# Dynamics of Public Opinion Evolution with Asymmetric Cognitive Bias

Yanbing Mao, Naira Hovakimyan, and Tarek Abdelzaher Y. Mao and N. Hovakimyan
are with the Department of Mechanical Science and Engineering, University of
Illinois at Urbana–Champaign, Urbana, IL 61801, USA (e-mail: {ybmao,
nhovakim}@illinois.edu).T. Abdelzaher is with the Department of Computer
Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
(e-mail: zaher@illinois.edu).This work was supported in part by DOD
HQ00342110002, DARPA W911NF-17-C-0099, AFOSR FA9550-15-1-0518, and NSF
CNS-1932529.

###### Abstract

In this paper, we propose a pubic opinion model with incorporation of
asymmetric cognitive bias: confirmation bias and negativity bias. We then
investigate the generic modeling guidance of capturing asymmetric confirmation
bias and negativity bias. A numerical examples is provided to demonstrate the
correctness of asymmetric cognitive bias model.

###### Index Terms:

Asymmetric confirmation bias, asymmetric negativity bias, social networks.

##  I Introduction

While opinion evolution models have always been an active research area,
recently with the wide use of social media [1], in conjunction with automated
news generation with the help of artificial intelligence technologies [2], it
has gained a vital importance in studying misinformation spread and
polarization. In this regard, confirmation bias plays a key role. Confirmation
bias broadly refers to cognitive bias towards favoring information sources
that affirm existing opinion [3]. It is well understood that confirmation bias
helps create “echo chambers” within networks, in which misinformation and
polarization thrive, see e.g., [4, 5]. Recently, Abdelzaher et al. in [6] and
Xu et al. in [1] reveal the significant influence of consumer preferences for
outlying content on opinion polarization in the modern era of information
overload. Meanwhile, Lamberson and Stuart in [7] suggest that negative
information, which is far away from expectations, is more “outlying.”
Motivated by these discoveries, the negativity bias, which refers to a
tendency to be more attentive and/or responsive to a unit of negative
information than to a unit of positive information [7], should not be ignored
in the study of information spread in social networks or public opinion
evolution.

In recent few years, the Hegselmann-Krause model [8, 9, 1] and the like-minded
social influence [10, 11, 12] are widely employed to capture confirmation
bias. However, the considered models therein cannot fully capture the
asymmetric bias, which hinders their applications in many real social problems
where humans hold asymmetric confirmation bias. Motivated by the problems, we
first propose a pubic opinion evolution model which explicitly takes
asymmetric confirmation bias and negativity bias into account. We then
investigate the generic modeling of asymmetric cognitive bias.

##  II Preliminaries

###  II-A Notation

The social system is composed of $n$ individuals in a social network. The
interaction among individuals is modeled by a digraph
$\mathfrak{G}=(\mathbb{V},\mathbb{E})$, where $\mathbb{V}$ =
$\left\\{\mathrm{v}_{1},\ldots,\mathrm{v}_{n}\right\\}$ is the set of vertices
representing the individuals, and
$\mathbb{E}\subseteq\mathbb{V}\times\mathbb{V}$ is the set of edges of the
digraph $\mathfrak{G}$ representing the influence structure.

###  II-B Pubic Opinion Evolution Model

We propose the following opinion evolution model (adopted from [10]) with
asymmetric cognitive bias: confirmation bias and negativity bias.

$\displaystyle{x_{i}}(k+1)$
$\displaystyle={\alpha_{i}}(k){s_{i}}+\sum\limits_{j\in\mathbb{V}}c_{ij}(x(k))x_{j}(k),~{}~{}~{}i\in\mathbb{V}.$
(1)

Here we clarify the notations and variables:

  * •

${x_{i}}\\!\left({k}\right)\in[-1,1]$ is individual $\mathrm{v}_{i}$’s
opinion, ${s_{i}}\in[-1,1]$ is her subconscious bias, which is based on
inherent personal characteristics (e.g., socio-economic conditions where the
individual grew up and/or lives in) [13].

  * •

The state-dependent influence weight $c_{ij}(x(k))\geq 0$ is proposed to
capture individual’s conjunctive confirmation bias and negativity bias, which
is written as

$\displaystyle c_{ij}(x(k))$
$\displaystyle=\left({1-\beta_{i}}\right)\overline{c}({{{\bar{x}}_{i}}(k),{x_{j}}(k)})$
$\displaystyle\hskip
51.21504pt+\beta_{i}\underline{c}\left({{x_{i}}(k),{x_{j}}(k)}\right),~{}i\\!\in\\!\mathbb{V}$
(2)

where $\beta_{i}\in[0,1]$, $\underline{c}\left({{x_{i}}(k),{x_{j}}(k)}\right)$
is used to describe confirmation bias, while
$\overline{c}\left({{{\bar{x}}_{i}}(k),{x_{j}}(k)}\right)$ describes
negativity bias with ${{\bar{x}}_{i}}(k)$ denoting individual
$\mathrm{v}_{i}$’s sensed expectation from her neighbors, defined as the mean
of her neighbors’ opinions, i.e.,

$\displaystyle{{\bar{x}}_{i}}(k)\triangleq\frac{1}{{\sum\limits_{l\in\mathbb{V}}{{{w_{il}}}}}}\sum\limits_{j\in\mathbb{V}}{{{w_{ij}}}{x_{j}}(k)}.$
(3)

  * •

$\alpha_{i}(k)\geq 0$ is the “resistance parameter” of individual
$\mathrm{v}_{\mathrm{i}}$ on her subconscious bias. To guarantee
$x_{i}(k)\in[-1,1]$ for $\forall k\in\mathbb{N}$ and $\forall i\in\mathbb{V}$,
it is determined in the sufficient and necessary condition:

$\displaystyle{\alpha_{i}}(k)+\sum\limits_{j\in\mathbb{V}}{{c}_{ij}(x(k))x_{j}(k)}=1,~{}~{}\forall
i\in\mathbb{V}.$ (4)

##  III Asymmetric Cognitive Bias Modeling

To simplify the presentation without loss of generality, we refer
${x_{i}}(k)$, ${\bar{x}_{i}}(k)$ and ${x_{j}}(k)$ to ${x_{i}}$,
${\bar{x}_{i}}$ and ${x_{j}}$, respectively, in this section.

###  III-A Literature Review

We first use the examples of confirmation bias to present the modeling issues
of symmetric bias. In recent few years, the Hegselmann-Krause model [8], i.e.,

$\displaystyle\left\\{\begin{array}[]{l}\\!\\!\\!x\left({k+1}\right)=A\left({x\left(k\right)}\right)x\left(k\right).\\\
\\!\\!\\!{\left[{A(x(k))}\right]_{i,j}}=\left\\{\begin{array}[]{l}\\!\\!>0,~{}{\underline{\varepsilon_{i}}}\leq{x_{i}}(k)-{x_{j}}(k)<{\overline{\varepsilon_{i}}}\\\
\\!\\!=0,~{}\text{otherwise}\end{array}\right.\end{array}\right.$ (9)

and the state-dependent influence weights, i.e.,

$\displaystyle\underline{c}({x_{i}},{x_{j}})={\beta_{i}}-{\gamma_{i}}\left|{{x_{i}}-x_{j}}\right|,~{}~{}{\beta_{i}}\geq{\gamma_{i}}\geq
0$ (10)

are widely used in [9, 1, 10, 11, 12] to capture confirmation bias. However,
the model (9) with asymmetric level of confidence (i.e.,
$|{\underline{\varepsilon_{i}}}|\neq{\overline{\varepsilon_{i}}}$) cannot
fully capture the asymmetric bias while the model (10) can only capture the
symmetric bias, which however hinder their applications in many real social
problems where humans hold asymmetric confirmation bias. For example, we
suppose the topic being discussed is “ _COVID-19 Is a Hoax_ ”, then $-1$ and
$1$ correspond to individual $\mathrm{v}_{i}$ completely opposing and
supporting the claim, respectively. As a consequence, $x_{j}\in[-1,0)$ means
the opinion of supporting position $-1$, while $x_{j}\in(0,1]$ means the
opinion of supporting position 1. We suppose that the opinions forwarded by
two individuals are $x_{j}=-0.3$ and $x_{h}=0.5$, and an individual
$\mathrm{v}_{i}$’s opinion is $x_{i}=0.1$. By that logic, and according to
confirmation bias, $\mathrm{v}_{i}$ should place more weight on the opinion
$x_{h}=0.5$ than the opinion $x_{j}=-0.3$, since both $x_{h}=0.5$ and
$x_{i}=0.1$ are supporting position 1 and the difference is their supporting
degree. Yet
$\left|{{x_{i}}-{x_{j}}}\right|=\left|{{x_{i}}-{x_{h}}}\right|=0.4$, which
according to (10) and (9), respectively, implies
$\underline{c}({x_{i}},{x_{j}})=\underline{c}({x_{i}},{x_{h}})$ and
${\left[{A(x)}\right]_{i,j}}={\left[{A(x)}\right]_{i,h}}$ if
$0.4<\min\\{{\overline{\varepsilon_{i}}},-{\underline{\varepsilon_{i}}}\\}$.
We thus conclude that the more realistic _asymmetric_ confirmation bias is not
captured by both the models (10) and (9).

Building on DeGroot model [14], Dandekar et al. in [15] proposed the opinion
polarization dynamics with biased assimilation. We now examine if the model
can capture the interested asymmetric confirmation bias in a simple scenario
as considered in [15], where the social network consists of only two
individuals: $\mathrm{v}_{i}$ and $\mathrm{v}_{j}$. The proposed opinion
dynamics in this scenario is
${x_{i}}\left({t\\!+\\!1}\right)\\!=\\!\frac{{{w_{ii}}{x_{i}}\left(t\right)+{{\left({{x_{i}}\left(t\right)}\right)}^{{b_{i}}}}{w_{ij}}{x_{j}}\left(t\right)}}{{{w_{ii}}+{{\left({{x_{i}}\left(t\right)}\right)}^{{b_{i}}}}{w_{ij}}{x_{j}}\left(t\right)+{{\left({1-{x_{i}}\left(t\right)}\right)}^{{b_{i}}}}{w_{ij}}\left({1-{x_{j}}\left(t\right)}\right)}}$,
where $x_{i}(t),x_{j}(t)\in[0,1]$. If individual $\mathrm{v}_{i}$’s opinion at
time $t$ is neutral, i.e., $x_{i}(t)\\!=\\!0.5$. In light of the model, we
have
${x_{i}}\left({t\\!+\\!1}\right)\\!=\\!\frac{{0.5{w_{ii}}+{{\left({0.5}\right)}^{{b_{i}}}}{w_{ij}}{x_{j}}\left(t\right)}}{{{w_{ii}}+{{\left({0.5}\right)}^{{b_{i}}}}{w_{ij}}}}$,
which indicates that regardless of individual $\mathrm{v}_{j}$’s opinion
$x_{j}(t)$, individual $\mathrm{v}_{i}$ puts the same influence weight on
$x_{j}(t)$ at time $t+1$. Therefore, we can conclude the proposed polarization
dynamics cannot fully capture the symmetric confirmation bias in the
simplified scenario.

###  III-B Asymmetric Cognitive Bias Conditions

The modeling challenge moving forward is _How to capture the asymmetric
cognitive bias?_ In this section, we provide the generic modeling conditions
to address the challenge.

####  III-B1 Asymmetric Confirmation Bias

It is well understood

  * •

confirmation bias happens when a person gives more weight to evidence that
confirms their beliefs and undervalues evidence that could disprove it [16],

  * •

both polarization and homogeneity are the results of the conjugate effect of
confirmation bias and social influence [9, 17],

motivated by which, to capture asymmetric confirmation bias, we require the
influence weights $\underline{c}(x_{i},x_{j})\geq 0$ in (2) to satisfy

$\displaystyle\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}}),$
$\displaystyle\hskip
31.2982pt\text{if}~{}|x_{j}-x_{i}|<|{{x_{d}}-x_{i}}|,~{}x_{j}\cdot x_{d}>0,$
$\displaystyle\hskip
42.67912pt\text{or}~{}|{x_{j}-{x_{i}}}|=|{{{x_{d}}}-{x_{i}}}|,~{}{x_{i}}\cdot
x_{j}>{x_{i}}\cdot{x_{d}},$ $\displaystyle\hskip
42.67912pt\text{or}~{}|{x_{j}-{x_{i}}}|=\zeta(x_{j},x_{d},x_{i})|{{{x_{d}}}-{x_{i}}}|,~{}x_{i}\cdot
x_{j}<0,$ $\displaystyle\hskip 56.9055ptx_{i}\cdot
x_{d}>0~{}\text{and}~{}0<\zeta(x_{j},~{}x_{d},x_{i})<1,$ (11a)
$\displaystyle\underline{c}({0,x_{j}})=\underline{c}({0,{x_{d}}}),\hskip
3.98337pt~{}\text{if}~{}x_{j}=-{x_{d}}.$ (11b)

How the proposed condition (11) can capture the asymmetric confirmation
bias are explained in the following remarks.

###### Remark 1

The condition $x_{j}\cdot x_{d}>0$ included in the first item of the condition
(11a) means $x_{j}>0$ & $x_{d}>0$, or $x_{j}<0$ & $x_{d}<0$. In light of the
expression, the first item indicates that for the two sensed opinions that are
both supporting the position -1 or 1, the individual will put larger influence
weight on the closer opinion with hers.

