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Plasmons in quantum materials

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Plasmons in quantum materials

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Content Plasmons in Quantum Materials
by
Yuling Guan
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
August 2022
Copyright 2022 Yuling Guan
Dedication
Dedication to my father and mother who give me the best education in all areas.
ii
Acknowledgements
The five years PhD life is one of the most meaningful journal for me. I arrived at Los Angeles
in the summer of 2017 with expectation and nervousness. It was my first time to experience this
famous city. In my mind at that moment, PhD life will full of challenging and difficulties. Time
flies so fast and it comes to 2022. Looking back to the past five years in the end of my PhD journal,
I would appreciate it a lot to many people that helped and encouraged me. I cannot finish my
academic program without their support.
Firstly, I am grateful to my academic supervisor, Prof. Stephan Haas for supporting and guiding
me during my whole research time since the beginning of my PhD program. Stephan is a super
kind professor and full of wisdom. He always helps me start the research projects with meaningful
and accurate motivations as well as useful literatures. They give me a good start and guide me in
the beginning of the projects. During my PhD research program, I usually met with Stephan once
or twice every week. In the meeting, we discussed the research plan and the data that I calculated.
He gave me a lot of suggestions vary from how to improve the methods to what can we calculate
to verify the results. We made great progress in the discussion every week, which speed up the
research process and finally help up finish the excellent papers. Working with Stephan is very
happy and comfortable. I learned a lot from him not only for academy but also for how to be a
good researcher.
Also, I am very grateful to my co-supervisor Prof. Satish Kumar and Prof. Sven Koenig in
the Computer Science department. I started to collaborate with them at 2018 and we have worked
together in many projects. Coding takes an important role in computational condensed matter
physics and they are very professional experts in this area. In our discussion, Satish and Sven gave
iii
me meaningful directions and many useful ideas, which lead the way of how to finish the research
projects. Also, Satish helped me a lot in writing papers, we usually worked together and produced
papers with good quality.
Besides my supervisors, I also want to say thank you to other committee members of my
defense and my qualifying exam. They are Prof. Rosa Di Felice, Prof. Lorenzo Campos Venuti,
Prof. Aiichiro Nakano and Prof. Wei Wu. Their insightful comments and suggestions are very
helpful for my thesis. Also, the questions they asked helped me think deeply in my research
projects and let me understand some knowledge related.
I want to thank my collaborators and colleagues during my research projects. Dr. Zhihao
Jiang has taught and helped me a lot during the projects. He is an expert in plasmon problems
and topological insulators from one-dimension to three-dimension. From him I learned random
phase approximation, screened Coulomb interaction, collective excitations and the properties of
topological insulators with different symmetries. Ang Li is a PhD candidate in Computer Science
department, we cooperated a lot in the research programs for years and usually work together to
make the code works. It is very happy to work with him.
Further more, I feel thankful to all of the teachers for their classes. The fundamental of physics
and mathematics is really important for the research. I have learned a lot from the classes, not only
the formulas and concepts but also the deep physical principle under them.
The most important people that I want to thank are my parents. My parents give me the best
education in the world from my childhood. During my growth, my father always told me the health
is the most important thing for everyone and it will be worthless if I get many achievement but don’t
have a healthy body. Our family went to hiking or swimming every weekend and we usually had
trips to other cities in the holiday when I was at my hometown. The healthy body provides me
enough energy when I facing large workload until now. For study, my parents encouraged me to
pursue anything that I interested and told me never give up when meet difficulties. Every time
when I facing tough problems, they always willing to guide me to resolve them. Instead of telling
me how to solve the questions, they always motivate me to think by my self and explore my own
iv
ideas. I felt interested in physics when I was in middle school, my parents helped me find good
teachers as well as the literature, which benefit me a lot for many years.
In the last, I want to thank and respect all the people that are seeking the truth of physics. Each
small step makes human being understand the world better in the past thousand years.
v
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables viii
List of Figures ix
Abstract xi
Chapter 1: Introduction 1
Chapter 2: Control of Plasmons in One-Dimensional Topological Insulators via Local
Perturbations 5
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Topology in Quantum Materials . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 One-Dimensional Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . 9
2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Random Phase Approximation in Real Space Representation . . . . . . . . 12
2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Plasmonic excitations in a decorated one-dimensional metallic chain . . . . 15
2.3.2 Plasmonic excitation in the SSH model . . . . . . . . . . . . . . . . . . . 17
2.3.3 Effects of added diatomic molecules on plasmons in the SSH chain . . . . 21
2.3.4 Plasmonic excitations in the mirror-SSH model . . . . . . . . . . . . . . . 24
2.3.5 Effects of added diatomic molecules in the mirror-SSH chain . . . . . . . . 25
2.3.6 Induced field energy of the perturbed SSH model . . . . . . . . . . . . . . 29
Chapter 3: Plasmons inZ
2
Topological Insulators 30
3.1 Two-Dimensional Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Two-Dimensional Su-Schrieffer-Heeger Model . . . . . . . . . . . . . . . 30
3.1.2 Haldane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.3 Kane-Mele Model andZ
2
Topology . . . . . . . . . . . . . . . . . . . . . 31
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Momentum Space Coulomb Interaction . . . . . . . . . . . . . . . . . . . 34
3.2.2 Random Phase Approximation in Momentum Space . . . . . . . . . . . . 35
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vi
3.3.1 Plasmonic excitations in the honeycomb lattice . . . . . . . . . . . . . . . 36
3.3.2 Plasmonic excitations in the Kane-Mele model on a nano-ribbon . . . . . . 38
3.3.3 The effect to edge states by inner parameter . . . . . . . . . . . . . . . . . 41
3.3.4 Effects of a Zeeman field in the Kane-Mele model . . . . . . . . . . . . . 42
3.3.5 Real space plasmonic excitations in diamond-shaped nanoflakes . . . . . . 46
Chapter 4: FastPivot and Inverse Problem: Plasmons Design 49
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Idea of Design and Artificial Intelligent . . . . . . . . . . . . . . . . . . . 49
4.1.2 Atoms-up design in manmade nanostructure . . . . . . . . . . . . . . . . . 50
4.1.3 The prediction of plasmonic excitations . . . . . . . . . . . . . . . . . . . 53
4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Local Search Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 FastPivot Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Warm up : Atoms-up Design . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 Plasmonic excitations Design . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 5: Conclusion 65
References 67
Appendices 72
A Full Pseudo Code for Plasmons Design Problem . . . . . . . . . . . . . . . . . . . 73
Own Publications 80
vii
List of Tables
4.1 Density of State Design Problem: Effect of Size and Dimension . . . . . . . . . 60
4.2 Plasmon Design Problem of all EELSs: Effect of Size and Dimension . . . . . 63
5.1 Plasmon design problem: Physical Parameters . . . . . . . . . . . . . . . . . . 74
5.2 Plasmon design problem: Algorithmic Parameters . . . . . . . . . . . . . . . . 74
5.3 Plasmon design problem: Variable . . . . . . . . . . . . . . . . . . . . . . . . . 75
viii
List of Figures
2.1 Topology equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Illustration of SSH model and mirror SSH model . . . . . . . . . . . . . . . . 10
2.3 Energy spectrum levels of SSH model and mirror SSH model . . . . . . . . . . 12
2.4 Plasmons in decorated metallic chain . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 EELS of 1D SSH model non-trivial phase and edge modes . . . . . . . . . . . . 18
2.6 1D SSH model non-trivial phase plasmons with only topological susceptibility 19
2.7 Induced energy spectrum and edge modes of 1D SSH model topological phase. 20
2.8 Topological EELSs for 1D SSH model with perturbation at different positions 22
2.9 Dependence of plasmons on the internal hopping . . . . . . . . . . . . . . . . . 23
2.10 EELS of 1D mirror-SSH model . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.11 mirror-SSH model weak interface with perturbation . . . . . . . . . . . . . . . 26
2.12 mirror-SSH model strong interface with perturbation . . . . . . . . . . . . . . 27
2.13 Energy spectrum for 1D SSH model with external electrical field . . . . . . . . 28
3.1 Illustration of honeycomb zigzag nano-ribbon . . . . . . . . . . . . . . . . . . 32
3.2 Energy spectrum of Kane-Mele model nano-ribbon . . . . . . . . . . . . . . . 33
3.3 Energy bands and plasmon of graphene nano-ribbon . . . . . . . . . . . . . . 37
3.4 EELS of Kane-Mele model in different phases . . . . . . . . . . . . . . . . . . 39
3.5 EELS and charge density of Kane-Mele model at K=π/8L . . . . . . . . . . . 40
3.6 EELS of Kane-Mele model with different unit size . . . . . . . . . . . . . . . . 41
3.7 Energy spectrum of Kane-Mele model with Zeeman field . . . . . . . . . . . . 43
3.8 Momentum-space EELS of Kane-Mele model with Zeeman field . . . . . . . . 43
ix
3.9 EELS and charge density of small gap Kane-Mele model with Zeeman field . . 44
3.10 EELS and charge density of big gap Kane-Mele model with Zeeman field . . . 45
3.11 Structure and wave function of Kane-Mele model in nanoflake . . . . . . . . . 46
3.12 Plasmonic charge density of Kane-Mele model in nanoflake . . . . . . . . . . . 47
4.1 STM schematic and picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 DoS corresponding to configurations of atoms . . . . . . . . . . . . . . . . . . 51
4.3 Comparison of Local Search and FastPivot algorithms in DoS Problem . . . . 60
4.4 Comparison of Monte Carlo and FastPivot algorithms in plasmon Problem . . 63
x
Abstract
Plasma oscillations are collective excitations of electron arising from long-range Coulomb inter-
action and the quanta for it is called plasmons. Motivated by analyzing the stability of many-
body collective excitations of topological insulators, we use a fully quantum mechanical approach
to demonstrate control of plasmonic excitations in prototype models of topological insulators by
molecule-scale perturbations. Strongly localized surface plasmons are present in the host systems,
arising from the topologically non-trivial single-particle edge states. A numerical evaluation of the
random phase approximation (RPA) equations for the perturbed systems reveals how the positions
and the internal electronic structure of the added molecules affect the degeneracy of the locally
confined collective excitations, i.e., shifting the plasmonic energies of the host system and chang-
ing their spatial charge density profile. In particular, we identify conditions under which significant
charge transfer from the host system to the added molecules occurs. Furthermore, the induced field
energy density in the perturbed topological systems due to external electric fields is determined.
Besides the one-dimensional topological insulators, we also theoretically analyze the collec-
tive plasmon excitations in two-dimensional Z
2
topological insulators using the RPA numerical
method. In the Kane-Mele model, the quantum spin Hall (QSH) phase is in a time reversal in-
variant electronic state with a bulk gap andgapless edge states. The QSH state is a state of matter
that has a quantized spin Hall conductance and a vanishing charge Hall conductance. We con-
sider a nano-ribbon structure with zigzag edges. Here, strongly localized plasmonic excitations are
observed, with induced charge density distribution confined to the top/bottom boundaries in the
non-trivial QSH phase. We demonstrate that the time reversal symmetry and theZ
2
topological
phase can be destroyed by an Zeeman field, thus changing the plasmonic edge states. Furthermore,
xi
we demonstrate that the edge modes are observed in the real space diamond-shaped nanoflake
which have all of its boundaries zigzag edges.
As well as exploring the plasmonic excitation in different topological insulators, we also dis-
cuss the plasmon in a perspective of inverse problem. The laws of physics are usually stated using
mathematical equations, allowing us to accurately map a given physical system to its response,
for example, getting plasmon spectrum from materials. However, when building systems, we are
often faced with the inverse question: How should we design a physical system that produces a
target response, i.e., design the system which has the target plasmon spectrum? In this paper, we
present a novel algorithm, called FastPivot, for solving such inverse problems. FastPivot starts
with a system state and invokes alternating forward and backward passes through the system vari-
ables. In the forward pass, it leads the current state of the system to its response. In the backward
pass, a small amount of information is allowed to percolate from the target response back to the
system variables. Upon convergence, FastPivot produces good quality solutions efficiently. We
demonstrate the success of FastPivot on the inverse problem of placing atoms in a bounded region
using a scanning tunneling microscope to achieve target responses in the density of states(DoS)
or electron energy loss spectrum(EELS). We also compare FastPivot to Monte Carlo methods and
analyze various empirical observations.
xii
Chapter 1
Introduction
Topological insulators (TIs) are characterized by a gapped bulk energy spectrum and symmetry-
protected conducting surface states. Prominent examples of TIs include the integer quantum Hall
effect[1–7], the two-dimensional (2D) quantum Hall insulator[8–10], which can host chiral edge
currents, as well as three-dimensional (3D) TIs whose topological surface states are formed by
massless Dirac fermions [11–13]. TIs can also be realized in one spatial dimension, for example,
in the paradigmatic example of the Su-Schrieffer-Heeger (SSH) model[14]. In the topologically
non-trivial phase, the SSH chain has two localized single-electron edge states on its boundaries.
Implementations of topological wave guides based on the SSH model include quantum emitters or
superconducting qubits that are locally coupled to photonic TIs [15–18]. Furthermore, collective
excitations, such as plasmons, in the SSH chain and other TIs have been studied [19–25]. Here,
strongly localized plasmon modes were observed on the boundaries of the SSH chain, which can
be traced back to the localized single-particle edge states.
One essential property of edge states in TIs is their robustness against disorder. For example,
quantum Hall currents are immune to back-scattering from any surface impurity[26]. A topological
classification for such defects was identified, and the bulk-boundary correspondence was general-
ized, relating these topological classes to protected gapless modes at the defect. [27]. Single-
electron topological states are well protected if their symmetry is preserved. However, whether
the collective excitations in TIs are protected, completely or partially, is still an interesting and not
1
fully resolved question. Unlike single-electron states, collective excitations are correlated phenom-
ena routed in interactions[28–30]. It has recently been shown that the plasmon edge modes in the
SSH model are less robust against global hopping disorder than their constituent single-electron
edge states[24], which is mainly due to the screening effect from the bulk electronic bands [14,
31]. Impurities will occur in TIs[32–34]. In this work, we focus on the effects of local impurities
on plasmonic excitations in the SSH model and a mirrored variant model (mSSH) connecting two
SSH chains with distinct indices of topological invariant. Specifically, the local impurities here
are modeled by diatomic molecules that can be placed at any position close to the unperturbed 1D
host material. While these local inpurities are not expected to change the single-electron spectrum
of the host material drastically due to topological protection, they can bring about other phenom-
ena, such as electron tunneling and Coulomb coupling between the impurity molecule and the host
material. As discussed below, we clearly observe such effects of local impurities on the local-
ized plasmons excited at the boundaries (domain walls) of the 1D SSH (mSSH) model, which are
strongly dependent on the positions of impurity molecules. As a reference point, we also study
local impurities in proximity to a simple 1D metallic chain which hosts only extended plasmon
modes propagating in the bulk. In both cases, topological and trivial, we discuss how control of
impurity positions can be used as a tuning knob for manipulating the plasmonic excitations.
The spin Hall effect has been predicted for doped semiconductors with spin-orbit coupling[35,
36] and detected experimentally in n-GaAs and n-In
0.07
Ga
0.93
As films [37], as well as in a 2D
hole layer system[38]. Furthermore, a quantized spin Hall effect has been proposed, with degen-
erate quantum Landau levels arising from the gradient of mechanical strain, which are quantized
in units of 2
e
4π
[5]. Kane and Mele predicted that such a quantum spin Hall (QSH) state exists for
a model of graphene which consists of two copies of Haldane’s model[39], with spin degrees of
freedom[40] causing a topologically non-trivial energy band due to the spin-orbit coupling. QSH
effects can be described by using a momentum space winding number to characterize the topolog-
ical insulator with spin Hall transport carried by helical edge states[41]. The QSH state does not
violate T symmetry and indicates the absence of non-zero Chern invariants in QSH insulators.
2
One essential property ofZ
2
order topological insulators and the protected QSH phase is the
presence of gapless edge states which cross the Fermi level. Z
2
invariants describe pairs of four
spin-split bands related by time reversal and connected with the integer invariants that underlie
the integer quantum Hall effect[42]. Also, the two pairs of gapless edge states within the bulk
gap match the two zero points of the Bloch function on the complex-energy Riemann surface
(RS)[43, 44]. Following Laughlin’s and Halperin’s arguments[45, 46], these QSH edge states are
characterized by a topological index I
s
which distinguishes the difference in the winding numbers
of the spin-resolved edge states crossing the holes of the complex-energy RS[47].
While the QSH effect andZ
2
topological insulators have been studied extensively, there has
not yet been a detailed investigation of collective excitations in these systems. In this paper, we
analyze the plasmonic excitations in the Kane-Mele (KM) model, including plasmonic edge states
in ribbon structures. Here, we observe the localized collective modes only when the system is in
a topological non-trivial regime, i.e., the QSH phase. We identify these collective edge modes as
’Z
2
topological plasmons’ of the QSH phase. Furthermore, we discuss how these plasmonic edge
modes are affected by the bulk electronic states, Coulomb interactions, and by the application of
an external Zeeman field. Moreover, we investigate plasmons in a diamond nanoflake structure
where the topological edge states of the Kane-Mele model are realized as collocation excitations.
Besides plasmonic excitation, there are many laws of physics usually be stated using math-
ematical equations at different spatiotemporal scales. These formulas and equations allow us to
accurately map a given physical system to its response like we calculate the corresponding plas-
mon from Hamiltonian and structure of materials. Other examples like, the mathematical equations
that describe gravity allow us to predict the trajectories of planets in distant galaxies; and the math-
ematical equations that describe Faraday’s law of electromagnetic induction allow us to predict the
amount of energy generated by wind turbines. Similarly, at a microscopic scale, the Boltzmann
equation of statistical mechanics allows us to analyze hysteresis in magnetic materials [48, 49]; and
modern theories allow us to calculate the density of states (DoS) for a given placement of atoms
of a specific kind in a bounded region on a substrate. Despite the usefulness of mathematics in
3
“forward” reasoning, i.e., from the system state to its response, we often face the inverse question
while building systems: How should we design a physical system that produces a target response?
Such inverse problems arise at both macroscopic and microscopic scales. At macroscopic scales,
inverse problems can involve the design of circuits [50], Carnot engines [51], radio-frequency (RF)
antennas [52], or aircraft with aerodynamic optimizations [53].
In this paper, we will discuss the physics and methodologies of inverse problems at microscopic
scales. For example, as we discussed in the previous paragraph, inverse problems can involve the
placement of atoms in a bounded region using a scanning tunneling microscope (STM) to achieve
target responses in the density of states. Also, plasmonic excitations [54] is another physical re-
sponse that is meaningful to research it’s inverse problem since STM will also help. In material
science engineer, the plasmonic excitation in specific frequency or energy region for both trivial
and non-trivial materials is an important topic for exploring. The forward pass, of course, is get-
ting the plasmonic response from the materials’ properties. The development of modern devices
provides the possible way to design the material in specific condition such as placing atoms on the
specific surface, which allows us to achieve the target plasmonic excitations in experiments, i.e.
getting the excitations with purpose amplitude in the external energy that we want. The approach
of designing the man-made materials with goal plasmons will help the experiments in the way of
providing the idea guide.
4
Chapter 2
Control of Plasmons in One-Dimensional Topological Insulators
via Local Perturbations
2.1 Background
2.1.1 Topology in Quantum Materials
Topology is a concept in mathematics that connected with the properties of a geometric object that
are preserved and invariant under any continuous deformation. An example is that a square can be
deformed into a rectangle (or we can say topological equal to), and also can be deformed into a
circle. However, cannot be deformed to a shape of 8. Hence a square is topologically equivalent
to a rectangle and circle, but different from a shape 8 (Fig 2.1). In mathematics, topology has
many subfields of research, general topology that considers local properties of spaces, algebraic
topology that use algebraic objects to consider global properties, etc. Also, some other branches
in mathematics use topology for calculation, such as differential equations, dynamic systems and
Riemann geometry. Not only in mathematics, topology can also be applied in many fields like
biology system, computer science (topological data analysis) and condensed matter physics that
we will discuss in this paper.
In the field of many-body condensed matter physics, different with normal metals and insu-
lators, there exists a class of compounds that have electronic band structures that are topological
5
Figure 2.1: Topology equivalence. A square is topologically equivalent to a rectangle and circle,
but different from a shape 8.
distinct which we can call them topological quantum materials[55]. The topological non-trivial
electronic structure and states of these materials rise many interesting topics such as quantum Hall
effects, spin-momentum locking and manifesting as high mobility. There are different quantum
materials have the topological non-trivial phase such as topological insulators, Weyl semi-metals,
twisted graphene and related two-dimensional Chern magnetic insulators. In this and following
chapter, we will focus on both single particle and many-body collective properties of topological
insulators from one-dimensional materials to two-dimensional materials in calculation.
Both fully gapped and gapless topological insulators can be classified in terms of different
symmetries. In this paragraph, we review how different symmetries are implemented in fermionic
system with annihilation and creation operators
n
ˆ ψ
I
, ˆ ψ
†
I
o
I=1,...,N
that satisfy the canonical anti-
commutation
n
ˆ ψ
I
, ˆ ψ
†
J
o
=δ
IJ
. The second quantized Hamiltonian if fermions can be written as:
ˆ
H = ˆ ψ
†
I
H
IJ
ˆ ψ
J
≡ ˆ ψ
†
H ˆ ψ (2.1)
where the expression of H is under Einstein’s convention of summation. and H
IJ
is the matrix
for system. Since any symmetry transformation can be represented by an operator on the Hilbert
space, we can denote the operator
ˆ
U and the example of line transformation is
ˆ ψ
I
→ ˆ ψ
′
I
:=
ˆ
U ˆ ψ
I
ˆ
U
− 1
= U
J
I
ˆ ψ
J
(2.2)
6
In this section we focus on the non-spatial symmetries which does not act on the spatial part of the
collective indices. Here are several classical symmetries[56]:
1. Time-reversal symmetry: Time reversal operator
ˆ
T is an anti-unitary operator that acts on
fermion creation and annihilation operators.
ˆ
T ˆ ψ
I
ˆ
T
− 1
=(U
T
)
J
I
ˆ ψ
J
,
ˆ
T i
ˆ
T
− 1
=− i (2.3)
So, we say a system is time-reversal if it meets two constraints: (a)
ˆ
T
ˆ
H
ˆ
T
− 1
=
ˆ
H; (b)T
preserves the canonical anticommutator.
2. Particle-hole symmetry: Particle hole operator
ˆ
C is a unitary transformation as:
ˆ
C ˆ ψ
I
ˆ
C
− 1
=(U
∗ C
)
J
I
ˆ ψ
†
J
(2.4)
It flips the sign of U(1) charge ˆ c
ˆ
Q ˆ c
− 1
=− ˆ
Q and leads to U
†
C
H
T
U
C
=− H. Also, in the
particle-hole symmetric system
ˆ
H,
ˆ
C
ˆ
H
ˆ
C
− 1
=
ˆ
H.
3. Chiral symmetry: Chiral symmetry is the combination of
ˆ
T and
ˆ
C , which means one can
have a situation where both
ˆ
T and
ˆ
C are broken but the combination is satisfied
ˆ
S =
ˆ
T · ˆ
C .
When we apply
ˆ
S on fermion we will have
ˆ
S ˆ ψ
I
ˆ
S
− 1
=(U
C
U
T
)
J
I
ˆ ψ
†
J
and if we apply it on
Hamiltonian we have
ˆ
S
ˆ
H
ˆ
S
− 1
=
ˆ
H. After the simplification, we can illustrate the chiral
symmetry condition:
ˆ
S :{H,U
S
}= 0, U
2
S
= U
†
S
U
S
= 1 (2.5)
After the discussion of symmetry, we can move to the tenfold classification of gapped topolog-
ical insulators and topological superconductors as well as the defects, in terms of bulk topological
invariants, which is listed below:
7
1. Primary series for s even: Chern number: In the non-chiral classes of gapped topological
insulators (s is even), theZ-classified topology is characterized by the Chern number which
is well defined only when d+ D is even:
Ch
n
=
1
n!

