}$ (Eq. 9),
the _Switch_ could not respond fast enough to context changes, resulting in
slightly weaker performance.

6\. Switch with only discount factor: This quick-change factor could not do
the job by itself as it is susceptible to noisy patterns, performing even
worse than row 5.

7\. Without switching transient: The transient channels is always on and may
capture irrelevant patterns outside the HOI-context, leading to a drop in the
performance.

8\. Without switch loss: We study the role of the switch’s direct supervision
(Sec. 3.7) by setting $\lambda_{switch}$ to 0, taking $\mathcal{L}_{switch}$
out of Eq. 11. This unsupervised switch only relies on weak gradient flowing
back from prediction loss and delivers significantly weakened performance.

9\. Without multistage training: Without such a training procedure (Sec. 3.7),
the model suffers from accumulating losses during early epochs and capture
less accurate motion pattern, resulting in a decrease in the performance.

###  4.3 Motion forecasting on Bimanual Action Dataset

While WBHM is rich in patterns and is well-suited for large scale analysis,
the Bimanual Action is more realistic, as its 2D RGB data are more available.
Due to the increased ambiguity of the 2D data, we use longer observed (2s/20
steps) and shorter predict lengths (1s/10 steps).

Quantitative evaluation: The average errors measured in pixel are reported in
Tab. 3. Consistent with the results in WBHM dataset, PTD outperforms other
SOTAs in both human and object motion forecasting. The errors at each time
steps are also detailed in Supp, Sec. 3. Visual analysis of PTD’s operation in
this dataset (similar to that of WBHM in Fig. 5 and Fig. 6) is provided in
Supp, Sec 4.

| Arm Keypoints | Hand | BoxObj
---|---|---|---
Zero-Velocity | 12.11 | 21.52 | 7.02
Running avg. 2 | 12.80 | 22.19 | 7.30
GRU [35] | 12.37 $\pm$ 0.4 | 20.80 $\pm$ 0.9 | 7.04 $\pm$ 0.0
STS-GCN [40] | 11.85 $\pm$ 0.5 | - | -
Motion-Mixer [6] | 11.68 $\pm$ 0.2 | - | -
CRNN-OPM [8] | 12.81 $\pm$ 0.1 | 21.66 $\pm$ 0.2 | 7.16 $\pm$ 0.1
CRNN-OPM-LI [8] | 11.97 $\pm$ 0.3 | 20.13 $\pm$ 0.2 | 7.05 $\pm$ 0.1
PTD (Ours) | 10.94 $\pm$ 0.2 | 18.80 $\pm$ 0.1 | 6.81 $\pm$ 0.0
Table 3: The avg. errors (pixel) on Bimanual Action dataset from five
different runs.

###  4.4 Empirical Complexity Analysis

We did an empirical analysis on the model size and observed that _PTD has a
comparable number of parameters to other methods_. Detailed numeric analysis
is shown in Supp, Sec 2. This fact confirms that the good performance of PTD
is caused by the new multi-mechanism scheme without the negative trade-off in
computation cost.

##  5 Discussion

In this work, we have introduced a new concept of Persistent-Transient Duality
to model the multi-mechanism nature of human behavior in interaction with
objects. The duality is implemented into a parent-children network that
demonstrates its effectiveness and generalizability through extensive
evaluations in HOI-M forecasting.

Given the ubiquity of persistent-transient relationship in human behavior and
the genericity of PTD design, the model can be readily extended to other
applications including _pedestrian trajectory prediction_ (as demonstrated in
Supp., Sec 9) and potentially _social interaction modeling_ , and _human-
machine collaboration_. Future development may include more fluid dynamics
between the two processes and allowing multiple transient instances.

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# Synthesis and Anisotropic Magnetism in Quantum Spin Liquid Candidates
$A$YbSe2 ($A$ = K and Rb)

Jie Xing  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Liurukara D. Sanjeewa  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  Missouri Research Reactor, University of Missouri,
Columbia, Missouri 65211, USA  Department of Chemistry, University of
Missouri, Columbia, MO 65211, USA  Andrew F. May  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  Athena S. Sefat  Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

Quantum spin liquid (QSL) state in rare-earth triangular lattice has attracted
much attention recently due to its potential application in quantum computing
and communication. Here we report the single-crystal growth synthesis, crystal
structure characterizations and magnetic properties of $A$YbSe2 ($A$= K, Rb)
compounds. The X-ray diffraction analysis shows that $A$YbSe2 ($A$ = K and Rb)
crystallizes in a trigonal space group, $R$-3$m$ (No. 166) with Z = 3.
$A$YbSe2 possesses a two-dimensional (2D) Yb-Se-Yb layered structure formed by
edged-shared YbSe6 octahedra. The magnetic properties are highly anisotropic
for both title compounds and no long-range order is found down to 0.4 K,
revealing the possible QSL ground state in these compounds. The isothermal
magnetization exhibits one-third magnetization plateau when the magnetic
fields are applied in $ab$-plane. Heat capacity is performed along both
$ab$-plane and $c$-axis, and features the characteristic dome for triangular
magnetic lattice compounds as a function of magnetic fields. Due to the change
of the interlayer and intralayer distance of Yb3+, the dome shifts to low
fields from KYbSe2 to RbYbSe2. All these results indicate the $A$YbSe2 family
presents unique frustrated magnetism close to the possible QSL and noncolinear
spin states.

