rightsretained 978-1-4503-6819-3/20/04 https://doi.org/10.1145/3334480.XXXXXXX

# App Limits Bar: A Progress of App Limits for Overcoming Smartphone Overuse

Yixian Wu

School of Artificial Intelligence  Southeastern University  Nanjing, 210096,
Jiangsu, China<EMAIL_ADDRESS>
(2020)

###### Abstract

Smartphone overuse has many negative effects on human beings. The function App
Limits in our phones, which belongs to behavior reinforcement strategies for
overcoming problematic smartphone overuse, constrains users by shutting them
completely out of an app after a certain period of time. While it has
effectiveness to some extent, lacking procedural detection of problematic
behavior always brings about anxiety and depression for users. We proposed App
Limits Bar (ALB) to progress the traditional App Limits, aiming to mitigate
the negative feelings about using App Limits and enhance users’ abilities to
manage their time on addictive apps. Meanwhile, we took a three-phase user
study to answer whether ALB helps users be more content with using the app
limits function and whether ALB performs better than traditional App Limits in
terms of reducing usage time and open times on addictive apps. Future work
will study the best shape of the edge screen for visualization when the front
screen faces us.

###### keywords:

Smartphone Overuse Strategies; Mobile App Addiction; App Limits

††conference: CHI’20, April 25–30, 2020, Honolulu, HI, USA

<ccs2012> <concept> <concept_id>10003120.10003121</concept_id>
<concept_desc>Human-centered computing Human computer interaction
(HCI)</concept_desc> <concept_significance>500</concept_significance>
</concept> <concept>
<concept_id>10003120.10003121.10003125.10011752</concept_id>
<concept_desc>Human-centered computing Smartphone overuse</concept_desc>
<concept_significance>300</concept_significance> </concept> <concept>
<concept_id>10003120.10003121.10003122.10003334</concept_id>
<concept_desc>Human-centered computing User studies</concept_desc>
<concept_significance>100</concept_significance> </concept> </ccs2012>

[500]Human-centered computing Human computer interaction (HCI) [300]Human-
centered computing Smartphone overuse [100]Human-centered computing User
studies

##  1 Introduction

Excessive and problematic use of smartphones, also referred to as smartphone
addiction, has potentially harmful effects. Research indicates that
musculoskeletal impairment [2], poor academic performance [12], anxiety and
depression [10] as well as poor sleep quality [5] are all negative
consequences of smartphone overuse. In recent years, some smartphones allow
users to set limited usage time to apps (_e.g._ , Huawei Digital Balance and
Iphone App Limits) in order to overcome smartphone addiction. Findings from
prior works indicate that smartphone use is reduced as a result of apps
regulating use [4]. Nevertheless, notification prompt informing users of
remaining time only at the last few minutes may frustrate users if they wasted
previous limited time on meaningless contexts of that app. That’s because it
is obviously late at this moment for them to take action to change their
behaviors. Take WeChat as an example, meaninglessly browsing friends’ updates
without consciousness occupies a lot during no notification period. Therefore,
with the arrival of notification implying that only 5 minutes are left for
use, users might feel regretted about their previous improper use of this app
and feel anxious about being refused to receive and deal with a more important
subsequent message from this app. To solve the mental pressure issue on using
the existing app limits function and to change their problematic use of
addictive apps, we propose App Limits Bar (ALB) to upgrade the traditional App
Limits.

App Limits Bar (ALB) is designed to progress the function App Limits in our
mobile phones. ALB can present usage time or remaining time for use during the
whole using process, which helps users manage their limited use time better.
Meanwhile, we propose to have the left edge of the phone to be a display
screen for ALB considering the limited screen of smartphones. In addition,
signals representing compulsive open times in the using process are also
visualized as Ding _et al._ [8] proved that compulsive open time is also a
good indicator of app addiction.

##  2 Related Work

There are three [4] categories of corrective measures for problematic
smartphone overuse: information-enhancing strategies, capacity-enhancing
strategies and behavior reinforcement strategies. The proposed ALB belongs to
the mixture of the latter two strategies.

Several information-enhancing strategies include direct warnings [3],
documentaries and public campaigns emphasizing the severity and risk of
problematic smartphone use [6] and educational programs and guidelines for the
young [7]. The limitation of information-enhancing strategies is that it
probably won’t influence those who have already fully informed about the
negative effects of smartphone overuse [11]. In addition, it is difficult for
users to quit their addictive behavior voluntarily only via information-
enhancing strategies [4].

Behavior reinforcement strategies refrain users from accessing smartphone
after a certain period of time or by making smartphone use more difficult
(_e.g._ , [7]). The existing function App Limits is designed based on this
logic. While it does restrict smartphone overuse to some extent, App Limits
still contains a few imperfections. For instance, lacking procedural detection
causes inappropriate management of available usage time, and thus brings about
regrets and depression. Inspired by the logic of capacity-enhancing strategies
which aims at reinforcing self-disciplinary and rational management abilities
[11], tools are supposed to build in features that timely detect the
problematic behavior [13] for users to make adjustments. Our proposed App
Limit Bar applies that.

In terms of progress bar design, we apply Harrison _et al._ ’s visually
augmented progress bar [9] that makes processes appear 11 % faster visually,
prompting users to feel urgent in the last several minutes for quick
preparation of withdrawing themselves from the current app.

In addition, similar work that Wang _et al._ [14] conducted compared point-of-
choice prompt with the always-on progress bar on PC for sedentary behavior
change, which has a reference influence on our study design.

##  3 Prototype

Figure 1: Appearance of App Limits Bar (ALB).

[6pc]  Left: visualization of usage time (green bar) and remaining time (blue
bar). Middle: visualization of open times (two imaginary lines). Right:
visually augmented bar.

In order to test the feasibility of our App Limits Bar (ALB), we built a
prototype device using the following components:

  * 1)

An Iphone 11 or other IOS smartphone devices that contain App Limits function.

  * 2)

A piece of external, curved and multi-touch screen connected with the phone.
The display screen is modified to half-terete shape, whose radius is 2
millimeters long, and is matched with the left edge of the phone (see Figure
1).

###  3.1 Progress Bar

To achieve ALB function, we are required to develop a progress bar on this
external and multi-touch screen, which includes the following functions:

  * 1)

Record the usage time and open time of the addictive apps. A green bar is
designed for recording the usage time while a blue bar is for the left time.
The default setting is that the green bar is on the blue one (see Figure 3).
Meanwhile, a designed imaginary line is visualization of the open time (see
Figure 3 (middle)).

  * 2)

A visually augmented bar (see Figure 3 (right)) is to be displayed when the
last 10 minutes left.

  * 3)

A horizontal black bar will be displayed when users slide down the screen
versus the bar will be canceled when users slide up (see Figure 3 and Figure
4).

  * 4)

To set blue bar is on the green bar, tap twice on the blue bar and vice versa.

##  4 Study

This user study is designed to test the effect of ALB. We are intended to
answer the following questions by this study:

  * 1)

Dose ALB make users reduce the usage time and open times cost on addictive
apps compared with no limits and traditional App Limits?

  * 2)

Dose ALB reduce the frequency of choosing "no limits today" compared with the
traditional App Limits?

  * 3)

Do users have more positive feelings about using ALB than traditional App
Limits?

Question 1 & 2 are designed to explain whether users manage their limited
usage time better than using traditional App Limits. Question 3 is designed to
figure out whether ALB reduces negative feelings about using app limits
function for users.

###  4.1 Participant Selection

We recruited 12 participants aged 18 to 22 who were college students. Of these
participants, half were male and half were female to avoid gender bias. All of
them have experienced using App Limits on their phones in order to reduce the
time spent on addictive apps.

###  4.2 Tutorials of How to use ALB

Before using ALB, we were required to assist participants to get familiar with
all the ALB functions:

Figure 2: Gesture that evokes touch on the left edge screen.

  * •

Show participants how to set a limited usage time for a certain app using IOS
App Limits function.

Figure 3: Set a horizontal black bar to time from zero.

  * •

Open the limited app and browser it for a while. We need to explain that the
green bar represents the usage time and the blue bar shows the left time.
Meanwhile, participants are supposed to be taught that in order to display a
countdown, they need to hold the phone sideways, making the left edge screen
upward (see Figure 2), and then tap the blue bar twice. If they want to
recover to show the usage time, hold the phone as mentioned before and tap the
green bar twice. In addition, participants should be informed that once they
open the app, there will be an imaginary line in the position of the usage
time progress bar.

  * •

Participants can make the progress bar time from zero instead of continuous
accumulation by holding the phone sideways, making the left edge screen
upward, and then sliding down on the edge screen with a figure to set a
horizontal black bar (see Figure 3). In terms of recovery, slide up to delete
the latest bar (see Figure 4).

  * •

When the blue bar remains last 10 minutes, this 10-minute part will be full of
all the edge screen and display final countdown (see Figure 3 (Right)). The
pure blue bar will become a visually augmented bar to make users feel urgent.
In the end, the point-of-choice prompt will display on the whole screen to
inform users that the available usage time is over. Participants can choose
either "OK" or "no limits today". If it is "no limits today", participants
will be allowed to use this app with no time limits for the rest of the day.

Figure 4: Slide up to delete the latest horizontal black bar.