i
2π

n
Z
BZ
d
× M
D
Tr(F
n
) (2.6)
n=(d+ D)/2 andF is the Berry curvature. Chern insulator which in two-dimensional
lattice and have non-zero Chern number was illustrate and exhibits the quantum anomalous
Hall (QAH) effect[39, 57]. In experiment, the Chern insulators were also realized in many
materials[58–60].
2. Primary series for s odd: The winding number: The winding number topological invari-
ant ν can be defined only in the presence of chiral symmetry, i.e., {H(k,r),U
S
}= 0, with
U
2
S
= 1. The examples of winding number are Chern-Simons invariant[61–64], 1D class
AIII polyacetylene[14], 3D class DIII
3
He− B[65–67].
3. The first Z
2
descendant for s even: The first and second descendants of the topological
phases are characterized by aZ
2
invariant. The first Z
2
descendant typologies are character-
ized by:
CS
2n− 1
=
Z
BZ
d
× M
D
Q
2n− 1
∈
1
2
Z, (2.7)
So the Z
2
topology will be trivial (non-trivial) when CS
2n− 1
is an integer (half integer).
There are also some class of this topological insulators: class D in d = 1[68, 69] and class
AII in d = 3[11, 42].
4. The second Z
2
descendant for s even: Fu and Kane proposed the topological invariant
which applies to the second Z
2
descendant for non-chiral symmetry classes and called it
Fu-Kane (FK) invariant:
FK
n
=
1
n!

i
2π

n
Z
BZ
d
1/2
× M
D
Tr(F
n
)− I
∂BZ
d
1/2
× M
D
Q
2n− 1
(2.8)
8
where n=(d+ D)/2 and the FK topological invariant has values inZ
2
={0,1}.
5. The first Z
2
descendant for s odd: Class DIII in d = 2 belongs to this branch. TheZ
2
The
Z2 topological number can thus be expressed as[67]:
W =
∏
K
Pf[q( K)]
p
det[q( K)]
(2.9)
6. The secondZ
2
descendant for s odd: TheZ
2
topology is trivial if CS
2n− 1
in eq. 2.7 is even,
and nontrivial if CS
2n− 1
is odd. When d = 1, the CS integral has the value in full integers
and can be simplified into Z
2
invariant that relies only on momenta k= 0,π[70]:
(− 1)
ν
=
Pf[q(π)]
Pf[q(0)]
p
det[q(0)]
p
det[q(π)]
(2.10)
Hence the model in non-trivial when the pairing∆ in the Hamiltonian D(k)=− t sinkσ
1
− i[∆+ u(1− cosk)]σ
2
is negative.
2.1.2 One-Dimensional Su-Schrieffer-Heeger Model
Then, we come to the easiest topological insulator: the one-dimensional SSH model [14], which
describes the one-dimensional lattice with spinless fermions hopping and the hopping amplitudes
is staggered, as shown in Fig.2.2(a) . The chain consists N unit cells, and there are two sites in
each single cell: sublattice A and sublattice B. The Hamiltonian can be written as
ˆ
H = t
1
N
∑
m=1
(|m,B⟩⟨m,A|+ H.c.)
+t
2
N− 1
∑
m=1
(|m+ 1,A⟩⟨m,B|+ H.c.)
(2.11)
here N is the number of unit cells. m,B and m,A indicate the states of the chain which label the
atom sublattice. t
1
and t
2
are intracell and intercell hopping. In this paper, we take the hopping t
to be real and non-negative value. The SSH chain has bulk and boundary, the bulk is the central
9
Figure 2.2: Illustration of SSH model and mirror SSH model.(a) Illustration of the simple SSH
model with staggered hopping on the bipartite tight-binding chain. (b) Illustration of the mirror
SSH model on the tight-binding chain with inversion at the middle. (c) Illustration of the SSH
model with an additional molecule above the chain and connected with one site on the chain, the
position of the connected site varies form the edge to the bulk. (d) Illustration of the mirror SSH
model with an additional molecule above the chain and connected with one site on the chain, the
position of the connected site varies form the bulk to the edge.
part of the chain and the boundaries are the two edges. The physics in the bulk and boundary are
independent, for simplicity we take periodic boundary conditions(PBC) and the bulk Hamiltonian
will be [31].
ˆ
H
bulk
=
N
∑
m=1
(t
1
|m,B⟩⟨m,A|+
t
2
|(m mod N)+ 1,A⟩⟨m,B|)+ H.c.
(2.12)
In momentum space, the Hamiltonian
ˆ
H
k
defined as
ˆ
H
k
=

k

ˆ
H
bulk

k

; |k⟩=
1
√
N
N
∑
m=1
e
imk
|m⟩ (2.13)
We define
ˆ
H
k
= d d d(k) ˆ σ ˆ σ ˆ σ and the winding number of the bulk momentum spaceν is[31]
ν =
1
2π
Z
π
− π

˜
d d d(k)
d
dk
˜
d d d(k)

z
dk, (2.14)
10
We want to calculateν of bulk momentum space from bulk Hamiltonian
ˆ
H(k)[31]
ˆ
H
k
=



0 h(k)
h
∗ (k) 0



; h(k)= d
x
(k)− id
y
(k), (2.15)
So it can be written as the integral of the complex logarithm function of h(k)
ν =
1
2πi
Z
π
− π
dk
d
dk
logh(k), (2.16)
The bulk topological invariant is defined as an integer value that can characterize an insulating
Hamiltonian. In the SSH model, it is the winding numberν which can only be 0 or 1, depending
on t
1
and t
2
. When intracell hopping dominates, t
1
> t
2
, we obtainν= 0 and the system is in trivial
phase. When t
1
< t
2
, we obtainν = 1 and the system is in topological phase. The boundary of the
two phases is the line t
1
= t
2
, where the bulk gap closes at some k.
Suppose that the number of zero-energy states on sublattice A (Fig.2.2(a)) is N
A
and on sublat-
tice B is N
B
, we can make a conclusion that the net number of edge states at the left end N
A
− N
B
is
also a topological invariant. From the bulk-boundary correspondence, we can get that in the open
boundary SSH model chain, the topology can be recognized by the number of zero-energy edge
states N
es
= 2ν. We can get the properties of zero-energy by the energy spectrum of the SSH model
in trivial phase (t
1
= 1.25eV > 0.75eV = t
2
) and topological phase (t
1
= 0.75eV < 1.25eV = t
2
)
with 52-site in Fig.2.3(a). In trivial phase, the band gap is observed and N
es
= 0, whereas two de-
generate zero-energy states are observed (in the band gap) in topological phase and N
es
= 2. These
two degenerate zero-energy states are the results of the chiral symmetry of the SSH model.
11
Figure 2.3: Energy spectrum levels of SSH model and mirror SSH model. (a) The spectrum lev-
els of the SSH model in a 52-site open chain. There are two zero-energy edge states in topological
phase (t
1
= 0.75eV < 1.25eV = t
2
) but no edge state in trivial phase (t
1
= 1.25eV > 0.75eV = t
2
).
(b) The energy spectrum levels of the mirror SSH model in a 55-site open chain. There are one
zero-energy localized state in strong interface and three zero-energy localized states in weak inter-
face.
2.2 Method
2.2.1 Random Phase Approximation in Real Space Representation
In order to study the plasmonic excitations in TIs, we use the real-space random phase approx-
imation (RPA) which has been introduced in previous studies [71–73]. One major advantage of
this approach is that we can directly obtain the real-space charge oscillation pattern for each single
plasmon mode, as will be explained below.
We first evaluate the non-interacting charge susceptibility function in the atomic basis via[74]
[χ
0
(ω)]
ab
= 2
∑
i, j
f(E
i
)− f(E
j
)
E
i
− E
j
− ω− iγ
ψ
∗ ia
ψ
ib
ψ
∗ jb
ψ
ja
, (2.17)
where the independent variable ω is the excitation frequency in units of eV . We consider finite
lattices with open boundaries. Therefore, a and b run over all atomic lattice sites. E
i
andψ
i
are the
12
electronic eigenenergy and eigenstates of the i− th level, which is here obtained by diagonalizing
the model Hamiltonian. f(·) is the Fermi function, which in the following is approximated by a
step function, namely, by its zero temperature limit. The factor 2 accounts for the spin degeneracy.
γ = 0.01 eV is a finite broadening.
Electron-electron Coulomb interactions are then introduced into the calculation on the RPA
level. To do this, we evaluate the bare Coulomb interaction matrix V
ab
in the same atomic basis.
To avoid a divergence, we use the Ohno potential[75] with a proper cutoff parameter ∆. The
Coulomb interaction between two sites⃗ r
a
and⃗ r
b
is therefore given by
V
ab
=
e
2
4πε
env
p
|⃗ r
a
− ⃗ r
b
|
2
+∆
2
, (2.18)
with ε
env
being the dielectric constant of the background environment. In our calculation, we set
∆ to be 1
˚
A, which is smaller than the 1D lattice spacing d = 3
˚
A. The real-space RPA dielectric
response function (a matrix) is then calculated from[76]
ε
RPA
(ω)= I− Vχ
0
(ω). (2.19)
We identify the plasmonic excitations from the electron energy loss spectrum (EELS), which is
defined by [77, 78]
EELS(ω)= max
n

− Im

1
ε
n
(ω)