††preprint: AIP/123-QED

##  I Introduction

The quantum spin liquid (QSL) state in geometric frustrated materials attracts
wide interests due to the future application of quantum entanglement. Sadoc
and Mosseri (1999); Moessner and Ramirez (2006) The highly entangled spins in
frustrated magnetic lattices prevent the breaking of any symmetry, even at
zero temperature. Anderson (1973); Mila (2000); Balents (2010) Besides the
proposed quantum magnets with spin-1/2 transition metal, the rare-earth based
materials are also a rich ground for frustrated magnetism. Gardner, Gingras,
and Greedan (2010); Broholm  _et al._ (2020); Li, Gegenwart, and Tsirlin
(2020); Sarkis  _et al._ (2020); Pajerowski  _et al._ (2020); Rau and Gingras
(2018); Xing  _et al._ (2020) As an example, Yb-based triangular magnetic
lattice structures have been proposed as effective spin $S_{\mathrm{eff}}$ =
1/2 materials at low temperature due to the crystal electric field (CEF)
effect. Wu  _et al._ (2019a, b); Agrapidis, van den Brink, and Nishimoto
(2019); Hester  _et al._ (2019); Li, Lu, and Chen (2017); Li  _et al._ (2018);
Li, Wang, and Chen (2016) One Yb3+ based triangular lattice material,
YbMgGaO4, has been proposed as a QSL candidate by the magnetization, heat
capacity, neutron scattering, muon spin relaxation and thermal conductivity
measurements. Li  _et al._ (2015a, b, 2016); Shen  _et al._ (2016); Paddison
_et al._ (2017); Zhang  _et al._ (2018); Li  _et al._ (2017); Steinhardt  _et
al._ (2019); Li (2019) YbMgGaO4 crystallizes in high symmetry (trigonal
$R$-3$m$) by creating a perfect Yb3+ triangular magnetic lattice. Non-magnetic
Mg2+ and Ga3+ ions separate the Yb3+ triangular magnetic layers by 8.4 Å and
the distance between the two Yb3+ ions within the layer is 3.4 Å, which
provides well 2D character for YbMgGaO4. However, due to the similar ionic
radii of Mg2+ and Ga3+, both Mg2+ and Ga3+ tend to mix occupy in the
structure. Since Mg2+ and Ga3+ have two different charges, this can affect the
crystal electric field splitting of the Yb3+ site compared to the perfect
structure. Consequently, this type of intrinsic Mg/Ga disorder between the
Yb3+ layers could perturb the spin liquid ground state. Li  _et al._ (2015a);
Paddison  _et al._ (2017); Zhang  _et al._ (2018); Shen  _et al._ (2018); Zhu
_et al._ (2017, 2018); Kimchi, Nahum, and Senthil (2018) Therefore, it is
important not only to study new Yb3+ based 2D geometrically frustrated
materials, but also to develop rational synthetic pathways targeting novel Yb-
based 2D compounds without site disorder, particularly ones that enable the
effect of QSL ground state by varying the intralayer and interlayer distances
of the Yb3+ magnetic lattice.

