University of Southern California Dissertations and Theses

Superconductivity in low-dimensional topological systems

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Superconductivity in low-dimensional topological systems

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Content Superconductivity in low-dimensional topological systems
by
Ying Wang
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
(CHEMISTRY)
December 2023
Copyright 2023 Ying Wang

Acknowledgements
I would like to thank my supervisor Stephan Haas for his dedicated support and
guidance. Stephan continuously encouraged me and was always enthusiastic to assist in any way he could throughout the research projects. His encouragement and
patience helped me establish the confidence to do research. Meanwhile, his advice
about general life and careers boosted my journey to mental/emotional vitality. I
feel incredibly fortunate to have Stephan as my supervisor for the duration of my
PhD.
I would also like to thank Prof. Anuradha Jagannathan for her support in my
research projects throughout my Ph.D. journey. Her opinions and advice are always
sharp, professional, and comprehensive. Her encouragement made me feel warm.
I would like to thank Dr. Gautam Rai for his patient help especially when I started
my first project in this group and for all his suggestions during our collaboration.
His encouragement made me feel more confident.
I would like to express my sincere gratitude to all my lab mates and collaborators
who enabled my research to be possible.
Many thanks to all the professors for their high-quality teaching and guidance.
ii

I am also grateful to all my friends and fellow Ph.D. students who made my life in
Los Angeles not so boring.
I would like to thank my parents for their love and support.
I also want to thank myself for not giving up halfway through my PhD.
Finally, I want to thank my fianc´e, Yangyang Wan. Your love has taught me how to
love myself. Thank you for loving me. Thank you for being there with me.
iii

Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1:
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2:
Bardeen–Cooper–Schrieffer theory . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Cooper pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Calculations at zero temperature . . . . . . . . . . . . . . . . . . . . 5
2.3 Calculations at finite temperature . . . . . . . . . . . . . . . . . . . . 9
2.4 Tight-binding model and Bougoliubov de Gennes equations . . . . . . 10
2.4.1 tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.2 Bogoliubov de Gennes transformation . . . . . . . . . . . . . . 12
2.5 Gauge symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 3:
Basic topology in condensed matter physics . . . . . . . . . . . . . . . . . 15
3.1 Introduction to topological insulators . . . . . . . . . . . . . . . . . . 16
3.1.1 Topological invariants . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Topological invariants in one dimensional SSH model . . . . . 17
3.1.3 Chiral symmetry in the SSH model . . . . . . . . . . . . . . . 20
3.1.4 Higher-order topological insulators . . . . . . . . . . . . . . . 22
3.2 Topological systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
iv

Chapter 4:
Edge and corner superconductivity in two-dimensional Su-Schrieffer-Heeger
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1 Two-dimensional Su-Schrieffer-Heeger model and higher-order
topological insulators . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2 Boundary states . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 Order parameter results . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Finite temperature results . . . . . . . . . . . . . . . . . . . . 35
4.3 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.1 Calculation for periodic boundary conditions . . . . . . . . . . 37
4.3.2 Calculation for open boundary conditions . . . . . . . . . . . . 39
4.4 Related works and applications . . . . . . . . . . . . . . . . . . . . . 43
Chapter 5:
Superconductivity in the Fibonacci chain . . . . . . . . . . . . . . . . . . . 45
5.1 Backgruonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3.1 Order Parameter and critical temperature at half-filling . . . . 52
5.3.2 Scaling of critical temperature with U . . . . . . . . . . . . . 54
5.3.3 The order parameter as a function of band filling . . . . . . . 55
5.3.4 filling-U phase diagram . . . . . . . . . . . . . . . . . . . . . . 58
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 6:
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 7:
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.1 Model analysis in momentum space . . . . . . . . . . . . . . . . . . . 64
7.2 Effects of intra-cellular next-nearest-neighbor hopping . . . . . . . . . 67
7.3 Properties of the convergents of the golden ratio . . . . . . . . . . . . 67
7.4 Quadratic scaling of the superconducting gap with the modulation
strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.5 Average order parameter as a function of temperature . . . . . . . . . 70
7.6 Intensity distribution of local density of states in perpendicular space
and energy space under strong modulation limit . . . . . . . . . . . . 72
v

7.7 Critical temperature at strong modulation limit . . . . . . . . . . . . 73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
vi