, (2.20)
for each single frequency ω, where ε
n
is the n-th eigenvalue of ε
RPA
(ω). The EELS is peaked at
plasmon frequencies. Letting M be the selected index that maximizes− Im[1/ε
n
(ω
p
)] at a plasmon
frequency ω
p
, we can then simultaneously obtain the real-space charge distribution pattern of
13
the corresponding plasmon mode ω
p
by using the eigenstate ψ
M
(ω
p
) of the dielectric matrix,
multiplied by the non-interacting susceptibility matrixχ
0
(ω
p
)[77], namely,
ρ
0
(ω
p
)=χ
0
(ω
p
)ψ
M
(ω
p
). (2.21)
Here,ρ
0
(ω
p
) is the induced charge density vector in the atomic basis, representing the plasmonic
eigenmode at frequencyω
p
. Similarly, we can define the second EELS (2nd EELS) at any single
frequency ω by selecting the second maximum of− Im
h
1
ε
n
(ω)
i
among all eigenvalues{ε
n
(ω)}.
A peak in the 2nd EELS indicates a degenerate mode (at least two-fold), which is due to the
symmetry of the underlying model. In fact, for a two-fold degenerate plasmon mode, the 1st EELS
and the 2nd EELS coincide, indicating a degenerate subspace furnished by degenerate eigenvectors
of the dielectric matrixε
RPA
(w). In general, one can define the third, forth, etc., EELS in the same
manner. In this paper, we focus on 1D models, for which the 1st EELS and 2nd EELS are usually
sufficient for the analysis.
While the EELS yields the plasmonic eigenmodes of a system, it does not give any information
of the system’s response to a specific external field that is applied in experiment to excite the
system. In this case, we need to calculate the induced charge distribution due to a specific external
field φ
ext
(ω) via
ρ
ind
(ω)=χ
RPA
(ω)φ
ext
(ω), (2.22)
using now the (interacting) RPA charge susceptibility function
χ
RPA
(ω)=[I− χ
0
(ω)V]
− 1
χ
0
(ω). (2.23)
The induced potential, induced electric field, and the induced field energy density in real space
can then be determined from ρ
ind
(ω). By integrating the induced field energy density over the
full space of consideration, we obtain the frequency-dependent induced energy spectrum, U
ind
(ω),
which also peaks at the plasmon frequencies [24].
14
2.3 Results and Discussion
2.3.1 Plasmonic excitations in a decorated one-dimensional metallic chain
Figure 2.4: Plasmons in decorated metallic chain (a) Illustration of a 54-site decorated metallic
chain with hopping t= 1.0 eV,
˜
t= 0.5 eV and t
′
= 2.0 eV (b) Energy dispersion of a homogeneous
metallic chain in momentum space. (c) Blue: EELS of a finite homogeneous metallic chain with
open boundary conditions. Red: EELS of the same open-ended metallic chain with additional
diatomic molecules like (a). (d)-(g) Real-space charge density modulation for (d) a low-energy
plasmon in the pure metallic chain. (e) a high-energy plasmon in the pure metallic chain. (f)
a low-energy plasmon in the decorated chain. (g) a confined low-energy plasmon in the bulk at
ω = 3.878 eV of the decorated chain.
15
Before discussing plasmons in topological insulators, let us first consider a homogeneous one-
dimensional (1D) metallic chain (MC) with open boundaries as a benchmark, described by the
real-space tight-binding Hamiltonian,
ˆ
H = t
M− 1
∑
n=1
(n+ 1n+ H.c.)+µ
M
∑
n=1
nn, (2.24)
where M is the number of the atoms in the chain, t is the nearest-neighbor hopping parameter,
and µ is the chemical potential. We explore the plasmonic excitations in this simple model by
calculating the EELS for a finite chain wit M = 54 sites, t = 1.0 eV and µ =-1.0 eV , which is
shown by the blue line in Fig. 2.4(c). The EELS is made of a (quasi-)continuum of plasmonic
excitations in the frequency range betweenω = 0 eV andω = 7.28 eV. Furthermore, we observe
a pseudo-gap at around ω ≈ 6 eV, which arises from finite size effects. For periodic boundary
condition, this pseudo-gap is absent, and we have verified numerically that with increasing chain
length it gradually disappears. Fig. 2.4(b) shows the momentum space dispersion of the pure
metallic chain, which corresponds to the pure EELS shown in Fig. 2.4(c). In the high energy
region, the plasmon dispersion curve in momentum space becomes flat, leading to the van-Hove
singularity observed at the maximum excitation energy. The low-energy plasmonic excitations
arise from two-particle processes combining distant electrons that are energetically close to the
Fermi surface, whereas the high-energy plasmons arise from scattering electrons that are in close
proximity to each other. All plasmons in this pure 1D MC are bulk modes. Two representative
examples of different excitation energies are shown in Figs. 2.4(d) and (e) with their real-space
charge modulation patterns plotted. We can see the low-energy mode [Fig. 2.4(d)] displays a longer
wavelength, corresponding to slowly propagating waves. In contrast, the high-energy mode shows
a much shorter wavelength, characterized by a rapidly oscillating charge modulation pattern. The
enhanced phase space spectral density at high energies leads to the van Hove singularity, arising
from two-particle scattering processes, connecting single electron states from the band minimum
to those at the band maximum.
16
Next, we investigate a decorated metallic chain (DMC) by adding four diatomic molecules
with equal spacing to the homogeneous host system, as illustrated in Fig. 2.4(a). Specifically,
these four molecules are located above sites 4,19,34 and 49 of the open-ended M = 54-site host
chain, respectively. They are aligned vertically, with an internal hopping t
′
= 2.0 eV between the
two atoms of the molecule, and a small tunneling hopping
˜
t= 0.5 eV between the molecule and the
chain. The red line in Fig. 2.4(c) shows the corresponding EELS of the DMC. Compared with the
EELS of the pure MC, we see that for frequencies below≈ 2 eV and above the pseudo-gap at≈ 6 eV, the spectrum of DMC is almost unaffected by the decorating molecules. In the intermediate
frequency regime between 4 eV to 6 eV, the spectrum structure of the pure MC is approximately
preserved, with slightly shifted energies. In this regime, molecules are far off resonance. There
is little charge tunneling between the molecules and the host MC. However, the spectrum within
the energy windowω∈{2.7 eV,4 eV} is strongly affected due to the added molecules. Moreover,
the molecules are on resonance and are excited as well, leading to significant charge transfer due
to the interactions between the decorating molecules and the host 1D MC (see e.g. Fig. 2.4(f)).
Furthermore, a pattern of oscillating charges between nearby decorating molecules is observed. In
Fig. 2.4(g), we find another interesting mode, where the charge density in the 1D MC is confined
due to the addition of the molecules. These two modes are not observed in the unperturbed host
MC, and they both occur in the most affected energy regime for the chosen parameter set.
2.3.2 Plasmonic excitation in the SSH model
Let us now turn to a prototype symmetry-protected topological insulator, i.e., the SSH chain [14],
described by a one-dimensional lattice Hamiltonian of spinless fermions with staggered hopping
parameters, as illustrated in Fig. 2.2(a). There are two atoms in each primitive cell, labeled by A
and B.
In Fig. 2.3(a), we show the electronic energy spectra of the SSH chain with 52 sites in both
the topologically non-trivial sector (t
1
= 0.75 eV< 1.25 eV= t
2
) and in the trivial sector (t
1
=
1.25 eV> 0.75 eV= t
2
). In the former case we observe two zero-energy edge states in the gap,
17
Figure 2.5: EELS of 1D SSH model non-trivial phase and edge modes (a) EELS of the 52-
site open-ended SSH model in the topologically non-trivial sector (t
1
= 0.75 eV< 1.25 eV= t
2
).
(b) and (c): charge density modulations of the two localized plasmons in the first EELS at an
intermediate and at a high frequency, only observed in the topological sector.
whereas in the latter case there are no edge states. Due to the bulk-boundary correspondence, these
topological properties can also be identified via the bulk winding number W. We now focus on the
SSH model in the topologically non-trivial sector (t
1
= 0.75 eV and t
2
= 1.25 eV) and analyze its
plasmonic excitations. The EELS is shown in Fig. 2.5, where we observe two continua (mainly
consisting of bulk modes) separated by an energy gap, as well as two isolated modes in the gap at
ω= 4.471 eV andω= 7.344 eV, labeled by p
1
and p
2
. The real-space charge density modulations
of these two modes are shown in Figs. 2.5(b) and (c). We observe that they are both localized at the
ends of the chain. Furthermore, the charge distribution of the higher frequency mode [Fig. 2.5(c)]
is more strongly localized than the lower frequency mode [Fig. 2.5(d)]. We also point out that
these two modes are both two-fold degenerate in the pure SSH model, as indicated by the peaks
in both the 1st EELS and the 2nd EELS. Below, we will further study the effects of molecular
perturbations on mode degeneracy.
18
Figure 2.6: 1D SSH model non-trivial phase plasmons with only topological susceptibil-
ity (a) EELS using only the topological charge susceptibility χ
topo
0
in the topological sector
(t
1
= 0.75 eV < 1.25 eV = t
2
) of the SSH model on an open-ended 52-site chain. (b) to (g):
charge density modulations of the three degenerate localized plasmonic excitations at different fre-
quencies. The edge modes observed inχ
full
0
are preserved when onlyχ
topo
0
is considered.
In previous work, localized plasmons in open-ended TIs have been shown to originate from
the topological electronic edge states [24]. This was demonstrated by decomposing the full charge
susceptibilityχ
full
0
into bulk and topological surface contributions, namely,
∑
i, j
...
|{z}
χ
full
0
=
∑
i∈TS
∑
j/ ∈TS
··· +
∑
i/ ∈TS
∑
j∈TS
| {z }
χ
topo
0
··· +
∑
i, j/ ∈TS
...
| {z }
χ
bulk
0
, (2.25)
where TS is the set of the topological zero-energy edge states in the bulk gap. The spectrum
of χ
topo
0
preserves the plasmonic edge modes observed before in the full EELS spectrum, along
with their degeneracies and their localized character in the real space. (see Fig. 2.5 and Fig. 2.6).
This allows us to focus on χ
topo
0
instead of χ
full
0
for an isolated examination of these localized
edge plasmon modes. Below we calculate the EELS of the SSH model, using only χ
topo
0
, and
denote the resulting spectrum by EELS
topo
(ω), which is shown in Fig. 2.6(a). As expected, there
19
Figure 2.7: Induced energy spectrum and edge modes of 1D SSH model topological phase.
(a) Induced energy spectrum in the 52-site open-ended SSH chain in the topological sector (t
1
=
0.75 eV< 1.25eV = t
2
), subject to a linear external electrical field. (b), (c) and (d): charge density
modulation of the mode corresponding to the three excitations highlighted in (a).
is no bulk plasmonic continuum in the spectrum because of removal of the bulk contributions,
χ
bulk
0
. EELS
topo
(ω) shows three peaks atω = 2.038 eV, 2.526 eVand3.846 eV. Each mode has
a two-fold degeneracy of different parities: odd parity and even parity, and they all have localized
charge distributions [see Figs. 2.6(b) to (g), up: even parity; down: odd parity]. Also, similar to
the EELS
full
(ω) with full susceptibility, the edge plasmons are more strongly localized at higher
frequencies, resembling the patterns shown in Fig. 2.5. Furthermore, note that even though the
bulk susceptibility does not contribute to the localization of the edge plasmons, it still affects the
excitation energies of these modes.
We also calculate the induced energy spectrum U
ind
(ω) of the SSH model in response to a spe-
cific external electromagnetic field, which can be directly compared to experiments. Here we con-
sider a linear external electric potential applied to a finite SSH chain in the topologically non-trivial
sector. The resulting spectrum shown in Fig. 2.7(a) contains three main peaks at the exactly same
frequencies as those obtained from EELS
topo
(ω) [Fig. 2.6(a)], which confirms the eigen-modes
indicated EELS. However, The induced charge distributions of these modes in Figs. 2.7(b)-(d) are
odd functions in the real space, whereas their even-parity partners from the EELS [Figs. 2.6(b)-(d)]
20
become inactive now. This is expected because, under the linear potential, only modes with odd
parity are excited.
2.3.3 Effects of added diatomic molecules on plasmons in the SSH chain
The topological SSH model studied above hosts both bulk plasmons and localized plasmonic edge
modes [24]. While the former show no essential difference from the propagating modes in the
1D MC in Sec. 2.3.1, the localized edge plasmons respond differently to molecular perturbations.
Here we introduce diatomic molecules in the vicinity of the SSH chain and study their effects
on the plasmonic excitations. Specifically, we place a single diatomic molecule above the SSH
chain and gradually change its position from the edge to the center of the chain, as illustrated in
Fig. 2.2(c). The tunneling hopping between the molecule and the SSH chain is denoted by
˜
t, with
˜
t < t
1
,t
2
). The internal hopping in the molecule is denoted as t
′
, with t
′
> t
1
,t
2
. Such a situation
could be experimentally realized by atoms attached on an STM tip or by tip atoms themselves
when scanning over a sample, such as described in [79]. Here, we focus on studying the effects of
the perturbing molecule at various positions on the plasmonic edge modes.
We first consider the effect of a diatomic molecule in proximity to one of the ends of the SSH
chain [Fig. 2.2(c)], which is expected to have maximum impact on the plasmonic edge modes.
Fig. 2.8(a) shows the EELS of such a perturbed SSH chain, together with the unperturbed case for
comparison. We find that the molecular perturbation on the edge site removes the degeneracy of
all three modes observed in the pure host system. Due to the additional molecule attached to one
edge site, the two ends of the chain are no longer equivalent. Therefore, they now each host edge
modes with slightly different energies. For instance, in Figs. 2.8(e) and (f) we show the real-space
charge distribution patterns for the two highest energy modes [labeled as p
1
and p
2
in Fig. 2.8(a)].
They originate from a degenerate pair at ω = 3.486 eV of the host model. In the presence of the
molecular perturbation applied to the left end of the open chain, the mode localized on the right
end of the chain remains at the same frequency as before because of its far distance away from the
perturbing molecule, whereas the edge mode at the left end of the chain now has a slightly shifted
21
Figure 2.8: Topological EELSs for 1D SSH model with perturbation at different positions
(a) to (d): Topological EELSs in the SSH chain (t
1
= 0.75 eV< 1.25 eV= t
2
) with one added
diatomic molecule at different positions, X
m
, which means the molecule is connected with the x
th
site on the chain like Fig. 2.2(c). The connection hopping
˜
t = 0.5 eV and t
′
= 2.0 eV. (e) and (f):
charge density modulation of the modes corresponding to the two high-frequency excitations in
(a).
energy. Naturally, plasmons that are localized close to the left end of the chain are mostly affected
by this local perturbation.
As we gradually move the perturbing molecule from the left end to the center of the chain,
the effects on the plasmonic edge modes become less pronounced. Quantitatively, this depends
on the localization length of the plasmonic edge modes. As mentioned above, the highest-energy
mode is most localized. The two lower-energy modes are slightly more extended [Figs. 2.6(b)-
(g)]. When the molecule is being moved towards the center of the chain, the highest-energy mode
first becomes unaffected to the perturbation, then followed by the lower-energy ones. In Figs. 2.8
22
Figure 2.9: Dependence of plasmons on the internal hopping Dependence of the energy of the
topological plasmons and of the charge transferred from the host to the diatomic molecule on the
internal hopping t
′
, when the molecule is connected with an edge atom of the topological SSH
chain (t
1
= 0.75 eV< 1.25 eV= t
2
). E
L
(Red) is the highest energy excitation on the left end
of the chain (see Fig. 2.8(f)). E
R
(Blue) is the highest energy excitation on the right end of the
chain (see Fig. 2.8(e)). W
m
is the percentage weight of the charge transferred from the chain to the
molecule.
(b)-(d) we show the variation of the EELS, as the molecular perturbation is gradually moved to
the center. In detail, we can see that when the perturbing molecule is moved onto the 3
rd
site
away from the the chain edge, the highest-energy mode is already not affected. However, the two
lower-energy modes are still affected by the perturbation, as we can see from the split peaks. When
the perturbing molecule is on the 7
th
site of the chain, the second-highest-energy mode becomes
unaffected as well [Fig. 2.8(c)]. Finally, when the molecule is on the 15
th
site of the chain, which
is quite deep into the bulk, all three modes are unaffected. In this case, the full EELS is almost the
same as for the unperturbed host system.
In the above calculations, the internal hopping t
′
was fixed to 2.0 eV , which is larger than the
hopping in the chain. Here, we would like to examine how the the excitation energies change when
we modify t
′
. When we consider only the topological susceptibility, the excitations are localized
at the two ends of the chain. In Fig. 2.9 we see that the excitation on the right end remains constant
at about E
R
= 3.84 eV when the perturbing molecule is connected to the left end. However, the
23
excitation at the left end, E
L
, shifts to higher energies when t
′
is increased because a higher energy
is required to excite the molecule with a larger internal energy gap t
′
. Furthermore, we find that
the charge transferred from the chain to the molecule also changes with t
′
. W
m
in Fig. 2.9 shows
the percentage weight of the charge on the molecule compared to the charge in the entire system
(host chain plus perturbation molecule). We observe that the relative weight on the molecule drops
from about 16% to about 8% when t
′
is increased, i.e. there is less charge transfer from the host
to the molecule. Hence, the internal electronic structure of the added molecule affects its ability
to hybridize with the host system. When the internal hopping (t
1
and t
2
) inside the SSH chain is
fixed, if the molecule has a larger internal energy gap, i.e. larger t
′
, then it is more protected from
hybridization with the host system.
We conclude that a local perturbation can affect the topologically originated plasmonic edge
modes in the SSH model only significantly when it is sufficiently close to the edge of the chain.
In other words, the edge plasmon modes in the topological SSH model are very robust against
local perturbations, which is different from bulk plasmon modes. It has been shown before that
these modes are also quite stable when subjected to global random noise in the bulk hopping
parameters[24].
2.3.4 Plasmonic excitations in the mirror-SSH model
In addition to the SSH model, let us also inspect plasmons in the mirror-SSH model, which is
a variant of the SSH model by reflecting the chain about its center [Fig. 2.2(b)]. This mirror-
SSH (mSSH) model is inversion symmetric with respect to the mirror interface located at the
middle point of the chain, which also hosts localized zero-energy state(s) depending on the hopping
characteristics at the interface. In Fig. 2.3(b), the energy spectra of the strong interface mSSH
model (t at the center is 1.25 eV) and the weak interface mSSH model (t at the center is 0.75 eV)
are displayed, where we observe one zero-energy state and three zero-energy states, respectively.
We calculate the EELS of the mSSH model plasmons by using only the “topological part” of the
susceptibilityχ
topo
0
, defined in the same manner as for the SSH model. Fig. 2.10(a) shows the EELS
24
Figure 2.10: EELS of 1D mirror-SSH model (a) EELS of the open-ended mirror-SSH chain
(M=53) with weak mirror interface hopping (t
1
= 0.75eV < 1.25eV = t
2
) and on an open-ended
55-site chain with strong mirror interface hopping (t
1
= 1.25eV > 0.75eV = t
2
). (b) and (c): charge
density modulation of the modes corresponding to the excitations at the weak mirror interface. (d)
and (e): charge density modulation of the modes corresponding to the excitations at the strong
mirror interface.
of the mSSH model for both the strong mirror interface (hopping at the mirror is t
1
= 1.25 eV) and
the weak mirror interface (hopping at the mirror is t
2
= 0.75 eV). In each case, we observe a set of
(non-degenerate) excitations that are all strongly localized around the mirror interface. We show
the real-space charge modulations of some typical excitations in Figs. 2.10 (b)-(e). As we can see
here, the modes around the strong interface and the weak interface have different parities. Similar
to what we observed before in the SSH model, the modes with higher energies are more strongly
localized.
2.3.5 Effects of added diatomic molecules in the mirror-SSH chain
Here we introduce a single molecular perturbation into the mSSH model, starting from the central
site (the mirror interface) of the chain and moving towards one end, as shown in Fig. 2.2(d). We
focus on studying its effects on the localized plasmons around the interface. Unlike the edge
25
Figure 2.11: EELSs for 1D mirror-SSH model weak interface with perturbation at different
positions (a) Comparison of the EELS of the weak interface mirror-SSH model (t
1
= 0.75 eV<
1.25 eV= t
2
) with M=53 sites in the presence of a perturbation diatomic molecule placed at dif-
ferent positions (green line: connected with the 26
th
site; orange line: connected with the 20
th
site)
and the model without perturbation (red line). (b) - (e): charge density modulations of the modes
corresponding to the excitations highlighted in (a).
plasmons observed in the SSH model, the interface plasmons here are non-degenerate even in the
unperturbed case. So, there is no degeneracy splitting effect. We will, however, observe other
interesting effects due to the perturbation.
Fig. 2.11(a) shows the perturbed EELS of the weak-interface mSSH model, together with the
unperturbed one (red line) for comparison. The spectra are similar to the SSH model discussed in
section (c), in the sense that the effects from the perturbation are weakened when the perturbing
molecule is gradually shifted away from the charge concentration area of the localized plasmons.
When the added molecule is connected to the 26
th
site, which is the center site of the 53-site
chain, all of the excitations change their positions (green line) because of significant charge transfer
between the chain and the molecule. Figs. 2.11(b) and (c) show the charge distributions of two
typical excitations in (a). Here, we observe that there is little charge transfer from the chain to
the molecule in the high energy excitation, but more significant charge transfer to the molecule
26
Figure 2.12: EELSs for 1D mirror-SSH model strong interface with perturbation at different
positions (a) Comparison of the EELS of the strong interface mirror-SSH model (t
1
= 1.25 eV>
0.75 eV= t
2
) in the presence of a diatomic perturbation molecule placed at different positions
(green line: connected with the 26
th
site; orange line: connected with the 20
th
site) and the model
without perturbation (blue line). (b) - (e): charge density modulations of the modes corresponding
to the excitations indicated in (a).
connected to the center site for the lower energy mode. The reason for this is that the internal
hopping magnitude of the molecule is t
′
= 2 eV, which is closer to the frequency of the low energy
excitation. However, when we move the perturbation from the center towards the edge, the EELS
curve will coincide with the original curve (red line) in the high and intermediate energy regimes,
since the charge is more concentrated at the center in the higher energy modes [Fig. 2.11(e)], so
the interaction between the chain and the molecule substantially vanish. In this case, there is still
charge transfer in lower energy modes, and the charge distribution will not be symmetrical anymore
[Fig. 2.11(d)].
Next, we discuss the strong interface case, whereby the hopping at the center mirror interface
is stronger in magnitude. Fig. 2.12(a) shows a comparison of the strong interface with (red and
orange line) or without (blue line) the molecular perturbation. For the strong interface, since the
plasmon is also concentrated in the center we observe analogous spectra as for the weak interface.
27
Figure 2.13: Energy spectrum for 1D SSH model with external electrical field Energy spec-
trum, induced by a linear external electrical field, of a 52-site non-trivial topological SSH chain
(t
1
= 0.75 eV< 1.25 eV= t
2
) with one diatomic perturbation molecule located at different posi-
tions, calculated using only the topological susceptibilityχ
topo
0
.
Specifically, the effect of the perturbation molecule on topological surface plasmons in the mirror-
chain decreases when we move it from the chain center towards the chain ends, especially in the
higher energy regime. Whenω is larger than 2.5 eV, the blue curve and orange curve coincide with
each other, which means there is little effect when the molecular perturbation is connected with the
20th site than connected with the 26
th
site. In the low energy region, however, we also observe an
excitation shift caused by charge transfer (Fig. 2.12(d)) because the plasmonic excitation expands
to the 20
th
site of the chain. However, when the molecular perturbation is connected with the 26
th
of the chain (in the center), all of the excitations of the host system shift from low energy to high
energy, and plasmonic charge is transferred to the molecule [Figs. 2.12(b) and (c)].
28
2.3.6 Induced field energy of the perturbed SSH model
In Sec. 2.3.2, we discussed the induced energy spectrum of the SSH chain and analyzed the spec-
trum in Fig. 2.7. We observe three excitations at exactly the same positions as in Fig. 2.6. In
Sec. 2.3.3, the effects of local perturbation were considered. Here, we analyze the induced energy
spectrum of the topological SSH model in the presence of a diatomic molecular perturbation, sub-
ject to an external linear electrical field. As a reference, the blue dashed line in Fig. 2.13(a) shows
the spectrum without perturbation that we have already discussed. Figs. 2.13 (a)-(d) reveal that the
effects of the perturbation to SSH model in the presence of an external electrical field are similar
to the EELS spectrum, i.e., mostly affecting the degeneracy with respect to the position. As the
perturbing molecule is gradually moved from the left end (0
th
side) towards the center of the bulk
(15
th
site), the effects on the plasmonic edge modes become less pronounced. The highest-energy
excitation first becomes unaffected to the perturbation, as seen in Fig. 2.13(b), and then the lower
energy modes recover. When the perturbation moves sufficiently deep into the bulk, the energy
spectrum recovers to the same shape of the non-perturbed system.
29
Chapter 3
Plasmons inZ
2
Topological Insulators
3.1 Two-Dimensional Topological Insulators
In the previous chapter, we introduced the one-dimensional topological insulator SSH model and
its robustness. In this chapter, we will analyze the plasmonic excitations in the topological insu-
lators with another invariant : Z
2
topological invariant. Before the plasmon, we first discuss the
background of several two-dimensional topological insulators.
3.1.1 Two-Dimensional Su-Schrieffer-Heeger Model
The one-dimensional SSH model has a unit cell with two sites: A and B. We can extend the model
to a square lattice with 2× 2 sites in the unit cell with intracell hopping t
1
and intercell hopping t
2
,
so the momentum space Hamiltonian is:
H
12
=H
34
= t
1
exp{ik
x
L
2
}+t
2
exp{− ik
x
(L
1
− L
2
)}+ H.C.
H
13
=H
24
= t
1
exp