###  4.3 Study Design

In order to answer the research questions above, we ran the study in three
phases, which cost three weeks in total:

The first phase is to help participants figure out which apps they are
addicted to. We provided an Iphone 11 for each participant and took away their
own phones for six days (week 1 from Monday to Saturday). Each participant had
downloaded their commonly-used apps on the provided phones ahead of the
experiment. During the first phase (week 1 from Monday to Saturday), they used
this new phone without any limit. Usage time and open time frequency per day
for each app were recorded in the built-in settings of their Iphone. When the
first phase was over, each participant was required to review their usage time
per day for each app and chose several addictive apps for themselves on the
following Sunday.

Finding the specific addictive apps for each participant, phase two and phase
three are designed to compare the effect of ALB with the traditional app limit
function.

As a preparation for the second phase, participants were asked to use the
built-in App Limit function to restrict all the addictive apps they selected
in the first phase during the following six days (week 2 from Monday to
Saturday). While the participants can set time for each addictive app
according to their own will, the restricted time per day for each addictive
app should be less than the maximum daily usage time of that app in the first
phase. Usage time per day, open time frequency per day, and how many times
they choose "no limits today" in these six days were recorded. What needs to
be emphasized is that participants cannot be informed that they are going to
be recorded the chosen frequency of "no limits today" in the experimental
days.

Prior to the last phase, we assisted all the participants to get familiar with
ALB (see _Tutorials of How to use ALB_) on Sunday of week 2. In the last phase
(week 3 from Monday to Saturday), each participant were asked to use the ALB
to restrict all the addictive apps (NOTE. The restricted usage time and apps
were the same as the second phase). Usage time per day, open time frequency
per day, and how many times they choose "no limits today" in these six days
were also recorded. In addition, that we are going to record the chosen
frequency of "no limits today" in these six days cannot be known by
participants either.

On the last Sunday, each participant was interviewed by us, answering the
following questions:

  * 1)

Do you feel less frustrated and anxious when the limited time is up during
phase three compared with phase two? (Yes or No)

  * 2)

Do you think the App Limits Bar helps you reduce meaningless usage time on
addictive apps compared with the traditional App Limits function? (Yes or No)

  * 3)

Do you think your management and self-regulation abilities on app addiction
have improved during phase three compared with phase two? (Yes or No)

###  4.4 Data Collection and Processing

For each participant, we drew the line chart of usage time per day of each app
distributed in the six days from three phases for the convenience of
comparison. Open times per day for each app were made the same statistics as
the usage time. The chosen frequencies of "no limits today" from three phases
were also compared. In addition, we also did statistics on the answers to each
interview question.

##  5 Limitation

For our design, we mainly focused on ALB’s effectiveness. Therefore, edge
screen design has not been well explored. Further work will research whether
the half-terete edge screen is able to display ALB best when the front screen
faces us compared with other possible shape design on the edge screen.

## References

  * [1]
  * [2] Abdullah M. Alsalameh, Mohammad Harisi, Muath A. Alduayji, Abdullah Almutham, and Farid M. Mahmood. 2019.  Evaluating the relationship between smartphone addiction/overuse and musculoskeletal pain among medical students at Qassim University.  Journal of Family Medicine and Primary Care 8 (2019), 2953 – 2959.
  * [3] S. Barnes, A. Pressey, and E. Scornavacca. 2019.  Mobile ubiquity: Understanding the relationship between cognitive absorption, smartphone addiction and social network services.  Comput. Hum. Behav. 90 (2019), 246–258.
  * [4] Peter André Busch and S. McCarthy. 2020.  Antecedents and consequences of problematic smartphone use: A systematic literature review of an emerging research area.  Computers in Human Behavior (2020), 106414.
  * [5] A. Chang, D. Aeschbach, J. Duffy, and C. Czeisler. 2014.  Evening use of light-emitting eReaders negatively affects sleep, circadian timing, and next-morning alertness.  Proceedings of the National Academy of Sciences 112 (2014), 1232 – 1237.
  * [6] B. Chen, F. Liu, S. Ding, X. Ying, L. Wang, and Yufeng Wen. 2017.  Gender differences in factors associated with smartphone addiction: a cross-sectional study among medical college students.  BMC Psychiatry 17 (2017).
  * [7] Kyung-Seu Cho and J. Lee. 2017.  Influence of smartphone addiction proneness of young children on problematic behaviors and emotional intelligence: Mediating self-assessment effects of parents using smartphones.  Comput. Hum. Behav. 66 (2017), 303–311.
  * [8] Xiang Ding, J. Xu, G. Chen, and Chenren Xu. 2016.  Beyond Smartphone Overuse: Identifying Addictive Mobile Apps. In CHI EA ’16.
  * [9] C. Harrison, Z. Yeo, and S. Hudson. 2010.  Faster progress bars: manipulating perceived duration with visual augmentations. In CHI.
  * [10] Joshua Harwood, J. Dooley, A. Scott, and R. Joiner. 2014.  Constantly connected - The effects of smart-devices on mental health.  Comput. Hum. Behav. 34 (2014), 267–272.
  * [11] Hyeokkoo Eric Kwon, Hyunji So, S. Han, and Wonseok Oh. 2016.  Excessive Dependence on Mobile Social Apps: A Rational Addiction Perspective.  Inf. Syst. Res. 27 (2016), 919–939.
  * [12] A. Lepp, J. Barkley, and Aryn C. Karpinski. 2014.  The relationship between cell phone use, academic performance, anxiety, and Satisfaction with Life in college students.  Comput. Hum. Behav. 31 (2014), 343–350.
  * [13] C. Shin and Anind K. Dey. 2013.  Automatically detecting problematic use of smartphones. In UbiComp ’13.
  * [14] Y. Wang and Harald Reiterer. 2019.  The Point-of-Choice Prompt or the Always-On Progress Bar?: A Pilot Study of Reminders for Prolonged Sedentary Behavior Change.  Extended Abstracts of the 2019 CHI Conference on Human Factors in Computing Systems (2019).

## References

  * [1]
  * [2] Abdullah M. Alsalameh, Mohammad Harisi, Muath A. Alduayji, Abdullah Almutham, and Farid M. Mahmood. 2019.  Evaluating the relationship between smartphone addiction/overuse and musculoskeletal pain among medical students at Qassim University.  Journal of Family Medicine and Primary Care 8 (2019), 2953 – 2959.
  * [3] S. Barnes, A. Pressey, and E. Scornavacca. 2019.  Mobile ubiquity: Understanding the relationship between cognitive absorption, smartphone addiction and social network services.  Comput. Hum. Behav. 90 (2019), 246–258.
  * [4] Peter André Busch and S. McCarthy. 2020.  Antecedents and consequences of problematic smartphone use: A systematic literature review of an emerging research area.  Computers in Human Behavior (2020), 106414.
  * [5] A. Chang, D. Aeschbach, J. Duffy, and C. Czeisler. 2014.  Evening use of light-emitting eReaders negatively affects sleep, circadian timing, and next-morning alertness.  Proceedings of the National Academy of Sciences 112 (2014), 1232 – 1237.
  * [6] B. Chen, F. Liu, S. Ding, X. Ying, L. Wang, and Yufeng Wen. 2017.  Gender differences in factors associated with smartphone addiction: a cross-sectional study among medical college students.  BMC Psychiatry 17 (2017).
  * [7] Kyung-Seu Cho and J. Lee. 2017.  Influence of smartphone addiction proneness of young children on problematic behaviors and emotional intelligence: Mediating self-assessment effects of parents using smartphones.  Comput. Hum. Behav. 66 (2017), 303–311.
  * [8] Xiang Ding, J. Xu, G. Chen, and Chenren Xu. 2016.  Beyond Smartphone Overuse: Identifying Addictive Mobile Apps. In CHI EA ’16.
  * [9] C. Harrison, Z. Yeo, and S. Hudson. 2010.  Faster progress bars: manipulating perceived duration with visual augmentations. In CHI.
  * [10] Joshua Harwood, J. Dooley, A. Scott, and R. Joiner. 2014.  Constantly connected - The effects of smart-devices on mental health.  Comput. Hum. Behav. 34 (2014), 267–272.
  * [11] Hyeokkoo Eric Kwon, Hyunji So, S. Han, and Wonseok Oh. 2016.  Excessive Dependence on Mobile Social Apps: A Rational Addiction Perspective.  Inf. Syst. Res. 27 (2016), 919–939.
  * [12] A. Lepp, J. Barkley, and Aryn C. Karpinski. 2014.  The relationship between cell phone use, academic performance, anxiety, and Satisfaction with Life in college students.  Comput. Hum. Behav. 31 (2014), 343–350.
  * [13] C. Shin and Anind K. Dey. 2013.  Automatically detecting problematic use of smartphones. In UbiComp ’13.
  * [14] Y. Wang and Harald Reiterer. 2019.  The Point-of-Choice Prompt or the Always-On Progress Bar?: A Pilot Study of Reminders for Prolonged Sedentary Behavior Change.  Extended Abstracts of the 2019 CHI Conference on Human Factors in Computing Systems (2019).


# Nonlinear Anderson localization in Toda lattices

Motohiko Ezawa  Department of Applied Physics, University of Tokyo, Hongo
7-3-1, 113-8656, Japan

###### Abstract

We study the Anderson localization in nonlinear systems by taking a nonlinear
transmission line realizing the Toda lattice. It is found that the randomness
in inductance induces the Anderson localization in the voltage propagation.
Furthermore, the nonlinearity enhances the Anderson localization. They are
understood in terms of attractive interactions to form a soliton in a
nonlinear system. Our results will be applicable in general to the Anderson
localization in nonlinear systems that have solitons.