List of Figures
2.1 (a) Explanation of the origin of attraction in Cooper pairs. (b) Cooper
pairs in s-wave superconductors . . . . . . . . . . . . . . . . . . . . . 4
3.1 Schematic of the SSH model. Blue and orange dots represent lattice
sites. Each unit cell contains one blue dot and one orange dot. The
sample unit cell is circled by the solid-dot line. Hopping amplitudes
are intra-cellular hopping tA (solid line) and inter-cellular hopping tB
(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 (a)-(d): Dispersion relations of SSH model under different tA/tB ratios. (f)-(j): bulk Hamiltonian in momentum space shown on dx − dy
plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Schematic of the Fibonacci chain model. Blue dots represent lattice
sites. Hopping amplitudes are tA (solid line) and tB (dashed line).
Atom sites are weaker hopping amplitude tA on both sides. Molecule
sites have different hopping amplitudes on left and right hand sides. 23
4.1 Two finite systems showing a) a non-trivial case b) and the trivial case
(black dots represent lattice site, t1 is represented by blue thinner
bond, t2 is green thicker bond). (c) DOS for the OBC non-trivial
lattice. Bands in this DOS plot are labelled by B (bulk), E (edge) and
the peak at E = 0 includes a contribution C (corner). (d) DOS for the
OBC trivial case. PBC lattice has the same DOS as the OBC trivial
lattice when the latter has enough unit cells. In these calculations,
r = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Spatial variation of the local superconducting order parameters in a
finite 2D Su-Schrieffer-Heeger lattice for different band fillings (a) 1/4-
filling (in the edge band), (b) 0.362-filling (in the central bulk band)
and (c) 1/2-filling. Parameters used: r = 0.1 and V = 1. . . . . . . . 30
4.3 Plots of the LDOS of corner, edge, and bulk sites for different band
fillings. Parameters used: r = 0.1 and V = 1. . . . . . . . . . . . . . . 31
vii

4.4 Plots of the T = 0 order parameters versus V with fitting curves.
(a) ∆ for PBC (up-triangle with dash line), ∆bulk for OBC (downtriangle with solid line); (b) ∆corner for OBC (circle with solid line).
Points indicate numerical results, and the continuous lines are fits to
the analytical expressions (see text). . . . . . . . . . . . . . . . . . . 32
4.5 (a) Plot of the log of local order parameter ln(∆i) versus distance to
the edge under V = 1 and r = 0.25, showing the exponential decay
and the odd-even oscillation (see text). (b) Plot of localization length
ξ as a function of 1/| ln r|. The points show the values of ξ for different
ratios of r, while the line shows the expected theoretical dependence.
r = 0.1, 0.15, 0.2, 0.25, 0.3, 0.333 from left to right. . . . . . . . . . . . 33
4.6 PBC critical temperatures plotted vs. V for ratio values of 0.0, 0.1,
0.5, 0.9, 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7 Inset: T-dependence of ∆corner and ∆bulk for OBC and of ∆ for r = 0.5
and V = 1 lattice. The main figure shows ∆bulk versus T. The low
temperature region T < Tc,P BC is colored grey, and the tail of the bulk
OP has been fitted to the expression given in the text. . . . . . . . . 34
5.1 (a) cut-and-project method of generating the Fibonacci chain: Projecting selected sites of the 2D square lattice (within the blue strip) on
to a line with slope equal to the golden ratio generates the Fibonacci
chain. The chain linked by ‘A’s and ‘B’s is shown near the bottom
of this figure. Atom sites (defined in Sec. 5.2) are marked by green
in the Fibonacci chain. Global density of states in a 610-site approximant of the Fibonacci tight-binding chain without (b) and with (c)
the attractive Hubbard U. When an attractive Hubbard interaction
is introduced, a superconducting gap appears in the density of states
at the Fermi level, indicated by the arrows in (b). The modulation
strength w = 0.2 in (b) and (c). . . . . . . . . . . . . . . . . . . . . 47
viii