Among the most recently studied Yb3+ triangular magnetic lattices, the
delafossite structure (112-structure type) $ARQ_{2}$ ($A$ = Li, Na, K, Rb, Cs,
Tl, Ag, Cu; $R$ = rare earth; $Q$ = O, S, Se, Te) attracts widespread
attention in search of QSL candidates. Liu  _et al._ (2018); Xing  _et al._
(2019a, b); Zangeneh  _et al._ (2019); Scheie  _et al._ (2020); Bordelon  _et
al._ (2021) The high symmetry of $ARQ_{2}$ structures is reported to have
different packing structures, where rare earth ions form 2D triangular
lattice. However, depending upon the $A$-site cation, $ARQ_{2}$ structure
could crystallize in different space groups. Xing  _et al._ (2019a) Further,
magnetic properties of $ARQ_{2}$ compounds can be systematically tuned by
either changing $A$-site cation (interlayer-interactions), number of 4$f$
electrons in the rare earth ions or by changing the type of $Q$-site
(intralayer interactions). Therefore, $ARQ_{2}$ structures serve as an
experimental touchstone to identify novel quantum magnetic ground states such
as QSL. Liu  _et al._ (2018); Xing  _et al._ (2019a, b); Zangeneh  _et al._
(2019); Scheie  _et al._ (2020); Gao  _et al._ (2020); Xing  _et al._ (2021)
The presence of $S_{\mathrm{eff}}$ = 1/2 ground state of Yb3+ based
NaYb$Q_{2}$ structures at low temperature region is proposed by electron spin
resonance, magnetization and neutron scattering measurements.Baenitz  _et al._
(2018); Bordelon  _et al._ (2020); Zhang  _et al._ (2021a); Sichelschmidt  _et
al._ (2020); Guo  _et al._ (2020); Zhang  _et al._ (2021b); Schmidt  _et al._
(2021) Further, no long-range order has been observed in heat capacity at zero
field down to mK temperatures at low magnetic fields. Liu  _et al._ (2018);
Ranjith  _et al._ (2019a, b); Xing  _et al._ (2019c); Ding  _et al._ (2019);
Ma  _et al._ (2020); Ferreira  _et al._ (2020); K. M. Ranjith and Ph.
Schlender and Th. Doert and M. Baenitz ; Iizuka  _et al._ (2020); Zhang  _et
al._ (2020a) Quantum disordered or QSL ground state is proposed in NaYbO2
Ranjith  _et al._ (2019a); Ding  _et al._ (2019); Bordelon  _et al._ (2019)
and the signature of the magnetic excitation continuum has been found in
NaYbSe2 from 0.1 to 2.5 meV in inelastic neutron scattering (INS) measurement,
suggesting the ground state of NaYbSe2 is a QSL with a spinon Fermi surface.
Dai  _et al._ (2021) The continuum-like excitation with short-range order has
been found in CsYbSe2, indicating it may be located close to the critical
boundary between QSL and 120∘ phase.Xie  _et al._ (2021) The external magnetic
fields could perturb the zero-field magnetic ground state and induce the long-
range order where the characteristic 1/3 magnetization plateau can be observed
in isothermal magnetization. Ranjith  _et al._ (2019b); Xing  _et al._ (2019c)
These field-induced phase transitions reveal the unique magnetic phase diagram
in Yb3+ based 112 compounds. Maksimov  _et al._ (2019); Yamamoto, Marmorini,
and Danshita (2014) Moreover, superconductivity has been reported in NaYbSe2
(Tc $\sim$8 K) under high pressure. Jia  _et al._ (2020); Zhang  _et al._
(2020b)

These variations in the structures and the magnetic properties have motivated
us to undertake a comprehensive study on $A$YbSe2 ($A$ = K, Rb, Cs), where
large $A$ ions could avoid the possible mixed occupancy of $A$/Yb. In the
first step of our investigation in this series, we have studied the magnetic
properties of CsYbSe2 using ac and dc magnetic susceptibility and heat
capacity down to 0.4 K.Xing  _et al._ (2019a, c) Additionally, our INS on
CsYbSe2 single crystals suggest that this system could provide a natural
realization of QSL ground stateXie  _et al._ (2021). For this purpose, we
extend our study to investigate structure-property correlation of KYbSe2 and
RbYbSe2. In this paper, we discuss the single crystal growth, structure
characterization and magnetic properties of KYbSe2 and RbYbSe2. To our
knowledge, there have not been any reported studies of the magnetic properties
of KYbSe2 and RbYbSe2, however, synthesis and the crystal structure of KYbSe2
has been reported by Gray et al. Gray, Martin, and Dorhout (2003) Single
crystal X-ray confirmed that both KYbSe2 and RbYbSe2 crystallizes in $R$-3$m$
space group. The magnetization and heat capacity are performed down to 0.4 K
when the applied magnetic field is along the $ab$-plane and $c$-axis. Both
KYbSe2 and RbYbSe2 do not exhibit long-range ordering above 0.4 K suggesting
the QSL ground state. Field-induced magnetic transitions are found, and heat
capacity form a dome shape in the $H$-$T$ phase diagram. The 1/3 magnetization
plateau is observed in both KYbSe2 and RbYbSe2 when magnetic field is applied
along the $ab$ plane. Finally, this study allows us to establish the empirical
relationship between the intralayer/interlayer space of Yb3+ ions and the
magnetic properties of $A$YbSe2 ($A$ = Na, K, Rb and Cs) family of compounds.

##  II EXPERIMENTAL SECTION

Millimeter-sized hexagonal shape KYbSe2 and RbYbSe2 single crystals,
synthesized by a two-step method, are shown in Fig. 1(a). In the first step, K
or Rb chunk (99.9%, Alfa Aesar), and Yb ingot (99.9%, Alfa Aesar) and Se shot
(99.999%, Alfa Aesar) were loaded by the stoichiometric ratio into an alumina
crucible and sealed it in a quartz tube. The ampoules were slowly heated to
850 ∘C and cooled down to room temperature in the furnace. After the first
reaction, a red color powder was recovered from the crucibles. In the second
step, the resulting powder was loaded into quartz tube with mass ratio 1:10
corresponding flux (KCl or RbCl). After that, tubes were sealed again. The
final reaction was 850 ∘C for 2 weeks. Finally, the reactions were quenched by
water. The red plate-like single crystals (Fig. 1(a)) were collected by
washing with de-ionized water using suction filtration method. A similar
method was employed to synthesize single crystals of both NaYbSe2 and KLuSe2,
which were used as references in our measurements.