###### Remark 2

We note that the second item in the condition (11a) includes the case:
$\underline{c}({x_{i},{u}_{d}})>\underline{c}({x_{i},{x_{d}}})$, if
$|{x_{j}-x_{i}}|=|{{{x_{d}}}-x_{i}}|$, $x_{j}\cdot x_{i}>0$ and
${{x_{d}}}\cdot x_{i}<0$. If $x_{i}>0$, we have implying $x_{j}>0$ and
${x_{d}}<0$. We here conclude that in this case, although the two opinions
$x_{j}$ and $x_{d}$ has the same distance with individual opinion $x_{i}$,
i.e., $|{x_{j}-x_{i}}|=|{{{x_{d}}}-x_{i}}|$, individual puts larger influence
weight on $u_{d}$, i.e.,
$\underline{c}({x_{i},x_{j}})>\underline{c}({x_{i},{x_{d}}})$, since both of
them support the position $1$, and vice verse as $x_{i}<0$. Therefore, the
second item in the condition (11a) captures the asymmetric confirmation bias
when $x_{j}\cdot x_{i}>0$ and ${{x_{d}}}\cdot x_{i}<0$.

###### Remark 3

The second item in the condition (11a) also includes the case:
$\underline{c}({x_{i},x_{j}})>\underline{c}({x_{i},{x_{d}}})$, if
$|{x_{j}-x_{i}}|=|{{{x_{d}}}-x_{i}}|$ and $x_{j}\cdot x_{i}>{{x_{d}}}\cdot
x_{j}>0$. Taking $x_{i}<0$ as an example, we have $x_{j}<{x_{d}}<0$, by which
this case implies that although $x_{j}$ and ${x_{d}}$ has the same distance
with the opinion $x_{i}$ and all of them support the position $-1$, the
individual $\mathrm{v}_{i}$ puts larger influence weight on $x_{j}$ since
$x_{i}$ and $x_{j}$ are closer to the supporting position of $-1$ than
${x_{d}}$.

###### Remark 4

Taking $x_{i}>0$ as an example and considering $0<\zeta(x_{j},x_{d},x_{i})<1$,
the third item in the condition (11a) implies that it is possible that
$\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}})$, if $x_{i}<0$,
$x_{d}<0$ and $x_{j}>0$. This case means although both $\mathrm{v}_{i}$ and
$\mathrm{v}_{d}$ support the position $-1$, while $\mathrm{v}_{j}$ supports
the position $1$, the individual puts larger influence weight on $x_{j}$ than
$x_{d}$ when the ratio of their supporting-degree differences is larger than a
threshold, i.e,
$\frac{{\left|{x_{d}-x_{i}}\right|}}{{\left|{x_{j}-x_{i}}\right|}}\geq\frac{1}{\zeta(x_{j},x_{d},x_{i})}>1$.

###### Remark 5

The condition (11b) means that if individual’s opinion is neutral, i.e.,
$x_{i}=0$, she will put identical influences on her sensed opinions that have
the same distance with hers.

####  III-B2 Asymmetric Negativity Bias

In this subsection, we present the conditions pertaining to capturing
asymmetric negativity bias. Lamberson and Soroka in [7] revealed that

  * •

negative information, which is far away from expectations, is more “outlying”
(which motivates the sensed expectation $\bar{x}_{i}(k)$ in (2) and (3)),

  * •

the negativity bias refers to a tendency that is more attentive and/or
responsive to a unit of negative information than to a unit of positive
information.

Motivated by the discoveries and inspired by (11), to capture asymmetric
negativity bias, we require the influence weights
$\overline{c}\left({{\bar{x}_{i}},x_{j}}\right)\geq 0$ in (2) to satisfy

$\displaystyle\overline{c}({{\bar{x}_{i}},x_{j}})>\overline{c}({{\bar{x}_{i}},{x_{d}}}),$
$\displaystyle\hskip
28.45274pt\text{if}~{}|x_{j}-\bar{x}_{i}|>|{{x_{d}}-\bar{x}_{i}}|,~{}x_{j}\cdot
x_{d}>0,$ $\displaystyle\hskip
36.98866pt\text{or}~{}|{x_{j}-\bar{x}_{i}}|=|{{{x_{d}}}-\bar{x}_{i}}|,~{}\bar{x}_{i}\cdot
x_{j}<\bar{x}_{i}\cdot{x_{d}},$ $\displaystyle\hskip
36.98866pt\text{or}~{}|{{x_{d}}-\bar{x}_{i}}|\leq\breve{\zeta}(x_{j},x_{d},\bar{x}_{i})|{{x_{j}}-\bar{x}_{i}}|,~{}\bar{x}_{i}\cdot
x_{j}>0,$ $\displaystyle\hskip 56.9055pt\bar{x}_{i}\cdot
x_{d}<0~{}\text{and}~{}0<\breve{\zeta}(x_{j},x_{d},\bar{x}_{i})<1,$ (12a)
$\displaystyle\overline{c}({0,x_{j}})=\overline{c}({0,{x_{d}}}),\hskip
10.81218pt\text{if}~{}x_{j}=-{x_{d}}.$ (12b)

###  III-C Asymmetric Cognitive Bias Modeling Guidance

In this paper, we construct $\underline{c}\left({{x_{i}},x_{j}}\right)$ and
$\overline{c}({{\bar{x}_{i}},x_{j}})$ to have the following general forms:

$\displaystyle\underline{c}({x_{i}},x_{j})$
$\displaystyle={\underline{g_{i}}}({{\underline{f_{i}}}({x_{i}})-{\underline{f_{i}}}(x_{j})})\geq
0,$ (13) $\displaystyle\overline{c}({\bar{x}_{i}},x_{j})$
$\displaystyle={\overline{g_{i}}}({{\overline{f_{i}}}({\bar{x}_{i}})-{\overline{f_{i}}}(x_{j})})\geq
0.$ (14)

We next present the sufficient and necessary conditions of the models (13) and
(14) on satisfying (11) and (12), respectively, which will work as a guidance
of modeling the asymmetric confirmation bias and negativity bias.

###### Theorem 1

The influence weight $\underline{c}\left({{x_{i}},x_{j}}\right)$ given in (13)
satisfies (11) if and only if

$\displaystyle{\underline{g_{i}}}({{\underline{f_{i}}}(x_{i})\\!-\\!{\underline{f_{i}}}x_{j}})~{}\text{is
strictly decreasing w.r.t. the distance}$ $\displaystyle\hskip
152.22241pt|{{\underline{f_{i}}}({{x_{i}}})\\!-\\!{\underline{f_{i}}}({x_{j}})}|,$
(15a) $\displaystyle{\underline{f_{i}}}(x_{j})~{}\text{is strictly increasing
w.r.t. $x_{j}$},$ (15b)
$\displaystyle{\underline{f_{i}}}({{x_{i}}})\\!<\\!\frac{{{\underline{f_{i}}}({x_{j}})\\!+\\!{\underline{f_{i}}}({{{x}_{d}}})}}{2},\\!~{}\text{if}\\!~{}|{x_{j}\\!-\\!{{x}_{i}}}|\\!=\\!|{{{x}_{d}}\\!-\\!{{x}_{i}}}|,x_{j}\\!>\\!{{x}_{d}}$
$\displaystyle\hskip 177.54535pt\text{and}\\!~{}{x_{i}}\\!>\\!0,$ (15c)
$\displaystyle{\underline{f_{i}}}({{x_{i}}})\\!>\\!\frac{{{\underline{f_{i}}}({{x_{j}}})\\!+\\!{\underline{f_{i}}}({{{x}_{d}}})}}{2},\\!~{}\text{if}\\!~{}|{x_{j}\\!-\\!{{x}_{i}}}|\\!=\\!|{{{x}_{d}}\\!-\\!{{x}_{i}}}|,x_{j}\\!<\\!{{x}_{d}}$
$\displaystyle\hskip 176.97617pt\text{and}\\!~{}{x_{i}}\\!<\\!0,$ (15d)
$\displaystyle\underline{{f_{i}}}(0)\\!=\\!\frac{{{\underline{f_{i}}}(x_{j})\\!+\\!{\underline{f_{i}}}({-x_{j}})}}{2}.$
(15e)

###### Proof:

We first prove the condition (15) is a sufficient condition.

#### Sufficient Condition

Without loss of generality, we let $x_{i}\geq x_{j}>0$. It follows from (15b)
that
$|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{j}}})}|={{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{j}}})}$.
If $u_{d}$ decreases to $x_{d}>0$, we then have
$|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({x_{j}})}|<|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{d}}})}|$
and $|x_{i}-x_{j}|<|x_{i}-x_{d}|$. Considering (15a) and (13), we have
$\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}})$. If $u_{d}$ can
increase to $x_{d}$ such that $x_{d}-x_{i}$ $>$ $x_{i}-x_{j}\geq 0$, we obtain
from (15b) that
$|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{j}}})}|<|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{d}}})}|$
and $|x_{i}-x_{j}|<|x_{i}-x_{d}|$. Considering (15a) and (13), we then have
$\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}})$. We thus conclude
that the conjunctive conditions (15a) and (13) imply that

$\displaystyle\\!\\!\\!\\!\underline{c}(x_{i},x_{j})\\!>\\!\underline{c}({x_{i},\\!{x_{j}}}),\\!\\!~{}\text{if}\\!~{}|x_{d}\\!-\\!x_{i}|\\!>\\!|x_{i}\\!-\\!x_{j}|,x_{j}\\!>\\!0,x_{d}\\!>\\!0.$
(16)

In the case of $0>x_{i}\geq x_{j}$, following the same steps to derive (16),
we have

$\displaystyle\\!\\!\\!\\!\underline{c}(x_{i},x_{j})\\!>\\!\underline{c}({x_{i},\\!{x_{d}}}),\\!\\!~{}\text{if}\\!~{}|x_{d}\\!-\\!x_{i}|\\!>\\!|x_{i}\\!-\\!x_{j}|,x_{j}\\!<\\!0,x_{d}\\!<\\!0.$
(17)

The results (16) and (17) indicate that the conjunctive conditions (15a) and
(15b) result in the first item in (11a).

We now consider the condition

$\displaystyle|{x_{j}-{{x}_{i}}}|=|{{{x}_{d}}-{{x}_{i}}}|~{}\text{and}~{}x_{j}{x_{i}}>{{x}_{d}}{x_{i}}.$
(18)

If $x_{i}>0$, (18) implies that $x_{j}-{x_{i}}={x_{i}}-{{x}_{d}}>0$ and
$x_{j}>{{x}_{d}}$, which follows from (15c) that
$|{{\underline{f_{i}}}({{x_{j}}})-{\underline{f_{i}}}({{x_{i}}})}|={\underline{f_{i}}}({{x_{j}}})-{\underline{f_{i}}}({{x_{i}}})$
$<$
$|{{\underline{f_{i}}}({{x_{i}}})-{\underline{f_{i}}}({{{x}_{d}}})}|={\underline{f_{i}}}({{x_{i}}})-{\underline{f_{i}}}({{{x}_{d}}})$.
We then can obtain from (15a) and (13) that

$\displaystyle\\!\\!\\!\\!\underline{c}(x_{i},x_{j})\\!\\!>\\!\\!\underline{c}({x_{i},\\!{x_{d}}}),\\!\\!~{}\text{if}\\!~{}|x_{d}\\!-\\!x_{i}|\\!\\!=\\!\\!|x_{i}\\!-\\!x_{j}|,x_{j}\\!\\!>\\!{x_{d}},x_{i}\\!\\!>\\!0.$
(19)

If $x_{i}<0$, following the same steps to derive (19), we have

$\displaystyle\\!\\!\\!\\!\underline{c}(x_{i},x_{j})\\!\\!>\\!\\!\underline{c}({x_{i},\\!{x_{d}}}),\\!\\!~{}\text{if}\\!~{}|x_{d}\\!-\\!x_{i}|\\!\\!=\\!\\!|x_{i}\\!-\\!x_{j}|,x_{j}\\!\\!<\\!{x_{d}},x_{i}\\!\\!<\\!0.$
(20)

The results (19) and (20) means that the conjunctive conditions (15a)-(15d)
result in the second item in (11a).

Let us consider the condition

$\displaystyle x_{i}\cdot\hat{u}_{d}<0~{}\text{and}~{}x_{i}\cdot x_{d}>0.$
(21)

If $x_{i}>0$, the condition (21) implies that $x_{j}<0$ and $x_{d}>0$. Without
loss of generality, we let $x_{i}<x_{d}-x_{i}$. In the light of (19) and (16),
we thus have

$\displaystyle\underline{c}(x_{i},x_{j})$
$\displaystyle<\underline{c}({x_{i},{x_{d}}}),~{}~{}\text{if}~{}|x_{i}-x_{j}|=|x_{i}-x_{d}|$
$\displaystyle\underline{c}(x_{i},0)$
$\displaystyle>\underline{c}({x_{i},{x_{d}}}),~{}~{}\text{if}~{}0<x_{i}<x_{d}-x_{i},$

which indicates that there exist $x_{j}<0$ such that
$|x_{i}-x_{d}|>|x_{i}-x_{j}|$ and
$\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}})$. Therefore, we
conclude there exists an $x_{j}$ such that

$\displaystyle\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}}),~{}~{}\text{if}~{}|x_{i}-x_{j}|<|x_{i}-x_{d}|,x_{i}>0,$
$\displaystyle\hskip 156.49014ptx_{j}<0,x_{d}>0.$ (22)

If $x_{i}<0$, following the same steps to derive (22), we have conclude that
there exists an $x_{j}$ such that

$\displaystyle\underline{c}(x_{i},x_{j})>\underline{c}({x_{i},{x_{d}}}),~{}~{}\text{if}~{}|x_{i}-x_{j}|<|x_{i}-x_{d}|,x_{i}<0,$
$\displaystyle\hskip 156.49014ptx_{j}>0,x_{d}<0.$ (23)

The results (22) and (22) means that the conjunctive conditions (15a)-(15d)
result in the third item in (11a).