5.2 (a) Superconducting order parameter in real space and (b) in perpendicular space for a chain with 610 sites at half-filling. Panels (a)-(d)
share all the same calculation parameters, chain length is 610, modulation strength w = 0.2 and BCS pairing strength U = 1.3t. (c) Local
density of states (LDOS) at the four different sites marked in (b) with
different values of the order parameter. The width of the central gap
is the same in the LDOS for all sites. (d) coherence peak height in
LDOS vs. the order parameter at the corresponding site. The order
parameter at a given site is positively correlated with the height of
the coherence peak in the local density of states for that same site.
(e) Dependence of the critical temperature on the square of the modulation strength for a chain with 610 sites and BCS pairing strength
U = 1.3t. The data points within the red frame in the main figure
is zoomed in in the inset, which shows a clear linear scaling between
critical temperature and the square of the modulation strength. In
the inset of panel (e), w values are 0.0,0.1,0.2,0.3,0.4 from left to right. 49
5.3 Order parameter vs. temperature for a chain of length 610 and attraction U = 1.3t at two modulation strengths (a) w = 0.2 and (b)
w = 1.5. Each column of data at a specific temperature represents all
the order parameter values of the Fibonacci chain. . . . . . . . . . . 56
5.4 (a) Plots of ln kBTc versus ln U for varying values of w. Calculations
are done for a chain of 610 sites. This indicates a power law dependence between the critical temperature and attraction strength. (b)
Plots of gap width ln ∆g vs. ln U showing that they are proportional,
with a w-dependent coefficient of proportionality. . . . . . . . . . . . 57
5.5 (a) Local order parameter distribution in conumber scheme for BCS
attraction U=0.9t at various chemical potentials for a chain of 2584
sites. (b) Average order parameter as a function of the BCS attraction
U and the chemical potential µ. . . . . . . . . . . . . . . . . . . . . 59
7.1 (a) Band structure of the periodic 2D SSH model on a square lattice. Here, the high symmetry points are M(
π
2a
,
π
2a
), X(
π
2a
, 0.0), and
Γ(0.0, 0.0). (b) Band structure of the SSH model on a ribbon with
periodic boundary conditions in one direction and open boundaries
in the other direction. Here, the path in momentum space is onedimensional, and the high symmetry points are Γ(0.0) and M(
π
2a
).
We use r = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
ix

7.2 (a) Illustration of the 2D SSH model, including intra-cellular nextnearest-neighbor hoppings. t1 is represented by the thin blue bonds,
whereas t2 depicted by the thick green bonds. The intra-cellular nextnearest-neighbor (NNN) hopping, tNNN is represented by the orange
bonds. (b) Band structure of the periodic 2D SSH model with NNN
hoppings. Here, the high symmetry points are M(
π
2a
,
π
2a
), X(
π
2a
, 0.0),
and Γ(0.0, 0.0). (c) Band structure of the 2D SSH model on a ribbon
structure with NNN hopping. Here, we used t1 = 0.4, t2 = 1.6, and
tNNN = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.3 (a)-(d) The square of average order parameter at different temperatures near its corresponding Tc value under different modulation
strengths. The dots represent the data point. The curve is the fitting
curve following the formula ∆2
avg = const. × (Tc − T) (which is the
same as ∆avg = const.×
√
Tc − T). The Fibonacci approximant length
is 610 and the attraction value is 1.3t. . . . . . . . . . . . . . . . . . 71
7.4 LDOS intensity distribution as a function of energy and conumber
indexing. The darler color represents the higher LDOS intensity. Reproduced with permission from [72]. . . . . . . . . . . . . . . . . . . 72
7.5 Plots of gap width ∆g vs. kBTc showing that they are proportional,
with a w-dependent coefficient of proportionality. . . . . . . . . . . . 74
x

Abstract
Fundamental research in low-dimensional topological systems is an exciting and
rapidly evolving field, offering numerous avenues for exploration. We focus on the
superconducting properties at zero and finite temperatures in basic low-dimensional
topological models, such as the two dimensional Su-Schrieffer-Heeger (SSH) model
and the Fibonacci chain. We show analytically and by numerical diagonalization
that the superconductivity on surface sites can be tuned by the electron filling. The
properties of the system are controlled by a central parameter which enters the
Hamiltonian, the hopping ratio. The scaling relations involving superconducting
observables–such as the superconducting order parameter, the spectral gap width,
and the critical temperature are investigated. These scaling relations are determined
by the singularity strength of the electronic spectrum. The critical temperature can
be uniform or site-dependent, depending on the specific model. When the critical
temperature distribution is not homogeneous, a novel proximity effect arises when
the temperature is in between the higher and the lower critical temperature, here
the sites with larger superconducting order parameter induce a nonzero tail of the
superconducting order parameter in nearby sites. In contrast, the uniform spectral
xi

gap width in the quasiperiodic Fibonacci chain is a corollary of the fact that there
is only one critical temperature in this system.
xii