ik
y
L
2

+t
2
exp

− ik
y
(L
1
− L
2
)

+ H.C.
(3.1)
here L
1
and L
2
are the intracell and intercell nearest neighbor distance constant. Two-dimensional
SSH model has time reversal and inversion symmetry. The Berry curvature only exists at C
4ν
invariant points in the whole Brillouin zone, which is|k
x
|=

k
y

. So the zero integration results
a vanishing Chern number. The two-dimensional SSH model will have a non-trivial topological
30
phase and has topological edge states when t
1
> t
2
[80]. Also, the edge localized plasmonic excita-
tions appear in the two-dimensional SSH lattice.[25].
3.1.2 Haldane Model
Haldane model [39] is a Chern insulator on the honeycomb lattice that is the first example of
quantum anomalous Hall (QAH) effect[81]. The momentum space Hamiltonian is:
H(k)= H
0
(k)+ Mσ
z
+ 2t
2∑
i
σ
z
sin(k· b
i
) (3.2)
where H
0
is the Bloch Hamiltonian or can say the nearest-neighbor tight-binding Hamiltonian for
graphene:
H
0
(k)=



0 h(k)
h
†
(k) 0



(3.3)
with h(k)= t
1
∑
i
exp(ik· a
i
). Here a
i
is the vector in the honeycomb lattice that connect nearest
neighbors. t
2
is a complex next-nearest neighbor hopping term and the last term in Eq.3.2 breaks
the time reversal symmetry.
3.1.3 Kane-Mele Model andZ
2
Topology
In this chapter, we focus on the Kane-Mele model[2] which has aZ
2
topological invariant, i.e., we
consider a honeycomb lattice in the presence of intrinsic spin-orbit coupling. The corresponding
Hamiltonian can be written as:
H =t
∑
⟨i j⟩
c
†
i
c
j
+ it
SO ∑
⟨⟨i j⟩⟩
ν
i j
c
†
i
s
z
c
j
+ it
R∑
⟨i j⟩
c
†
i

s× ˆ
d
i j

z
c
j
+t
v∑
i
ξ
i
c
†
i
c
i
.
(3.4)
Here, c
†
i
=

c
†
i↑
,c
†
i↓

is the spinor notation[82], where c
†
iσ
creates an electron at the ith site. The
notation⟨i j⟩ and⟨⟨i j⟩⟩ represents the nearest and next nearest neighbor pairs in the lattice. The
31
first term is a nearest neighbor hopping term with amplitude t on the honeycomb lattice - regular
graphene only has this term.
Figure 3.1: Illustration of honeycomb zigzag nano-ribbon (a) Honeycomb lattice and the relative
signν
i j
of the direction between next nearest neighbor sites. (b) Nano-ribbon structure with zigzag
edges, consisting of sublattices sites A and B in the unit cell. The structure has finite length in the
X direction (perpendicular to edge) and periodic boundary conditions in Y the direction (parallel
with edge).
The second term is the mirror symmetry intrinsic spin-orbit coupling involving next nearest
neighbor sites. The signν
i j
depends on the sublattice and can be written as
ν
i j
=(2/
√
3)

ˆ
d
1
× ˆ
d
2

z
, (3.5)
where i and j are next nearest neighbor sites, the vectors
ˆ
d
1
and
ˆ
d
2
are unit vectors along the two
bonds that traverse from the jth site to the ith site. Fig.3.1(a) illustrates the exact sign of ν
i j
in
the honeycomb. s
z
is the electron spin Pauli matrix. t
SO
is the amplitude for spin-orbit interaction
and can be determined by calculations [83–86]. The third term is the Rashba coupling between
nearest neighbor sites with amplitude t
R
. It violates the z→− z mirror symmetry and arises due to
a perpendicular electric field or interactions with a substrate. Its amplitude can be experimentally
measured [87, 88]. The last term is a staggered sublattice potential (ξ
i
=± 1). In Fig.3.1(b), the
A-site (white circle) and B-site (black circle) in the unit cell correspond to ξ
i
= 1 and ξ
i
=− 1.
32
This term violates the symmetry under twofold rotations in the plane and plays an important role
in phase transitions.
Figure 3.2: Energy spectrum of Kane-Mele model nano-ribbon Energy spectrum of the Kane-
Mele model ribbon with zigzag edges. (a): Insulating phase with t
v
= 0.4t, (b): Quantum spin Hall
(QSH) phase with t
v
= 0.1t. (c) and (d): (Rotated by 90 degrees) Probability density (|ψ|
2
) for the
two edge modes in a zigzag nano-ribbon (ZNR) Kane-Mele model spin-up sites in the unit cell at
momentum K= 1.01π/L.
In this section, to explore the edge topological invariant, we study the Kane-Mele model in
the structure of a graphene ribbon with zigzag edges as shown in Fig.3.1(b). The nano-ribbon
is periodic in the Y direction and has finite width in X direction, also, it has two edges in the X
direction: on the top and bottom. Kane and Mele illustrated that for t
R
= 0, there is an energy gap
with magnitude

6
√
3λ
SO
− 2λ
v

between bulk states, and the system will be an insulator when
λ
v
> 3
√
3λ
SO
, but in the quantum spin Hall phase (QSH) otherwise[2]. We diagonalize Eq.3.13
numerically and show the energy dispersions for two phases in Fig.3.2. In the calculation, the
spin-orbit coupling hopping is chosen as t
SO
= 0.05t, and the Rashba coupling t
R
is set to zero.
33
To characterize the two different phases, we calculate the energy spectrum of the zigzag ribbon
geometry with different staggered potentials, t
v
= 0.4t and t
v
= 0.1t, as shown in Figs.3.2(a) and
(b). In the spectrum, we find that both phases have a bulk energy gap. In the QSH phase, edge
states transverse the energy gap in pairs. In contrast, in the insulating phase, there are no gapless
edge states. Analytical calculations have predicted edge states in the ribbon unit cell [47]. To
illustrate the existence of such edge states in the KM model, we plot the probability density (the
square of the absolute value of the wave function|ψ|
2
) of spin-up sites in the unit cell for the KM
model in a zigzag edge ribbon. We find two edge states in Figs.3.2(c) and (d), which are localized
on opposite edges. Down spins also have two edge states, which makes it clear that the four edge
state branches between the bulk are: (1) spin up localized at top; (2) spin up localized at bottom;
(3) spin down localized at top; and (4) spin down localized at bottom.
3.2 Method
3.2.1 Momentum Space Coulomb Interaction
Since the structure we analyze is the nano-ribbon and has periodic boundary condition in one
direction but open-ended in another, we should use the modified Coulomb potential to match the
geometry in calculation. we consider electron-electron Coulomb interaction which is also Fourier
transformed along the y-direction, namely, V(q
y
). Using the same basis and assuming that the
Coulomb interaction only depends on the real-space coordinates of two sites, we have
V
µµ
′(q
y
)=