#### Necessary Condition

Given the form (13), it is straightforward to verify the condition (11b) is
equivalent to (15e). For the rest of proof, we consider contradiction, i.e,
assuming (15a)–(15d) do not hold, the condition (11a) does not hold as well.
We assume that (15) does not hold, i.e.,
$\underline{g_{i}}({{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}(x_{j})})$
is nondecreasing
w.r.t.$|{{\underline{f_{i}}}({{x_{i}}})-{\underline{f_{i}}}({{x_{j}}})}|$. We
let $x_{i}\geq x_{j}>0$, and thus have
$|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{j}}})}|={{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{j}}})}$.
If $x_{j}$ decreases to $x_{d}>0$, we then have
$|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{j}}})}|<|{{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}({{x_{d}}})}|$
and $|x_{i}-x_{j}|<|x_{i}-x_{d}|$. Considering (15a) and (13), we have

$\displaystyle\underline{c}(x_{i},x_{j})\leq\underline{c}({x_{i},{x_{d}}}).$
(24)

We now consider the case that
$\underline{g_{i}}({{\underline{f_{i}}}(x_{i})-{\underline{f_{i}}}(x_{j})})$
is strictly non-decreasing
w.r.t.$|{{\underline{f_{i}}}({{x_{i}}})-{\underline{f_{i}}}({{x_{j}}})}|$ and
${\underline{f_{i}}}(\cdot)$ is strictly non-increasing w.r.t. $c$. Let us set
$0<x_{d}<x_{j}<x_{i}$. We thus have $|x_{i}-x_{d}|>|x_{i}-x_{j}|$,
$|{{\underline{f_{i}}}({{x_{i}}})-{\underline{f_{i}}}({{x_{d}}})}|\geq|{{\underline{f_{i}}}({{x_{i}}})-{\underline{f_{i}}}({{x_{j}}})}|$
and (24). Following the same analysis method, we can conclude that if the
conditions (15a)–(15d) do not hold, the (11a) does not hold as well. ∎

###### Theorem 2

$\overline{c}\left({{\bar{x}_{i}}(k),{x_{j}}(k)}\right)$ given in (14)
satisfies (12) if and only if

$\displaystyle{\overline{g_{i}}}({{\overline{f_{i}}}({{\bar{x}_{i}}})\\!-\\!{\overline{f_{i}}}({{x_{j}}})})~{}\text{is
strictly increasing w.r.t. the distance}$ $\displaystyle\hskip
149.37697pt|{{\overline{f_{i}}}({{\bar{x}_{i}}})\\!-\\!{\overline{f_{i}}}({{x_{j}}})}|,$
(25a) $\displaystyle{\overline{f_{i}}}(x_{j})~{}\text{is strictly increasing
w.r.t. $x_{j}$},$ (25b)
$\displaystyle{\overline{f_{i}}}({{\bar{x}_{i}}})\\!<\\!\frac{{{\overline{f_{i}}}({x_{j}})\\!+\\!{\overline{f_{i}}}({{{x}_{d}}})}}{2},\\!~{}\text{if}\\!~{}|{x_{j}\\!-\\!{\bar{x}_{i}}}|\\!=\\!|{{{x}_{d}}\\!-\\!{\bar{x}_{i}}}|,x_{j}\\!>\\!{{x}_{d}}$
$\displaystyle\hskip 177.54535pt\text{and}\\!~{}{\bar{x}_{i}}\\!>\\!0,$ (25c)
$\displaystyle{\overline{f_{i}}}({{\bar{x}_{i}}})\\!>\\!\frac{{{\overline{f_{i}}}({{x_{j}}})\\!+\\!{\overline{f_{i}}}({{{x}_{d}}})}}{2},\\!~{}\text{if}\\!~{}|{x_{j}\\!-\\!{\bar{x}_{i}}}|\\!=\\!|{{{x}_{d}}\\!-\\!{{x}_{i}}}|,x_{j}\\!<\\!{{x}_{d}}$
$\displaystyle\hskip 176.97617pt\text{and}\\!~{}{\bar{x}_{i}}\\!<\\!0,$ (25d)
$\displaystyle{\overline{f_{i}}}(0)\\!=\\!\frac{{{\overline{f_{i}}}(x_{j})\\!+\\!{\overline{f_{i}}}({-x_{j}})}}{2}.$
(25e)

###### Proof:

The proof steps completely follow those of Theorem 15, it is thus omitted. ∎

Figure 1: Influence weight function
$\underline{c}(x_{i},x_{j})=0.6-0.011(\tanh(x_{i})-\tanh(x_{j}))^{2}$ captures
the asymmetric confirmation bias.

###  III-D Numerical Example

Theorems 15 and 25 provides the guides to construct the models of asymmetric
confirmation bias and negativity bias, respectively. The models are not
unique. Take the confirmation bias as an example, its models include
$\underline{c}({{x_{i}},{x_{j}}})={\chi_{i}}-{\gamma_{i}}({\tanh({{x_{i}}})-\tanh({{x_{j}}})})^{2}$,
$\underline{c}({{x_{i}},{x_{j}}})={\chi_{i}}-{\gamma_{i}}|{x_{i}^{3}-x_{j}^{3}}|$,
with $\gamma_{i}>0$, and among many others. We now use
$\underline{c}(x_{i},x_{j})=0.6-0.011(\tanh(x_{i})-\tanh(x_{j}))^{2}$ as one
numerical example to demonstrate its checkable properties, observing its
values shown in Figure 1, we can verify that

  * •

If individual $\mathrm{v}_{i}$’s opinion is neutral, i.e., $x_{i}=0$, she puts
the identical influence weights on the opinions that have same distance with
hers.

  * •

If individual $\mathrm{v}_{i}$ supports the position $1$ with $x_{i}=0.2$, for
the two opinions $0.8$ and $-0.4$ that have the same distance with
$x_{i}=0.2$, she puts larger influence weight on $0.8$, since $0.8$ and $0.2$
are in the same supporting domain.

  * •

If individual $\mathrm{v}_{i}$ supports the position $1$ with $x_{i}=0.2$, for
the two opinions $0.8$ and $-0.1$, she puts larger influence weight on $-0.1$,
since although $0.2$ and $0.8$ are in the same supporting domain while $-0.1$
is in the opposing domain, $0.8$ has much larger distance with $x_{i}=0.2$
compared with $-0.1$.

  * •

If individual $\mathrm{v}_{i}$ supports the position $1$ with $x_{i}=0.4$, for
the two opinions $0.6$ and $0.2$ that have the same distance with $x_{i}=0.4$,
she puts larger influence weight on $0.6$, since all of them are in the same
supporting domain but both $0.6$ and $0.4$ stick more to the supporting domain
compared with $0.2$.

## References

  * [1] C. Xu, J. Li, T. Abdelzaher, H. Ji, B. K. Szymanski, and J. Dellaverson, “The paradox of information access: On modeling social-media-induced polarization,” _arXiv:2004.01106_ , 2020.
  * [2] P. Giridhar and T. Abdelzaher, “Social media signal processing,” _Social-Behavioral Modeling for Complex Systems_ , pp. 477–493, 2019.
  * [3] R. S. Nickerson, “Confirmation bias: A ubiquitous phenomenon in many guises,” _Review of General Psychology_ , vol. 2, no. 2, pp. 175–220, 1998.
  * [4] D. M. Lazer, M. A. Baum, Y. Benkler, A. J. Berinsky, K. M. Greenhill, F. Menczer, M. J. Metzger, B. Nyhan, G. Pennycook, D. Rothschild _et al._ , “The science of fake news,” _Science_ , vol. 359, no. 6380, pp. 1094–1096, 2018.
  * [5] A. Kappes, A. H. Harvey, T. Lohrenz, P. R. Montague, and T. Sharot, “Confirmation bias in the utilization of others’ opinion strength,” _Nature Neuroscience_ , vol. 23, no. 1, pp. 130–137, 2020.
  * [6] T. Abdelzaher, H. Ji, J. Li, C. Yang, J. Dellaverson, L. Zhang, C. Xu, and B. K. Szymanski, “The paradox of information access: Growing isolation in the age of sharing,” _arXiv:2004.01967_ , 2020.
  * [7] P. Lamberson and S. Soroka, “A model of attentiveness to outlying news,” _Journal of Communication_ , vol. 68, no. 5, pp. 942–964, 2018.
  * [8] R. Hegselmann and U. Krause, “Opinion dynamics and bounded confidence models, analysis, and simulation,” _Journal of artificial societies and social simulation_ , vol. 5, no. 3, 2002.
  * [9] M. Del Vicario, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, “Modeling confirmation bias and polarization,” _Scientific reports_ , vol. 7, no. 1, pp. 1–9, 2017.
  * [10] Y. Mao, E. Akyol, and N. Hovakimyan, “Impact of confirmation bias on competitive information spread in social networks,” _to appear in IEEE Transactions on Control of Network Systems, DoI: 10.1109/TCNS.2021.3050117._
  * [11] Y. Mao, N. Hovakimyan, T. Abdelzaher, and E. Theodorou, “Social system inference from noisy observations,” _submitted to IEEE Transactions on Control of Network Systems (Special Issue on Dynamics and Behaviors in Social Networks), arXiv:2012.03075_.
  * [12] Y. Mao and E. Akyol, “On inference of network topology and confirmation bias in cyber-social networks,” _IEEE Transactions on Signal and Information Processing over Networks_ , vol. 6, pp. 633–644, 2020.
  * [13] N. E. Friedkin and E. C. Johnsen, “Social influence and opinions,” _Journal of Mathematical Sociology_ , vol. 15, no. 3-4, pp. 193–206, 1990\.
  * [14] M. H. DeGroot, “Reaching a consensus,” _Journal of the American Statistical Association_ , vol. 69, no. 345, pp. 118–121, 1974.
  * [15] P. Dandekar, A. Goel, and D. T. Lee, “Biased assimilation, homophily, and the dynamics of polarization,” _Proceedings of the National Academy of Sciences_ , vol. 110, no. 15, pp. 5791–5796, 2013.
  * [16] I. Noor, “Confirmation bias,” _Simply Psychology_ , 2020, https://www.simplypsychology.org/confirmation-bias.html, accessed 2020-08-25.
  * [17] M. Del Vicario, A. Bessi, F. Zollo, F. Petroni, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, “The spreading of misinformation online,” _Proceedings of the National Academy of Sciences_ , vol. 113, no. 3, pp. 554–559, 2016.


# Giant and Reversible Electronic Structure Evolution in a Magnetic
Topological Material EuCd2As2

Yang Wang1,2♯, Cong Li1,2♯, Taimin Miao1,2♯, Shuai Zhang1,2♯, Yong Li1,2♯,
Liqin Zhou1,2, Meng Yang1,2,3, Chaohui Yin1,2, Yongqing Cai1,2, Chunyao
Song1,2, Hailan Luo1,2, Hao Chen1,2, Hanqing Mao1, Lin Zhao1,2,4, Hanbin
Deng1,2, Yingkai Sun1,2, Changjiang Zhu1,2, Fengfeng Zhang5, Feng Yang5,
Zhimin Wang5, Shenjin Zhang5, Qinjun Peng5, Shuheng Pan1,2,4,6, Youguo
Shi1,2,3, Hongming Weng1,2,4, Tao Xiang1,2,4,7, Zuyan Xu5 and X. J.
Zhou1,2,4,7∗
1Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3Center of Materials Science and Optoelectronics Engineering, University of
Chinese Academy of Sciences, Beijing 100049, China
4Songshan Lake Materials Laboratory, Dongguan 523808, China
5Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China
6CAS Center for Excellence in Topological Quantum Computation, University of
Chinese Academy of Sciences, Beijing 100190, China
7Beijing Academy of Quantum Information Sciences, Beijing 100193, China
♯These people contributed equally to the present work.
∗Corresponding authors<EMAIL_ADDRESS>
(August 26, 2022)

The electronic structure and the physical properties of quantum materials can
be significantly altered by charge carrier doping and magnetic state
transition. Here we report a discovery of a giant and reversible electronic
structure evolution with doping in a magnetic topological material. By
performing high-resolution angle-resolved photoemission measurements on
EuCd2As2, we found that a huge amount of hole doping can be introduced into
the sample surface due to surface absorption. The electronic structure
exhibits a dramatic change with the hole doping which can not be described by
a rigid band shift. Prominent band splitting is observed at high doping which
corresponds to a doping-induced magnetic transition at low temperature (below
$\sim$15 K) from an antiferromagnetic state to a ferromagnetic state. These
results have established a detailed electronic phase diagram of EuCd2As2 where
the electronic structure and the magnetic structure change systematically and
dramatically with the doping level. They further suggest that the transport,
magnetic and topological properties of EuCd2As2 can be greatly modified by
doping. These work will stimulate further investigations to explore for new
phenomena and properties in doping this magnetic topological material.