Chapter 1
Introduction
Low-dimensional systems have been a subject of extensive research for many years,
primarily due to their distinct quantum confinement effects, the emergence of unique
electronic and optical states, and their enhanced adaptability for engineering applications. Topological systems offer a wide array of advantages, including the presence
of protected edge states and robustness against disorder. The topology of a system is
determined by its dimension and symmetry [1]. The interplay between interactions
and topology make low-dimensional topological systems a fertile ground for both
fundamental scientific research and innovative technological applications. This includes research such as identifying and characterizing new topological phases, gaining
insights into the nature of topological phase transitions, investigating the behavior
of edge states in topological insulators and topological superconductors, deepening
our understanding of interactions and correlations, and the creation of various topological devices, such as topological transistors, topological qubits, and topological
1

sensors.
Superconductivity, which was first observed in mercury in 1911 [2], remains a captivating and exotic state of matter that continues to intrigue scientists. Unraveling
the mechanisms behind superconductivity and its associated properties has significantly enhanced our understanding of quantum phase transitions and prompted the
emergence of unconventional electronic states. The examination of superconductivity within low-dimensional topological systems enhances our comprehension of these
electronic structures and the behavior of condensed Cooper pairs within these unique
electronic environments.
In this dissertation, we concentrate on s-wave superconductivity in two models:
one is periodic with open boundary conditions and another is quasiperiodic with
closed boundary conditions. Our aim is to gain a comprehensive understanding
of edge modes, superconductivity distribution patterns, the scaling dependence of
superconductivity, and critical temperatures. It’s worth noting that although we
specifically consider s-wave pairing in these basic models, similar phenomena and
properties are expected to emerge when extending these models to accommodate
other types of superconducting pairings.
2

Chapter 2
Bardeen–Cooper–Schrieffer theory
In this chapter, we briefly introduce the theoretical grounds of BCS theory. The
reference sources are [3–5]
2.1 Cooper pairs
BCS theory assumes that superconductivity arises from attractive Cooper pair interaction, which is mediated by electron-phonon interaction. A Cooper pair is composed
of a pair of coupled electrons with opposite spins and equal but opposite momenta.
These pairs are from the electrons with k > kF , i.e., with energies in excess of the
fermi energy. When the energy of the attractive potential outweights the excess
kinetic energy, superconductivity forms.
The origin of the attraction in Cooper pairs can be explained as follows. Normally,
electrons repel each other due to the Coulomb repulsion. However, an instability of
normal states in the presence of an attractive interaction arises even if the attraction
3

�↑ -�↓
Spin singlet pair (s-wave)
Conventional superconductivity
Cooper pair
Electron-phonon interaction mediated effective attraction
within an electron pair
Figure 2.1: (a) Explanation of the origin of attraction in Cooper pairs. (b) Cooper
pairs in s-wave superconductors
is very weak [6]. This sparked the inquiry into the contribution of the underlying
interaction coupled to electrons. It was Leon Cooper who first proposed the idea that
vibrations within the lattice, i.e., phonons, could interact with electrons, leading to
the formation of an effective attractive force between them. The attraction can be
understood as follows. When an electron moves in an positively charged ion-cores
lattice, it attracts the ion cores and leads to a lattice distortion. This distortion
causes the local area accumulate a positive net charge. Then, this positive net
charge attracts another electron. The second electron overcomes the repulsion with
the first electron, and they form a weak bounded pair. The bounded electron pair
can move as an entity through the lattice. These Cooper pairs are time-reversed
momentum pairs. In the course of this dissertation, we exclusively focus on s-wave
superconductivity involving Cooper pairs with opposite spins (singlet). See Fig. 2.1
for graphic illustration.
4

2.2 Calculations at zero temperature
The wave function of a pair of electrons can be described by a pair wave function
ϕ(r1 − r2). In order to treat N electrons and keep the electrons grouped in pairs, we
generalize the single pair wave function into the following form:
ϕN (r1, r2, ..., rN ) = ϕ(r1 − r2)ϕ(r3 − r4) · · · ϕ(rN−1 − rN ) (2.1)
By minimizing the energy of wavefunciton, we can get the ground state ϕN . For each
pair of electrons, they have opposite spins ↑↓. The antisymmetry of function ϕN is
introduced by operator A:
ϕN = Aϕ(r1 − r2)ϕ(r3 − r4) · · · ϕ(rN−1 − rN )(1 ↑ 2 ↓)(3 ↑ 4 ↓) · · · (N − 1 ↑ N ↓)
(2.2)
For convenience of calculation, we perform a Fourier transform on the pair wave
function ϕ(r1 − r2),
ϕ(r1 − r2) = X
k
gke
ik·(r1−r2)
, (2.3)
ϕN =
X
k1
··· X
kN/2
gk1
···gkN/2Aeik1(r1−r2)
···e
ikN/2(rN−1−rN)
(1 ↑)(2 ↓)···(N −1 ↑)(N ↓)
(2.4)
5