2e
2
∗ K
0
(|q
y
||x− x
′
|)/κ, if⃗ τ̸=
⃗
τ
′
,
U
0
/κ, if⃗ τ =
⃗
τ
′
.
(3.6)
Here K
0
(·) is the zeroth modified Bessel function of the second kind. This form of the Coulomb
interaction is referred from [89]. For the onsite interaction, we introduce a parameter U
0
to avoid
divergence. Additionally,κ is the dielectric constant of the background medium.
34
3.2.2 Random Phase Approximation in Momentum Space
In order to study plasmonic excitations, we calculate the dielectric function within the random
phase approximation (RPA)[71–73]. The RPA dielectric function has traditionally been introduced
in the momentum space for homogeneous electron gas[25, 72], as well as recently been formulated
fully in the real space to study non-translational-invariant systems [24, 90]. In this paper, we
mainly focus on the ribbon structure of a topological system, which is periodic in the longitudinal
direction while finite-sized in the transverse direction (Fig.3.1(b)). Therefore, we adopt a combined
treatment of both the real-space and the momentum space, as introduced below.
Generically, as indicated in Fig. 3.1 (b), the unit cell spans the entire width of the ribbon (along
X-direction) and is repeated periodically along the Y -direction. We denote the position and the
spin of the i-th atom in the unit cell as⃗ τ
i
and⃗ σ
i
. By Fourier transformation along the Y -direction,
we can construct a tight-binding basis{µ≡ ⃗ τ⊗ σ,⃗ τ =⃗ τ
1
,⃗ τ
2
,...,⃗ τ
N
andσ =+,−} for each k
y
,
suppose that there are N atoms in the unit cell. Using this, we can express the non-interacting
polarization function as a matrix, whose element indexed byµ andµ
′
is given by [74]
[χ
0
(ω,q
y
)]
µµ
′ =
1
V
∑
k
y
,n,n
′
f(E
k
y
n
)− f(E
k
y
+q
y
n
′
)
ω+ iγ+ E
k
y
n
− E
k
y
+q
y
n
′
ψ
k
y
nµ
(ψ
k
y
nµ
′
)
∗ (ψ
k
y
+q
y
n
′
µ
)
∗ ψ
k
y
+q
y
n
′
µ
′
,
(3.7)
where ω is the frequency, q
y
is the momentum transfer in the y-direction. E
k
y
n
and ψ
k
y
nµ
are the
eigenenergy and the µ-th component of the eigenfunction of the state with band index n and mo-
mentum k
y
. They can be solved from the tight-binding Hamiltonian H(k
y
) written in the same
basis above. f(·) is the Fermi-Dirac distribution function, whose zero temperature limit is applied
in all calculations in this paper. Moreover, we introduce a finite broadening parameter γ which is
set to be 0.01 eV.
We can then derive the dielectric matrix within RPA as [76]
ε
RPA
(ω,q
y
)= I− V(q
y
)χ
0
(ω,q
y
), (3.8)
35
form which we further extract the electron energy loss spectrum (EELS) by
EELS(ω,q
y
)= max
i

− Im

1
ε
i
(ω,q
y
)

, (3.9)
ε
i
(ω,q
y
) is the i-th eigenvalue of ε
RPA
(ω,q
y
). Plasmonic excitations are identified as peaks in
EELS. Meanwhile, the real-space charge distribution patterns of plasmon modes can also be ob-
tained utilizing the dielectric matrixε
RPA
(ω,q
y
). The method is similar to the one reported in [24]
for the full real-space RPA calculation. Suppose that ε
m
(ω,q
y
) is the selected eigenvalue for a
plasmon mode, the the real-space charge distribution pattern of this mode can be obtained by
ρ
0
(ω,q
y
)=χ
0
(ω,q
y
)ξ
m
(ω,q
y
) (3.10)
withξ
m
(ω,q
y
) is the eigenvector corresponding toε
m
(ω,q
y
). In fact, we can also obtain the second
EELS (2nd-EELS) by selecting the second maximum in Eqn. 3.9, which will provide information
for a degenerate plasmon mode. For most situations discussed in this paper, we just perform the
calculation of (the first maximum) EELS.
3.3 Results and Discussion
3.3.1 Plasmonic excitations in the honeycomb lattice
We analyze plasmons in a graphene-structured model with a simple zigzag nano-ribbon (ZNR)
structure as a benchmark, considering the tight-binding Hamiltonian,
ˆ
H = t
∑

c
+
i
c
j
+ h.c., (3.11)
where< i, j> represents nearest neighbor sites on the honeycomb lattice, and t is the correspond-
ing hopping parameter. In Fig.3.3 we show results for a nanoribbon with 16 atoms in the unit
cell. We first inspect the single particle energy bands, shown in Fig.3.3(a), and wave functions
36
at particular momenta, shown in Figs.3.3(b) and (c). The finite size of the nanoribbon produces
confinement of the electronic states in the regions near the Dirac points. The two bands in the gap
overlap to form a state around K/L=π. By plotting the combined wave functions, we find that
these zero-energy states in Fig.3.3(c) localize at the edges of the strip-like unit cell, whereas the
other, higher energy, wave functions at the same momentum are confined to the bulk. The two
bands of the localized edge states that occur between K and K’ in Fig.3.3(a) are affected twofold
by the finite width, i.e., they merge and are slightly offset from zero[91].
Figure 3.3: Energy bands and plasmon of graphene nano-ribbon (a): Single particle energy
spectrum of the tight-binding nano-ribbon with zigzag edges. (b)-(c): Probability density of two
wave functions (|ψ|
2
) at momentum K =π/L. (d): Electron energy loss spectrum (EELS) of the
ribbon structure in the first Brillouin zone with periodic boundary in the X direction (parallel to
the edges) and open boundaries in the Y direction (perpendicular to edges).
We now examine plasmonic excitations in this basic model and calculate the EELS along the
Y-direction in momentum space while accounting for the spin degeneracy. Fig.3.3(d) shows the
37
EELS in the first Brillouin zone, where we observe clusters of plasmonic branches in different
energy regions. At high energies (ω > 5eV), there are 8 plasmonic bands corresponding to the
8 A-sites in the unit cell. Similarly, there are also 8 plasmonic bands at intermediate energies
(ω∈{2eV,5eV}), including an arrow-shaped band at the exact center of the first Brillouin zone.
In the low energy region (ω < 2eV), we observe a continuum of excitations as well as several
quasi-bands. By analyzing the single particle energy structure and the EELS of the ribbon, we
conclude that the bulk energy bands and the geometry of the model are the original source of the
high and intermediate energy plasmonic excitations. In contrast, the low energy continuum is due
to the edge modes. We will study next the related case ofZ
2
topological insulators in the following
section.
3.3.2 Plasmonic excitations in the Kane-Mele model on a nano-ribbon
We now turn to analyze collective excitations in Kane-Mele ribbon structure. As discussed in
Fig.3.2, the model has two topologically distinct phases whose key difference is reflected by ex-
istence of conducting edge states crossing the bulk band gap. In the insulating phase, there is no
surface state within the bulk gap, but in the QSH phase there are four low-energy edge bands. We
show corresponding EELS for both scenarios in Fig. 3.4 in which plasmon dispersions can be eas-
ily identified. We observe gapped, high-energy plasmons dispersions in both phases, which in fact
also appear in the pure graphene-like model with only the first term in the Hamiltonian 3.13 is kept
( see Appendix ??). This is not surprising since these plasmons are bulk modes that are essentially
stem from transitions between gapped bulk energy bands. The number of these bulk branches is
equal to the number of A-sites in the unit cell. In contrast, the gapless low-energy plasmon disper-
sion which comes conducting edge states only appears in the non-trivial QSH phase (Fig.3.4(b)),
while absent in the trivial insulating phase (Fig.3.4(c)). Therefore, the gapless low-energy plasmon
is a clear indication of the topological non-trival QSH phase of the Kane-Mele model.
We demonstrate that the low-energy gapless plasmons observed above are indeed edge modes
by investigating their real-space charge modulation patterns in Fig.3.5. Here we apply a method
38
Figure 3.4: EELS of Kane-Mele model in different phases (a): Electron energy loss spectrum
(EELS) of the Kane-Mele model on a zigzag nano-ribbon in the Quantum Spin Hall (QSH) phase.
(b): Low-energy EELS of QSH phase, t
v
= 0.1t. (c): Low-energy EELS in the topologically trivial,
insulating phase, t
v
= 0.4t.
of decomposing the full polarization function χ
0
(denoted as χ
full
0
) into the bulk part (involving
only bulk electron bands) and the edge part (involving only edge electron bands, denoted as χ
edge
0
below). Such a decomposition has been introduced in [90] in order to disentangle bulk-states and
edge-states contributions to plasmons in a topological non-trivial system. Fig. 3.5(a) shows the
EELS(ω,q
y
) for the QSH phase at the momentum q
y
=π/8L, using χ
full
0
and χ
edge
0
respectively.
By comparing the spectra obtained from full χ
full
0
and χ
edge
0
, we can see that χ
edge
0
nicely, qual-
itatively reproduce the low-energy part (ω < 2.5eV) of the full spectrum with full polarization
χ
full
0
considered. This indicates that low-energy plasmonic excitations are dominated by electronic
39
Figure 3.5: EELS and charge density of Kane-Mele model at K=π/8L (a): Degenerate elec-
tron energy loss spectrum (EELS) of the Kane-Mele model in the QSH phase at momentum
K=π/8L with only edge energy bands susceptibility χ
edge
0
and with full energy bands suscep-
tibilityχ
f ull
0
. (b) and (c): charge density modulations of the plasmonic excitation at the degenerate
peaks p
1
and p
2
(d) and (e): charge density modulations of the plasmonic excitation at the degen-
erate peaks p
3
and p
4
.
edge states. We plot the real-space charge modulation patterns of a few typical modes p
1,2
and p
3,4
indicated in the full EELS. The plots are shown in Figs. 3.5 (b) to (e). Fig. 3.5 (b) and Fig. 3.5 (c)
represent the charge densities of the excitations p
1
and p
2
in the low energy region. These charge
modulations obviously show a concentration on the two edges of the ribbon unit cell and thus con-
firm that p
1,2
are energetically degenerate collective edge modes which have opposite charges on
the two ends of the ribbon. This matches the discussion in of energy band spectrum, where we
found that the edge state energy bands in the bulk gap are two spin-up bands and two spin-down
40
bands. For comparison, we also plot the charge densities of the excitations with the highest ener-
gies: p
3,4
in Figs. 3.5 (d) and (e). These show a rapidly modulating multi-pole distribution in the
bulk of the ribbon unit cell and also have opposite charges for the two degenerate modes.
3.3.3 The effect to edge states by inner parameter
In this section, furthermore, we explore the influence for the Kane-Mele model plasmonic edge
modes by different parameters including Coulomb interaction and geometry. Firstly, to analytically
predict the plasmonic branches, we approximate the edge energy bands as fully independent, with
the energy dispersion E
k,up
= v
f
k and E
k,down
=− v
f
k for unidirectional up and down propagation
of the edge modes. Also, we use the uniform V(q)= 2e
2
K
0
(σ
C
|q|)/κ to instead the Coulomb
interaction in Eq.3.6 for approximation and σ
C
is the effective cross section which indicates the
crossover region from logarithmic to linear dispersive behavior. By differentiating EELS(ω,q)=
− Im1/ε(ω,q), the RPA results for the plasmon dispersion is [25]:
ω(q)=

2e
2
K
0
(σ
c
|q|)
π
+ 2t

|q| (3.12)
Figure 3.6: EELS of Kane-Mele model with different unit size Electron energy loss spectrum
(EELS) of the Kane-Mele model in low-energy region ([0, 2] eV ) with (a): 8 atoms in unit cell.
(b): 32 atoms in unit cell.
We plot the edge plasmon with different parameters in Fig.3.6, we can find the effects when
we tune the size of robbon unit cell and the background dielectric. Comparing with Fig.3.6(a),
41
Fig.3.6(b) shows that when the nano-ribbon has a wider size, the edge mode branch close to q=
0 is lower and more crooked since the bulk of the ribbon structure affects more on the edge,
which makes the effectiveσ
c
larger and thus, increase the Coulomb scattering cross section. This
observation matched the predicted dispersion Eq.3.12. Also, the number of minor edge branches
increases as the ribbon become wider and there are more A-site atoms.
3.3.4 Effects of a Zeeman field in the Kane-Mele model
In this section, we analyze the behavior of Kane-Mele model in the presence of an external Zeeman
field. Zeeman fields break the time-reversal symmetry, which will thus break the Z
2
topological
phase but can protect the higher-order topological phase [92]. To explore the effects of Zeeman
field to Kane-Mele model, especially for plasmonic excitations, we add the magnetic field to the
nano-ribbon with zigzag edges. The Hamiltonian of the modified Kane-Mele model is:
H
Z
= H+t
B∑
i
c
†
i
B· sc
i
. (3.13)
Here, H is the Hamiltonian in Eq.3.13 and B is the Zeeman field vector. t
B
is the hopping
strength. We will see that magnetic fields applied perpendicular to the plane (Z-direction) or in the
plane (XY-direction) have different effects. If the Zeeman field is along the direction perpendicular
to the plane, B=(0,0,Bz). The in-plane Zeeman field is B=(B
x
,B
y
,0).
We set t
B
= 0.15t and the single particle energy dispersions are shown in Fig.3.7 for two
different Zeeman field directions. From these spectra, we observe that B
z
changes the energy
level of the bulk bands as well as the gapless edge states. It splits spin-up and spin-down bands and
changes the distance between the pairs of edge states. In contrast, in the presence of a Zeeman field
along the ˆ x direction, the pairs of spin-helical gapless edge states disappear and the time-reversal
symmetry is broken. Thus the QSH phase is destroyed by in-plane Zeeman fields.
Next, we explore the effects of Zeeman fields on the plasmonic edge modes. We calculate the
EELS in the low-energy region ([0, 2] eV ) and within half of the first Brillouin zone ([0, π/L]) for
42
Figure 3.7: Energy spectrum of Kane-Mele model with Zeeman field Single particle energy
spectrum of the Kane Mele model nano-ribbon with zigzag edges: (a) with a perpendicular Zeeman
field, t
B
B= 0.15t ˆ z; (b) with an in-plane Zeeman field, t
B
B= 0.15t ˆ x. (c) and (d): (rotated by 90
degrees) Probability densities (|φ|
2
) of the crossing points in (a) of the two edge modes.
Figure 3.8: Momentum-space EELS of Kane-Mele model with Zeeman field Electron energy
loss spectrum (EELS) of the Kane-Mele model nano-ribbon in the low-energy region ([0, 2] eV )
with (a): Zeeman field in the Z direction B = (0,0,B
z
). (b): Zeeman field in the X direction
B=(B
x
,0,0).
both cases, Fig.3.8. The narrow unit cell has 8 sites. Compared with the plasmonic excitations in
the absence of a magnetic field, we find that the perpendicular-plane Zeeman field does not destroy
the edge modes but the in-plane Zeeman field does. In Fig.3.8(a), we observe edge modes close to
the origin (ω = 0eV , K = 0), which is in the same regime where edge states are observed in the
absence of a Zeeman field (in Fig.3.4(b)). In Fig.3.8(b), as the in-plane Zeeman field destroys the
time-reversal symmetry, there is no band traversing the gap between valence band and conduction
band. Furthermore, in this case there are no plasmonic edge modes in the EELS close to the origin.
43
Figure 3.9: EELS and charge density of small gap Kane-Mele model with Zeeman field (a):
Degenerate EELS of the Kane-Mele model in the QSH phase at momentum K=π/8L with Z
direction Zeeman field t
B
B= 0.15t ˆ z. Spin-orbit interaction t
SO
= 0.05t. (b)(c): Charge density
modulation of the edge state plasmonic excitation at degenerate peek p
1
, p
2
. (d): The dependency
of the charge weight (percentage) on the edge with the Zeeman field B
z
for peak p
1,2
.
We now start to analyze the effects of the Z-direction magnetic field on edge plasmons by
plotting the degenerate EELS and the edge states charge distribution at K=π/8L with external
field B
z
= 0.15t in Fig.3.9. Same as Fig.3.5, we plot both of the excitation spectrum with full
susceptibility and with only edge susceptibility in Fig.3.9(a), which let us confirm that the edge
bands only contributes to the low energy plasmonic excitation even applies the external magnetic
field. When comparing with Fig.3.5(a), we can conclude that the Zeeman field in perpendicular
direction doesn’t influence the bulk mode excitations but only affect the edge modes ( ω < 2eV ).
One essential effect for the edge mode is that the charge distributions of the edge mode become
less localized in Fig.3.9(b) and (c). To demonstrate this points, we calculate the dependence of the
percentage weight of the charge amount on the edges as a function of B
z
(see Fig.3.9(d)) and show
that the weight decreases when the magnetic field becomes stronger. The external field splits two
44
pairs of edge mode energy bands that transverse the gap and changes the crossing points of them
with the Fermi level, see in Fig.3.7(a). The wave function delocalizetion of the edge mode bands
(Fig.3.9(c)(d)) that cross through Fermi level leads to the change in the plasmonic distribution.
This finding can provide a possible method to tune the strength of localization of the edge mode
excitations in Kane-Mele model experimentally.
Figure 3.10: EELS and charge density of big gap Kane-Mele model with Zeeman field (a):
EELS of the Kane-Mele model in QSH phase at momentum K=π/8L with different Zeeman
field in Z direction. Spin-orbit interaction t
SO
= 0.15t. (b)(c): Charge distribution modulation
of the edge state plasmonic excitation at peek p
1
, p
2
for B
z
= 0.15t. (d): The dependency of the
charge weight (percentage) on the edge with the Zeeman field B
z
for peak p
1,2
.
To focus on the influence of Zeeman field on edge states and reduce the effects from bulk, we
use a relatively stronger spin-orbit interaction t
SO
which can increase the bulk gap size (see energy
band spectrum with Bz= 0.15t in the inset of Fig.3.10(a)). When looking at the the energy bands
that traverse the bulk gap, we can find that the stronger spin-orbit hopping decreases the shift of
edge modes near Fermi level comparing with Fig.3.7(a). This change will simultaneously decease
the effect of delocalization and makes the plasmon edge state more localized when experienced
45
Zeeman field (see Fig.3.10(b)(c)). The quantitative analyze in Fig.3.10(d) shows that the concen-
tration on the edges is still decreasing as the Zeeman field getting stronger but the charge weight
remains much more higher than the case with weaker spin-orbit (see in Fig.3.9(d)). In the EELSs,
the peaks of the edge state plasmonic excitations (low energy region) shift to lower energies with
increasing magnetic field, but the bulk plasmons ( ω> 1.5eV ) stay basically unchanged. The mag-
netic field perpendicular to the ribbon plane splits the energy of spin-up and spin-down, thus, shift
the collective excitation energies.
3.3.5 Real space plasmonic excitations in diamond-shaped nanoflakes
In the above sections, we discussed the plasmonic excitations of the Kane Mele-model in nano-
ribbons. In this section, we explore plasmons in a purely real space geometry, i.e., diamond-shaped
nanoflake structures with four zigzag boundaries, as shown in Fig.3.11(a).
Figure 3.11: Structure and wave function of Kane-Mele model in nanoflake (a) Structure of
a diamond-shaped honeycomb lattice nanoflake with zigzag edges. (b) Probability density of the
single particle edge modes on the zigzag boundaries of the Kane-Mele model at energies± 0.075
eV .
Examining the single particle wave functions of this model with t
SO
= 0.2t and t
v
= 0, we find
edge states at ultra-low energies, shown in Fig.3.11(b). Next, we study the plasmonic excitations
on the nanoflake. To perform the calculation in real space, we modify the dielectric function to
ε
RPA
(ω)= I− Vχ
0
(ω)., where V is the real space Coulomb interaction matrix:
46
V
ab
=