The electronic structure and physical properties of a material can be
manipulated by varying the crystal structure, magnetic structure and the
concentration of charge carriers. The doping of charge carriers can cause
significant change of electronic structures and resultant physical properties
like insulator-metal transition and emergence of superconductivityMImadaRMP ;
PLeeRMP . EuCd2As2 is a novel magnetic topological materialGYHua_PRB2018 ;
LLWang_PRB ; CWNiu_PRB ; JZMa_AM ; JZMa_SA . A number of topological states
including antiferromagnetic topological insulatorsJZMa_AM , a triple-point
magnetic topological semimetalGYHua_PRB2018  and a quantum anomalous Hall
insulatorCWNiu_PRB  can be produced in EuCd2As2 by manipulating its magnetic
structures combined with rotational or inversion symmetry breaking. In
particular, when EuCd2As2 is in the inter-layer antiferromagnetic state, it is
expected to host only one pair of Dirac points at the Fermi levelGYHua_PRB2018
. When the ferromagnetism is induced along the c axis, it can generate a
single pair of Weyl points that is an ideal platform to study the unique
properties of Weyl semimetalsLLWang_PRB ; JZMa_SA ; Soh_PRB2019 . EuCd2As2 has
therefore become a fertile playground for studying the interplay between
magnetism and topology. The magnetic properties of EuCd2As2 originate from the
Eu layers and there have been extensive studies on manipulating its magnetic
structure by temperatureArtmann_Chem ; Schellenberg_Chem ; MCRahn_PRB ;
JZMa_SA ; JZMa_AM ; Jo_PRB ; YXu_PRB ; YWang_CPL2021 , applied magnetic
fieldSoh_PRB2019 ; YXu_PRB  and the chemical substitutionSanjeewa_PRB .
However, little is known about the doping induced effects on the electronic
structure, magnetic structure and topological properties of this topological
material.

Here we report the discovery of a giant and reversible electronic structure
evolution in EuCd2As2. By high-resolution laser-based angle-resolved
photoemission spectroscopy (ARPES) measurements, we directly observed the
Fermi surface, and the band structure of EuCd2As2 experienced a dramatic
change with doping that is induced by the surface absorption/desorption. We
also observed a prominent band splitting at high doping corresponding to a
doping-induced magnetic transition at low temperature (below $\sim$15 K) from
an antiferromagnetic state to a ferromagnetic state.

Initially, we found that the Fermi surface of EuCd2As2 exhibits a dramatic
change with time, as shown in Fig. 1. The corresponding band structure
evolution with time is shown in Fig. 2 (More complete Fermi surface and band
structure evolutions with time are shown in Fig. S1 in Supplementary
MaterialsSM ). The drastic Fermi surface change with time shows up in terms of
the shape, the size and the number of Fermi surface sheets (Fig. 1c). The
schematic Fermi surface for several typical dwell times, based on the detailed
Fermi surface mapping analysis (Fig. S2 in Supplementary MaterialsSM ) and the
band structure analysis, is shown in Fig. 1d. At the beginning, when EuCd2As2
is cleaved and measured at 250 K, the measured Fermi surface is a point at
$\bar{\mathrm{\Gamma}}$ (first panel in Fig. 1c). After the sample was quickly
cooled down to 3 K and immediately measured, the Fermi surface remains a tiny
circle (second panel in Fig. 1c and first panel in Fig. 1d). The tiny hole
pocket increases in its size with the dwell time up to 3 hours (third panel in
Fig. 1c and second panel in Fig. 1d). At the dwell time of 6 hours, the
observed Fermi surface consists of two pockets: a smaller hexagon-shaped sheet
and another larger petal-like sheet (fourth panel in Fig. 1c). These two hole-
like Fermi surface sheets get larger with time up to the dwell time of 12
hours (sixth panel in Fig. 1c and third panel in Fig. 1d). Further Fermi
surface splitting occurs with time and at the time of 24 hours, four Fermi
surface sheets can be clearly observed (ninth panel in Fig. 1c and fourth
panel in Fig. 1d). These four Fermi surface sheets increase in their size with
time and at the dwell time of 81 hours, the outer two Fermi surface sheets get
so large that they are approaching the measurement window (second last panel
in Fig. 1c and last panel in Fig. 1d). After that, when the sample is warmed
up to 250 K, the measured Fermi surface (last panel in Fig. 1c) recovers to a
single point at $\bar{\mathrm{\Gamma}}$, similar to the Fermi surface measured
at 250 K before the cycle (first panel in Fig. 1c). When the sample is quickly
cooled down again to 3 K, similar dramatic evolution of the Fermi surface with
time can be observed. The Fermi surface can always be recovered to a single
point at $\bar{\mathrm{\Gamma}}$ whenever the sample is warmed up to 250 K.
The observed Fermi surface evolution with time at 2$\sim$3 K was repeated
several times on different samples. We find that the trend of the electronic
structure evolution with time is the same but the speed of the evolution
process varies between different samples and between different cycles even for
the same sample.

Figure 2 shows the corresponding band structure evolution of EuCd2As2 with
time measured at 3 K. The band structure also exhibits a significant change
with time in terms of the band position and the number of the observed bands.
When the sample was cleaved and measured at 250 K, two main bands are observed
in the measurement energy window (marked as $\alpha$ and $\beta$ bands in the
first panel of Fig. 2a). After the sample was quickly cooled down to 3 K, the
overall band structure shifts upward by $\sim$70 meV, accompanied by the
splitting of the $\alpha$ band (second panel in Fig. 2a). Till the dwell time
of 6 hours, the two split bands are fully separated (marked as B1 and B2 bands
in the fourth panel of Fig. 2a). When the dwell time reaches 15 hours, the
original B1 and B2 bands start to further split and at the time of 24 hours,
four well separated bands can be clearly observed (marked as B3, B4, B5 and B6
bands in the ninth panel of Fig. 2a). These four hole-like bands increase in
their Fermi momenta with the further increase of the dwell time and after 39
hours, the outer band (B3) gets out of the detection window. On the other
hand, the $\beta$ band experiences a more complicated evolution with the dwell
time. Before the dwell time of 9 hours, this single $\beta$ band lowers its
energy position with the increasing time by $\sim$100 meV (second to fifth
panels in Fig. 2a). After 9 hours, the energy position of this $\beta$ band
shifts continuously upwards by $\sim$260 meV up to 81 hours. The $\beta$ band
also starts to split after the dwell time of 18 hours and eventually two well-
split bands can be observed at 81 hours (marked as B7 and B8 bands in the last
second panel of Fig. 2a). When the sample was warmed up to 250 K again, the
two original $\alpha$ and $\beta$ bands can be recovered as seen in the last
panel in Fig. 2a.

In order to keep track on the band structure evolution with the dwell time, in
Fig. 2d and 2e, we plot the change of momentum distribution curves (MDCs) at
the Fermi level and at the binding energy of 100 meV, respectively. From these
figures, the initial splitting of the original $\alpha$ band into B1 and B2
bands, further splitting of the B1 and B2 bands into four B3-B6 bands, and the
continuous Fermi momentum increase of these bands with the increasing time can
be clearly visualized. The splitting of the original $\beta$ band into B7 and
B8 bands and the corresponding momentum increase with the increasing time can
also be clearly observed in Fig. 2e. The energy position change of the $\beta$
band can be clearly seen from the photoemission spectra (energy distribution
curves, EDCs) at the $\bar{\mathrm{\Gamma}}$ point in Fig. 2f. The top of the
$\beta$ band exhibits a non-monotonic variation with time: it first shifts to
higher binding energy during the dwell time of 0$\sim$9 hours and then moves
to lower binding energy in the time range of 9$\sim$81 hours. We notice that,
during the time of 0$\sim$9 hours, the $\alpha$ and $\beta$ bands shift with
time along the opposite energy direction. This clearly indicates that the band
shift with the dwell time is not a rigid band shift.

We have also carried out electronic structure evolution experiments on
EuCd2As2 at 25 K (Fig. S3 in Supplementary MaterialsSM ) and 150 K (Fig. S4 in
Supplementary MaterialsSM ). At the temperature of 25 K which is higher than
the antiferromagnetic transition temperature of EuCd2As2 at 9.3 K, obvious
electronic structure evolution with time is also observed. It involves both
the Fermi surface topology change and the band structure position shift (Fig.
S3), similar to those we observed at 3 K (Fig. 1 and Fig. 2). Even at 150 K,
the electronic structure evolution can still be observed (Fig. S4). In this
case, it mainly involves the energy position shift of the two $\alpha$ and
$\beta$ bands.

We have measured the electronic structure of the fully-modified EuCd2As2 at
different temperatures. This is realized by keeping the EuCd2As2 sample for a
sufficiently long time at a low temperature and then slowly changing to other
temperatures; at each temperature, the sample stayed for 3 hours when the
ARPES measurement was carried out. Fig. 3 shows the measured Fermi surface and
band structure at several typical temperatures in the low temperature regime
between 2$\sim$30 K; another independent measurement in a larger temperature
range between 3$\sim$250 K is shown in Fig. S5 in Supplementary MaterialsSM .
For the fully-modified EuCd2As2, its electronic structure exhibits a dramatic
change with temperature, particularly in the low temperature range between
2$\sim$30 K.

For the modified sample at 2 K, the band splitting is fully realized with four
bands (B3, B4, B5 and B6) clearly observed near the Fermi level (first panel
in Fig. 3b). This band splitting persists up to 9.3 K, the antiferromagnetic
transition temperature of EuCd2As2 (second panel in Fig. 3b). Above 9.3 K, the
intensity of the B3, B4 and B5 bands quickly gets weaker and becomes nearly
invisible above 20 K (25 K and 30 K panels in Fig. 3b). Correspondingly, the
Fermi surface topology changes from four Fermi surface sheets at 2 K and 9.3 K
to a single main sheet at 25 K and 30 K as seen from Fig. 3a.

From our detailed temperature-dependent measurements in a wide temperature
range of 3$\sim$250 K (Fig. S5 and Fig. S6 in Supplementary MaterialsSM ), we
find that there are three distinct temperature regions where the electronic
structures of the modified EuCd2As2 behave quite differently. Fig. 3c shows
the stacked second-derivative MDCs at the binding energy of 25 meV from the
band structures measured at different temperatures (Fig. S5b in Supplementary
MaterialsSM ). The stacked second-derivative MDCs at other binding energies
(Fig. S6 in Supplementary MaterialsSM ) give similar results. In the
temperature region I of 3$\sim$30 K, as described before, the electronic
structure exhibits a dramatic change with temperature in terms of the band
position and the number of the observed bands. In the second temperature
region II of 30$\sim$135 K, the observed electronic structure shows little
change with temperature, keeping similar band position and the same number of
bands. When moving to the third temperature region III of 135$\sim$250 K, the
measured electronic structure shows a significant change with temperature
again. The position of the B1 band and the $\beta$ band keeps shrinking with
the increasing temperature.

In order to keep track on the band position change with temperature, we show
in Fig. 3e the EDCs at the $\bar{\mathrm{\Gamma}}$ point measured on the
modified EuCd2As2 at different temperatures obtained from the measurements in
Fig. S5. Two EDC peaks are observed, originating from the $\gamma$ band and
$\delta$ band, as shown in Fig. S7 in Supplementary MaterialsSM . In the
temperature range of 2$\sim$30 K, the position of these two bands moves toward
the Fermi level with increasing temperature. Their positions exhibit little
change with temperature in the range of 30$\sim$135 K. Upon further increase
of the temperature, the position of these two bands shifts to higher binding
energies. The temperature evolution of the EDCs at $\bar{\mathrm{\Gamma}}$ is
consistent with the MDC evolution in Fig. 3c in terms of three distinct
temperature regions. We note that the overall band shift with temperature is
not monotonic. Furthermore, it is not a rigid band shift because the magnitude
of the band position shift between the $\gamma$ band and the $\delta$ band is
different and the number of the observed bands changes significantly at
different temperatures.

The observation of such a dramatic electronic structure evolution with time in
EuCd2As2 is rare. The first question to ask is whether this electronic
structure change is associated with the sample surface or the entire bulk. To
this end, we measured the time-dependence of the resistivity and heat capacity
of EuCd2As2 at different temperatures and these bulk properties do not exhibit
obvious change with time even at a low temperature of 5 KYWang_CPL2021 . This
is in a strong contrast to the dramatic electronic structure evolution we have
observed. It indicates that the observed electronic structure change is mainly
associated with the sample surface since ARPES is a surface sensitive
technique. The second question is whether there is a structure change with
time in EuCd2As2. The crystal structure of EuCd2As2 exhibits little change
with time from the time-dependent X-ray diffraction (XRD) measurement at a low
temperatureYWang_CPL2021 . By performing time-dependent scanning tunneling
microscope (STM) (Fig. S8 in Supplementary MaterialsSM ) and low energy
electron diffraction (LEED) (Fig. S9 in Supplementary MaterialsSM )
measurements on EuCd2As2, we find that there is no surface structure change
with time at low temperature. These results demonstrate that the electronic
structure evolution observed in EuCd2As2 is not due to bulk or surface
structure change with time.