##  I Introduction

The Anderson localization is a prominent phenomenon inherent to disordered
systemsAnderson , where the system exhibits an insulating behavior due to
disorders. It is caused by the interference of wave functions due to multiple-
scattering paths under a random potential. In the strong scattering limit, the
wave functions are completely trapped by a disordered medium. The Anderson
localization has been discussed in various systemsCutler ; Gia ; Roati ; Segev
including nonlinear systemsPiko ; Lahini . It is pointed out that the Anderson
localization is destructed due to a nonlinearity in the case of the discrete
Anderson nonlinear Schrödinger equationPiko . On the other hand, the
enhancement and delocalization of the Anderson localization are reported in
photonic latticesLahini . It is an intriguing problem to make a further
investigation how the nonlinearity affects the Anderson localization.

Electric circuits attract renewed interest in the context of topological
physicsTECNature ; ComPhys ; Hel ; Lu ; YLi ; EzawaTEC ; Research ; Zhao ;
EzawaLCR ; EzawaSkin ; Garcia ; Hofmann ; EzawaMajo ; Tjunc ; Lee  based on
the observation that a circuit Laplacian may be identified with a Hamiltonian.
Furthermore, the telegrapher equation is rewritten in the form of the
Schrödinger equation, and thus describes the behavior of a quantum walkerQWalk
; EzawaUniv ; EzawaDirac . One of the merits of electric circuits is that we
can naturally introduce nonlinear elements into circuits. Hence, electric
circuits present an ideal playground to study nonlinear physics. There are
also some studies on topological physics in nonlinear electric circuitsKot ;
Sone ; TopoToda .

The Toda lattice is a celebrated nonlinear exactly solvable model possessing
soliton solutionsToda ; Toda2 ; Toda3 . It is realized in a nonlinear
transmission line consisting of variable capacitance diodes and
inductorsHirota ; Singer ; Yemele ; Yemele2 ; Pelap05 ; Houwe ; Nakajima , as
illustrated in Fig.1(a). Recently, it is shown that the dimerized Toda lattice
has a topological phaseTopoToda  which is a nonlinear extension of the Su-
Schrieffer-Heeger model.

In this paper, we study an interplay between the Anderson localization and
nonlinear effects taking an instance of a nonlinear transmission line
realizing the Toda lattice. We introduce randomness into the inductance of the
transmission line and investigate the dynamics of the voltage propagation
starting from one node. Namely, we analyze a propagation dynamics of the
voltage by giving a delta-function type pulse at one node. The diffusion
process strongly depends on the strength of the disorder. It is found that the
voltage propagation shows a localization behavior when disorders are strong
enough, which is the nonlinear Anderson localization in the nonlinear
transmission line. Furthermore, it is found that the nonlinearity enhances the
Anderson localization.

We interpret these phenomena as caused by the presence of the attractive
nonlinear force in a nonlinear system which has a soliton. Indeed, in a
soliton system, there is an attractive force in order to prevent a soliton
from decaying. Without the attractive force, the waves with the different
momenta propagate with different velocities, resulting in the destruction of a
soliton. This attractive force maintains the wave packet starting from the
delta function even though it is not a soliton solution. Moreover, once one of
the wave components is trapped to a randomness, the other components are also
trapped, which enhances the localization in the nonlinear system. Then, it is
natural that the localization is strong when the nonlinearity is strong. Our
results will be applicable in general to the nonlinear Anderson localization
in nonlinear systems having solitons.

##  II Toda lattice

We review the Toda model. The Toda Hamiltonian is defined byToda ; Toda2

$H=\sum_{n}\frac{m}{2}\left(\frac{dy_{n}}{dt}\right)^{2}+\Phi\left(r_{n}\right),$
(1)

where $r_{n}=y_{n+1}-y_{n}$, and $\Phi\left(r_{n}\right)$ is the Toda
potential,

$\Phi\left(r_{n}\right)\equiv\frac{a}{b}e^{-br_{n}}+ar_{n}-\frac{a}{b}.$ (2)

A nonlinear differential equation follows from Eq.(1),

$m\frac{d^{2}y_{n}}{dt^{2}}=f_{n-1}-f_{n},$ (3)

where $f_{n}$ is the force,

$f_{n}=-\frac{d}{dr_{n}}\Phi\left(r_{n}\right).$ (4)

The Toda equation (3) is rewritten in the form of

$m\frac{d^{2}}{dt^{2}}\log\left(1+\frac{f_{n}}{a}\right)=b\left(f_{n+1}+f_{n-1}-2f_{n}\right),$
(5)

which is well realized by a transmission line with variable-capacitance
diodesHirota : See Fig.1(a) and Eq.(15).

The right hand side of Eq.(5) is identical to the tight-binding hopping
Hamiltonian between the nearest neighbor sites,

$H=\sum_{n}M_{nm}\left|\psi_{n}\right\rangle\left\langle\psi_{m}\right|,$ (6)

where $M_{nm}=b\left(\delta_{n-1,m}+\delta_{n+1,m}-2\delta_{nm}\right)$ with
$b$ the hopping parameter.

Figure 1: Illustration of a transmission line made of (a) nonlinear elements
realizing the inhomogeneous Toda lattice and (b) linear elements realizing the
telegrapher equation. The inductance indicated by magenta is random. Each node
is grounded via a variable-capacitance diode and a capacitor in (a) and (b),
respectively.

##  III Inhomogeneous Toda Lattice

We generalize the Toda lattice to an inhomogeneous Toda latticeTopoToda ,
because the hopping becomes node-dependent in the presence of randomness in
the transmission line. We show the transmission line corresponding to the
inhomogeneous Toda lattice in Fig.1(a), which consists of variable-capacitance
diodes and inductors. The Kirchhoff law is given by

$\displaystyle L_{n}\frac{dJ_{n}}{dt}=$ $\displaystyle v_{n}-v_{n+1},$ (7)
$\displaystyle\frac{dQ_{n}}{dt}=$ $\displaystyle J_{n-1}-J_{n},$ (8)

where $v_{n}$ is the voltage, $J_{n}$ is the current and $Q_{n}$ is the charge
at the node $n$, while $L_{n}$ is the inductance for the inductor between the
nodes $n$ and $n+1$, as illustrated in Fig.1(a). The Kirchhoff law is
summarized in the form of the second-order differential equation,

$\displaystyle\frac{d^{2}Q_{n}}{dt^{2}}=$
$\displaystyle\frac{dJ_{n-1}}{dt}-\frac{dJ_{n}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{L_{n-1}}\left(V_{n-1}-V_{n}\right)-\frac{1}{L_{n}}\left(V_{n}-V_{n+1}\right),$
(9)

where we have introduced a new variable $V_{n}$ by $v_{n}=V_{0}+V_{n}$.

The capacitance is a function of the voltage $V_{n}$ in the variable-
capacitance diode, and it is well given byNakajima

$C\left(V_{n}\right)=\frac{Q\left(V_{0}\right)}{F_{0}+V_{n}-V_{0}},$ (10)

where $F_{0}$ is a constant characteristic to the variable-capacitance diode.
Especially, we have

$C\left(V_{0}\right)=\frac{Q\left(V_{0}\right)}{F_{0}}.$ (11)

The charge is given by

$\displaystyle Q_{n}=$ $\displaystyle\int_{0}^{V_{n}}C\left(V\right)dV$
$\displaystyle=$ $\displaystyle
Q\left(V_{0}\right)\log\left[1+\frac{V_{n}}{F_{0}}\right]+\text{const.}$ (12)

A closed form of the differential equation for $V_{n}$ follows from Eqs.(9)
and (12),

$\displaystyle Q$
$\displaystyle\left(V_{0}\right)\frac{d^{2}}{dt^{2}}\log\left[1+\xi\frac{V_{n}}{V_{1}}\right]$
$\displaystyle=\frac{V_{n+1}}{L_{n}}-\left(\frac{1}{L_{n-1}}+\frac{1}{L_{n}}\right)V_{n}+\frac{V_{n-1}}{L_{n-1}},$
(13)

where we have introduced a dimensionless nonlinearity parameter

$\xi\equiv V_{1}/F_{0}.$ (14)

Small $\xi$ implies weak nonlinearity, while large $\xi$ implies strong
nonlinearity. It is an inhomogeneous generalization of the Toda equation.

Figure 2: (a) Wave pattern of the voltage $V_{n}$ at a fixed time $t$ with
$\xi=0.1$. The vertical axis is the voltage in the unit of $V_{1\text{. }}$(b)
$n_{\text{peak}}^{2}$ as a function of the nonlinearity $\xi$ at a fixed time
$t$. We have set $Q\left(V_{0}\right)=L=1$.

We first study the homogeneous case $L_{n}=L$. Eq.(13) is reduced to

$Q\left(V_{0}\right)\frac{d^{2}}{dt^{2}}\log\left[1+\xi\frac{V_{n}}{V_{1}}\right]=\frac{1}{L}\left(V_{n+1}-2V_{n}+V_{n-1}\right),$
(15)

which is the Toda equation (5) with $m=Q\left(V_{0}\right)$, $a=F_{0}$ and
$b=1/L$. We set an initial condition,

$V_{n}\left(0\right)=V_{1}\delta_{n,n_{1}},$ (16)

where the voltage is nonzero only at the node $n_{1}$.

Figure 3: Dynamics of the voltage propagation. (a1)$\sim$(d1) $\delta=0$,
(a2)$\sim$(d2) $\delta=0.4$, and (a3)$\sim$(d3) $\delta=0.8$. (a1)$\sim$(a3)
Linear wave propagation corresponding to $\xi=0$. We have set (b1)$\sim$(b3)
$\xi=0.1$, (c1)$\sim$(c3) $\xi=1$ and (d1)$\sim$(d3) $\xi=10$. The vertical
axis is the dimensionless voltage $x_{n}$.