e
2
/κ|r
a
− r
b
| a̸= b
U
0
/κ a= b
(3.14)
with the same U
0
in eq.3.6 and
[χ
0
(ω)]
ab
=
1
V
∑
i, j
f(E
i
)− f

E
j

ω+ iγ+ E
i
− E
j
× ψ
∗ ia
ψ
ib
ψ
∗ jb
ψ
ja
.
(3.15)
the real-space charge susceptibility function. Here, the matrix has size 2N× 2N, and N is the
number of sites of the nanoflake, where each site has two spin of freedom. a and b label these
degrees of freedom and are in the range [0,2N]. The electron energy loss spectrum (EELS) and
induced charge densityρ
0
(ω) can be extracted from a similar formula as discussed in Sec.?? but
without the momentum part.
Figure 3.12: Plasmonic charge density of Kane-Mele model in nanoflake Real space plasmons
in the QSH phase of the Kane-Mele model on a diamond-shaped nanoflake at three different ener-
gies. t
SO
= 0.2t and t
v
= 0.
To analyze the collective excitations in this real-space Kane-Mele system, we determine the
plasmonic charge distributions on a diamond-shaped lattice with uniform lattice spacings, a =
1.42
˚
A, and show three chosen instances in Fig.3.12. Here, we observe that for low energies ω,
the induced plasmonic charge modulations are localized at the boundaries of the nanoflake, see
Figs.3.12(a) and (b). In contrast, the plasmons on a graphene lattice nanoflake do not have edge
modes but only corner modes. Also, we find that with increasing energy there the edge modes have
47
a faster spatial modulation, see Fig.3.12(b). Finally, Fig.3.12(c) shows a bulk mode plasmonic
excitation of the nanoflake in the higher energy regime.
48
Chapter 4
FastPivot and Inverse Problem: Plasmons Design
4.1 Background
4.1.1 Idea of Design and Artificial Intelligent
Physical law allows us to calculate response from a specific system. The development of computer
science may help up do the steps inversely. Generally speaking, inverse problems in computational
physics present the following challenges: (a) The inverse map from the response to the system is
only implicitly defined, i.e., we cannot input the target response into a mathematical equation; (b)
The inverse map is not unique; (c) The search space is very large; and (d) There is a substantial
involvement of complex numbers, especially at microscopic scales. Inverse problems are particu-
larly hard to solve when the map from the system to its response involves many cascading steps;
Each step may be hard to reverse or may introduce branching possibilities if its inverse is not
unique. For these reasons, inverse problems are fundamentally combinatorial in nature; Hence, we
can benefit from techniques developed in computer science, artificial intelligence (AI), machine
learning (ML), and operations research (OR).
Some inverse algorithms have been designed for specific inverse problems using data-driven
methods like neural networks [93–95]. However, such methods are mostly tailored to specific
problems and are not broadly applicable. In fact, it is difficult to train a neural network to learn the
inverse mapping of a step that computes the eigenvalues of a matrix satisfying a certain property
49
P. This is because it is hard to identify a unique matrix that satisfies P given only its eigenvalues.
So far, very few general principles have been developed for the design of inverse algorithms. The
Monte Carlo method [96] is one such general principle.
In this chapter, we present a novel algorithm, called FastPivot, for solving inverse problems.
FastPivot introduces a general principle that can be described as follows. We start with a system
state and invoke alternating forward and backward passes through the system variables. In the
forward pass, we calculate the system’s response from its current state. In the backward pass, we
percolate a small amount of information from the target response back to the system variables. The
inner loop implements several alternating forward and backward passes in the hope of convergence.
The outer loop keeps track of the best solution found so far and triggers algorithm termination
based on the availability of computational resources.
While the Monte Carlo method works on a discrete space, i.e., on a lattice, FastPivot works
on a continuous space and produces good quality solutions efficiently. We demonstrate its success
on the DoS design problem and plasmons design problem], i.e., the inverse problem of placing
atoms in a bounded region using an STM to achieve target responses in the DoS or plasmons. We
also compare FastPivot to the Monte Carlo method and analyze various empirical observations. In
general, we observe that FastPivot outperforms the Monte Carlo method in higher dimensions and
is better suited for continuous spaces.
4.1.2 Atoms-up design in manmade nanostructure
The STM is a type of microscope that can be used for imaging surfaces at the atomic level. It
does this by applying an electrical voltage to a very sharp metal wire tip that scans the surface
(substrate) very closely. It can also be used to manipulate individual atoms. In fact, the STM
can be used to place individual atoms at any location in a bounded region on a specific substrate.
IBM’s nanophysicists have even created a minute-long film with 242 stop-motion frames, called
“A Boy and His Atom”, by moving individual carbon atoms on a copper substrate. Fig 4.1 shows
the schematic and an actual STM.
50
Figure 4.1: STM schematic and picture The left panel, borrowed from Wikipedia, shows the
schematic of an STM. The right panel, borrowed fromhttps://tmi.utexas.edu/facilities/
instrumentation/scanning-tunneling-microscopes, shows an actual STM.
Figure 4.2: DoS corresponding to configurations of atoms . (a), (b), and (c) show three different
configurations with 4, 4, and 8 atoms, respectively. (d), (e), and (f) show the DoS corresponding
to (a), (b), and (c), respectively.
Because of its ability to place individual atoms at desired locations on a substrate, the STM
supports nanofabrication. In turn, nanofabrication is a cornerstone capability in nanotechnology
for creating nanostructures. Because of their size, nanostructures often have special properties
that macroscopic structures of the same material may not have. These special properties can be
exploited for various applications, including drug delivery and quantum dots.
51
In this context of using an STM for nanofabrication, one of the fundamental inverse problems
is the following: Given N atoms of a certain kind and a bounded region on a substrate, where
should we place these atoms to achieve a target response in the DoS? In order to understand this
question, we first need to understand the “forward” version of this problem: For a given placement
(configuration) of N atoms, what is the DoS that it determines? This determination is done as
described below.
First, we subscribe to the tight-binding model in solid-state physics. The tight-binding model [97,
98] is an effective approach for estimating the electronic band structure by using approximations
of the atomic wave function. Even though the tight-binding model is a single-electron model, it
provides good results for a wide variety of solids. We consider a non-periodical system, which has
the following Hamiltonian:
ˆ
H =− ∑
i, j
t
i, j

ˆ c
†
i
ˆ c
j
+ ˆ c
i
ˆ c
†
j

, (4.1)
where ˆ c
†
i
and ˆ c
i
represent the electron creation and annihilation operators, respectively, at the site
r
i
, and t
i, j
is the spatial decay long-range hopping term given by the power law:
t
i, j
=
t