We apply the Wigner-Jordan notation,
a
†
k1↑
a
†
−k1↓
· · · a
†
kN/2↑
a
†
−kN/2↓ϕ0, (2.5)
to represent the state described in 2.4, where ϕ0 is the vacuum state and a
†
kα is
creation operator which creates an electron in the state kα. The annihilation operator
akα can destruct an electron such that akαϕ0 = 0.
wave function ϕN takes the new form
ϕN =
X
k1
· · · X
kN/2
gk1
· · · gkN/2
a
†
k1↑
a
†
−k1↓
· · · a
†
kN/2↑
a
†
−kN/2↓ϕ0 (2.6)
For even easier management, consider the function
ϕ˜ = C
Y
k
(1 + gka
†
k↑
a
†
−k↓
)ϕ0, (2.7)
where C is the normalization constant and the product Q
k
extends over all the
plane wave states. It’s straightforward to see that ϕN is part of ϕ˜ by comparing the
formulas of ϕN and ϕ˜. Define vk
uk
= gk, and with the restriction v
2
k + u
2
k = 1, it’s easy
to incorporate the normalization constant C into the parenthesis.
ϕ˜ =
Y
k
(uk + vka
†
k↑
a
†
−k↓
)ϕ0 (2.8)
where u
2
k + v
2
k = 1. We can treat a
†
k↑
a
†
−k↓ϕ0 as the source of generating Cooper pairs.
Substitute ϕ˜ in to equation 2.9
⟨ϕ˜| H |ϕ˜⟩ − EF ⟨ϕ˜| N |ϕ˜⟩ (2.9)
6

It leads to
⟨ϕ˜| H − EFN |ϕ˜⟩ = 2X
k
ξkv
2
k +
X
kl
Vklukvkulvl (2.10)
Vkl = V ⟨l, −l| |k, −k⟩ (2.11)
Since v
2
k + u
2
k = 1, let uk = sin θk, vk = cos θk. ⟨ϕ˜| H |ϕ˜⟩ can be rewritten in the
following form,
⟨ϕ˜| H |ϕ˜⟩ = 2X
k
ξk cos θk
2 +
1
4
X
kl
sin 2θk sin 2θlVkl (2.12)
When the energy is minimized, we get
ξk tan 2θk =
1
2
X
l
Vkl sin 2θl (2.13)
Define
∆k = −
X
l
Vklulvl (2.14)
ϵk =
q
ξ
2
k + ∆2
k
(2.15)
Combine the three equations above, we get
tan 2θk = −
∆k
ξk
(2.16)
2ukvk = sin 2θk =
∆k
ϵk
(2.17)
v
2
k − u
2
k = cos 2θk = −
ξk
ϵk
(2.18)
7

∆k = −
X
l
Vkl
∆l
2(ξ
2
l + ∆2
l
)
1/2
(2.19)
For simplicity, the interaction is set in the following way,
Vkl =



-V if |ξk| ≤ ℏωD
0 otherwise
(2.20)
where V is a positive value. With the simplification above,



∆k = 0 if |ξk| > ℏωD
∆k = ∆ if |ξk| < ℏωD
(2.21)
∆ is independent of k. We are only interested in states with energy ∈ [EF −ℏωD, EF +
ℏωD]. We need to sum over all the ξl
in all directions within this energy range. This
step results in a density of states per unit energy N(ξl). ℏωD is usually much less
than EF , so N(ξl) can be replaced by the density of states at the Fermi level N(0).
Finally the order parameter ∆ can be written in the following formula.
∆ = N(0)V
Z ℏωD
−ℏωD
∆
dξ
2
p
∆2 + ξ
2
(2.22)
∆ can be calculated in a self-consistent way.
Based on equation 2.22, we get
∆ = 2ℏωDe
−1/N(0)V
(2.23)
if we only consider the weak coupling limit N(0)V ≪ 1
8

2.3 Calculations at finite temperature
When consider the calculation at non zero temperature, the self-consistent equation
corresponding to equation 2.19 is:
∆k = −
X
l
Vkl
∆l
2ϵl
[1 − 2f(ϵl)] (2.24)
where f(ϵl) is the Fermi-Dirac distribution, f(ϵl) = 1
1+e
ϵl
/kBT
. Then the integral form
is,
1 = N(0)V
Z ℏωD
0
dξ
p
ξ
2 + ∆2
[1 − 2f(
p
ξ
2 + ∆2
)] (2.25)
This equation indicates the relation between T and ∆. By setting ∆ = 0, the critical
temperature Tc can be calculated by the following way:
1 = N(0)V
Z ℏωD
0
dξ
ξ
tanh ξ
2kBTc
(2.26)
The order parameter value at zero temperature ∆(0) = 1.76kBTc. When T is below
Tc and close to Tc, ∆ ∝ 3.2kBTc[1 −
T
Tc
]
1/2
.
9