The Toda equation (15) is reduced to a linear wave equation by taking the
limit $\xi\rightarrow 0$ in (10),

$C\left(V_{0}\right)\frac{d^{2}V_{n}}{dt^{2}}=\frac{1}{L}\left(V_{n+1}-2V_{n}+V_{n-1}\right).$
(17)

Correspondingly, a nonlinear transmission line [Fig.1(a)] is reduced to a
linear transmission line [Fig.1(b)], which we have already studied in the
context of quantum walksQWalk .

##  IV Dynamics of the voltage propagation

We investigate the time evolution of the voltage $V_{n}$ by solving the Toda
equation (15) numerically together with the initial condition (16). In
Fig.2(a), we show the wave pattern of $V_{n}$ as a function of the node $n$ at
a fixed time, where the absolute value of the voltage takes the largest value
at the wave front whose position is indicated by $n_{\text{peak}}$. In
Fig.3(a1), (b1), (c1) and (d1), we show the time evolution of the voltage
propagation for $\xi=0$, $0.1$, $1$ and $10$, respectively. It shows a
diffusive motion of a wave, where the wave front is indicated in red. It is
found to be a linear function of the time $t$, $n_{\text{peak}}\varpropto t$.
Next, we have plotted the peak position $n_{\text{peak}}^{2}$ as a function of
$\xi^{-1}$ with all other variables fixed in Fig.2(b). It is clearly a linear
function of $\xi^{-1}$, or $n_{\text{peak}}\varpropto\xi^{-1/2}$.
Consequently, it follows that

$n_{\text{peak}}\varpropto\xi^{-1/2}t\varpropto\sqrt{F_{0}}t,$ (18)

when $L$ and $Q(L_{0})$ are fixed.

We note that the ripples are confined between two wave fronts and absent
outside of the wave fronts. This resembles the light cone, where the
information propagates with a finite velocity.

A prominent feature of the wave pattern reads as follows: There are many
ripples between two wave fronts in Fig.3(a1) and (b1). In particular,
Fig.3(a1) is for the voltage propagation in the linear limit, where the delta-
function like pulse disperses quickly because its Fourier components propagate
freely. This is approximately the case provided the nonlinearity is weak as in
Fig.3(b1) or in Fig.2(a). On the other hand, the ripples are weak and the two
wave fronts look as if they are two isolated solitary waves in strong
nonlinear systems as in Fig.3(d1).

##  V Toda soliton solution

It is well known that the Toda equation (15) has an exact soliton solution. By
assuming the Toda soliton starts from $n=1$ at $t=0$, it is given byToda ;
Toda2 ; Toda3

$V_{n}=F_{0}\sinh^{2}\kappa\,\text{sech}^{2}\left(\kappa n-\beta t\right),$
(19)

where

$\beta=\pm\sqrt{\frac{F_{0}}{LQ\left(V_{0}\right)}}\sinh\kappa=\pm\frac{1}{\sqrt{LC\left(V_{0}\right)}}\sinh\kappa,$
(20)

and $\kappa$ is a constant determining the width of the soliton. In Fig.4, we
show the time evolution of the voltage $V_{n}$ subject to the Toda equation
(15) for various $\kappa$. The soliton splits into two solitons and propagate
in two directions without tails.

It follows from Eq.(19) that the position $n_{\text{Toda}}$ of the Toda
soliton moves as

$n_{\text{Toda}}=\frac{\beta}{\kappa}t=\pm\sqrt{\frac{1}{LQ\left(V_{0}\right)}}\frac{\sinh\kappa}{\kappa}\sqrt{F_{0}}t,$
(21)

which is consistent with Eq.(18).

Let us make a qualitative argument with respect to the propagations of the
wave front and the Toda soliton. First, it is found that the propagation of
the Toda soliton for $\kappa=1$ in Fig.4(a) is quite similar to that of the
wave front for $\xi=10$ in Fig.3(d1). It seems that the Toda soliton becomes
the delta-function type as $\kappa\rightarrow\infty$, while the wave front
becomes the delta-function type as $\xi\rightarrow\infty$.

Figure 4: Propagation dynamics of the Toda soliton. (a) $\kappa=1$, (b)
$\kappa=0.2$ and (c) $\kappa=0.1$. The vertical axis is the voltage $V_{n}$.
We have set $C(V_{0})=0.1$ and $L=F_{0}=1$.

Figure 5: (a1)$\sim$(a4) Time evolution of the voltage at the initial node
$n_{1}$. The vertical axis is the voltage in the unit of $V_{1}$.
(b1)$\sim$(b4) The voltage at the initial node after enough time. Red color
indicates $\delta=0$, while cyan color indicates $\delta=1$. (c1)$\sim$(d4)
100 times average of time evolution of the standard deviation $\
S\left(t\right)$ for various disorder strength $\delta$ in the unit of
$V_{1}$. (d1)$\sim$(d4) The fitted standard deviation exponent $\alpha$ as a
function of disorder strength $\delta$. (a1)$\sim$(d1) Linear wave ($\xi=0$).
We have set (a2)$\sim$(d2) $\xi=0.1$, (a3)$\sim$(d3) $\xi=1$ and
(a4)$\sim$(d4) $\xi=10$. We have set $C(V_{0})=0.1$ and $L=1$.

Indeed, when the soliton width is small enough, we may approximate Eq.(19) as

$V_{n}=2F_{0}\sinh^{2}\kappa\,\delta(\kappa n-\beta t).$ (22)

The narrow soliton corresponds to the limit $\kappa\rightarrow\infty$, which
corresponds to the limit $\xi\rightarrow\infty$ to keep the voltage finite.

We summarize our observation with respect to the voltage propagation in
Fig.3(b1)$\sim$(d1) as follows. In a soliton system, there is an attractive
force in order to form a soliton. Without the attractive force, the waves with
the different momenta propagate with different velocities, resulting in the
destruction of a soliton. This attractive force maintains the wave packet
starting from the delta function even though it is not a soliton solution.

##  VI Effects of randomness

We introduce randomness into inductors uniformly distributing from $-\delta$
to $\delta$. As a result the inductance $L$ becomes node-dependent,
$L\rightarrow L_{n}$, where

$L_{n}=L\left(1+\eta_{n}\delta\right),$ (23)

with $\eta_{n}$ being a random variable ranging from $-1$ to $1$. Typical
values are $\delta=0.05\sim 0.2$ in electric circuits.

We now solve the inhomogeneous Toda equation (13) with the use of Eq.(23). We
show the time evolution of the voltage for various $\delta$ and $\xi$ in
Fig.3. We show the results with $\delta=0.4$ in Fig.3(a2)$\sim$(d2), where
there remains a localized component at the initial node $n_{1}$ although the
wave fronts are clearly seen. The voltage is almost localized at the initial
node $n_{1}$ for $\delta=0.8$, as shown in Fig.3(a3)$\sim$(d3).

Next we show the time evolution of the voltage at the initial node $n_{1}$ for
various $\xi$ in Fig.5(a1)$\sim$(a4). The voltage rapidly decreases in the
absence of the disorder , as shown by the red curves ($\delta=0$). The voltage
can be checked to vanish after enough time, although such a limit is not
displayed in these figures. On the other hand, the voltage remains nonzero
when the disorder is introduced as indicated by cyan curves ($\delta=1$). We
plot the voltage after enough time in Fig.5(b1)$\sim$(b4). The remnant voltage
is almost proportional to the disorder strength $\delta$. We note that there
is no significant dependence on $\xi$.

We define the standard deviation of the voltage propagation by

$S\left(t\right)\equiv\sqrt{\sum_{n}\left(n-n_{1}\right)^{2}V_{n}^{2}\left(t\right)},$
(24)

where $n_{1}$ is the initial node. We show the time evolution of the standard
deviation in Fig.5(c1)$\sim$(c4). It increases almost linearly as a function
of the time. It indicates a diffusion process of the quantum-walk type. We
approximate $S\left(t\right)$ by $S\left(t\right)\propto t^{\alpha}$ as a
function of $t$. We show the exponent $\alpha$ in Fig.5(d1)$\sim$(d4). It is
almost constant $\alpha=1$ for $\delta<0.4$ for $\xi=0$ and $\delta<0.1$ for
$\xi=0.1,1,10$, while it decreases gradually as the increase of $\delta$. The
exponent $\alpha$ decreases rapidly to $\alpha\simeq 0.5$ for strong
nonlinearity $\xi$, as shown in Fig.5(d4). It indicates that the nonlinearity
enhances the Anderson localization.

##  VII Discussion

We discuss why the Anderson localization enhances in the nonlinear system with
the Toda soliton. In the linear wave equation, the waves with the different
momenta propagate independently. On the other hand, the waves do not propagate
independently in nonlinear wave equations. Especially, there is such an
interaction in a soliton system that prohibits the soliton from decaying.
Hence, there must be a strong attractive force preventing a solitary wave from
diffusing. We interpret this phenomenon as caused by the presence of the
attractive nonlinear force forming a soliton in the Toda model. Once one of
the wave components is trapped to a randomness, the other components are also
trapped, which enhances the localization in the nonlinear system. Furthermore,
the localization is stronger when the nonlinearity is stronger. Our results
will be applicable to the nonlinear Anderson localization in other soliton
systems. It is interesting to study the Anderson localization in other
nonlinear systems.