Energy-dispersive spectroscopy analysis (EDS) was performed using a Hitachi
S-3400 scanning electron microscope equipped with an OXFORD EDX microprobe to
identify the elemental composition in single crystals and it confirmed the
molar ratio of K(Rb):Yb:Se is 1:1:2 as expected. X-ray single-crystal
diffraction (SXRD) data were collected using Bruker Quest D8 single-crystal
X-ray diffractometer equipped with Mo Kα radiation, $\lambda$ = 0.71073 Å. The
data were collected at 200 K using a nitrogen cold stream and crystals were
mounted in paratone oil. The crystal diffraction images were collected using
$\phi$ and $\omega$-scans. The diffractometer was equipped with an Incoatec
I$\mu$S source using the APEXIII software suite for data setup, collection,
and processing. ShelX was utilized for the structural solution and refinement
after data reduction via SMART-Plus; spherical absorptions corrections were
performed within ShelX after initial refinement. Twinning was taken into
account during refinement. Sheldrick (2015) Table 1 provides the detailed unit
cell parameters of $A$YbSe2 series and the selected bond distances are
summarized in Table 2. X-ray powder diffraction (PXRD) data was collected on a
PANalytical X’pert Pro diffractometer equipped with an incident beam
monochromator (Cu Kα1 radiation) at room temperature. Diffraction patterns of
the flat surface of $A$YbSe2 crystals were collected to confirm the 00$l$
orientation ($ab$-plane), as shown in Fig. 1(b).

Magnetic properties were measured using a Quantum Design (QD) Magnetic
Properties Measurement System (MPMS3). The magnetization below 2 K was
measured by the MPMS3 iHe3 option. The total mass of the measured crystals was
$\approx$3 mg. Temperature dependent heat capacity was measured in QD Physical
Properties Measurement System (PPMS) using the relaxation technique. The mass
of the single crystals for heat capacity was $\approx$1 mg.

Figure 1: (a) Single crystals of $A$YbSe2. (b) (00l) peaks of X-ray diffraction on the flat surface ($ab$-plane) of $A$YbSe2 single crystals. Table 1: Crystallographic data of $A$YbSe2 determined by single-crystal X-ray diffraction.  empirical formula |  NaYbSe2 Liu  _et al._ (2018) |  KYbSe2 |  RbYbSe2 |  CsYbSe2 Xing  _et al._ (2019a)
---|---|---|---|---
formula weight (g/mol) |  441.18 |  370.06 |  416.43 |  463.87
$T$, K |  298 |  200 |  200 |  298
crystal habitat |  red plates |  red plates |  red plates |  red plates
crystal dimensions, mm |  0.08$\times$0.02$\times$0.02 |  0.04$\times$0.02$\times$0.02 |  0.04$\times$0.02$\times$0.02 |  0.04$\times$0.02$\times$0.02
crystal system |  trigonal |  trigonal |  trigonal |  hexagonal
space group |  $R$-3$m$ (No.166) |  $R$-3$m$ (No.166) |  $R$-3$m$ (No.166) |  $P$63/$mmc$ (No.194)
$a$, Å |  4.0568(8) |  4.1149(5) |  4.1371(1) |  4.1484(4)
$c$, Å |  20.772(6) |  22.6911(4) |  23.6428(5) |  16.549(3)
volume, Å3 |  296.06 |  332.76(2) |  350.45(2) |  246.64(7)
$Z$ |  3 |  3 |  3 |  2
$D$ (calc), g/cm3 |  – |  5.540 |  5.920 |  6.246
$\mu$ (Mo Kα), mm-1 |  – |  38.200 |  45.730 |  40.788
$F$(000) |  – |  471 |  525 |  386
$T_{\mathrm{max}}$, $T_{\mathrm{min}}$ |  – |  0.1576-1.0000 |  0.1986-1.0000 |  0.1702-1.0000
$\theta$ range |  – |  5.392-38.789 |  2.58-38.72 |  2.46-30.40
reflections collected |  – |  3934 |  4222 |  5806
data/restraints/parameters |  – |  456/0/10 |  485/0/10 |  116/0/9
final $R$ [$I$> 2$\sigma$($I$)] $R_{\mathrm{1}}$, $R_{\mathrm{w2}}$ |  – |  0.0246/0.660 |  0.0178/0.0402 |  0.0549/0.01427
final $R$ (all data) $R_{\mathrm{1}}$, $R_{\mathrm{w2}}$ |  – |  0.0251/0.0663 |  0.0210/0.0424 |  0.0552/0.1428
GoF |  – |  1.078 |  1.092 |  1.023
largest diff. peak/hole, e/Å3 |  – |  1.454/-2.3631 |  3.401/-2.911 |  1.985/-3.949