The measured Fermi surface (Fig. 1c) and band structure (Fig. 2) of EuCd2As2
indicate that, during the evolution process, there is a continuous increase of
the charge carriers (holes) in the measured region. When EuCd2As2 is fully
modified, the hole doping level can reach up to $\sim$0.18 holes per unit cell
(Fig. S10 in Supplementary MaterialsSM ). A natural question to ask on the
origin of the dramatic electronic structure change is where the charge
carriers come from. There are two main possibilities. The first is related to
the intrinsic relaxation process in EuCd2As2, such as the long-time magnetic
relaxation with timeYWang_CPL2021  or some other unidentified relaxation
processes that causes such a dramatic electronic structure evolution with
time. The second possibility is related to an extrinsic surface
absorption/desorption process that causes a dramatic change of charge carriers
at the surface. This process can be further divided into a natural absorption
or a laser-assisted absorption where the laser may cause or enhance the
surface absorptionRJiang_PRB2012 . To pin down on the exact origin, we have
taken the measurements many times under different experimental conditions. We
find that, although the trend of the electronic structure evolution with time
is the same, the speed of the evolution varies between different samples and
between different cycles even for the same sample. In particular, we found
that the electronic structure evolution speed is closely related to the vacuum
level in the ARPES measurement chamber. Fig. 4 shows the electronic structure
change of EuCd2As2 with time measured under different experimental conditions
on the same sample. In this case, when the vacuum is good and the laser is off
in between the measurements, the sample experiences little electronic
structure change with time even after $\sim$27 hours, as shown in Fig. 4a and
4f. This is in a strong contrast to the results in Fig. 1 where a giant
electronic structure change already occurs after the similar time duration.
This difference can basically rule out the first possibility that the dramatic
electronic structure evolution with time observed in Fig. 1 is associated with
the intrinsic relaxation process in EuCd2As2, making the surface absorption a
more likely origin. When we maintained all the measurement conditions the
same, but only kept the laser always on the sample, we found that the
electronic structure started to show an obvious change with time, as shown in
Fig. 4b and 4g. This indicates that the laser played a role in the electronic
structure evolution process and it can be related to the laser-assisted
absorptionRJiang_PRB2012 . To confirm the surface absorption, we warmed up the
sample to 250 K to let its electronic structure recovered to its initial
state. In the meantime, the vacuum level in the chamber got worse during this
warming-up process. After the sample was quickly cooled down to 5 K, even when
the laser was turned off in between the measurements like the situation in
Fig. 4a, the electronic structure experiences an obvious change with time, as
shown in Fig. 4d and 4i. These measurements indicate that a natural absorption
process can also give rise to a dramatic electronic structure variation when
the vacuum level is relatively lower. When the natural absorption process
already produces a substantial change in the electronic structure, the laser-
assisted absorption may further slightly enhance the process, but its effect
may become less dominant, as shown in Fig. 4e and 4j. The results in Fig. 4
clearly indicate that the dramatic electronic structure evolution we observed
in Fig. 1 can be mainly attributed to the surface absorption/desorption
process. Since hydrogen is the major ingredient of the residual gas in our
ultra-high vacuum chamber, we suspect the charge carrier change on surface is
mainly due to hydrogen absorption/desorption although further work needs to be
done to pin down on the exact origin.

In order to understand the dramatic electronic structure evolution with hole
doping (Fig. 1 and Fig. 2) and the temperature evolution of the modified
EuCd2As2 (Fig. 3), we carried out detailed electronic structure calculations
of EuCd2As2 bulk (Fig. 5, Fig. S11 and Fig. S12 in Supplementary MaterialsSM
), EuCd2As2 slab (Fig. S13 in Supplementary MaterialsSM ) and EuCd2As2
monolayer (Fig. S14 in Supplementary MaterialsSM ) in different magnetic
states. For the undoped EuCd2As2 in the paramagnetic state at 250 K, ARPES
should measure the bulk electronic structure and the surface states if they
exist. When only the top layers are doped due to the surface absorption, it
may be more appropriate to understand the electronic structure from the band
structure calculations of the monolayers. In Fig. S15 in Supplementary
MaterialsSM , we showed the comparison between the measured band structure of
EuCd2As2 at 250 K and the calculated results of EuCd2As2 monolayer, bulk and
slab in the paramagnetic state. All the three calculated results are similar
in that they consist of three main branches of bands labeled as B1, B2 and
$\beta$ within the energy range of $\sim$1 eV below the Fermi level (Fig.
S15b, S15c and S15d). The measured band structure is consistent with the band
structure calculations in that three bands are observed labeled as $\alpha$
and $\beta$ with $\alpha$ consisting of two bands (B1 and B2 in Fig. S15b in
Supplementary MaterialsSM ). However, we find that neither of these
calculations can be fully consistent with the measured results in the
quantitative manner. This deviation may be due to some correlation effects
that are not properly considered in the calculations and further theoretical
efforts are needed to address this issue.

Fig. 5 highlights the calculated electronic structures of EuCd2As2 and their
comparison with the doping evolution of the measured electronic structures. In
the paramagnetic state, there are mainly three bands (B1, B2 and $\beta$) near
the Fermi level (Fig. 5n and 5o); the corresponding Fermi surface is only a
point (Fig. 5m). The measured band structures (Fig. 5b and 5c) and the Fermi
surface (Fig. 5a) of EuCd2As2 at 250 K in the paramagnetic state are
qualitatively consistent with the calculated results although the bands B1 and
B2 are nearly degenerate ($\alpha$ band in Fig. 5b and 5c) in this case. In
the initial stage of doping evolution, the measured bands (Fig. 5e and 5f)
exhibit a slight upward shift and an apparent B1 and B2 splitting; the overall
band structure and Fermi surface (Fig. 5d) are still consistent with the
calculated results in the paramagnetic state (Fig. 5m, 5n and 5o). In the
early stage of evolution at 2$\sim$3 K, the measured band structures (Fig. 5h
and 5i) show an obvious upward shift and the B1 and B2 bands become well
separated. The measured Fermi surface (Fig. 5g) consists of two pockets. These
results are in a good agreement with the calculations (Fig. 5p, 5q and 5r)
when the chemical potential is shifted by $\sim$0.3 eV. In the later stage of
doping evolution at 2$\sim$3 K, 6 bands appear near the Fermi level (B3-B8 in
Fig. 5k and 5l) and 4 Fermi surface sheets are observed (Fig. 5j). Such
electronic structures can not be understood if EuCd2As2 is in the paramagnetic
state or in the antiferromagnetic state (Fig. S11 in Supplementary MaterialsSM
). On the other hand, these electronic structures are rather consistent with
the calculated results with EuCd2As2 being in the ferromagnetic state and an
overall upward band shift of $\sim$0.4 eV (Fig. 5v, 5w and 5x). The analysis
of this electronic structure evolution indicates that, during the doping
evolution process, in addition to the significant band shift, there is also a
change of the magnetic state at low temperature from the initial
antiferromagnetic state at zero or low doping to ferromagnetic state at high
doping. This is consistent with the recent observation of band filling
controlled magnetism in EuCd2As2: the antiferromagnetic state is observed at
low doping while the ferromagnetic state is observed at high dopingJo_PRB .

The particular electronic structure evolution with temperature observed in the
fully modified EuCd2As2, as shown in Fig. 3, can be understood by combining
the doping change and the magnetic state change. In the temperature region I
of 3$\sim$30 K, the electronic structure exhibits a dramatic change with
temperature in both the band position and the number of the bands. Although
the hole doping may be gradually decreased due to surface desorption with
increasing temperature, such a doping change alone is hard to understand the
drastic electronic structure change in this temperature region. The magnetic
state of EuCd2As2 at low temperatures most likely plays an important role in
dictating its electronic structure. Coincidentally, this is the temperature
range where below $\sim$30 K the magnetic susceptibility shoots up and the
sample enters into the magnetic states, as shown in Fig. 3d. In the
temperature region II of 30$\sim$135 K, the band structure shows little change
with temperature. Above 30 K, EuCd2As2 enters into a paramagnetic state and
its electronic structure is mainly controlled by the doping level. The
increasing temperature tends to decrease the absorption but, in the meantime,
it causes the vacuum level in the ARPES chamber to become worse which tends to
increase the absorption. Therefore, the doping shows little change with
temperature in this temperature region because of these two competing
absorption/desorption processes involved. In the temperature region III above
$\sim$135 K, the surface desorption process due to the temperature increase
becomes dominant. The doping level keeps decreasing with increasing
temperature until nearly negligible surface absorption is left and the
EuCd2As2 sample recovers to its initial non-doped state at $\sim$250 K. The
reversible electronic structure evolution can thus be understood by this
surface absorption/desorption process.

Our present results provide a new perspective in understanding the previous
ARPES results on EuCd2As2JZMa_SA ; Soh_PRB2019 ; JZMa_AM ; Jo_PRB . Our
results clearly indicate that, when EuCd2As2 is undoped or only slightly
doped, the electronic structure in the paramagnetic state (250 K) (Fig. 4c and
4h) is similar to that observed at low temperature in the antiferromagnetic
state ($<$9.3 K) (Fig. 4a and 4f). It is consistent with the observations
reported in Ref.Soh_PRB2019  but quite different from the results in
Ref.JZMa_SA . It was reported in Ref.JZMa_SA  that the degeneracy of the Bloch
bands is already lifted in the paramagnetic state of EuCd2As2 which is
attributed to the presence of the quasi-static and quasi-long-range
ferromagnetic fluctuations. Our high resolution laser ARPES measurements
demonstrate unambiguously that such a lifting of the band degeneracy does not
occur in the entire measured temperature range of 2$\sim$250 K. On the other
hand, our present work asks for special caution to be exercised about surface
absorption in ARPES measurements of EuCd2As2 which might not be noticed
before. The band structures observed in Ref.JZMa_SA  are similar to those we
observed in EuCd2As2 with sufficient hole doping due to surface absorption.
These indicate that the band splitting observed in Ref.JZMa_SA  is more likely
due to hole doping from surface absorption rather than the ferromagnetic
fluctuation as they proposed.

In summary, we have discovered a giant and reversible electronic structure
evolution phenomenon in EuCd2As2. A large amount of hole doping is introduced
into the sample surface during the cooling process because of surface
absorption that dramatically changes the electronic structures. The doping
also causes a magnetic state transition at low temperature (below $\sim$15 K)
from an antiferromagnetic state at zero or low dopings to a ferromagnetic
state at high doping which give rise to band splitting in EuCd2As2. The
discovery of the dramatic electronic structure evolution process in EuCd2As2
due to surface absorption asks to reexamine the previous resultsJZMa_SA ;
JZMa_AM ; Jo_PRB . Our results have established a detailed electronic phase
diagram of EuCd2As2 where the electronic structure and the magnetic structure
change systematically and dramatically with the hole doping. They further
suggest that the transport, magnetic and topological properties of the surface
or bulk EuCd2As2 can be greatly modified. There are many feasible ways to
change the doping like aliovalent substitution, formation of vacancies or
interstitialsJo_PRB , or gating. The present work will motivate people to
explore for new phenomena and properties by doping the bulk EuCd2As2 or
EuCd2As2 thin films.

Methods
Sample Single crystals of EuCd2As2 were grown by Sn flux methodHPWang_PRB .
High purity elements of Eu, Cd, As, and Sn were put in an alumina crucible at
a molar ratio of 1:2:2:10 and sealed in a quartz tube under high vacuum. The
tube was heated to 1173 K, kept for 20 hours, and then slowly cooled to 773 K
at a rate of 2 K/hour. After that the single crystals were separated from the
Sn liquid in a centrifuge.

ARPES Measurements High-resolution ARPES measurements were performed at our
laboratory-based systems equipped with the 6.994 eV vacuum ultraviolet laser
and the time-of-flight electron energy analyzer (ARToF 10k by Scienta Omicron)
or a hemisphere analyzer (DA30L by Scienta Omicron)GDLiu_RSI2008 ; XJZ_Review
. The ARToF-ARPES system can cover two-dimensional momentum space ($k_{x}$,
$k_{y}$) simultaneously. It is also equipped with an ultra-low temperature
cryostat which can cool the sample down to a low temperature of 1.6 K. The
overall energy resolution is $\sim$1 meV and the angular resolution is
$\sim$0.1 degree. The Fermi level is referenced by measuring polycrystalline
gold which is in good electrical contact with the sample. The samples were
cleaved in situ and measured at different temperatures in ultrahigh vacuum
with a base pressure better than 5.0$\times$10-11 mbar.

STM Measurements Scanning tunneling microscope measurements were performed at
the home-made ultra-high vacuum (UHV) low temperature STM system. The samples
were cleaved at 4.2 K. During the measurements, the current and the V-bias
were set at $\sim$100 pA and $\sim$100 mV, respectively. The scanning range is
about 30 nm $\times$ 30 nm.