In conclusion, we have studied nonlinear Anderson localization in Toda
lattices, where we find that the nonlinearity enhances the localization. Our
results will excite further studies on the relevance of the nonlinearity to
the Anderson localization.

The author is very much grateful to M. Kawamura, S. Katsumoto and N. Nagaosa
for helpful discussions on the subject. This work is supported by the Grants-
in-Aid for Scientific Research from MEXT KAKENHI (Grants No. JP17K05490 and
No. JP18H03676). This work is also supported by CREST, JST (JPMJCR16F1 and
JPMJCR20T2).

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# Pseudo-Hermitian Dirac operator on the torus for massless fermions under the
action of external fields

Ö. Yeşiltaş  Department of Physics, Faculty of Science, Gazi University, 06500
Ankara, Turkey<EMAIL_ADDRESS>J. Furtado  Universidade Federal do
Cariri(UFCA), Av. Tenente Raimundo Rocha,
Cidade Universitária, Juazeiro do Norte, Ceará, CEP 63048-080, Brasil

###### Abstract

The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studied
for a massless fermion particle under the action of external fields. Using the
covariant approach based in general relativity, the Dirac operator stemming
from a metric related to the strain tensor is discussed within the Pseudo-
Hermitian operator theory. Furthermore, analytical solutions are obtained for
two cases, namely, constant and position-dependent Fermi velocity.

###### keywords:

Pseudo-Hermiticity , Torus , Dirac equation

##  1 Introduction

When gravity meets quantum theory, there is mutual incompatibility between
general relativity and quantum mechanics. In this sense, quantum gravity is
one of the most popular and essential topics which is being aimed to become a
working physical theory. Besides its complexities, there are fundamental
physics problems involving the interaction between an atom and the
gravitational field that can be examined with Dirac equation in a curved
spacetime where the spacetime curvature can change the phase of the wave
function and thus the curvature effect is restricted by the atomic spectrum.
In this context, among the fundamental quantum field investigations one can
highlight the studies such as the perturbations of the energy levels of an
atom in a gravitational field by Parker [1], particle creation [2], spinning
objects in a curved spacetime [3] and transformation techniques for the curved
Dirac equation in gravity into a Dirac equation in flat spacetime with the
exact solutions and scattering analysis [4, 5].

In the context of low energy physics, the importance of technological advances
can bring a new sight into Dirac’s theory and its symmetries. The growing
interest in two-dimensional materials such graphene is drawing more attention
to $(2+1)$ dimensional physics . In this point of view, the unique properties
of graphene, which is an atomic honeycomb lattice made of carbon atoms, has
opened a way in a wide spectrum of applications ranging from electronics to
optics and nanotechnology since its discovery [7]. A single layer graphene
presents no gap in the conductance band so that an electron in its surface is
governed by a linear dispersion relation, behaving as a relativistic massless
particle described by Dirac equation [8].

The topology of graphene requires $(2+1)$ dimensional Dirac equation which
allows the study curvature effects in the lattice [9] and it is pointed out
that curvature of the graphene changes the electron density of the states
[10]. Inherently, different geometries can bring different curvature effects
in the lattice and curvature can alter the electron density. Therefore, the
possibility of constructing new electronic devices based on curved graphene
structures has motivated the study of graphene in several curved surfaces,
such as Möbius-strip [11], ripples [12], corrugated surfaces [13], catenoid
[14], among others. The intrinsic curvature and strain effects are discussed
through $(2+1)$ dimensional Dirac equation in [15]. Graphene nanoribbons are
discussed using the long-wave approximation in [16]. Electronic structure of a
helicoidal graphene and the scattering states can be found in [17]. It is
important to highlight here that elegant methods in quantum mechanics, such as
supersymmetry, can be used to describe curvature effects on carbon
nanostructures [18, 19].

An important geometry intensively studied in the last years is the torus
surface. Curvature effects plays an important role in toroidal geometry [20].
Toroidal carbon nanotubes, also known as carbon nanotori, appears in
nanoelectronics, quantum computing, and biosensors [21, 22]. Considering an
electron governed by the Schrödinger equation, the curvature-induced bound-
state eigenvalues and eigenfunctions were calculated for a particle
constrained to move on a torus surface in [23]. Under these same
considerations the action of external fields was addressed in [24]. A charged
spin $1/2$ particle, governed by Pauli equation, moving along a toroidal
surface was studied in [25]. Exact solutions of $(2+1)$ Dirac equation were
first obtained in [26] using supersymmetric quantum mechanics for two cases,
constant and position-dependent Fermi velocity. As far as we know, the non-
constant Fermi velocity was first studied in [28], and the consideration of
position-dependent Fermi velocity could be thought as an effective way of
treating the lattice strain. As a natural continuation of the work [26], in
this paper we study the Dirac equation in $(2+1)$ dimensions on the toroidal
surface for a massless fermion particle under the action of external fields.
Using the covariant approach based in general relativity, the Dirac operator
stemming from a metric related to the strain tensor is discussed within the
Pseudo-Hermitian operator theory. Furthermore, analytical solutions are
obtained for two cases, namely, constant and position-dependent Fermi
velocity.

This paper is organized as follows: In section 2 we discuss the Dirac equation
on the torus and we decouple the left and right sectors of the spinor in order
to obtain two Klein-Gordon-like equations for the system for two cases,
namely, constant and position-dependent Fermi velocity. In section 3 we
present the pseudo-Hermitian operators as well as the pseudo-supersymmetry for
the system in both cases. In section 4, the point canonical transformation are
used in order to obtain the solutions. The conclusions are given in section 5.

##  2 Dirac equation on the torus

Condensed matter physics has been witnessed an important evolution in the
study of massless fermions on the surface of graphene which devotes the
interest of the community of both condensed matter and relativity theorists.
The massless Dirac equation, written as

$i\gamma^{\mu}\partial_{\mu}\psi=E\psi,$ (1)

describes the dynamics of a low energy electron in a flat surface of graphene.
Here $\gamma^{\mu}$ are Dirac matrices. Moreover, the Dirac equation can be
generalized to the curved spacetime in terms of covariant derivatives,
vierbein fields and spin connection as [16]

$[i\gamma^{\mu}(\partial_{\mu}-\Gamma_{\mu}+ieA_{\mu})+\Delta]\Psi=0,$ (2)

where $\Gamma_{\mu}$ is the spin connection, $A_{\mu}$ is the gauge field,
$\Delta$ is the position-dependent energy gap [29, 30], and

$\Psi=\left(\begin{array}[]{cc}\Psi_{1}\\\ \Psi_{2}\end{array}\right)$ (3)

is the spinor which includes electron’s wave-functions near the Dirac point.
The Dirac matrices $\gamma^{\mu}$ in curved spacetime satisfy the Clifford
algebra, so that,

$\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu},$ (4)

and

$\gamma^{\mu}(x)=e^{\mu}_{i}\bar{\gamma}^{i}.$ (5)

Here $g^{\mu\nu}$ is the metric tensor and the tetrad(vierbein) frames field
is defined as

$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}$ (6)

where $\eta_{ab}=diag(1,-1,-1)$. The Greek and Roman letters correspond to
global and local indices respectively. Additionally, the metric for the torus
surface is given by

$ds^{2}=(dx^{0})^{2}-a^{2}dv^{2}-(c+a\cos v)^{2}du^{2}.$ (7)

In the metric given above, the inner radius of the torus is $c$, the outer
radius is shown by $a$, $c\neq a$ and $u,v\in[0,2\pi)$. Besides, the angle
going round the big sweep of the torus from $0$ to $2\pi$ is $u$ and the angle
going around the little waist of the torus through the same interval is $v$,
as you can see in figure (1). In (2), we can use the spin connection formula
which is

Figure 1: Torus section

$\Gamma_{\mu}=\frac{1}{2}S^{ab}e_{a}^{\nu}g_{\rho\nu}D_{\mu}e_{b}^{\rho}$ (8)

where $x^{1}=v=x,~{}~{}x^{2}=u$. Moreover, the $S^{ab}$ spin matrix and
covariant derivatives on zweibeins are

$\displaystyle S^{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[\bar{\gamma}^{a},\bar{\gamma}^{b}],$ (9)
$\displaystyle D_{\nu}e^{a}_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\nu}e_{\mu}^{a}+e_{\nu}^{b}\omega^{a}_{b\mu}-e_{\lambda}^{a}\Gamma^{\lambda}_{\nu\mu}.$
(10)

In the tetrad formalism [31], a set of $n$ independent vector fields are
defined as

$e_{a}=e^{\mu}_{a}\partial_{\mu},~{}~{}~{}~{}e^{a}=e_{\mu}^{a}dx^{\mu},$ (11)

where a vierbein is identified as the coefficients $e_{a}^{\mu}$. In [10], the
Christoffel symbols $\Gamma^{\lambda}_{\mu\sigma}$ were given in terms of the
variable $R(x)=c+a\cos x$. Then, the nonvanishing components of the
Christoffel symbols are:

$\Gamma^{2}_{12}=-\frac{a\sin
x}{R(x)},~{}~{}\Gamma^{1}_{22}=\frac{1}{a}R(x)\sin x$ (12)

hence $\Gamma_{2}$ can be obtained as,

$\Gamma_{2}=\frac{a}{2}R(x)\sin x\gamma_{1}\gamma_{2}.$ (13)

We also note that the vierbeins read as

$e^{i}_{\mu}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&a&0\\\ 0&0&R(x)\\\
\end{array}\right).$ (14)

Using (11), (13) and (2), one can obtain

$\left[\frac{i}{V_{F}}\bar{\gamma}^{0}\frac{d}{dt}+\frac{i}{a}\bar{\gamma}^{1}\left(\frac{d}{dx}-\frac{a^{2}}{2}\sin
x+ieA_{x}(x)\right)+\bar{\gamma}^{2}\left(\frac{i}{R(x)}\frac{d}{du}-\frac{e}{R(x)}A_{u}(x)\right)\right]\Psi+\Delta\Psi=0,$
(15)

where the Dirac matrices in flat spacetime $\bar{\gamma}^{i}$ are written in
terms of Pauli matrices $\sigma^{i}$ as

$\bar{\gamma}^{0}=\sigma^{3},\bar{\gamma}^{1}=-i\sigma^{2},\bar{\gamma}^{2}=-i\sigma^{1}.$
(16)

We also note that $V_{F}$ stands for the Fermi velocity. Thus, we get

$H_{D}\Psi(X)=i\frac{d}{dt}\Psi(X)$ (17)

where

$H_{D}=-\frac{1}{a}\sigma^{1}\left(\frac{d}{dx}-\frac{a^{2}}{2}\sin
x+ieA_{x}(x)\right)+\sigma^{2}\left(-\frac{1}{R(x)}\frac{d}{du}-ia\frac{e}{R(x)}A_{u}(x)\right),$
(18)

and $X=x^{\mu}$.