r
i
− r
j

q
. (4.2)
Here, t is a constant, and r
i
is the position vector of atom i, for 1≤ i≤ N. Moreover, q is the
power decay parameter that reflects an algebraic variation of the overlap integral with inter-atomic
separation. It is specific to the material and is measurable via experiments [99].
Given the position vectors r
1
,r
2
...r
N
of the N atoms, the steps required to determine the DoS
are as follows:
1. Determine the Hamiltonian H as the N× N matrix with H
i, j
= t
i, j
.
2. Determine the eigenvalues e
1
,e
2
...e
N
of H.
3. Determine the DoS by placing uniform Lorentzian functions of appropriate height and width
centered at each of the eigenvalues.
52
Essentially, the DoS describes the proportion of states occupied by the system at each energy
level. It is peaked at the eigenvalues of H. Fig 4.2 shows the DoS for three different configurations
of atoms.
The inverse problem, of interest in this chapter, is to find the position vectors r
1
,r
2
...r
N
in a
bounded region given the target eigenvalues of H, i.e., the positions of the peaks in the target DoS.
4.1.3 The prediction of plasmonic excitations
The benefit of STM that can place atoms on the surface of some materials also gives us a possible
way to design the particles that can achieve our target plasmonic responses at specific energies and
can be meaning for nanofabrication engineer. The ”forward pass” of the problem is getting the
properties of plasmonic excitations for the specific structure and Hamiltonian, which includes the
energy or frequency of the excitations and the strength of them. From the experimental method
of STM nanofabrication, the inverse problem of plasmon is the following: Given N atoms of
a certain kind (certain hopping and interaction) and a bounded region on a two-dimensional or
three-dimensional substrate, where should we place these atoms to achieve a target response in the
plasmonic excitation spectrum?
We design the inverse problem of plasmon also by real-space random phase approximation(RPA)
as we talked in Chapter 2. Instead of the Hamiltonian of topological insulators, we use the simple
tight-binding model Eq. 4.1 same as in the last section for the calculation of charge susceptibility
function Eq. 3.7. Also, the real-space Coulomb interaction can be obtained by Eq. 2.18 from the
location of atoms. Then, the plasmonic excitation can be derived by the random phase approxima-
tion.
Given the position vectors r
1
,r
2
...r
N
of the N atoms, the steps required to determine the
electron energy loss spectrum (EELS) are as follows:
1. Determine the Hamiltonian H as the N× N matrix with H
i, j
= t
i, j
.
2. Determine the Coulomb interaction V as the N× N matrix with V
a,b
.
53
3. Determine the eigenvalues e
1
,e
2
...e
N
and eigenvectorsφ
1
,φ
2
...φ
N
of H.
4. Calculate the charge susceptibility function[χ
0
(ω)]
ab
.
5. Calculate the real-space RPA dielectric response functionε
RPA
(ω) and diagonalize it to get
electron energy loss spectrum (EELS).
6. Result the plasmonic excitations by location the peaks’ energy position and amplitude.
The inverse problem of plasmon design, is to find the position vectors r
1
,r
2
...r
N
in a bounded
region given the target plasmonic excitation pairs(ω
1
,EELS
1
),(ω
2
,EELS
2
)...(ω
M
,EELS
M
).
4.2 Method
In this section, we introduce the two approaches to solve the inverse problems: Monte Carlo
method (a traditional local search method) and the FastPivot method. We will illustrate how to
apply the methods to design density of state for the long-range tight-binding model by presenting
and explaining the pseudocodes. The pseudocodes of these two approaches for solving plasmon
design problem are in the Appendix.
4.2.1 Local Search Approach
We firstly provide a description of the Local Search (Monte Carlo) algorithms for our inverse
problems.
Let the target eigenvalues be λ
1
≥ λ
2
...λ
N
. Of course, all these eigenvalues are real since
the Hamiltonian is always a symmetric square matrix. For position vectors r
1
,r
2
...r
N
of the N
atoms in a K-dimensional space, let R denote the K× N position matrix with columns r
1
,r
2
...r
N
.
We note that R represents a configuration of the N atoms in K-dimensional space. We also note
that K is usually 2 or 3. Let H be the Hamiltonian corresponding to a position matrix R; and
let e
1
≥ e
2
...e
N
be its eigenvalues. R can be assigned a score to measure how close it is to
54
achieving the target eigenvalues. One such score is the Root Mean Square Error (RMSE) given by
score(R)=
q
∑
N
i=1
(e
i
− λ
i
)
2
N
.
The Local Search algorithm works on a discretization of the continuous space and is inherently
limited by the resulting discretized grid. It uses the Metropolis-Hastings importance sampling
procedure [96] in its inner loop and generally works as follows.
In the outer loop, the algorithm runs a user-specified number of trials. In the inner loop, i.e.,
in each trial, the algorithm starts by generating a position matrix R
0
by choosing the position
vector of each atom uniformly at random in the bounded region. The corresponding Hamiltonian
H
0
is also computed. Subsequently, with each increment in the iteration number i, the algorithm
updates the current position matrix R
i
and computes the corresponding Hamiltonian H
i
. R
i+1
is
constructed from R
i
after the consideration of moving a randomly chosen atom from its current
position to a neighboring empty position on the discretized grid to obtain R
′
. If score(R
′
)<
score(R
i
), the move is accepted and R
i+1
is set to R
′
. If not, the move is accepted with probability
e
score(R
i
)− score(R
′
)
k
B
T
, where k
B
is the Boltzmann constant and T is a “temperature” that starts high and
decreases according to an annealing rule as the iteration number increases. The inner loop is
repeated for a user-specified maximum number of iterations before the next trial of the outer loop
is initiated. Upon termination, the algorithm returns the position matrix with the lowest recorded
score.
4.2.2 FastPivot Approach
The FastPivot algorithm works on a continuous space and does not have the inherent limitations of
a discretization. The pseudocode for the FastPivot algorithm is presented in Algorithm 1. Algo-
rithm 1 uses Algorithm 2, Algorithm 3, and Algorithm 4 as helper functions.
The input to Algorithm 1 consists of: (a) the number of particles N, (b) the target eigenvalues
λ
1
≥ λ
2
...λ
N
, (c) the number of dimensions K, (d) the bounded region of space specified by
B
1
,B
2
...B
K
and defined by the K-dimensional orthotope between the farthest corners (0,0...0)
and (B
1
,B
2
...B
K
), and (e) the power used in the Hamiltonian terms q. In addition to these, the
55
Algorithm 1: FastPivot: An algorithm that embeds N particles in a K-dimensional
space to achieve target eigenvalues of their interaction Hamiltonian.
Input: the number of particles, N; the target eigenvalues,λ
1
≥ λ
2
...λ
N
; the number of
dimensions, K; the bounding length on each dimension, B
1
,B
2
...B
K
; the power
used in the Hamiltonian terms, q; the seed position matrix, R
0
; the sampling range
vector,⃗ σ.
1 maximum number of iterations of the inner loop,MaxSteps; maximum number of
iterations of the outer loop,MaxTrials; cutoff improvement parameter in the
inner loop,γ; threshold error parameter in the outer loop,ε. Output: K× N
predicted position matrix, R.
2 LetΛ be a diagonal matrix of the target eigenvaluesλ
1
≥ λ
2
...λ
N
.
3 Let R be a K× N matrix with columns r
1
,r
2
...r
N
.
4 bestScore← ∞.
5 bestR← Null.
6 for trial= 1,2...MaxTrials do
7 for i= 1,2...N do
8 Pick r
i
to be a K-dimensional vector, where its k-th coordinate, for 1≤ k≤ K, is
chosen uniformly at random from the interval
[min(0,r
0
i
[k]− ⃗ σ[k]),max(B
k
,r
0
i
[k]+⃗ σ[k])].
9 H← ComputeHamiltonian(R,q).
10 currentScore← Score(H,Λ).
11 previousScore← ∞.
12 step← 1.
13 while (step≤ MaxSteps or currentScore− previousScore≤− γ) and
(currentScore>ε) do
14 previousScore← currentScore.
15 R← EigenChange(R,H,Λ,K,q).
16 H← ComputeHamiltonian(R,q).
17 currentScore← Score(H,Λ).
18 step← step+ 1.
19 if currentScore< bestScore then
20 bestScore← currentScore.
21 bestR← R.
22 if bestScore≤ ε then
23 Break.
24 return bestR.
user can also choose to input: (f) an initial guess for the position matrix R
0
, and (g) an allowed
perturbation on it using the sampling range vector⃗ σ. The output of Algorithm 1 is a predicted
position matrix R intended to achieve the target eigenvalues.
56
Algorithm 2:EigenChange: A gradient-descent algorithm that updates the position ma-
trix based on the target eigenvalues.
Input: K× N position matrix, R; the Hamiltonian, H; the target diagonal eigenvalue
matrix,Λ; the number of dimensions, K; the power used in the Hamiltonian terms,
q.
1 number of updates,τ; learning rate,η. Output: modified position matrix, R.
2 Let V× E× V
− 1
be the eigendecomposition of H, where E is a diagonal matrix of
non-increasing eigenvalues, and the columns of V represent the corresponding
eigenvectors.
3 D← V× Λ× V
− 1
.
4 for update= 1,2...τ do
5 R
′
← R.
6 for i= 1,2...N do
7 grad← 2∑
j̸=i
(D
i, j
+
1
∥r
′
j
− r
′
i
∥
q
)(
− q
∥r
′
j
− r
′
i
∥
q+2
)(r
′
i
− r
′
j
).
8 r
i
← r
′
i
− ηgrad.
9 return R.
Algorithm 3: ComputeHamiltonian: An algorithm that computes the Hamiltonian
given the position matrix and the nature of the pairwise interactions.
Input: position matrix, R; the power used in the Hamiltonian terms, q.
Output: the Hamiltonian, H.
1 Construct an N× N matrix H with the(i, j)-th term defined as follows:
2 if i= j then
3 H
i, j
← 0.
4 else
5 H
i, j
←− 1
∥r
i
− r
j
∥
q
.
6 return H.
Algorithm 4: Score: A scoring function for measuring the difference between current
eigenvalues and target eigenvalues.
Input: the Hamiltonian, H; the target diagonal eigenvalue matrix,Λ.
Output: a score, RMSE.
1 E← a diagonal matrix of non-increasing eigenvalues of H.
2 RMSE← q
∑
N
i=1
(E
i,i
− Λ
i,i
)
2
N
.
3 return RMSE.
Algorithm 1 also uses several parameters. It uses MaxTrials for the maximum number of
iterations of the outer loop and MaxSteps for the maximum number of iterations of the inner
57
loop. It also uses a cutoff improvement parameter γ in the inner loop to recognize quiescence
and a threshold error parameter ε in the outer loop to recognize convergence to a high-quality
configuration.
In lines 1-4, Algorithm 1 performs initialization. It stores the target eigenvalues in a diagonal
matrixΛ withΛ
i,i
set to λ
i
, for 1≤ i≤ N. It also initializes bestR and bestScore, which are in-
tended to record the best configuration encountered during runtime and its corresponding score,
respectively. In lines 5-22, the algorithms runs its outer loop for a maximum of MaxTrials itera-
tions. In lines 12-17, the algorithm runs its inner loop for a maximum of MaxSteps iterations. In
lines 6-11, the outer loop sets up the initialization for the inner loop. In lines 18-22, the outer loop
processes the results of the inner loop. In line 23, the algorithm returns the best position matrix
recorded across all iterations of the outer loop.
In lines 6-7, the outer loop initializes a position matrix by picking each of its columns, i.e.,
the position vectors of the N atoms, at random. Each position vector is restricted to be in the
K-dimensional orthotope between the farthest corners (0,0...0) and (B
1
,B
2
...B
K
). This can be
done by picking the k-th coordinate of each position vector r
i
to be within the interval [0,B
k
],
for 1≤ k≤ K. If the user specifies an area of interest, this interval can be narrowed down to an
interval of length 2⃗ σ[k] centered around the coordinate of interest in that dimension r
0
i
[k]. In lines
8-9, the outer loop computes the Hamiltonian and its score for the initialized position matrix. In
lines 18-20, the outer loop checks whether the inner loop just encountered a position matrix that is
better than the current best. If so, it updates bestR and bestScore. In lines 21-22, the outer loop
checks whether the algorithm has converged to a high-quality configuration with a score no greater
than the user-specified threshold error parameter ε. If so, no further iterations of the outer loop are
deemed necessary.
In lines 12-17, the inner loop continues only while “reasonable” progress is being made. That
is, if the improvement in score between consecutive iterations is no greater than a user-specified
cutoff parameterγ, we declare quiescence and break out of the inner loop. The inner loop makes
calls to Algorithm 2, Algorithm 3, and Algorithm 4. Algorithm 3 computes the Hamiltonian H
58
for a given position matrix R according to the specified value of q. Algorithm 4 computes the
score of H, corresponding to a position matrix R, by comparing its eigenvalues against the target
eigenvalues. In particular, it returns the RMSE between the lists of eigenvalues sorted in non-
increasing order.
In lines 15-16, the inner loop makes a forward pass, i.e., it computes the score of a position
matrix via computing the Hamiltonian. In line 14, the inner loop makes a critical backward pass
using Algorithm 2. The objective is to percolate information from the target eigenvalues back to the
position matrix. While this “reversal” of steps is the essence of the inverse problem and is already
known to be hard, the main idea is based on the observation that it becomes easy in the context of
the most recently completed forward pass. In line 1, Algorithm 2 computes an eigendecomposition
of H, where E is a diagonal matrix of non-increasing eigenvalues, and the columns of V represent
the corresponding eigenvectors. In line 2, it uses this eigendecomposition to reconstitute a new
Hamiltonian D by merely replacing the diagonal matrix of eigenvalues E with the diagonal matrix
of the target eigenvaluesΛ. Such a reversal step from the target eigenvalues to a new Hamiltonian
with the same eigenvalues is enabled only because of the most recently completed forward pass
that yields H (and consequently its eigenvectors).
Ideally, Algorithm 2 also needs to execute a reversal step from the new Hamiltonian D to a
new position matrix R. Although doing this reversal directly is already deemed to be hard, it be-
comes slightly easier in the context of the most recently completed forward pass since the current
values of the position vectors r
1
,r
2
...r
N
are available. However, this is not completely straight-
forward either. A second effective idea used in the backward pass is based on the observation that
it suffices for us to make only small updates in the direction of the reversal, since doing so across
many forward and backward passes achieves the same overall effect. In lines 3-7, Algorithm 2
does precisely this using τ steps of gradient descent with respect to the position vectors aimed at
minimizing the Frobenius norm of D− H. The updates use a user-specified learning rate η.
59
Figure 4.3: Comparison of Local Search and FastPivot algorithms in DoS Problem (a) and (b)
show the DoS and eigenvalues, respectively, for the configurations returned by the Local Search
and FastPivot algorithms in comparison with the target DoS and eigenvalues on a test instance with
8 atoms in a 2D bounded region of space. (c) and (d) show the DoS and eigenvalues, respectively,
for the configurations returned by the Local Search and FastPivot algorithms in comparison with
the target DoS and eigenvalues on a test instance with 8 atoms in a 3D bounded region of space.
Algorithm N= 4 N= 8 N= 16 N= 32 N= 64
Local Search 0.1283 0.3126 0.5007 0.9842 0.7322
FastPivot 0.0596 0.2958 0.6120 1.9073 2.5399
Local Search 0.1382 0.0949 0.1412 0.1701 0.2323
FastPivot 0.0540 0.0947 0.1324 0.1396 0.1840
Table 4.1: Density of State Design Problem: Effect of Size and Dimension, A comparison in
2D (first two rows) and 3D (last two rows).
4.3 Results and Discussion
4.3.1 Warm up : Atoms-up Design
In this section, we present our empirical results and analyses. Numerical experiments were con-
ducted on a 3.6 GHz AMD Ryzen 5 3600 6-core CPU with 16 GB RAM. All algorithms were
implemented in Python 3.6. We used the tight-binding model and placed atoms in either a 2D or
3D bounded region of space to achieve the target DoS eigenvalues. To facilitate a fair comparison
60
of the Local Search and FastPivot algorithms, we gave both algorithms the same number of inner
loop and outer loop iterations.
All our test instances were “reverse engineered”. That is, they were generated as follows. We
first pick a random configuration of the N atoms in the bounded region of space. We compute
the Hamiltonian using the tight-binding model and then its eigenvalues where the DoS is known
to peak. We provide these eigenvalues as input to the Local Search and FastPivot algorithms and
challenge them to reconstruct the original configuration. Of course, the algorithms are also free
to construct any other configuration that may have the same eigenvalues. The reverse engineering
merely validates each test instance and assures us that the DoS is achievable.
Fig 4.3 shows the comparative performances of the Local Search and FastPivot algorithms on
two example test instances: Each with 8 atoms but one in 2D and one in 3D. It is easy to see that
FastPivot outperforms the Local Search algorithm in these cases.
Table 4.1 shows the comparative performances of the Local Search and FastPivot algorithms
on a suite of test instances. Each entry in the table represents the RMSE values averaged over
the same 20 test instances supplied to both the algorithms. The test instances are categorized
according to different values of N. The top two rows correspond to test instances that use a 2D
bounded region of space; and the bottom two rows correspond to test instances that use a 3D
bounded region of space. The general observation is that FastPivot marginally outperforms the
Local Search algorithm on: (a) 3D test instances, and (b) 2D test instances when N is small.
Although thLocal Search algorithm marginally outperforms FastPivot on 2D test instances
when N is large, this does not automatically imply that the Local Search algorithm is preferred on
such instances. This is because, interestingly, FastPivot marginally outperforms the Local Search
algorithm on 2D test instances when N is small. This observation can be exploited to design a
“hierarchical” version of the FastPivot algorithm for larger values of N. A comparison of the pro-
posed hierarchical version of FastPivot against the Local Search algorithm is delegated to future
work.
61
4.3.2 Plasmonic excitations Design
After the discussion of the density of states design problem, we now turn to design the plasmonic
excitation. The calculation is also based on the tight-binding model with 2D or 3D bounded re-
gion of space. As the many-body interaction collective excitation, plasmon calculation is much
more complex even though we consider only real-space as in Chapter 2. There are two types of
plasmonic prediction :
(1) since the electron energy loss spectrum (EELS) is related with the internal energy and we
usually focus on the 1st EELS , the input parameter will contain the energy window [ω
s
,ω
e
], the
length of the region and the number of discrete energies that be contained in the window is decided
by the input. The pending energy region and the corresponding excitation is in the shape of pairs:
[(ω
1
, eels
1
), (ω
2
, eels
2
)... (ω
m
, eels
m
)], in which the eelss here represent the 1st EELS. The peaks
in the region have different amplitude which we want to make our predicted peaks in the correct
position as well as the amplitude. So, the score in the plasmonic design problem is the sum of bias
for each pairs (ω, eels), our target is to minimize the score.
(2) we can also design the system and predict all of the target EELSs, i.e., 1st EELS, 2nd EELS
... Nth EELS for a specific energy ω. In this case, each EELS will be count, which means all of the
eigenvalues of the dielectric matrixε
RPA
in Eq.2.19 make the same contribution to the prediction.
Similar with the atoms-up design problem, we will first take a random placement of atoms and
calculate all of the EELS at a specific energy as the target, then start the prediction and design. The
score for the second type plasmon design problem will be the sum of bias for the target EELSs and
predicted EELSs.
In the prediction, we apply both the Local Search approach and FastPivot approach to the
plasmonic problem and compare their accuracy and speed. Firstly, we test our methods by a small
case discrete energy region : [(ω
1
, eels
1
)... ] = [(0, 0), (1, 0.5), (2, 0), (3, 0), (4, 0)] which with one
peak atω= 1 and show the target and predictions in Fig. 4.4(a). The plot contains the target EELS
and EELSs predicted by the two methods. In this test case, both of the approaches can predict the
position of the excitation as well as the strength of it. The prediction proves that our framework
62
Figure 4.4: Comparison of Monte Carlo and FastPivot algorithms in plasmon Problem (a)
and (b) show the EElSs, respectively, for the predicted results returned by the Local Search and
FastPivot algorithms in comparison with the target EELS on a test instance with 8 atoms in a 2D
bounded region of space.
Algorithm N= 4 N= 8 N= 16 N= 32
Local Search 0.5828 1.2725 1.7001 2.4278
FastPivot 0.6683 0.6206 0.6856 0.9371
Local Search 1.5600 1.4027 1.8000 4.0994
FastPivot 0.7005 0.6981 0.76784 0.7889
Table 4.2: Plasmon Design Problem of all EELSs: Effect of Size and Dimension, A comparison
in 2D (first two rows) and 3D (last two rows).
works well in the plasmon design problem and can design the plasmonic spectrum in the target
energy region. We also test a relatively larger case in Fig. 4.4(b) for [(ω
1
, eels
1
) ... ] = [(0, 0),
(1, 0), (2, 0.05), (3, 0.2), (4, 0.05), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0)] which has an excitation peal
at ω = 3. The plots also shows the accuracy of both the approaches which can precisely get the
correct position and amplitude of the plasmonic excitation.
Then, we test the second type design problem by list the score table for different number of
atoms: N= 2,4,...64 and different dimension D= 2 and 3 on the specific energy ω= 2.0eV in the
Table 4.2. In this table, we can definitely found that FastPivot has a better score than Local Search
approach in any size and dimension. Even when the size increased, Local Search method cannot
get an accurate prediction because the increasement of the search space, but FastPivot can achieve
63
the prediction with better score. This test prove the highlight of FastPivot approach in plasmonic
design problem, it will have a good perform in the larger size and higher dimension.
The results of the accuracy for the plasmon design approaches show the comparison of the
methods and the benefit of FastPivot. Also, the two types of prediction will be useful in different
cases. The first type problem is the most typical design problem that we face and is meaningful
for engineer. Usually, in material science area, people are curious about the energy of the exci-
tations and the corresponding strength for a specific material. If we know the Hamiltonian of the
material, the idea of inverse problem can provide the way to arrange the material atoms and get the
excitations at any energy that we want also with the proposal strength. The whole energy region
is very wide so we can focus on a small region of energy and design the excitation inside. By
using the design tools we propose, people can get the material which has the plasmonic excitation
in the specific energy region with the specific amplitude. The second type prediction provides the
possible way to design the whole EELSs of the system and can help us get more information.
64
Chapter 5
Conclusion
In the first chapter of this paper, we have examined the effects of added diatomic molecules on the
plasmonic modes of 1D SSH model and its mirrored invariant, comparing them to the benchmark
of a simple metallic chain. By analysis of the electron energy loss excitation spectrum and of the
real space charge distributions of the plasmon modes, we conclude that the position of the local
perturbation is the key parameter to control the plasmons in 1D TIs. When the perturbation is on
or near the edges of the topological insulator, the plasmonic excitations in the topologically non-
trivial regime, i.e., their degeneracy and their charge distribution, will be significantly affected. In
contrast, the plasmonic excitations become less affected when the local perturbation is far from
the edges. We also identified conditions under which charge transfers from the host chain to the
added molecules. Here, the internal hopping t
′
within the diatomic molecule plays an important
role. It will be interesting to further analyze analogous effects of perturbing molecules in higher-
dimensional topological systems that harbor dispersive surface modes, such as the two-dimensional
SSH model and graphene, which may become localized due to the impurities.
In the second chapter, we have examined the plasmonic excitations of the Kane-Mele model
which hasZ
2
topological order, comparing them to the benchmark case of a graphene nano-ribbon.
We have studied the electron energy loss excitation spectrum in momentum space and the charge
distributions of the plasmon modes within a unit cell, with the screened momentum-space Coulomb
interaction implemented on the RPA level. Two continua of bulk branches as well as localized plas-
monic edge states are observed in the non-trivial quantum spin Hall phase, showing a collective
65
response composed of single-particle bulk energy bands and edge states. However, as collective ex-
citations, plasmons also display significant differences from single-particle states. Their excitation
energy and localization are affected by the geometry and by the Coulomb interactions. Notably,
these localized plasmonic excitations in Kane-Mele model nano-ribbon can be selected by spin,
which provides a promising experimentally control. Furthermore, we have analyzed the effects of
external Zeeman fields on the system in both perpendicular and parallel directions, which changes
the plasmon charge distribution and breaks the topological quantum spin Hall phase, i.e., opens an
edge gap. Finally, we have studied the Kane-Mele model on nanoflakes and observed plasmonic
edge modes. It will be interesting to further analyze the effects of the Rashba term in Kane-Mele
model as well as the effects of geometry perturbations. Also, we would like to further explore
plasmons in the three-dimensional Bernevig-Hughes-Zhang (BHZ) mode[6] which has two elec-
tron bands and two hole bands.
In the last chapter, we presented FastPivot, an algorithm for solving inverse problems. FastPivot
starts with a system state and invokes alternating forward and backward passes through the system
variables. In the forward pass, it leads the current state of the system to its response like density of
states (DoS) or plasmon excitation. In the backward pass, a small amount of information is allowed
to percolate from the target response back to the system variables. We demonstrated the success of
FastPivot in the context of using an STM for nanofabrication. We addressed an important problem:
Given N atoms of a certain kind and a bounded region on a substrate, where should we place these
atoms to achieve a target response in the DoS or plasmon spectrum? We showed that FastPivot
produces good quality solutions efficiently for this problem. We also compared FastPivot to Monte
Carlo methods and observed two advantages of FastPivot: (1) It outperforms Monte Carlo methods
in higher dimensions; and (2) It solves the problem on continuous spaces.
66
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72
Appendices
A Full Pseudo Code for Plasmons Design Problem
73
Quantity Description Type
N the number of particles non-negative integer
K the number of dimensions non-negative integer
B
k
, 1≤ k≤ K the bounding length on dimension k non-negative real number
ω frequency non-negative real number
eels electron energy loss for a specific ω non-negative real number
q the power used in the Hamiltonian terms real number
e the magnitude of the charge of a single elec-
tron
non-negative real number
ε
b
the background dielectric constant non-negative real number
U
0
the onsite Coulomb interaction parameter in
vacuum
non-negative real number
γ the finite broadening constant non-negative real number
f() the Fermi function approximated by a zero-
temperature step function
f(x) = 1/(1+ e
x− x
0
k
B
T
), where
x
0
is the Fermi level, k
B
is the
Boltzmann constant, and T is
the temperature
Table 5.1: Physical Parameters
Quantity Description Type
MaxTrials Algorithm 5: maximum number of iterations
of the outer loop
non-negative integer
MaxSteps Algorithm 5: maximum number of iterations
of the inner loop
non-negative integer
C Algorithm 5: cutoff improvement parameter
in the inner loop
non-negative real number
ε Algorithm 5: threshold error parameter in
the outer loop
non-negative real number
η Algorithm 10: learning rate non-negative real number
τ Algorithm 11: infinitesimal increment non-negative real number
Table 5.2: Algorithmic Parameters
74
Quantity Description Type
λ
m,1
,λ
m,2
...λ
m,N
the target eigenvalues for m-
thω
complex numbers
Λ
m
the diagonal matrix of
λ
m,1
,λ
m,2
...λ
m,N
N× N complex matrix
R=