Band Structure Calculations Band structure calculations were performed on
EuCd2As2 bulk, EuCd2As2 slab and EuCd2As2 monolayer. For the bulk
calculations, we calculated the electronic structure of EuCd2As2 in 5
different magnetic states, i.e., the paramagnetic state, the ferromagnetic
state with the in-plane and out-of-plane magnetizations, and the inter-layer
anti-ferromagnetic state with the in-plane and out-of-plane magnetizations.
For the antiferromagnetic state, we made a 1$\times$1$\times$2 supercell to
get consistent with the c axis magnetic order. We calculated the electronic
structure with the Vienna $ab$ $initio$ Simulation package
(VASP)GKresse_PRB1996 ; John_PRL1996 . The projector-augmented-wave (PAW)
methodPEBlochl_PRB1994 ; GKresse_PRB1999  with the Perdew-Burke-Ernzerhof
(PBE) exchange-correlationJohn_PRL1996  functional was used. In the
paramagnetic state, we did not take the $f$ electrons of Eu in our pseudo-
potential. For the ferromagnetic and the antiferromagnetic state, we took the
$f$ electrons into consideration and added the on-site $U$ as 5 eV. We set a
13$\times$13$\times$7 $k$-mesh for the primitive unit cell (paramagnetic state
and ferromagnetic state) and a 13$\times$13$\times$4 $k$-mesh for the
supercell (antiferromagnetic state). In all cases, we set the plane wave
energy cutoff as 500 eV. We also calculated the constant energy contours at
different binding energies. For these calculations, we constructed tight-
binding Hamiltonians using the WANNIER90 packageGPizzi_JPCM2020 , and
calculated the constant energy contours by WANNIERTOOLS packageQSWu_CPC2018 .
For the slab calculations, we constructed a 1$\times$1$\times$8 supercell and
add a vacuum layer larger than 15 $\AA$. For the monolayer calculations, the
Vienna $ab$ $initio$ simulation package (VASP) based on the density functional
theory (DFT) was used. The vacuum layer of 20 $\AA$ along the (001) direction
was used to avoid the interaction between nearest monolayers. The Perdew-
Burke-Ernzerhof (PBE) exchange-correlation function with the generalized
gradient approximation (GGA) $+$ $U$ method was used to treat the localized
4$f$ orbitals of Eu atomJohn_PRL1996 ; HJKulik_PRL2006 , and the $U$ parameter
was selected to be 5 eV. The cutoff energy was set at 500 eV and the Brillouin
zone (BZ) integral was sampled by 9$\times$9$\times$1 $k$-mesh. The spin-
orbital coupling (SOC) was taken into account in all calculations.

## References

  * (1) M. Imada, A. Fujimor and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998).
  * (2) P. A. Lee, N. Nagaosa and X-G Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006).
  * (3) G. Y. Hua, S. Nie, Z. Song, R. Yu, G. Xu and K. Yao, Dirac semimetal in type-IV magnetic space groups, Phys. Rev. B 98, 201116(R) (2018).
  * (4) L. L. Wang, N. H. Jo, B. Kuthanazhi, Y. Wu, R. J. McQueeney, A. Kaminski and P. C. Canfield, Single pair of Weyl fermions in the half-metallic semimetal EuCd2As2, Phys. Rev. B 99, 245147 (2019).
  * (5) C. Niu, N. Mao, X. Hu, B. Huang and Y. Dai, Quantum anomalous Hall effect and gate-controllable topological phase transition in layered EuCd2As2, Phys. Rev. B 99, 235119 (2019).
  * (6) J. Z. Ma, S. M. Nie, C. J. Yi, J. Jandke, T. Shang, M. Y. Yao, M. Naamneh, L. Q. Yan, Y. Sun and A. Chikina _et al_., Spin fluctuation induced Weyl semimetal state in the paramagnetic phase of EuCd2As2, Sci. Adv. 5, eaaw4718 (2019).
  * (7) J. Ma, H. Wang, S. Nie, C. Yi, Y. Xu, H. Li, J. Jandke, W. Wulfhekel, Y. Huang and D. West _et al_., Emergence of nontrivial low-energy Dirac Fermions in antiferromagnetic EuCd2As2, Adv. Mater. 32, 1907565 (2020).
  * (8) J. R. Soh, F. de Juan, M. G. Vergniory, N. B. M. Schr$\ddot{o}$ter, M. C. Rahn, D. Y. Yan, J. Jiang, M. Bristow, P. Reiss and J. N. Blandy _et al_., Ideal Weyl semimetal induced by magnetic exchange, Phys. Rev. B 100, 201102(R) (2019).
  * (9) A. Artmann, A. Mewis, M. Roepke and G. Michels, AM2X2-verbindungen mit CaAl2Si2-struktur. XI. Struktur und eigenschaften der verbindungen ACd2X2 (A: Eu, Yb; X: P, As, Sb), Z. Anorg. Allg. Chem. 622, 679 (1996).
  * (10) I. Schellenberg, U. Pfannenschmidt, M. Eul, C. Schwickert and R. P$\ddot{o}$ttgen, A 121Sb and 151Eu M$\ddot{o}$ssbauer spectroscopic investigation of EuCd2X2 (X = P, As, Sb) and YbCd2Sb2, Z. Anorg. Allg. Chem. 637, 1863 (2011).
  * (11) M. C. Rahn, J.-R. Soh, S. Francoual, L. S. I. Veiga, J. Strempfer, J. Mardegan, D. Y. Yan, Y. F. Guo, Y. G. Shi and A. T. Boothroyd, Coupling of magnetic order and charge transport in the candidate Dirac semimetal EuCd2As2, Phys. Rev. B 97, 214422 (2018).
  * (12) Y. Xu, L. Das, J. Z. Ma, C. J. Yi, S. M. Nie, Y. G. Shi, A. Tiwari, S. S. Tsirkin, T. Neupert and M. Medarde _et al_., Unconventional transverse transport above and below the magnetic transition temperature in Weyl semimetal EuCd2As2, Phys. Rev. Lett. 126, 076602 (2021).
  * (13) N. H. Jo, B. Kuthanazhi, Y. Wu, E. Timmons, T.-H. Kim, L. Zhou, L.-L. Wang, B. G. Ueland, A. Palasyuk and D. H. Ryan _et al_., Manipulating magnetism in the topological semimetal EuCd2As2, Phys. Rev. B 101, 140402(R) (2020).
  * (14) Y. Wang, Cong Li, Yong Li, X. Zhou, W. Wu, R. Yu, J. Zhao, C. Yin, Y. Shi and C. Jin _et al_., Long-time magnetic relaxation in antiferromagnetic topological material EuCd2As2, Chin. Phys. Lett. 38, 077201 (2021).
  * (15) L. D. Sanjeewa, J. Xing, K. M. Taddei, D. Parker, R. Custelcean, C. dela Cruz and A. S. Sefat, Evidence of Ba-substitution induced spin-canting in the magnetic Weyl semimetal EuCd2As2, Phys. Rev. B 102, 104404 (2020).
  * (16) See Supplemental Material at [http://link.aps.org/supplemental/10.1103/PhysRevB.106.085134] for detailed electronic structure evolution of EuCd2As2 with the measurement time at 3 K; identification of the measured Fermi surface for the modified EuCd2As2; electronic structure evolution of EuCd2As2 with the measurement time at 25 K; band structure evolution of EuCd2As2 with the measurement time at 150 K; detailed temperature-dependent electronic structure of the modified EuCd2As2; two-dimensional images showing the second-derivative MDCs at different temperatures; detailed analysis of the band structure measured on the modified EuCd2As2 at different temperatures; time-dependent STM measurements of EuCd2As2 at low temperature; time-dependent LEED measurements of EuCd2As2 at low temperature; estimation of the doping level in the modified EuCd2As2; calculated band structure of EuCd2As2 bulk in different magnetic states; calculated Fermi surface and constant energy contours of EuCd2As2 in the ferromagnetic state with an in-plane magnetization; calculated band structure of EuCd2As2 slab in paramagnetic state; calculated band structures of EuCd2As2 monolayer in different magnetic configurations; comparison of the measured and calculated band structures of EuCd2As2 in paramagnetic state.
  * (17) R. Jiang, L.-L. Wang, M. Huang, R. S. Dhaka, D. D. Johnson, T. A. Lograsso and A. Kaminski, Reversible tuning of the surface state in a pseudobinary Bi2(Te-Se)3 topological insulator, Phys. Rev. B 86, 085112 (2012).
  * (18) H. P. Wang, D. S. Wu, Y. G. Shi and N. L. Wang, Anisotropic transport and optical spectroscopy study on antiferromagnetic triangular lattice EuCd2As2: An interplay between magnetism and charge transport properties, Phys. Rev. B 94, 045112 (2016).
  * (19) G. D. Liu, G. Wang, Y. Zhu, H. Zhang, G. Zhang, X. Wang, Y. Zhou, W. Zhang, H. Liu and L. Zhao _et al_., Development of a vacuum ultra-violet laser-based angle-resolved photoemission system with a super-high energy resolution better than 1 meV, Rev. Sci. Instrum. 79, 023105 (2008).
  * (20) X. J. Zhou, S. He, G.-D. Liu, L. Zhao, L. Yu and W. Zhang, New developments in laser-based photoemission spectroscopy and its scientific applications: a key issues review, Rep. Prog. Phys. 81, 062101 (2018).
  * (21) G. Kresse and J. Furthm$\ddot{u}$ller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * (22) J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996).
  * (23) P. E. Bl$\ddot{o}$chl, Projector augmented-wave method, Phys. Rev. B 50,17953 (1994).
  * (24) G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  * (25) G. Pizzi, V. Vitale, R. Arita, S. Bl$\ddot{u}$gel, F. Freimuth, G. G$\acute{e}$ranton, Marco Gibertini, D. Gresch, C. Johnson and T. Koretsune _et al_., Wannier90 as a community code: new features and applications, J. Phys. Condens. Matter 32, 165902 (2020).
  * (26) Q. S. Wu, S. N. Zhang, H.-F. Song, M. Troyer and A. A. Soluyanov, Wanniertools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018).
  * (27) H. J. Kulik, M. Cococcioni, D. A. Scherlis and N. Marzari, Density Functional Theory in Transition-Metal Chemistry: A Self-Consistent Hubbard U Approach, Phys. Rev. Lett. 97, 103001 (2006).

Acknowledgement
We thank Changjiang Yi for the EuCd2As2 crystal growth and Xianxin Wu for the
theoretical discussions. We thank financial support from the National Natural
Science Foundation of China (Nos. 11888101, 11974404, 11674369, 11925408,
U2032204 and 11227903), the National Key Research and Development Program of
China (Nos. 2021YFA1401800, 2017YFA0302900, 2017YFA0302901, 2017YFA0302903,
2018YFA0305600 and 2018YFA0305700), the Strategic Priority Research Program
(B) of the Chinese Academy of Sciences (No. XDB33000000 and XDB28000000), the
Youth Innovation Promotion Association of CAS (No. 2017013), the Beijing
Natural Science Foundation (No. Z180008), Beijing Municipal Science and
Technology Commission (No. Z191100007219013, Z181100004218007 and
Z191100007219011), the Research Program of Beijing Academy of Quantum
Information Sciences (No. Y18G06), the K. C. Wong Education Foundation (No.
GJTD-2018-01) and the Synergetic Extreme Condition User Facility (SECUF).

Author Contributions
Y.W., C.L., T.M.M., S.Z. and Y.L. contributed equally to this work. X.J.Z.,
Y.W. and C.L. proposed and designed the research. Y.L. and M.Y. contributed to
EuCd2As2 crystal growth. Y.W., C.L., T.M.M., C.H.Y., Y.Q.C., C.Y.S., H.L.L.,
H.C., L.Z., F.F.Z., F.Y., Z.M.W., S.J.Z., Q.J.P., Z.Y.X. and X.J.Z.
contributed to the development and maintenance of Laser ARPES systems. S.Z.,
L.Q.Z., H.M.W. and T.X. contributed to the band structure calculations and
theoretical discussions. H.B.D., Y.K.S., C.J.Z., H.Q.M. and S.H.P. contributed
to STM measurements. Y.W., C.L. and T.M.M. carried out the ARPES experiment
with the assistance from C.H.Y., Y.Q.C., C.Y.S., H.L.L. and H.C.. Y.W., C.L.,
T.M.M. and X.J.Z. analyzed the data. Y.W., C.L. and X.J.Z. wrote the paper
with H.M.W. and T.X.. All authors participated in discussion and comment on
the paper.