##  III RESULTS AND DISCUSSION

###  III.1 Crystal Structure

Single crystal X-ray structure investigation reveals that both KYbSe2 and
RbYbSe2 adopt the trigonal $R$-3$m$ space group compare to $P$63/$mmc$ in
CsYbSe2. The detailed crystallographic information of $A$YbSe2 ($A$ = Na, K,
Rb, Cs) is given in Table 1. The crystal structure is built from 2D Se–Yb–Se
layers where $A$-site cations reside in between the layers. These 2D Se–Yb–Se
layers packed along the $c$-axis as displayed in Fig. 2, however, the packing
of Se–Yb–Se layers and $A$-site cations are different between the $R$-3$m$ and
$P$63/$mmc$ space groups. The interlayer distance is mainly controlled by the
size of the $A$-site cation. Table 2 summarizes the bond distances and angles
of $A$YbSe2 series of compounds, including the interlayer distances. For
example, in NaYbSe2, the interlayer distance is 6.97 Å while in CsYbSe2 it is
8.24 Å. This is mainly due to the larger size of the Cs+ ion compared to the
Na+ ion. The interlayer distances for KYbSe2 and RbYbSe2 are 7.56 and 7.80 Å,
respectively. The 2D Yb–Se–Yb layers are constructed along the $ab$-plane from
edged sharing YbSe6 units, as displayed in Fig. 2. In both structure types,
Yb3+ ions form an equilateral triangle with Se2- sitting in the threefold
symmetry. It is noteworthy to mention that the size of the Yb3+ equilateral
triangle represent the $a$-axis of the structures and this is the intralayer
distance between the Yb3+ ions within the 2D lattice. As pointed in Fig. 2,
there is a significant variation between the intralayer distances between
NaYbSe2 and CsYbSe2. It is 4.057 and 4.148 Å for NaYbSe2 and CsYbSe2,
respectively. Additionally, Yb–Se–Yb bond angle also changes significantly
throughout the $A$YbSe2 series. As pointed in Fig. 2, Yb–Se–Yb bond angles are
91.94∘ and 93.73∘ for NaYbSe2 and CsYbSe2, respectively. These subtle
differences in intra- and interlayer distances, and bond angles clearly effect
the magnetic properties of this series of compounds, as we discuss in the next
section.

Table 2: The selected bond distances (Å) and angles (∘) of $A$YbSe2 series. |  NaYbSe2 Liu  _et al._ (2018) |  KYbSe2 |  RbYbSe2 |  CsYbSe2 Xing  _et al._ (2019a)
---|---|---|---|---
Yb–Se |  4.0568 |  4.1149(5) |  4.1371(1) |  4.1484(4)
Yb–Se–Yb |  91.14 |  93.13(2) |  93.53(2) |  93.77(2)
Interlayer Yb–Yb |  6.90 |  7.56 |  7.80 |  8.27
Table 3: The effective magnetic moment ($\mu_{\mathrm{eff}}$) and Curie-Weiss temperature ($\theta_{\mathrm{CW}}$), and Van Vleck term $\chi_{\mathrm{vv}}$ of KYbSe2 and RbYbSe2 obtained from the fit at high temperatures (HT) of $200-250$ K, and at low temperatures (LT) of $4-10$ K. |  $\mu_{\mathrm{eff}}$[$\mu_{\mathrm{B}}$] |  $\theta_{CW}$[K] |  $\mu_{\mathrm{eff}}$[$\mu_{\mathrm{B}}$] |  $\theta_{CW}$[K] |  $\chi_{\mathrm{vv}}$[emu/mol Oe]
---|---|---|---|---|---
| HT | HT | LT | LT | LT
KYbSe2 |  |  |  |  |
$H\|$_ab_ | 4.87 | -73.5 | 3.41 | -12.6 | 0.002
$H\|$_c_ | 5.08 | -40.3 | 0.65 | -4.7 | 0.027
RbYbSe2 |  |  |  |  |
$H\|$_ab_ | 4.44 | -57.9 | 3.45 | -15.0 | 0.005
$H\|$_c_ | 5.01 | -41.9 | 0.67 | -4.5 | 0.026