2.4 Tight-binding model and Bougoliubov de Gennes
equations
2.4.1 tight-binding model
When we study materials which are not conducting or are with some specific symmetry, the traditional free electron model is not applicable. We can explore how the
wave functions of electrons interact with each other when we bring them together in
the tight-binding (TB) model. The tight-binding method is an ideal candidate for
determining electronic and transport properties for a large-scale system. It describes
the system as Hamiltonian matrices with a manageable number of parameters.
When examining non-conductive materials or those with particular symmetries,
the conventional free electron model loses its applicability. We can delve into the
interactions between electron wave functions as atoms aggregate using the tightbinding model. The tight-binding approach stands as a excellent option for assessing
electronic and transport characteristics in large-scale systems. This method represents the system through real-space Hamiltonian matrices, articulated using a feasible set of parameters. When atoms are distant from each other, their electron wave
functions are localized and can be treated independently. As they approach, their
wave functions mix up and create bonding and antibonding states. When atoms
are collectively close enough and form crystalline lattice, the overlaps of electron
wave functions facilitate the electron delocalize on the lattice by inter-atomic tunneling, described by the overlap parameter −t, between the two states at different
10

sites. The general real space Hamiltonian of a particle moving on a lattice without
external potential applied on the lattice is:
H0 = −
X
⟨i,j⟩
tij c
†
iσcjσ + h.c. (2.27)
where ⟨i, j⟩ denote nearest neighbor sites i and j, while σ denotes spin. −tij is the
hopping integral between the ith and jth site. The lattice spacing is a. For simplicity, we consider the one dimensional case. If the lattice is periodic, translational
invariance exists. then the Bloch theorem applies, and the Hamiltonian can be diagonalized using plane waves, which are the eigen-basis of the translation operator.
In momentum space, the real space operator cn can be represented by
cn =
1
√
N
X
k
e
ikxn
ck, (2.28)
where xn = n × a. The H0 can be rewritten as
H0 = −
X
k
c
†
kσckσ(te−ika + t
∗
e
ika) = X
k
c
†
kσckσE(k), (2.29)
where we only consider the nearest neighbor hopping and assume t is real. The
dispersion relation in one dimensional periodic chain is:
E(k) = −2t cos ka (2.30)
This dispersion relation can be easily extended to two dimensional and three dimensional models, which we will discuss in Chapter 4.
11

2.4.2 Bogoliubov de Gennes transformation
For the convenience of considering systems with open boundaries or defects, we need
to apply real-space formulation. We use a standard mean field approximation to
write the following effective total Hamiltonian
HBdG =
X
iσ
(u
HF
i − µi)c
†
iσciσ −
X
σ
X
⟨i,j⟩
tij (c
†
iσcjσ + h.c.) +X
i
(∆ic
†
i↑
c
†
i↓ + h.c.)
(2.31)
where µi
is the chemical potential. The mean fields are the Hartree-Fock shift
u
HF
i = −V ⟨c
†
i↑
ci↑⟩ and the (real) local superconducting order parameters (OP)
∆i = V ⟨ci↑ci↓⟩. These quantities are determined self-consistently for different choices
of band filling and boundary conditions. They are site-independent in the infinite
lattice, and in finite systems with periodic boundary conditions, but become sitedependent when there are edges. The two mean-field terms must be computed selfconsistently,
∆i =
X
n
v
∗
inuin (1 − 2f(En, T)) (2.32)
ϵ
HF
i = U
X
n
|uin|
2
f(En, T) + |vin|
2
(1 − f(En, T)) (2.33)
where En are the positive eigenvalues, and (uin, vin) are the eigenvectors of the
Bogoliubov-de Gennes pseudo Hamiltonian operator corresponding to (2.31). f(En, T) =
1
exp(
En
kBT
)+1 is Fermi-Dirac distribution. The calculation determines the pairing order
parameter ∆i
for each site self-consistently as follows: an initial ansatz is made for
the BdG Hamiltonian using randomly chosen values for the OPs, ∆(0)
i
. The Hamil12