Figure 1: Giant and reversible Fermi surface evolution with time of EuCd2As2
at 3 K. (a) The crystal structure of EuCd2As2 with a space group
$\mathrm{P\bar{3}m1}$. The $\mathrm{Cd_{2}As_{2}}$ bilayers are sandwiched in
between two adjacent Eu layers. The upper panel represents a 3-dimensional
crystal structure where the unit cell is indicated by the orange lines. The
lower panel shows the projection of the 3-dimensional crystal structure along
the c axis. The orange rhombus represents the 2-dimensional unit cell. (b)
Three-dimensional Brillouin zone (BZ) of EuCd2As2 (black solid lines) and the
corresponding two-dimensional BZ projected on the (001) plane (gray solid
lines). The high symmetry points are indicated. (c) Fermi surface of EuCd2As2
measured on the (001) surface at 3 K at different dwell times. The sample was
first cleaved at 250 K with its Fermi surface measured (first panel in (c)).
Then the sample was quickly cooled down to 3 K and the Fermi surface was
measured at 3 K at different times. The laser is always on the sample surface
during the entire measurements. The Fermi surface was continuously measured up
to 81 hours and a dramatic change of the Fermi surface with time is observed.
Then the sample was warmed up to 250 K and the Fermi surface was measured at
250 K again (last panel in (c)). Detailed Fermi surface evolution with time is
shown in Fig. S1a in Supplementary MaterialsSM . The polarization of the laser
light is marked by a double arrow in the first panel. (d) Schematic Fermi
surface of several representative panels in (c) during the evolution process.
When the dwell time is longer than 24 hours (last two panels in (d)), some
parts of the outer Fermi surface sheets are out of the detection window and
are marked by dashed lines. Figure 2: Giant and reversible band structure
evolution with time of EuCd2As2 at 3 K. (a) Band structure of EuCd2As2
measured at 3 K at different dwell times. The bands are obtained from the same
measurements in Fig. 1c. The location of the momentum cut is marked in (b) by
a red line. These are the second derivative images with respect to the
momentum. The observed bands are labeled by $\alpha$, $\beta$ , $\gamma$,
$\delta$ and B1-B8. A significant change of the band structure is observed at
different measurement times. It was found that the band structure at 250 K
after cooling at 3 K (last panel in (a)) is similar to the one before cooling
(first panel in (a)). (b-c) Fermi surface (b) and constant energy contour at
the binding energy (EB) of 0.1 eV (c) after the dwell time of 24 hours at 3 K.
Detailed band structure evolution with time is shown in Fig. S1b in
Supplementary MaterialsSM . (d-e) Two-dimensional image showing the momentum
distribution curves (MDCs) at different measurement times. The MDCs are
obtained from (a) and Fig. S1 at the Fermi level (d) and the binding energy of
EB=0.1 eV (e). The observed bands are labeled by $\alpha$, $\beta$ and B1-B8.
(f) Photoemission spectra (energy distribution curves, EDCs) measured at 3 K
at the $\bar{\mathrm{\Gamma}}$ point after different dwell times. On each EDC,
the energy position of the $\beta$ band (first panel in (a)) is marked by an
empty circle.  Figure 3: Temperature-dependent electronic structure of the
modified EuCd2As2. (a) Fermi surface of the modified EuCd2As2 measured at
different temperatures. The sample was first kept at 30 K for a long time to
make its electronic structure nearly fully modified. Then the sample was
cooled down to different temperatures of 25 K, 20 K, 15 K, 9.3 K and 2 K. At
each temperature, the sample stayed for 3 hours when the ARPES measurement was
carried out. The polarization of the laser light is marked by a double arrow
in the last panel. (b) Corresponding band structures measured at different
temperatures. The location of the momentum cut is marked in the last panel of
(a) by a red line. These are the second derivative images with respect to the
momentum. Another independent and detailed measurement results over a wide
temperature range (3$\sim$250 K) are shown in Fig. S5 in Supplementary
MaterialsSM . The observed bands are labeled by B1-B8, $\beta$, $\gamma$ and
$\delta$ in the first and last panels. (c) Two-dimensional image showing the
second-derivative MDCs at different temperatures. The MDCs are obtained from
Fig. S5 at the binding energy of EB=0.025 eV. The observed bands are labeled
by $\alpha$, $\beta$ and B1-B6. The temperature evolution can be divided into
three regions: 3$\sim$30 K (I), 30$\sim$135 K (II) and 135$\sim$250 K (III).
(d) Magnetic susceptibility of EuCd2As2 measured with the magnetic field
parallel (blue line) and perpendicular (orange line) to the (001) plane. To
highlight the magnetic susceptibility at high temperature, the original data
(blue line) is multiplied 100 times and is shown in (d) by a purple line. (e)
EDCs measured at different temperatures at the $\bar{\mathrm{\Gamma}}$ point.
On each EDC, the energy position of the $\gamma$ band and $\delta$ band is
marked by circle and asterisk , respectively.  Figure 4: Electronic structure
evolution of EuCd2As2 with time measured under different experimental
conditions (a) Fermi surface measured at 5 K after different times marked at
the upper-right corner of each panel. Each measurement took 30 minutes. The
laser was turned off between the two consecutive measurements. The vacuum in
the chamber is better than 5.0$\times$10-11 mbar. (b) Fermi surface measured
at 5 K after different times. The process continued after the measurements in
(a) but, in this case, the laser was always on the sample during the
measurements. Each measurement took 3 hours. The vacuum in the chamber
remained better than 5.0$\times$10-11 mbar. (c) After the measurements in (b),
the sample was warmed up to 250 K and the vacuum in chamber went up to a peak
pressure of 3.0$\times$10-10 mbar and then started to drop down. The Fermi
surface was measured at 250 K at a pressure of 1.0$\times$10-10 mbar. (d) The
sample was quickly cooled down to 5 K and the vacuum in the chamber became
6.0$\times$10-11 mbar and gradually recovered with time. The Fermi surface was
measured immediately after the sample was cooled down to 5 K and then measured
after different times. Each measurement took 30 minutes. The laser was turned
off between the two consecutive measurements. The vacuum in the chamber became
better than 5.0$\times$10-11 mbar at the end of the measurements. (e) Fermi
surface measured at 5 K after different times. The process continued after the
measurements in (d) but, in this case, the laser was always on the sample
during the measurements. Each measurement took 3 hours. The vacuum in the
chamber kept better than 5.0$\times$10-11 mbar. (f-j) The corresponding band
structures measured in (a-e). The location of the momentum cut is marked by a
red line in (a). These are the second derivative images with respect to
momentum.  Figure 5: Comparison of the measured electronic structure of
EuCd2As2 with the band structure calculations. (a) Fermi surface of EuCd2As2
measured at 250 K (the same data as the first panel of Fig. 1c). The
corresponding band structures measured along
$\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{M}}$ and
$\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{K}}$ directions are shown in (b) and
(c), respectively. (d-f) Same as (a-c) but for the sample immediately measured
at 3 K after quickly cooling down from 250 K (the same data as the second
panel of Fig. 1c). (g-i) Same as (a-c) but for the sample that is cooled at 3
K for 9 hours (the same data as the fifth panel of Fig. 1c). (j-l) Same as
(a-c) but for the sample measured at 2 K after staying for a significantly
long time (the same data as the first panel of Fig. 3a). (m-o) Calculated
Fermi surface of EuCd2As2 in the paramagnetic state (PM) (m) and the
corresponding band structures along $\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{M}}$
(n) and $\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{K}}$ (o) directions. (p)
Calculated constant energy contour at the binding energy of 0.3 eV for
EuCd2As2 in the paramagnetic state. The corresponding band structures along
$\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{M}}$ and
$\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{K}}$ directions at high binding energies
of [0.25, 0.72] eV are shown in (q) and (r), respectively. (s-u) Same as (m-o)
but for EuCd2As2 in the ferromagnetic state with the in-plane magnetization
(FM-x) (for the details refer to Fig. S11 and Fig. S12 in Supplementary
MaterialsSM ). (v) Calculated constant energy contour at the binding energy of
0.39 eV for EuCd2As2 in the ferromagnetic state with the in-plane
magnetization (FM-x). The corresponding band structures along
$\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{M}}$ and
$\bar{\mathrm{\Gamma}}$-$\bar{\mathrm{K}}$ directions at high binding energies
of [0.34, 0.81] eV are shown in (w) and (x), respectively.


# SignSpeak: Open-Source Time Series Classification for ASL Translation

Aditya Makkar*
Cheriton School of Computer Science
University of Waterloo
Waterloo, ON
<EMAIL_ADDRESS>
&Divya Makkar*
Cheriton School of Computer Science
University of Waterloo
Waterloo, ON
<EMAIL_ADDRESS>
&Aarav Patel
Faculty of Engineering
University of Waterloo
Waterloo, ON
<EMAIL_ADDRESS>
&Liam Hebert
Cheriton School of Computer Science
University of Waterloo
Waterloo, ON
<EMAIL_ADDRESS>
All authors contributed equally

###### Abstract

The lack of fluency in sign language remains a barrier to seamless
communication for hearing and speech-impaired communities. In this work, we
propose a low-cost, real-time ASL-to-speech translation glove and an
exhaustive training dataset of sign language patterns. We then benchmarked
this dataset with supervised learning models, such as LSTMs, GRUs and
Transformers, where our best model achieved 92% accuracy. The SignSpeak
dataset has 7200 samples encompassing 36 classes (A-Z, 1-10) and aims to
capture realistic signing patterns by using five low-cost flex sensors to
measure finger positions at each time step at 36 Hz. Our open-source dataset,
models and glove designs, provide an accurate and efficient ASL translator
while maintaining cost-effectiveness, establishing a framework for future work
to build on.

##  1 Introduction

American Sign Language (ASL) is the most prominent sign language in North
America [1], yet as of 2021, only 0.15% of Americans are fluent in it [2].
This low figure causes significant challenges for hearing and speech-impaired
individuals, including limited access to education, opportunities and
essential services, leading to isolation and depression[3].

To address these barriers, prior work using optical-based methods has shown
strong results in translating images of ASL gestures to speech; however, they
are limited in real-world applicability[4, 5]. CNN and vision-based
transformer models necessitate using a camera pointed at a user’s hands while
signing, which is impractical in many contexts. Additionally, the use of
cameras also presents a privacy risk by capturing the user and surrounding
individuals while requiring considerable computing resources as frames must be
sent to a server. This is infeasible and limits the scope of optical-based ASL
translation within a real-world context.

Sensor-based models using embedded devices have been introduced to treat ASL
as a time-series multi-label classification problem to address the limitations
of optical systems. However, many of these datasets are private [6, 7] and
have not been trained on a well-practiced sign-based language such as ASL [8],
limiting their applicability. To address this, we introduce SignSpeak, an
open-source ASL dataset comprising of 7200 recordings of 36 classes. Our
dataset was recorded using five low-cost flex sensors, one for each finger,
for all letters and numbers in ASL. The scale of our dataset enables
researchers to test novel models on a dataset collected in an environment
aimed at real-world feasibility to progress ASL-to-speech efforts. We
extensively benchmarked various methods on this dataset, with our best result
achieving 92% categorical accuracy, which matches or exceeds previous ASL
time-series classification work [9].111 The GitHub codebase and dataset are
available at
https://github.com/adityamakkar000/ASL-Sign-Research
and https://doi.org/10.7910/DVN/ODY7GH respectively.

##  2 Related work

Previous work using a glove-based apparatus involves sensory devices such as
flex sensors and inertial measurement units (IMUs). Amin et al. [6] utilize
flex sensor gloves to capture 37 hand gestures (numbers 0-10 and letters A-Z)
using MLPs achieving 97.6% accuracy. However, a fundamental flaw limits real-
world applicability as the measurements are static and recorded at only one
point during the gesture. This fails to account for ASL’s dynamic nature since
each sign is a sequence of motions that must be continuously measured.
Furthermore, the dataset is closed-source, prohibiting others from building on
it.

Lee et al. [7] developed a glove taking continuous measurements of 6 inertial
measurement units, including an accelerometer, gyroscope, and magnetometer.
They report a 99.87% accuracy; however, this study presents a drawback: each
input is 10-15 seconds long and impractical for real-world signing, which is
performed at a significantly faster rate of 4 syllables per second [10].
Similar to the previous work, the dataset was not released publicly.

Tan et al. [5] developed a 28-sensor glove which recorded 63 data channels to
train an LSTM model.111Certain sensors measured multiple data channels. See
the referenced paper for more details. Multiple data channels and sensory
equipment significantly increase the glove’s cost, decreasing its
affordability. In addition, less sensory equipment can produce similar results
in commercial use. Králik and Šuppa [8] utilized a transformer architecture to
achieve over 99% accuracy on a synthetic glove-collected gesture dataset,
preventing its applicability to the ASL community.

We differ from previous work by introducing an open-source ASL dataset
measuring 5 flex sensor channels. It includes 200 samples for all alphanumeric
classes, allowing for a cost-effective and resource-efficient glove with broad
applicability for the ASL community.

##  3 Methodology

Figure 1: Circuit of data collection glove.

###  3.1 Data collection

For this study, a glove was constructed with five parallel flex sensors on
each finger in series with a 10,000$\Omega$ resistor. 5$V$ were applied and
measured across each sensor with an Arduino MEGA 2560. We recorded each
feature within the standard Arduino 10-bit range of [0, 1023]. Each gesture
was recorded at 36 Hz while ensuring that the sum of all flex sensor
measurements was below 5000 or 24.4$V$. This value was experimentally
determined, and indicates that the fingers were flexed (the sign being
performed), allowing for intentional data collection. We retain all gesture
recordings between 1.38 and 2.22 seconds (50-time to 80-time) steps to ensure
that accidental gestures were not added and that the gestures reflect
realistic signing patterns.

###  3.2 Model architecture

Each gesture recording contains $C=5$ channels and has a maximum time
dimension of $T=79$ with all input features 0-padded to ensure a consistent
batch size. We benchmarked RNN and Transformer-based time series models on the
SignSpeak dataset. In particular, we evaluated a 2-layer LSTM[11] and a
2-layer GRU[12] model, where the output from the last cell was fed into a
2-layer MLP Softmax classification layer. We then applied a dropout layer to
reduce overfitting[13].