###  III.2 Magnetic Properties of $A$YbSe2

The geometrical frustrated Yb3+ triangular lattices with different intralayer
and interlayer distances inspire us to investigate the relationship between
the Yb-based 112-typed structures and their magnetism. The orientation is
determined by the flat surface, where sharp 00l peaks present in Fig. 1(b).
Fig. 3(a-b) present the temperature dependence of magnetic susceptibility
$\chi$ =$M$/$H$, where $M$ is the magnetization and $H$ the applied field, and
the inverse magnetic susceptibility is also shown above 2 K along $ab$-plane
and $c$-axis under 1 T applied magnetic field. No long-range order or
anomalies are observed above 2 K. Large anisotropy between $H$||$ab$ and
$H$||$c$ appears below 50 K, with a magnitude ratio of approximately 3 at 2 K.
The temperature dependence of 1/$\chi$ exhibits a linear relation at high
temperature. When the temperature decreases below 100 K, the curves strongly
bend on both directions in KYbSe2 and RbYbSe2, indicating the systems go to
the low spin state of Yb3+ due to the CEF effect. The Curie-Weiss fit of the
magnetization at 200 K to 250 K is used in both directions for the two
compounds. The Curie Weiss fit including a temperature independent Van Vleck
term is used at low temperature region (4-10 K). We present the fitting
parameters in Table 3. The fitting results show comparable value in KYbSe2 and
RbYbSe2 on each direction, indicating the similar magnetic interaction of Yb3+
ions in these two structures. The negative Curie-Weiss temperatures in all
measurements indicate the antiferromagnetic interactions between Yb3+. The
effective moment from high temperature fitting is close to the moment of free
Yb3+ (4.54 $\mu_{\mathrm{B}}$). At low temperature, the effective moments
present anisotropic smaller values along these two directions when we consider
Van Vleck contribution. Although the precious g values need further electron
spin resonance and high magnetic field measurements to confirm, these fitting
results at low temperature regions hint at the possible anisotropic g values
in these two compounds.

Figure 2: Crystal structure comparison between the $R$-3$m$ in NaYbSe2 and
$P$63/$mmc$ in CsYbSe2. In $A$YbSe2 ($A$ = Na, K, Rb, Cs) series, replacing Na
with larger cations such as K, Rb, and Cs changes the intralayer distances,
Yb–Se–Yb bond angles, and the interlayer distances.

The low-T magnetization data of KYbSe2 is shown in Fig. 4(a) and no evidence
of long-range order is observed. However, a broad maximum is found below 2 K,
suggesting short-range interactions. This feature is similar to previous
report for CsYbSe2 and NaYbSe2, but different from NaYbS2. Ranjith  _et al._
(2019a, b); Xing  _et al._ (2019c) The maximum becomes broader with increasing
the magnetic field. A small jump appears at $\sim$0.7 K at 2.5 T, which
reveals the field-induced long-range magnetic order. The transition
temperature increases to maximum $\sim$1 K with field up to 3 T, then
decreases to 0.7 K at 6 T, and finally disappears above 0.4 K at 7 T. The
inset of Fig. 4(a) presents the temperature dependence of magnetization under
different fields along $c$-axis. At the low field, the magnetization increases
monotonously by decreasing the temperature. However, a maximum appears from 1
to 7 T, suggesting short-range interactions. Due to the large interlayer
distance, the long-range order needs a higher magnetic field than our
instrument limit of 7 T. The magnitude at $H$||$c$ is much smaller than that
at $H$||$ab$, which proves XY-like interactions between Yb3+. These low
temperature magnetization results indicate that KYbSe2 has the possible spin
liquid ground state with the effective spin 1/2 state at zero field, and
magnetic fields could induce magnetic transitions above 2 T at $H$||$ab$. A
similar result of RbYbSe2 is presented in Fig. 4(b), besides the detailed
transition temperature shifts to low temperature due to the slight expansion
in the structure.

Figure 3: Temperature dependence of magnetic susceptibility of (a) KYbSe2 or
(b) RbYbSe2 below 1.6 K, from 0.1 T to 7 T applied magnetic fields.  Figure 4:
Temperature dependence of magnetic susceptibility and inverse magnetic
susceptibility for (a) KYbSe2 and (b) RbYbSe2.

The isothermal magnetization of $A$YbSe2 ($A$= Na, K, Rb, Cs) is shown in Fig.
5(a). We could observe all four samples present the similar 1/3 magnetic
plateau when magnetic fields are applied in the easy-plane ($ab$-plane). From
the theoretical calculations, the 1/3 magnetic plateau is expected in the
easy-plane XXZ model in the quantum picture. Yamamoto, Marmorini, and Danshita
(2014) The inset of Fig. 5(a) presents the isothermal magnetization of KYbSe2
at $H$||[100] and $H$||[110] at 0.42 K. No clear difference is found in these
two directions. To show the phase boundaries more clearly, we plot the
d$M$/d$H$ vs $H$ in Fig. 5(b). The pink dash line shows the boundary between
three states near the plateau. The plateau shifts to the lower fields from
NaYbSe2 to CsYbSe2 due to the change of intralayer and interlayer interaction.
NaYbSe2 requires the highest magnetic field to excite the plateau at 0.42 K
and among the four samples, as it has the shortest distances between Yb3+ in
the triangular layers. KYbSe2 and RbYbSe2 follow with the same relation,
however, the difference between CsYbSe2 and RbYbSe2 is very small. This could
be due to the small differences between the interlayer distances of the Rb and
CsYbSe2 structures. Also, we cannot ignore the change of the Yb–Se–Yb stacking
sequence between the two structures, which could affect the local environment
of YbSe6 octahedra. Moreover, the two-peak feature in d$M$/d$H$ of KYbSe2 and
RbYbSe2 can be observed even at $\sim$1 K, suggesting the strong spin
fluctuation above the long-range order temperature.