###  2.1 Hermiticity

Next we look at the Hermiticity of $H_{D}$ by noting that $A_{x}(x)$ and
$A_{u}(x)$ are real functions. For the stationary states of the Dirac spinor
$\Psi(X)=\exp(iEt)\Psi(x,u)$, we have

$H_{D}\Psi(x,u)=\left(\frac{E}{V_{F}}-\Delta\right)\Psi(x,u).$ (19)

It can be seen that the operator $H_{D}$ in (18) is non-Hermitian, i.e.
$H_{D}\neq H^{{\dagger}}_{D}$, and the matrix representation of $H_{D}$ can be
given by

$H_{D}=\left(\begin{array}[]{cc}0&-\frac{1}{a}\frac{d}{dx}+\frac{i}{R(x)}\frac{d}{du}+W_{1}(x)-iW_{2}(x)\\\
-\frac{1}{a}\frac{d}{dx}-\frac{i}{R(x)}\frac{d}{du}+W_{1}(x)+iW_{2}(x)&0\\\
\end{array}\right)$ (20)

where

$\displaystyle W_{1}(x)$ $\displaystyle=$ $\displaystyle\frac{a}{2}\sin
x-\frac{ie}{a}A_{x}(x)$ (21) $\displaystyle W_{2}(x)$ $\displaystyle=$
$\displaystyle-ie\frac{a}{R(x)}A_{u}(x),$ (22)

and $W_{1}(x)\neq W^{*}_{1}(x),W_{2}(x)\neq W^{*}_{2}(x)$, and ∗ stands for
the complex conjugation. In case of imaginary $A_{x}(x)$ or $A_{x}=0$, $H_{D}$
becomes Hermitian. In the next, we will look at the properties of $H_{D}$ in
more detail.

Case 1: Real vector potential components and constant Fermi velocity

We can define the two component spinors

$\Psi(x,u)=\exp(iku)\left(\begin{array}[]{cc}\Psi_{1}(x)\\\
\Psi_{2}(x)\end{array}\right).$ (23)

Hence, we obtain

$\displaystyle-\Psi^{{}^{\prime\prime}}_{1}(x)+\sigma(x)\Psi^{{}^{\prime}}_{1}(x)+\rho^{+}(x)\Psi_{1}(x)$
$\displaystyle=$ $\displaystyle\epsilon^{2}a^{2}\Psi_{1}(x)$ (24)
$\displaystyle-\Psi^{{}^{\prime\prime}}_{2}(x)+\sigma(x)\Psi^{{}^{\prime}}_{2}(x)+\rho^{-}(x)\Psi_{2}(x)$
$\displaystyle=$ $\displaystyle\epsilon^{2}a^{2}\Psi_{2}(x),$ (25)

where $\epsilon=\frac{E}{V_{F}}-\Delta$ and

$\displaystyle\sigma(x)$ $\displaystyle=$ $\displaystyle a^{2}\sin
x-2ieA_{x}(x)$ (26) $\displaystyle\rho^{+}(x)$ $\displaystyle=$
$\displaystyle-ieA^{{}^{\prime}}_{x}(x)+\left(eA_{x}(x)+i\frac{a^{2}}{2}\sin
x\right)^{2}+\frac{a^{2}}{2}\cos x+\frac{(ka+a^{2}eA_{u}(x))^{2}}{R(x)^{2}}$
$\displaystyle+\frac{aeA^{{}^{\prime}}_{u}(x)}{R(x)}-\frac{kaR^{\prime}(x)}{R(x)^{2}}-\frac{a^{2}eA_{u}(x)R^{\prime}(x)}{R(x)^{2}}$
$\displaystyle\rho^{-}(x)$ $\displaystyle=$
$\displaystyle\rho^{+}(x)|_{(k\rightarrow-k,~{}~{}A_{u}(x)\rightarrow-
A_{u}(x))}.$

Case 2: Real vector potential components and position dependent Fermi velocity

Next, we look at the same Hamiltonian in (18) but $V_{F}$ is taken as
position-dependent function. It is interesting to consider a position-
dependent Fermi velocity, i.e., $V_{F}=V_{F}(x)$, since such dependence is an
effective way of considering effects of strain. The dependence of the Fermi
velocity as a function only of $x$ lies in the symmetry of the torus on the
angular variable, so that no dependence of $u$ is expected. Using (23),

$\left[\frac{i}{V_{F}(x)}\sigma^{0}\frac{d}{dt}+\frac{i}{a}\sigma^{1}\left(\frac{d}{dx}-\frac{a^{2}}{2}\sin
x+ieA_{x}(x)\right)+\sigma^{2}\left(\frac{i}{R(x)}\frac{d}{du}-a\frac{e}{R(x)}A_{u}(x)\right)+\Delta\hat{1}\right]\Psi=0,$
(27)

which leads to a couple of differential equations by taking $\Delta=0$ for the
simplicity,

$\displaystyle\frac{E_{n}}{V_{F}(x)}\Psi_{1}(x)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{a}\frac{\partial}{\partial x}+\frac{a^{2}}{2}\sin
x-ieA_{x}(x)-\frac{k}{R(x)}-\frac{aeA_{u}(x)}{R(x)}\right)\Psi_{2}(x)$ (28)
$\displaystyle\frac{E_{n}}{V_{F}(x)}\Psi_{2}(x)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{a}\frac{\partial}{\partial x}+\frac{a^{2}}{2}\sin
x-ieA_{x}(x)-\frac{k}{R(x)}-\frac{aeA_{u}(x)}{R(x)}\right)\Psi_{1}(x).$ (29)

Hence, a couple of second order differential equations can be obtained as

$\displaystyle-
V_{F}^{2}(x)\Psi^{\prime\prime}_{1}(x)+[-V_{F}V^{\prime}_{F}+(a^{2}\sin
x-2ieA_{x}(x)]V^{2}_{F})\Psi^{\prime}_{1}(x)+[F^{+}(x)V^{2}_{F}+G^{+}(x)V_{F}(x)V^{\prime}_{F}(x)]\Psi_{1}(x)$
$\displaystyle=$ $\displaystyle a^{2}E^{2}\Psi_{1}(x)$ (30) $\displaystyle-
V_{F}^{2}(x)\Psi^{\prime\prime}_{2}(x)+[-V_{F}V^{\prime}_{F}+(a^{2}\sin
x-2ieA_{x}(x)]V^{2}_{F})\Psi^{\prime}_{2}(x)+[F^{-}(x)V^{2}_{F}+G^{-}(x)V_{F}(x)V^{\prime}_{F}(x)]\Psi_{2}(x)$
$\displaystyle=$ $\displaystyle a^{2}E^{2}\Psi_{2}(x)$ (31)

where

$\displaystyle F^{+}(x)$ $\displaystyle=$ $\displaystyle
e^{2}A^{2}_{x}(x)+\frac{a^{2}}{2}\cos x-\frac{a^{4}}{4}\sin
x+\frac{(ka+a^{2}eA_{u}(x))^{2}}{R(x)^{2}}+ie(a^{2}A_{x}\sin
x-A^{\prime}_{x})+\frac{a^{2}}{R}eA^{\prime}_{u}(x)$
$\displaystyle+\frac{aR^{\prime}(x)((k+aeA_{u}(x)))}{R(x)^{2}}$ $\displaystyle
G^{+}(x)$ $\displaystyle=$ $\displaystyle-
ieA_{x}(x)+\frac{ak}{R(x)}+\frac{a^{2}eA_{u}(x)}{R(x)}+\frac{a^{2}}{2}\sin x$

and

$F^{-}(x)=F^{+}(x)|_{k\rightarrow-k,A_{u}(x)\rightarrow-
A_{u}(x)},~{}~{}~{}~{}G^{-}(x)=G^{+}(x)|_{k\rightarrow-k,A_{u}(x)\rightarrow-
A_{u}(x)}.$ (32)

Let us discuss now the pseudo-Hermitian operators in the present context.