tonian is diagonalized numerically, to find the eigenvalues En and corresponding
eigenvectors {un, vn}. New values of ∆i are computed using the expression in equation 5.3. These ∆(1)
i
are injected back into the BdG hamiltonian, and the calculation
is iterated until convergence is reached. All the calculations in the following have
been done under a fixed bandwidth condition. Results are reported in units of t, the
average hopping amplitude.
2.5 Gauge symmetry breaking
In terms of symmetry and phase transition, a symmetry is identified that remains
preserved on one side of the transition and is broken on the other side, with the
ordered phase consistently displaying lower symmetry. Symmetry can be quantified
by an order parameter. Order parameter can be treated as an expectation value of
an observable which is non zero in the symmetry breaking phase and is zero in the
symmetric phase [7].
As defined in the previous sections,
∆ ∝ ⟨Φ↓Φ↑⟩ (2.34)
Under a gauge transformation with gauge paramter α, electron creation and annihilation operators become
Φσ → e
ieαΦσ (2.35)
Φ
†
σ → e
−ieαΦ
†
σ
(2.36)
13

Then, the operator Φ↓Φ↑ transforms to
Φ↓Φ↑ → e
2ieαΦ↓Φ↑ (2.37)
∆ → e
2ieα∆ (2.38)
Thus, order parameter is not gauge invariant and is complex. Superconductors break
gauge symmetry.
14

Chapter 3
Basic topology in condensed
matter physics
Topological structures play a significant role in condensed matter physics, particularly in the study of materials and systems that exhibit unique electronic, magnetic,
and topological properties. These structures arise from the topology of electronic
bands and the behavior of quantum states. Topology is a branch of mathematics
that deals with the study of properties that are preserved under continuous deformations, such as stretching, bending, and shrinking, but not tearing or gluing. The
continuous deformation can be grasped by considering the following explanation. In
general Hamiltonians of many-particle systems with an energy gap separating the
ground state from the excited states, a continuous deformation can be defined as a
modification in the Hamiltonian which does not result in the closure of the bulk gap.
15

3.1 Introduction to topological insulators
Topological insulators (TIs) exhibit robust edge states in proximity to a vacuum,
making them highly attractive for applications in the fields of electronics and quantum computing. The existence of a nontrivial topological phase of the bulk may imply
the existence of boundary states that are topologically protected. These boundary
states are localized near the corners, edges, or surfaces of the material and exhibit
unique electronic properties when compared with ordinary states. A topologically
protected boundary state cannot exist without the presence of the topological bulk.
These boundary states are robust against local perturbations and can be removed
only by a perturbation that closes the gap of the bulk band structure or reduces its
symmetry. This connection between a nontrivial topology of the bulk band structure
and topologically protected boundary states is called bulk–boundary correspondence.
3.1.1 Topological invariants
A topological invariant in condensed matter systems is an observable that remains
unvaried under a adiabatic deformation of the Hamiltonian. Adiabatic deformation
means that the parameters of the Hamiltonian are continuously changed and while
fundamental symmetries of the Hamiltonian are maintained. Correspondingly, a
topological invariant is dependent on the symmetries and is well defined in the thermodynamic limit [8].These invariants often reveal fundamental characteristics of a
system that are robust against small perturbations. They help classify and differentiate different states of matter or physical systems based on their global properties,
16

�! �"
Figure 3.1: Schematic of the SSH model. Blue and orange dots represent lattice
sites. Each unit cell contains one blue dot and one orange dot. The sample unit
cell is circled by the solid-dot line. Hopping amplitudes are intra-cellular hopping tA
(solid line) and inter-cellular hopping tB (dashed line).
even when local details might change. We will explain a topological invariant by an
example in the next sub section.
3.1.2 Topological invariants in one dimensional SSH model
The Su-Schrieffer-Heeger (SSH) model describes electrons hopping on a one-dimensional
lattice, with two alternating hopping amplitudes, tA and tB , as shown in Fig. 3.1.
There are N unit cells in the chain and there are two different sites with each unit
cell. The intra cellular hopping is denoted by tA and the inter cellular hopping by
tB. The Hamiltonian can be written as:
H0 = −
X
N
n=1,σ
tAc
†
⟨n,1⟩σ
c⟨n,2⟩σ + tBc
†
⟨n,2⟩σ
c⟨n+1,1⟩σ + h.c. (3.1)
The dispersion relation for the Hamiltonian above is
ϵ1D(k) = q
t
2
A + t
2
B + 2tAtB cos ka = tB
√
1 + r
2 + 2r cos 2ka, (3.2)
with the hopping ratio r =
tA
tB
. Depending on the different ratio values, the band
structure exhibits various features, as shown in Fig. 3.1. When tA or tB is 0, the 1D
chain becomes a sequence of dimers. There are only two energies, ±E, corresponding
17