Figure 5: (a) Isothermal magnetization with $H$||$ab$ for $A$YbSe2 at 0.42 K,
KYbSe2 at 1 K, and RbYbSe2 at 1.2 K. The inset shows the overlapping
isothermal magnetization of KYbSe2 at $H$||[100] and $H$||[110], at 0.42 K.
(b) The differential of magnetization versus field of $A$YbSe2. The curves are
offset to present clearly.

Heat capacity has been performed on single crystals down to 0.4 K with
$H$||$ab$ and also $H$||$c$. The temperature dependence of heat capacity of
K/RbYbSe2 from 0.4 to 200 K at 0 T is shown in Fig. 6(a); key feature is the
broad peak from 0.4 to 10 K. This behavior reveals the short-range
interactions in the frustrated magnetic layers. No $\lambda$ anomaly is found
above 0.4 K, which is consistent with no long-range order in the magnetization
at low fields. The heat capacity of isostructural nonmagnetic KLuSe2 single
crystal is used as the phonon background. The temperature dependence of
entropy in K/RbYbSe2 at 0 T is presented in Fig. 6(b). The magnetic entropy
reaches a plateau of $\sim$5.8 J/mol K above 10 K. The slight deviation at 10
K between KYbSe2 and RbYbSe2 may be caused by slightly different interactions
between Yb3+ or phonon contribution. This value is close to Rln2 of spin 1/2
system, confirming the effective spin 1/2 of Yb3+ in this family.

The heat capacity is also measured when magnetic field is applied along
$ab$-plane and $c$-axis for both samples, as shown in Fig. 6(c-f). In these
measurements, the field-induced magnetic transition appears and increases to
the highest temperature $\sim$ 1 K with the largest peak at the moderate
field, then decreases to lower temperature with higher fields. Comparing to
the transition at $H$||$ab$, a much higher field is required at $H$||$c$ due
to the easy-plane feature. There are long tails above the transition
temperature, indicating the strong spin fluctuations from Yb3+. The shape of
the transition anomaly is comparable to the previous reports in NaYbSe2, and
much sharper than in CsYbSe2. Ranjith  _et al._ (2019b); Xing  _et al._
(2019c) It could be due to the spin correlations in the different space groups
of $A$YbSe2, where more 2D character can be found in CsYbSe2 compare to the
rest of the compounds in $A$YbSe2 series.

Figure 6: (a) Temperature dependence of heat capacity for KYbSe2, RbYbSe2, and
KLuSe2 under zero magnetic field above 0.4 K. (b) Temperature dependence of
entropy in KYbSe2 and RbYbSe2 at 0 T below 20 K. (c-f) Temperature dependence
of heat capacity for KYbSe2 (c,d) and RbYbSe2 (e, f) under different magnetic
fields when $H$||$ab$ or $H$||$c$. (g-h) Temperature dependent entropy of
KYbSe2 and RbYbSe2 under magnetic fields when the fields are applied in
$ab$-plane or $c$-axis.

Based on the heat capacity results, we calculate the entropy for all fields in
both samples and plot with the applied magnetic fields in Fig. 6(g-h). We use
power law fitting to estimate the magnetic entropy between 0 K and 0.4 K,
similar to the previous study in NaYbSe2. Ranjith  _et al._ (2019b) The
entropy releases up to only $\sim$1.2 J/mol K at the highest transition
temperature $\sim$ 1 K, which is close to 1/5 of Rln2 in spin-1/2 system. We
find a quasi-linear relation between the field-induced transition temperature
and released entropy at transition temperature, marked as green dashed line in
Fig. 6(g-h). Above the long-range order temperature, the release entropy
merged to $\sim$1.9 J/mol K at $\sim$1.1 K. As the transition temperature
varies in a relatively large temperature region with different fields in
$A$YbSe2, from $\sim$0.48 K to $\sim$0.96 K, much sharper $\lambda$ anomaly
appears and hinting that thermal fluctuation rarely related to the long tails
feature. Besides, the released entropy is also independent of magnetic field
directions based on the similar entropy vs. temperature behavior of $H$||$ab$
and $H$||$c$ in K/RbYbSe2, although the field-induced spin states are
different. We also compare the result of the previously reported CsYbSe2 and
find similar relation. Xing  _et al._ (2019c) All the results show the similar
relation between the released entropy and temperature under different magnetic
fields in this Yb family, indicating two dimensional magnetism in them and
weakly related to three dimensional stacking or interlayer interactions.