⃗ r
1
⃗ r
2
...⃗ r
N

the matrix of position vectors
along columns
K× N real matrix
H=Ψ× Ξ× Ψ
− 1
the Hamiltonian matrix N× N real matrix
E
1
,E
2
...E
N
the eigenvalues of H in de-
creasing order of their norms
real numbers
Ξ the diagonal matrix of
E
1
,E
2
...E
N
N× N real matrix
⃗ ψ
1
,⃗ ψ
2
...⃗ ψ
N
the eigenvectors of H corre-
sponding to the above eigen-
values
N× 1 real vectors
Ψ=

⃗ ψ
1
⃗ ψ
2
...⃗ ψ
N

the matrix of eigenvectors of
H along columns
N× N real matrix
χ
0
(ω
m
) the charge susceptibility ma-
trix
N× N complex matrix
ε
RPA
(ω
m
)= P
m
× ∆
m
× P
m
− 1
the dynamic dielectric matrix N× N complex matrix
δ
m,1
,δ
m,2
...δ
m,N
the eigenvalues of ε
RPA
(ω
m
)
in decreasing order of their
norms
complex numbers
∆
m
the diagonal matrix of
δ
m,1
,δ
m,2
...δ
m,N
N× N complex matrix
⃗ p
m,1
,⃗ p
m,2
...⃗ p
m,N
the eigenvectors of ε
RPA
(ω
m
)
corresponding to the above
eigenvalues
N× 1 complex vectors
P
m
=

⃗ p
m,1
⃗ p
m,2
...⃗ p
m,N

the matrix of eigenvectors of
ε
RPA
(ω
m
) along columns
N× N complex matrix
e
m
the first EELS of ω
m
real number
Table 5.3: Variables
75
Algorithm 5:FastPivotPlasmon: the FastPivot algorithm adapted to the Plasmon prob-
lem.
Input: the number of particles, N; the list of frequency and target EELS value pairs
[(ω
1
,eels
1
),(ω
2
,eels
2
)...(ω
M
,eels
M
)]; the number of dimensions, K; the
bounding length on each dimension, B
1
,B
2
...B
K
; the power used in the
Hamiltonian terms, q.
Parameters: the charge of a single electron, e; the background dielectric constant,ε
b
; the
onsite Coulomb interaction parameter in vacuum, U
0
; the Fermi function
approximated by a zero-temperature step function, f(); finite broadening, γ;
learning rate,η; maximum number of iterations of the inner loop,
MaxSteps; maximum number of iterations of the outer loop,MaxTrials;
cutoff improvement parameter in the inner loop, C; threshold error
parameter in the outer loop,ε.
Output: K× N predicted position matrix, R.
1 for m = 1, 2 . . . M do
2 Setλ
m,1
= 0+ i
1
eels
m
.
3 for j = 2, 3 . . . N do
4 Sample y
j
uniformly at random from the interval[0,min(eels
m
,1)].
5 Set x
j
=
q
1− y
2
j
.
6 λ
m, j
= x
j
+ iy
j
.
7 LetΛ
m
be an N× N diagonal matrix with diagonalλ
m,1
,λ
m,2
...λ
m,N
.
8 Let R be a K× N matrix with columns r
1
,r
2
...r
N
.
9 bestScore← ∞.
10 bestR← Null.
11 for trial= 1,2...MaxTrials do
12 for i= 1,2...N do
13 Pick r
i
to be a K-dimensional vector, where its k-th coordinate, for 1≤ k≤ K, is
chosen uniformly at random from the interval[0,B
k
].
14 ForwardPass().
15 currentScore← Score([(e
1
,eels
1
),(e
2
,eels
2
)...(e
M
,eels
M
)]).
16 previousScore← ∞.
17 step← 1.
18 while (step≤ MaxSteps or currentScore− previousScore≤− C) and
(currentScore>ε) do
19 BackwardPass().
20 ForwardPass().
21 currentScore← Score([(e
1
,eels
1
),(e
2
,eels
2
)...(e
M
,eels
M
)]).
22 step← step+ 1.
23 if currentScore< bestScore then
24 bestScore← currentScore.
25 bestR← R.
26 if bestScore≤ ε then
27 Break.
28 return R. 76
Algorithm 6:ForwardPass: a macro associated with forward calculations.
1 for a,b∈{1,2...N} do
2 Compute V
a,b
, the entries of the N× N Coulomb matrix V, as follows:
3 if a= b then
4 V
a,b
=
U
0
ε
b
.
5 else
6 V
a,b
=
e
2
4πε
b
∥r
b
− r
a
∥
.
7 for a,b∈{1,2...N} do
8 Compute H
a,b
, the entries of the N× N Hamiltonian matrix H, as follows:
9 if a= b then
10 H
a,b
← 0.
11 else
12 H
a,b
←− 1
∥r
b
− r
a
∥
q
.
13 Compute the eigenvectorsψ
1
,ψ
2
...ψ
N
and their corresponding eigenvalues E
1
,E
2
...E
N
for H. // All are real.
14 for m = 1, 2 . . . M do
15 for a,b∈{1,2...N} do
16 Computeχ
0
(ω
m
)
a,b
, the entries of the N× N charge susceptibility matrixχ
0
(ω
m
),
as follows:
17 χ
0
(ω
m
)
a,b
= 2∑
i, j
f(E
i
)− f(E
j
)
E
i
− E
j
− ω
m
− iγ
ψ
∗ i,a
ψ
i,b
ψ
∗ j,b
ψ
j,a
. // ψs are real, but χ
0
(ω
m
)
is not real.
18 Compute the N× N dynamic dielectric matrixε
RPA
(ω
m
)= I− V× χ
0
(ω
m
).
19 Let P
m
× ∆
m
× P
− 1
m
be the eigendecomposition ofε
RPA
(ω
m
), where∆
m
is a diagonal
matrix of eigenvalues in non-increasing order of their norms with entries
δ
m,1
,δ
m,2
...δ
m,N
, and the columns of P
m
represent the corresponding eigenvectors.
20 Let e
m
beFirstEELS(δ
m,1
,δ
m,2
...δ
m,N
).
Algorithm 7:EELS: Returns the EELS value of a complex number.
Input: a complex number x+ iy.
Output: the EELS value of x+ iy.
1 return
y
x
2
+y
2
.
Algorithm 8:FirstEELS: Returns the EELS value of a list of complex numbers.
Input: a list of complex numbers x
j
+ iy
j
, for 1≤ j≤ L.
Output: the EELS value of the list of complex numbers.
1 return max
L
j=1
EELS(x
j
+ iy
j
).
77
Algorithm 9:Score: Returns a score measuring how close the dynamic dielectric matrix
is to the target.
Input: [(e
2
,eels
1
),(e
2
,eels
2
)...(e
M
,eels
M
)]
Output: a score measuring closeness to the target.
1 return∑
M
m=1
(e
m
− eels
m
)
2
.
Algorithm 10:BackwardPass: a macro associated with backward adjustments.
1 for m = 1, 2 . . . M do
2 Replaceλ
m,1
byδ
m,1
if EELS(δ
m,1
)≈ eels
m
.
3 for i= 2,3...N do
4 Replaceλ
m,i
byδ
m,i
if EELS(δ
m,i
) is less than or equal to eels
m
.
5 Compute the modified dynamic dielectric matrix ε
′
RPA
(ω
m
)= P
m
Λ
m
P
− 1
m
.
6 Compute the modified charge susceptibility matrix χ
′
0
(ω
m
)= V
− 1
(I− ε
′
RPA
(ω
m
)).
7 Create the objective function O(E
1
,E
2
...E
N
;ψ
1
,ψ
2
...ψ
N
)=
∑
M
m=1
∑
a,b
∥χ
′
0
(ω
m
)
a,b
− 2∑
i, j
f(E
i
)− f(E
j
)
E
i
− E
j
− ω
m
− iγ
ψ
∗ i,a
ψ
i,b
ψ
∗ j,b
ψ
j,a
∥
2
.
8 for i= 1,2...N do
9 Re(E
′
i
)= Re(E
i
)− η· ApprxPartial(O,Re(E
i
)).
10 Im(E
′
i
)= Im(E
i
)− η· ApprxPartial(O,Im(E
i
)).
11 for a= 1,2...N do
12 Re(ψ
′
i,a
)= Re(ψ
i,a
)− η· ApprxPartial(O,Re(ψ
i,a
)).
13 Im(ψ
′
i,a
)= Im(ψ
i,a
)− η· ApprxPartial(O,Im(ψ
i,a
)).
14 Arrange E
′
1
,E
′
2
...E
′
N
in non-increasing order of their norms.
15 Let∆
′
be a diagonal matrix of E
′
1
,E
′
2
...E
′
N
.
16 Let the columns of Q
′
beψ
′
1
,ψ
′
2
...ψ
′
N
.
17 H
′
= Q
′
∆
′
Q
′− 1
.
18 Create the objective functionΩ(R,H
′
)=∑
i̸= j
∥− 1
∥r
i
− r
j
∥
q
− H
′
i, j
∥
2
.
19 for update= 1,2...τ do
20 for i= 1,2...N do
21 for j= 1,2...K do
22 r
′
i, j
= r
i, j
− η· ApprxPartial(Ω,r
i, j
).
23 R← R
′
.
78
Algorithm 11:ApprxPartial: Returns the approximate partial derivative of a function.
Input: a function, f(x); current value of x, v.
Parameters: infinitesimal parameter, τ.
Output: an approximate value of
∂ f
∂x
|
x=v
.
1 currentValue← f(x)|
x=v
.
2 incrementedValue← f(x)|
x=v+τ
.
3 return
incrementedValue− currentValue
τ
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79
Own Publications
1. Yuling Guan, Zhihao Jiang, Henning Schlomer and Stephan Haas. ”Plasmons in Z2 Topo-
logical Insulators.” https://arxiv.org/abs/2205.04062
2. Yuling Guan, Ang Li, Sven Koenig, Stephan Haas and T. K. Satish Kumar. ”FastPivot: An
Algorithm for Inverse Problems.” (Accepted to 2022 IEEE 18th International Conference on
Automation Science and Engineering).
3. Yuling Guan, Zhihao Jiang and Stephan Haas. ”Control of Plasmons in Topological Insula-
tors via Local Perturbations.” Physical Review B 104 (2021), 125425
4. Yuling Guan, Ang Li, Sven Koenig, Stephan Haas and T. K. Satish Kumar. ”Hysteresis in
Combinatorial Optimization Problems.” FLAIRS, vol. 34, Apr. 2021.
5. A. Li, Y. Guan, S. Koenig, S. Haas and T. K. Satish Kumar, ”Generating the Top K Solutions
to Weighted CSPs: A Comparison of Different Approaches,” 2020 IEEE 32nd International
Conference on Tools with Artificial Intelligence (ICTAI), 2020, pp. 1218-1223
80

Abstract (if available)

Abstract Plasma oscillations are collective excitations of electron arising from long-range Coulomb interaction and the quanta for it is called plasmons. Motivated by analyzing the stability of many-body collective excitations of topological insulators, we use a fully quantum mechanical approach to demonstrate control of plasmonic excitations in prototype models of topological insulators by molecule-scale perturbations. Strongly localized surface plasmons are present in the host systems, arising from the topologically non-trivial single-particle edge states. A numerical evaluation of the random phase approximation (RPA) equations for the perturbed systems reveals how the positions and the internal electronic structure of the added molecules affect the degeneracy of the locally confined collective excitations, i.e., shifting the plasmonic energies of the host system and changing their spatial charge density profile. In particular, we identify conditions under which significant charge transfer from the host system to the added molecules occurs. Furthermore, the induced field energy density in the perturbed topological systems due to external electric fields is determined. Besides the one-dimensional topological insulators, we also theoretically analyze the collective plasmon excitations in two-dimensional Z2 topological insulators using the RPA numerical method. In the Kane-Mele model, the quantum spin Hall (QSH) phase is in a time reversal invariant electronic state with a bulk gap andgapless edge states. The QSH state is a state of matter that has a quantized spin Hall conductance and a vanishing charge Hall conductance. We consider a nano-ribbon structure with zigzag edges. Here, strongly localized plasmonic excitations are observed, with induced charge density distribution confined to the top/bottom boundaries in the non-trivial QSH phase. We demonstrate that the time reversal symmetry and the Z2 topological phase can be destroyed by an Zeeman field, thus changing the plasmonic edge states. Furthermore, we demonstrate that the edge modes are observed in the real space diamond-shaped nanoflake which have all of its boundaries zigzag edges. As well as exploring the plasmonic excitation in different topological insulators, we also discuss the plasmon in a perspective of inverse problem. The laws of physics are usually stated using mathematical equations, allowing us to accurately map a given physical system to its response, for example, getting plasmon spectrum from materials. However, when building systems, we are often faced with the inverse question: How should we design a physical system that produces a target response, i.e., design the system which has the target plasmon spectrum? In this paper, we present a novel algorithm, called FastPivot, for solving such inverse problems. FastPivot starts with a system state and invokes alternating forward and backward passes through the system variables. In the forward pass, it leads the current state of the system to its response. In the backward pass, a small amount of information is allowed to percolate from the target response back to the system variables. Upon convergence, FastPivot produces good quality solutions efficiently. We demonstrate the success of FastPivot on the inverse problem of placing atoms in a bounded region using a scanning tunneling microscope to achieve target responses in the density of states(DoS) or electron energy loss spectrum(EELS). We also compare FastPivot to Monte Carlo methods and analyze various empirical observations.

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Asset Metadata

Creator Guan, Yuling (author)

Core Title Plasmons in quantum materials

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2022-08

Publication Date 07/22/2022

Defense Date 05/17/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag artificial intelligent,Coulomb interaction,edge state,electron,OAI-PMH Harvest,plasma,topological insulators

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan (committee chair), Campos Venuti, Lorenzo (committee member), Difelice, Rosa (committee member), Koenig, Sven (committee member), Kumar, Satish (committee member)

Creator Email yulinggu@usc.edu,yulingguan1217@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111373966

Unique identifier UC111373966

Legacy Identifier etd-GuanYuling-10922

Document Type Dissertation

Format application/pdf (imt)

Rights Guan, Yuling

Type texts

Source 20220722-usctheses-batch-961 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Repository Name University of Southern California Digital Library

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