$\mathbf{y_{0}}=\text{LSTM}(\mathbf{x})$ (1)
$\mathbf{y_{\textbf{output}}}=\text{SOFTMAX}(\text{MLP}(\text{MLP}(\text{LSTM}(\mathbf{y_{0}})^{(T)})))$
(2)

For the Stacked LSTM model, refer to eq. 1 and 2 where
$\mathbf{x}\in\mathbb{R}^{T\times C}\text{and
}\mathbf{y_{output}^{\top}\in\mathbb{R}^{\text{classes}}}$. Toro-Ossaba et al.
[9] presented a dense-LSTM network for EMG-ASL classification, showing that a
MLP projection before the RNN could achieve state-of-the-art results. This
work used a 2-layer MLP Softmax classifier following the dense-RNN unit.

$\mathbf{y_{0}}=[MLP(\mathbf{x}^{(0)}),MLP(\mathbf{x}^{(1)}),\ldots,MLP(\mathbf{x}^{(T)})]$
(3)
$\mathbf{y_{\textbf{output}}}=\text{SOFTMAX}(\text{MLP}(\text{MLP}(\text{LSTM}(\mathbf{y_{0}})^{(T)})))$
(4)

Eq. 3 and 4 describe the dense-LSTM. For a dense-stacked RNN, eq. 4 is
modified by composing the RNN function with itself for the input
$\mathbf{y_{0}}$. Each RNN gate used a Sigmoid activation, while MLPs used a
Tanh activation. The hidden size of the RNN cells was $h_{\text{RNN}}=64$, and
the dense and/or output MLP was $h_{\text{MLP}}=128$.

In recent literature, transformers have matched or exceeded SOTA benchmarks in
time-series classification. Králik and Šuppa [8] WaveGlove Encoder, based on
Transformers [14], have surpassed previous SOTA architectures on this task
[8]. Inspired by this architecture, we benchmark a slightly modified version
of WaveGlove on SignSpeak, adding a classification token ([CLS]) to the start
of the input as done with BERT[15]. The input is passed through a learnable
embedding and positional embedding table with the projected input being fed
into an Encoder [14] with layer normalization before the self-attention and
MLPs, as described by Dosovitskiy et al. [16]. The input format
$\mathbf{x}\in\mathbb{R}^{T\times C}$ represents 5-flex sensor channels across
time $T$, before being projected into a dimension $D=32$ with the sequential
nature encoded by the positional embedding. We utilized the GELU activation
function[17] and the number of layers was $L=5$. All Encoder and RNN
parameters were found through a Cartesian product hyperparameter sweep.

$\mathbf{y_{0}}=[\mathbf{x}^{\text{class}},\mathbf{x}\mathbf{E}_{\text{emb}}]+\mathbf{E}_{\text{pos\\_emb}}$
(5) $\mathbf{y_{l}}=\text{Encoder}(\mathbf{y_{l}}),\quad\text{where
}l=1,2,\ldots,L$ (6)
$\mathbf{y_{\text{output}}}=\text{SOFTMAX}(\text{MLP}(\text{LN}(\mathbf{y_{L}}^{(0)}))$
(7)

The encoder is described by eq.(5) - (7), where
$\mathbf{E}_{\text{emb}}\in\mathbb{R}^{C\times D},\
\mathbf{E}_{\text{pos\\_emb}}\in\mathbb{R}^{(T+1)\times D}$.

##  4 Results

Table 1: Model description and results on accuracy, F1-score, $\sigma_{accuracy}$ and $\sigma_{F1Score}$ Model | Parameters | Categorical Accuracy | F1Score | $\sigma_{accuracy}$ | $\sigma_{F1Score}$
---|---|---|---|---|---
Dense LSTM | 63K | 0.8348 | 0.8301 | 0.0110 | 0.0100
Dense GRU | 51K | 0.6692 | 0.6574 | 0.0565 | 0.0631
Stacked LSTM | 64K | 0.9167 | 0.9164 | 0.0067 | 0.0068
Stacked GRU | 51K | 0.9221 | 0.9218 | 0.0109 | 0.0110
Dense Stacked LSTM | 96K | 0.8876 | 0.8873 | 0.0194 | 0.0196
Dense Stacked GRU | 76K | 0.9192 | 0.9188 | 0.0079 | 0.0079
Encoder | 67K | 0.9136 | 0.8873 | 0.0078 | 0.0195

All models were trained with AdamW [18], with $B_{1}=0.9,B_{2}=0.999$, a
weight decay of 0.01, and a plateau learning rate decay on validation loss
with a patience of 20 epochs of 0.5 starting from 0.001 until a minimum
learning rate of 0.0001. RNNs were trained with a batch size of 64 for 15
minutes on an M2, and the Encoder was trained with a batch size of 256 for 15
minutes on a T4 GPU. All models used a 0.2 dropout probability. The metrics
used to evaluate all models were categorical accuracy and the F1-Score. We
utilized a stratified 5-fold validation and reported the standard deviation
and average result of the held-out folds.

The results in Table 1 indicate methods on private datasets do not generalize
to the SignSpeak dataset. This may be due to a reduction in the number of data
channels. This can be seen with a model such as the Transformer where a lack
of data channels reduces its performance. In particular, we found that simple
models such as a stacked GRU perform the best, whereas models such as a dense
LSTM proposed by Toro-Ossaba et al. [9] do not achieve near state-of-the-art
results. We believe that the potential for Transformer-based architectures can
be unlocked with more training data, which can then be further fine-tuned on
the SignSpeak dataset. Our leading RNN and Encoder models still maintain above
99.5% traditional accuracy, demonstrating performance on par with previous
studies.

Figure 2: Confusion matrix for the Encoder classes of ’A’ to ’M’ to highlight
the poorest performance.

Additionally, the Transformer architecture has the largest difference between
F1-score and accuracy, indicating a bias towards certain classes. Figure 2
displays the confusion matrix and it is evident the low accuracy is due to
specific classes such as ’E’ and ’L’. Specifically, the Encoder incorrectly
predicts ’L’ 36% of the time when the actual label is ’E’. Additionally in
ASL, these letters do not share the same features; ’E’ is very similar to a
letter such as ’A’. This indicates the model’s over-predicts between certain
classes and could be outlier patterns in the dataset. Analyzing this class
with stacked GRU and LSTM models, they predict ’L’ instead of ’E’ 16% and 13%
of the time, respectively. This indicates it is a learned bias of an Encoder
model but over-fitting is still present in all models.

##  5 Future work

The models presented in this study only required a moderate amount of
computing power to achieve 92% accuracy. In the future, leveraging more
powerful computing resources can enable the implementation of larger-scale
architectures to further enhance performance. Additionally, the gestures
chosen in this dataset (alphanumeric classes) reflect an extremely limited
subsection of real-world ASL; thus, future work is aimed at expanding the
dataset by collecting data and creating classes for phrases and actions that
resemble daily communication making the product viable for commercial use.
Lastly, while our measurements were recorded at 36 Hz, which is slower than
average ASL signing rates, we anticipate that using an improved MCU will allow
us to increase this frequency to 200 Hz, aligning with more realistic signing
speeds [10]. These advancements will expand on our existing research and
contribute to a more refined product that can facilitate the integration of
hearing and speech-impaired individuals into society.

##  6 Conclusion

In this study, we presented SignSpeak, an open-source dataset collected using
a custom low-cost glove architecture benchmarked on time-series classification
models to mimic real-time ASL translation. We found that a stacked GRU
achieves the strongest results on categorical accuracy. SignSpeak benefits
speech and hearing-impaired communities by providing a way to benchmark models
on a universal dataset. Our work on Signspeak can provide a foundation for
researchers to build upon our open-source dataset, leveraging supervised
learning techniques to deliver assistive and accessible technology to
communities in need.

## References

  * National Institute on Deafness and Other Communication Disorders [2021] National Institute on Deafness and Other Communication Disorders.  American sign language, 2021.  https://www.nidcd.nih.gov/health/american-sign-language#:~:text=ASL%20is%20expressed%20by%20movements,some%20hearing%20people%20as%20well.
  * Kumar et al. [2021] Rakesh Kumar, Vishal Goyal, and Lalit Goyal.  Comparative analysis of automatic sign language generation systems.  _Journal of Scientific Research_ , 65(5):226–235, 2021.
  * Verge [2023] Janine Verge.  _Reducing Barriers for People Living with Hearing Loss During In-Person Meetings_.  Canadian Audiology, 2023.  URL https://canadianaudiology.ca/wp-content/uploads/2023/04/CAA_Stay-Connected_In-Person_Booklet_01.pdf.
  * Abdullah et al. [2023] Atesam Abdullah, Nisar Ali, Raja Hashim Ali, Zain Ul Abideen, Ali Zeeshan Ijaz, and Abdul Bais.  American sign language character recognition using convolutional neural networks.  In _2023 IEEE Canadian Conference on Electrical and Computer Engineering (CCECE)_ , pages 165–169, 2023.  doi: 10.1109/CCECE58730.2023.10288799.
  * Tan et al. [2023] Chun Keat Tan, Kian Ming Lim, Roy Kwang Yang Chang, Chin Poo Lee, and Ali Alqahtani.  Hgr-vit: Hand gesture recognition with vision transformer.  _Sensors_ , 23(12), 2023.  ISSN 1424-8220.  doi: 10.3390/s23125555.  URL https://www.mdpi.com/1424-8220/23/12/5555.
  * Amin et al. [2023] Muhammad Saad Amin, Syed Tahir Hussain Rizvi, Alessandro Mazzei, and Luca Anselma.  Assistive data glove for isolated static postures recognition in american sign language using neural network.  _Electronics_ , 12(8), 2023.  ISSN 2079-9292.  doi: 10.3390/electronics12081904.  URL https://www.mdpi.com/2079-9292/12/8/1904.
  * Lee et al. [2020] Boon Giin Lee, Teak-Wei Chong, and Wan-Young Chung.  Sensor fusion of motion-based sign language interpretation with deep learning.  _Sensors_ , 20(21), 2020.  ISSN 1424-8220.  doi: 10.3390/s20216256.  URL https://www.mdpi.com/1424-8220/20/21/6256.
  * Králik and Šuppa [2021] Matej Králik and Marek Šuppa.  Waveglove: Transformer-based hand gesture recognition using multiple inertial sensors.  In _2021 29th European Signal Processing Conference (EUSIPCO)_ , pages 1576–1580, 2021.  doi: 10.23919/EUSIPCO54536.2021.9616000.
  * Toro-Ossaba et al. [2022] Alejandro Toro-Ossaba, Juan Jaramillo-Tigreros, Juan Tejada, Alejandro Peña, Alexandro López-González, and Rui Alexandre Castanho.  Lstm recurrent neural network for hand gesture recognition using emg signals.  _Applied Sciences_ , 12, 09 2022.  doi: 10.3390/app12199700.
  * Wilbur [2009] Ronnie B. Wilbur.  Effects of varying rate of signing on asl manual signs and nonmanual markers.  _Language and Speech_ , 52(2–3):245–285, Jun 2009.  doi: 10.1177/0023830909103174.
  * Hochreiter and Schmidhuber [1997] Sepp Hochreiter and Jürgen Schmidhuber.  Long short-term memory.  _Neural Comput._ , 9(8):1735–1780, nov 1997.  ISSN 0899-7667.  doi: 10.1162/neco.1997.9.8.1735.  URL https://doi.org/10.1162/neco.1997.9.8.1735.
  * Cho et al. [2014] Kyunghyun Cho, Bart van Merrienboer, Çaglar Gülçehre, Fethi Bougares, Holger Schwenk, and Yoshua Bengio.  Learning phrase representations using RNN encoder-decoder for statistical machine translation.  _CoRR_ , abs/1406.1078, 2014.  URL http://arxiv.org/abs/1406.1078.
  * Hinton et al. [2012] Geoffrey E. Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov.  Improving neural networks by preventing co-adaptation of feature detectors.  _CoRR_ , abs/1207.0580, 2012.  URL http://arxiv.org/abs/1207.0580.
  * Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  _CoRR_ , abs/1706.03762, 2017.  URL http://arxiv.org/abs/1706.03762.
  * Devlin et al. [2018] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  BERT: pre-training of deep bidirectional transformers for language understanding.  _CoRR_ , abs/1810.04805, 2018.  URL http://arxiv.org/abs/1810.04805.
  * Dosovitskiy et al. [2020] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby.  An image is worth 16x16 words: Transformers for image recognition at scale.  _CoRR_ , abs/2010.11929, 2020.  URL https://arxiv.org/abs/2010.11929.
  * Hendrycks and Gimpel [2016] Dan Hendrycks and Kevin Gimpel.  Bridging nonlinearities and stochastic regularizers with gaussian error linear units.  _CoRR_ , abs/1606.08415, 2016.  URL http://arxiv.org/abs/1606.08415.
  * Loshchilov and Hutter [2017] Ilya Loshchilov and Frank Hutter.  Fixing weight decay regularization in adam.  _CoRR_ , abs/1711.05101, 2017.  URL http://arxiv.org/abs/1711.05101.