Figure 7: (a-d) $H$-$T$ phase diagram of KYbSe2 and RbYbSe2 at $H$||$ab$ and
$H$||$c$. The red dashed line presents the possible phase boundary. (e-f) The
highest field induced transition temperature and corresponding magnetic field
versus inversed intralayer distance and interlayer distance in NaYbS2 and
Na/K/Rb/CsYbSe2.

At last, we make the $H$-$T$ phase diagram of KYbSe2 and RbYbSe2 based on the
magnetization and heat capacity results, shown in Fig. 7(a-d). The obtained
transition temperatures from the magnetization and heat capacity are
consistent in both samples. When the fields are applied in the ab-plane,
KYbSe2 presents dome shape long-range order, separated into three different
states by the solid green diamonds (the boundaries of the plateau), which can
be recognized as 120∘, collinear up-up-down and 2:1 coplanar spin
arrangements. When the magnetic field is applied along c-axis, a similar dome
feature is found at a higher field due to the easy-plane feature. The similar
phase diagram of RbYbSe2 is shown in Fig. 7(c-d). The half solid diamonds
above the long-range order indicate the strong spin fluctuation. The width
between the plateau boundaries is almost independent with the temperature,
suggesting thermal fluctuation is barely related to the plateau. We could find
the domes shift to the lower field from KYbSe2 to RbYbSe2 along both $ab$ and
$c$ direction. This is due to the large intralayer and interlayer distance in
RbYbSe2.

Considering the previous experimental results of $A$YbSe2 and this present
work, the field-induced transitions present clear anisotropic behavior in this
family. This could be related to the intralayer and interlayer distance of
magnetic Yb3+ ions. Based on this assumption, we obtain each highest
transition temperature $T_{\mathrm{m}}$ with the corresponding field
$H_{\mathrm{m}}$ in NaYbS2 and Na/K/Rb/CsYbSe2 series, Ranjith  _et al._
(2019b); Xing  _et al._ (2019c); Ma  _et al._ (2020) and plotted them against
the inversed intralayer distance (1/dintra) of Yb3+ and interlayer distance
(1/dinter), as shown in Fig. 7(e-f). Since the similar $S_{\mathrm{eff}}$=1/2
state of Yb3+ is in this family, the interaction energy should directly relate
to the magnetic fields and the temperature. Therefore, Hm and Tm present
similar behavior against structure parameters in these systems. As displayed
in Figure 7(e), both $H_{\mathrm{m}}$ and $T_{\mathrm{m}}$ increase
monotonously with 1/dintra or 1/dinter. It is noteworthy to point that, with
the smallest value of 1/dintra or 1/dinter in this family, CsYbSe2 largely
deviates from the rest of the structures, Fig 7(e-f). This behavior further
implies that intralayer and interlayer interactions between Yb3+ triangular
magnetic lattice of CsYbSe2 are strongly affected by the size of the Cs+ ion
and the packing sequence in the overall 2D structure.

##  IV Conclusion

The KYbSe2 and RbYbSe2 single crystals have been synthesized by flux method
and characterized by magnetization and heat capacity. There is no long-range
order in both samples above 0.42 K at zero field, revealing the possible spin
liquid ground state. Large magnetic anisotropy with easy-plane feature is
found in both samples. Magnetic field application can induce the long-range
order along both $ab$ and $c$ crystallographic directions, forming similar
dome features in $H$-$T$ phase diagram. The temperature dependent magnetic
entropy indicates the field-induced magnetic transitions and strong spin
fluctuations. The 1/3 magnetic plateau is found in each $A$YbSe2 compound when
the magnetic field is applied in the $ab$ plane. At last, we proposed phase
diagrams for K/RbYbSe2 single crystals and the empirical relation between
field-induced transition and intralayer/interlayer distance. All these results
indicate the $A$YbSe2 family has a effective spin-1/2 frustrated magnetism,
which is close to QSL and noncolinear spin states.

###### Acknowledgements.

The research at the Oak Ridge National Laboratory (ORNL) is supported by the
U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences
(BES), Materials Sciences and Engineering Division (MSE). This manuscript has
been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with
the US Department of Energy. The United States Government retains and the
publisher, by accepting the article for publication, acknowledges that the
United States Government retains a nonexclusive, paid-up, irrevocable,
worldwide license to publish or reproduce the published form of this
manuscript, or allow others to do so, for United States Government purposes.
The Department of Energy will provide public access to these results of
federally sponsored research in accordance with the DOE Public Access
Plan(http://energy.gov/downloads/doe-public-access-plan).

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