##  3 Pseudo-Hermitian Operators

###  3.1 Hilbert Space

Let $\mathfrak{H}$ be the Hilbert space and
$\mathcal{O}:\mathfrak{H}_{+}\rightarrow\mathfrak{H}_{-}$ a linear operator. A
class of non-Hermitian operators is the pseudo-Hermitian operators [32]
satisfying the similarity transformation given as

$\mathcal{O}^{{\dagger}}=\eta\mathcal{O}\eta^{-1}$ (33)

where $\eta$ is the invertible and linear operator. In the basic properties of
pseudo-Hermitian operators, one can remember that the eigenvalues of
$\mathcal{O}$ are either real or complex conjugate pairs and the operator
commutes with an invertible antilinear operator. If the operator is pseudo-
Hermitian, there are infinite number of $\eta$ which satisfy (33),
$\eta_{\pm}:\mathfrak{H}_{\pm}\rightarrow\mathfrak{H}_{\mp}$. Moreover, the
pseudo-adjoint of $\mathcal{O}$ is,
$\mathcal{O}^{\sharp}:\mathfrak{H}_{-}\rightarrow\mathfrak{H}_{+}$. And it is
given by

$\mathcal{O}^{\sharp}=\eta^{-1}_{+}\mathcal{O}\eta_{-}$ (34)

If $\mathfrak{H}_{+}=\mathfrak{H}_{-}$ and $\eta_{-}=\eta_{+}=\eta$ and a
quantum Hamilton operator $H$ is pseudo-Hermitian, i.e.

$H^{{\dagger}}=\eta H\eta^{-1}.$ (35)

This operator $H$ can be also factorible within the first order differential
operators $L_{1},L_{2}$:

$H=L_{1}L_{2},$ (36)

and $H_{S}$ is the partner operator which is given by

$H_{S}=L_{2}L_{{}_{1}},$ (37)

and we note that the adjoint of $H_{S}$ is $H^{{\dagger}}_{S}$. We can link
$H$ to $H^{{\dagger}}_{S}$ using the intertwining relation given below

$\bar{\eta}H=H^{{\dagger}}_{S}\bar{\eta}^{-1},$ (38)

where $\bar{\eta}=\eta_{2}\eta_{1}$. One can look at the proof of (38) in
[33]. The operators $H$, $H_{S}$, $H^{{\dagger}}_{p}$ satisfy the
relationships given below [33],

$\eta_{1}H=H_{S}\eta_{1},$ (39) $\eta_{2}H_{p}=H^{{\dagger}}_{S}\eta_{2}$ (40)

On the other hand, we may give the intertwining operator relations as below

$L_{1}H=H_{S}L_{1},~{}~{}~{}~{}L_{2}H_{S}=HL_{2}.$ (41)

We note that $L_{2}=L^{\sharp}_{1}$.

###  3.2 Pseudo-supersymmetry for the torus-Dirac system

####  3.2.1 Constant Fermi velocity

Let $\mathcal{H}_{S}$ be the Hamiltonian linked to the system given in (24)
and (25):

$\mathcal{H}_{S}=-\frac{d^{2}}{dx^{2}}+\sigma(x)\frac{d}{dx}+\rho^{(+,-)}(x).$
(42)

The Hermitian counterpart of (42) can be found by

$\mathcal{H}^{{\dagger}}_{S}\eta_{2}=\eta_{2}\mathcal{H}_{S},$ (43)

where

$\eta_{2}(x)=\frac{d}{dx}+A(x),$ (44)

here $A(x)$ is an unknown function which will be found using (43). We can show
the findings in order to satisfy (43) as

$\displaystyle A_{x}(x)$ $\displaystyle=$ $\displaystyle-\frac{ia^{2}\sin
x}{2e}$ (45) $\displaystyle A(x)$ $\displaystyle=$ $\displaystyle
C_{1}+\frac{a^{4}x}{4}-\frac{a^{2}}{2}\sin x-\frac{a^{4}\sin 2x}{8}.$ (46)

Then, the Hermitian partner of $\mathcal{H}_{S}$ can be found as,

$\mathcal{H}_{S}^{{\dagger}}=-\frac{d^{2}}{dx^{2}}+\frac{(ak+a^{2}eA_{u}(x))^{2}}{R(x)^{2}}+\frac{aeA^{\prime}_{u}(x)}{R(x)}-\frac{akR^{\prime}(x)}{R(x)^{2}}-\frac{a^{2}eA_{u}(x)R^{\prime}(x)}{R(x)^{2}}.$
(47)

One may also be interested in the exact solutions of (47). Substituting
$A_{u}(x)$ in the potential function of (47) as

$A_{u}(x)=C_{2}R(x)^{2}+C_{3}$ (48) $\displaystyle V_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{(ak+a^{2}eA_{u}(x))^{2}}{R(x)^{2}}+\frac{aeA^{\prime}_{u}(x)}{R(x)}-\frac{akR^{\prime}(x)}{R(x)^{2}}-\frac{a^{2}eA_{u}(x)R^{\prime}(x)}{R(x)^{2}}$
(49) $\displaystyle=$ $\displaystyle
a^{4}C^{2}_{2}e^{2}R(x)^{2}+2aC_{2}eR^{\prime}(x)-a^{2}C_{2}eR^{\prime}(x).$
(50)

Here, $C_{3}=-\frac{k}{ae}$ is used to get $V_{1}(x)$ in terms of $R(x)$ and
$R^{\prime}(x)$ above. Let us recall the potential model which is known as
Mathieu potential in the literature [34]

$U(x)=B^{2}\sin^{2}bx-2Bb(2c+1)\cos bx+\nu b^{2},~{}~{}B>0,~{}~{}b>0.$ (51)

Let us express $V_{1}(x)$ in the form of $U(x)$:

$V_{1}(x)=a^{4}(a^{2}+c^{2})C^{2}_{2}e^{2}+2a^{5}cC^{2}_{2}e^{2}\cos
x+eC_{2}a^{2}(a-2)\sin x-a^{6}C^{2}_{2}e^{2}\sin^{2}x.$ (52)

The term $\sin x$ doesn’t match with the model in [34]. We will find the exact
solutions of (52) in the next section. Now we continue with the pseudo-
Hermiticity properties of the problem. If we turn back to
$\mathcal{H}^{{\dagger}}_{S}$, let us factorise it and then, we obtain
$\mathcal{H}_{1}$ which is the supersymmetric partner Hamiltonian of
$\mathcal{H}_{S}$. Hence,

$\mathcal{H}^{{\dagger}}_{S}=A^{{\dagger}}A,~{}~{}A=\frac{d}{dx}+W(x),$ (53)

where

$W(x)=-\frac{i\sqrt{a-1}}{a}\sin x+\frac{i(a-2)}{2a},$ (54)

where $W(x)$ is the superpotential and the constants $C_{2}$ and $c$ shall
satisfy the following conditions

$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{a-1}}{a^{4}e}$ (55) $\displaystyle c$
$\displaystyle=$ $\displaystyle\frac{1}{2}\frac{a^{2}}{\sqrt{1-a}}.$ (56)

Since $c$ is the outer radius of the torus, it must be real number and this
brings a constraint for the inner radius $a<1$. Hence, the symmetry leads to a
condition on the torus parameters. We note that $\mathcal{H}_{1}$ can also be
obtained using $\mathcal{H}_{1}=AA^{{\dagger}}$:

$\mathcal{H}_{1}=-\frac{d^{2}}{dx^{{}^{2}}}+V_{1}(x)$ (57)

where $V_{1}$ can be obtained as

$V_{1}(x)=\frac{a-1}{a^{2}}\cos^{2}x+\frac{a-2}{a^{2}}\sqrt{a-1}\sin
x-\frac{i\sqrt{a-1}}{a}\cos x-\frac{1}{4},$ (58)

while $V(x)$ was obtained as

$V(x)=\frac{a-1}{a^{2}}\cos^{2}x+\frac{a-2}{a^{2}}\sqrt{a-1}\sin
x+\frac{i\sqrt{a-1}}{a}\cos x-\frac{1}{4}.$ (59)

Now let us obtain $\eta_{1}$ operator using

$\mathcal{H}_{1}=\eta_{1}^{-1}\mathcal{H}_{S}\eta_{1}.$ (60)

And we get,

$\eta_{1}(x)=\frac{i(2-a)}{2a}+\frac{i\sqrt{a-1}}{a}\sin x.$ (61)

We have constructed the pseudosupersymmetry of the system in (24) and (25).
Final effort shall be given in order to express $\mathcal{H}_{1}$ in the form
of (42).

####  3.2.2 position-dependent Fermi velocity

For the system given in (30) and (31), the intertwining operator is given by

$\eta_{2}(x)=\frac{d}{dx}+\frac{a^{4}}{16}+C_{2}+\frac{3}{4}a^{2}\sin
x-\frac{a^{4}}{32}\sin 2x.$ (62)

By the way, we can mention the Sturm-Liouville equation in (30) and search for
physical model. For the sake of simplicity, we will discuss the partner
Hamiltonian representations afterwards. Using (62), the Hermitian counterpart
of the Hamiltonian operator corresponding to (24) becomes

$\mathcal{H}^{{\dagger}}-\frac{d^{2}}{dx^{2}}+V_{eff}(x),~{}~{}~{}~{}\mathcal{H}^{{\dagger}}\Psi_{1}=\frac{E^{2}}{V^{2}_{F}}\Psi_{1}$
(63)

where

$\Psi_{1}(x)=\exp\left[\frac{1}{2}\int\left(2ieA_{x}(x)-a^{2}\sin x+\tan
x\right)dx\right]\phi(x),$ (64) $V_{eff}(x)=-\frac{V^{\prime
2}_{F}}{4V^{2}_{F}}+\frac{V^{\prime\prime}_{F}}{2V_{F}}+\frac{(A_{u}(x)ae+k)^{2}}{R(x)^{2}}-\frac{aeA^{\prime}_{u}(x)}{R(x)}+\frac{(k+aeA_{u}(x))R^{\prime}(x)}{R(x)^{2}}-\frac{kV^{\prime}_{F}}{R(x)V_{F}}-\frac{aeA_{u}(x)V^{\prime}_{F}}{R(x)V_{F}},$
(65)

and $A_{u}(x)=a_{2}R(x)-\frac{k}{ae}$. Using $V_{F}(x)=a\cos x$, the effective
potential $V_{eff}(x)$ becomes

$V_{eff}(x)=a^{2}a^{2}_{2}e^{2}-\frac{1}{2}+a_{2}ea\tan
x-\frac{1}{4}\tan^{2}x,$ (66)

which is known as trigonometric Rosen-Morse-II potential in the literature
[35], [36]. Let us highlighr here that the ansatz on the Fermi velocity as a
trigonometric cosine functions is a reasonable assumption due to the symmetry
of the system.

##  4  Solutions

####  4.0.1 constant Fermi velocity: approximate solutions

Let us consider the system below

$-\psi^{\prime\prime}(x)+(A+B\cos x+C\sin x+D\sin^{2}x)\psi(x)=0$ (67)

and using a point transformation $z=\exp(ix)$, it becomes

$z^{2}\psi^{\prime\prime}(z)+z\psi^{\prime}(z)+\left[A+\frac{D}{2}+\frac{B-iC}{2}z+\frac{B+iC}{2}\frac{1}{z}-\frac{D}{4}\left(z^{2}+\frac{1}{z^{2}}\right)\right]\psi(x)=0.$
(68)

Expanding the coefficient of derivative-free term near $z=1$ up to the third
order term gives

$z^{2}\psi^{\prime\prime}(z)+z\psi^{\prime}(z)+(A+B+iC(z-1)+\frac{1}{2}(B-iC-2D)(z-1)^{2})\psi(x)=0.$
(69)

Then, we apply the following transformation $z=\exp(-\alpha t)$ to get the
equation given by

$\psi^{\prime\prime}(t)+\alpha^{2}\left[A+B+iC(\exp(-\alpha
t)-1)+\frac{1}{2}(B-iC-2D)(\exp(-\alpha t)-1)^{2}\right]\psi(t)=0.$ (70)

For $C=0$, the potential is real and this also terminates the $\sin x$
function in the model. One can give the parameters of (68) in terms of
original potential parameters as

$\displaystyle A$ $\displaystyle=$ $\displaystyle
E/V_{F}-\Delta,~{}~{}~{}~{}\Delta=a^{4}(a^{2}+c^{2})C^{2}_{2},~{}~{}~{}~{}B=2ca^{5}C^{2}_{2}e^{2}$
(71) $\displaystyle C$ $\displaystyle=$ $\displaystyle
eC_{2}a^{2}(a-2),~{}~{}~{}~{}~{}D=-a^{6}C^{2}_{2}e^{2}.$ (72)

For the real eigenvalues $C$ should be pure imaginary which means that we can
select $C_{2}$ as $C_{2}\rightarrow iC_{2}$. From (55), it can be seen that
$a<1$. Now, the solutions of the model can be obtained without solving a
differential equation because the solutions are already obtained in [37, 38].
One can solve the eigenvalue equation below to get the real energies [37]:

$\left(\frac{E_{n}}{V_{F}}-\Delta\right)^{2}=-\frac{\alpha^{2}}{4}\left(2n+1-\frac{\alpha(2D-B-2C)}{\sqrt{D-\frac{B+C}{2}}}\right)^{2}.$
(73)

And wavefunctions are given by

$\psi_{1,n}(t)\sim z^{2\mu}e^{-z/\alpha}L^{4\mu}_{n}(2\gamma s),$ (74)

with $s=\alpha\sqrt{D-\frac{B+C}{2}}e^{-\alpha t}$ and
$\mu=\pm\frac{i\sqrt{E}}{2\alpha}$, $L_{a}^{c}(x)$ are the Laguerre
polynomials.

####  4.0.2 position-dependent Fermi velocity

$V_{eff}(x)$ in (87) is the element of the equation given below

$-\phi^{\prime\prime}(x)+V_{eff}(x)\phi(x)=\epsilon^{2}\phi(x).$ (75)

Using $\phi(x)=e^{-\frac{\alpha x}{2}}\phi_{1}(x)$, $z=\tan x$ and
$\phi_{1}(z)=(1+z^{2})^{\beta}\bar{\phi}_{1}(z)$, we get

$\displaystyle(1+z^{2})\bar{\phi}_{1}(z)$ $\displaystyle+$
$\displaystyle(-\alpha+2z(1+2\beta))\bar{\phi}_{1}(z)+$ (76)
$\displaystyle+\frac{1}{4(1+z^{2})}\left[(-4a_{2}e-8\alpha\beta)z+(1+4\beta)^{2}z^{2}-4C_{1}+\alpha^{2}+8\beta+4\epsilon^{2}\right]\bar{\phi}_{1}(z)=0.$

For the values of the constant $a_{2}$ and $\beta$ as

$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle-\frac{2\alpha\beta}{e}$
(77) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sqrt{-1-4C_{1}+\alpha^{2}+4\epsilon^{2}}.$ (78)

(76) becomes

$\displaystyle(1+z^{2})\bar{\phi}^{{}^{\prime\prime}}_{1}(z)$ $\displaystyle+$
$\displaystyle\left(-\alpha+z(2+\sqrt{-1-4C_{1}+\alpha^{2}+4\epsilon^{2}})\right)\bar{\phi}^{{}^{\prime}}_{1}(z)+$
(79)
$\displaystyle+\left(\epsilon^{2}+\frac{\alpha^{2}-4C_{1}}{4}+\frac{1}{2}\sqrt{-1-4C_{1}+\alpha^{2}+4\epsilon^{2}}\right)\bar{\phi}_{1}(z)=0.$

Now we can apply the new variable as $s=\frac{1-iz}{2}$ and get

$\displaystyle s(1-s)\frac{d^{2}\bar{\phi}_{1}(s)}{ds^{2}}$ $\displaystyle+$
$\displaystyle\left(-\frac{i\alpha}{2}+1+2\beta-2(1+2\beta)s\right)\frac{d\bar{\phi_{1}}}{ds}$
(80)
$\displaystyle-\left(\epsilon^{2}+\frac{\alpha^{2}-4C_{1}}{4}+\frac{1}{2}\sqrt{-1-4C_{1}+\alpha^{2}+4\epsilon^{2}}\right)\bar{\phi}_{1}(s)=0.$

(80) is the type of Hypergeometric differential equation in the literature
[39]

$s(1-s)w^{\prime\prime}(s)+[\gamma-(a+b+1)s]w^{\prime}(s)-abw(s)=0$ (81)

whose solutions are given by

$w(s)=c_{1}~{}~{}_{2}F_{1}(a,b;\gamma;s)+c_{2}s^{1-\gamma}~{}~{}_{2}F_{1}(a-\gamma+1,b-\gamma+1;2-\gamma;s).$
(82)

If we match (81) with (80), we get

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
1+2\beta-\frac{i\alpha}{2}$ (83) $\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{2}+2\beta+\frac{1}{2}\sqrt{5+16C_{1}-4\alpha^{2}+8\beta+16\beta^{2}-16\epsilon^{2}}$
(84) $\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{2}+2\beta-\frac{1}{2}\sqrt{5+16C_{1}-4\alpha^{2}+8\beta+16\beta^{2}-16\epsilon^{2}}$
(85)

Finally the solutions $\phi(x)$ become polynomials if $a=-n$ and have a form

$\phi(x)=N~{}\exp\left(-\frac{\alpha
x}{2}\right)(1+\tan^{2}x)^{\beta}~{}~{}_{2}F_{1}\left(a,b;\gamma;\frac{1-i\tan
x}{2}\right)$ (86)

And the energy eigenvalues can be given by

$\epsilon_{n}=\pm\sqrt{\frac{1}{2}(n+\mu+1)^{2}-\frac{1}{2}\frac{\nu}{(n+\mu+1)^{2}}}$
(87)

where $\mu$ and $\nu$ are the constants in terms of $\alpha,\beta,a_{2}$,
$n=0,1,..$.

##  5 Final Remarks

In this paper we have studied the Dirac equation in $(2+1)$ dimensions on the
toroidal surface for a massless fermion particle under the action of external
fields. Using the covariant approach based in general relativity, the Dirac
operator stemming from a metric related to the strain tensor is discussed
within the Pseudo-Hermitian operator theory.

We have initially obtained two coupled first-order differential equations
coupling the left and right sector of the Dirac spinor. The decoupling of
these equations renders two Klein-Gordon-like equations which were discussed
in two cases, namely, constant and position-dependent Fermi velocity.

The solution for both constant and position-dependent Fermi velocity cases
were analytically obtained. In case of constant Fermi velocity calculations,
we have obtained a condition on the inner radius $a<1$ and we have extended
the solutions of more general Mathieu potential whose solutions are given in
terms of Laguerre polynomials. In the next case, the position-dependent Fermi
velocity function is used to obtain the solutions in terms of hypergeometric
functions with a trigonometric Rosen-Morse II type potential.

The paper not only presents important properties about the dynamics of an
electron constrained to move on a torus surface under the action of external
fields but also opens up new possibilities of investigation. The thermodynamic
properties as well as electron-phonon interaction will be addressed in a
future work.

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