to the bonding/anti-bonding energies. If tA > tB, a gap emerges in the band structure. As tB increases, when tA = tB, the gap closes; when tA < tB, the gap opens up
again. These graphical representations imply that in the presence of a Hamiltonian
with unequal values of tA and tB (i.e., staggering), the dispersion exhibits a gap,
indicating an insulating state. Only when tA = tB does the gap close, allowing for
the presence of states at arbitrarily low energies above the Fermi level, classifying
the case with tA = tB as a metallic state.
By checking the dispersion relation, it seems that tA > tB and tB > tA show
the same bulk features. The topological features of 1D SSH model are reflected
by eigenvectors. The hopping pattern in this model is staggered. When tA ̸= tB,
c2n+1 = c1e
iknb, c2n+2 = c2e
iknb, where b = 2 × a. The Hamiltonian 2.29 can be
blocked into 2 × 2 submatrix H(k)
H0 = H(k) ⊕ H(k) ⊕ H(k) ⊕ H(k) · · · ⊕H(k) (3.3)
H(k) = d(k) · σ (3.4)
σ = (σx, σy, σz) are the Pauli matrices.
dx(k) = ℜtA + |tB| cos kb + arg (tB) (3.5)
dy(k) = −ℑtA + |tB|sin kb + arg (tB) (3.6)
dz(k) = 0 (3.7)
with tB = |tB|e
i arg tB
18

with eigenvalues
ϵ(k) = ±
q
d
2
x + d
2
y + d
2
z = ±
p
|tA|
2 + |tB|
2 + 2|tA||tB| cos (kb + arg (tA) + arg (tB))
(3.8)
and eigenvectors
|±⟩ =


±e
−iϕ(k)
1

 (3.9)
with tan ϕ(k) = dy/dx
As the wave number k moves from 0 to 2π in the Brillouin zone, the path is a
closed circle of radius tB on the dx − dy plane, centered at (tA,0). and it needs to
avoid the origin, to describe an insulator. The loop’s topology can be described by an
integer known as the bulk winding number, denoted as ’ν’, which we will explain in
the next subsection 3.1.3. This integer quantifies how many times the loop encircles
the origin within the dx − dy plane. For example, in Fig. 3.2 (f) and (i), ν = 0; in
Fig. 3.2 (g) and (j), ν = 1, while in Fig. 3.2(h), the winding number is undefined.
The winding number of SSH model is 0 in trivial phase and 1 in nontrivial phase.
The winding number has been widely used as an invariant for diagnosing topological phases in one-dimensional chiral-symmetric systems [9], which will be explained
in the following.
19

2
0
2
E
(a)
tA = 1,tB = 0
0
k
2
0
2 (b)
tA < tB
0
k
2
0
2 (c)
tA = tB = 1
0
k
2
0
2 (d)
tA > tB
0
k
2
0
2 (e)
tA = 0,tB = 1
0
k
(f) dx
dy
-1 1
(g) dx
dy
(h) dx
dy
(i) dx
dy
(j) dx
dy
-1 1
Figure 3.2: (a)-(d): Dispersion relations of SSH model under different tA/tB ratios.
(f)-(j): bulk Hamiltonian in momentum space shown on dx − dy plane.
3.1.3 Chiral symmetry in the SSH model
Chiral symmetry requires that the Hamiltonian H of the system satisfies the following
relation with the chiral symmetry operator Γ,
ΓHΓ
† = −H (3.10)
Chiral symmetry has the following properties: 1) the chiral symmetry operator is
unitary and hermitian, i.e., ΓΓ† = Γ2 = 1; 2) Γ is local; 3) the chiral symmetry is
robust.
• Chiral symmetry is also called sublattice symmetry. Another way to define the
chiral symmetry operator is to use the orthogonal sublattice projectors PA and
20

PB,
PA =
1
2
(I + Γ) (3.11)
PB =
1
2
(I − Γ) (3.12)
where PAPB = 0 and PA + PB = I. Sublattice symmetry means that there is
no transition from any site on one sublattice to any site no the same sublattice
[8].
PAHPA = PBHPB = 0H = PAHPB + PBHPA (3.13)
• Chiral symmetry results in a symmetric energy spectrum. For a eigenstate Φ
of Hamiltonian H
HΦ = EΦ (3.14)
Apply the chiral symmetry,
HΓΦ = −ΓHΦ (3.15)
= −EΓΦ (3.16)
ΓΦ is another eigenstates of H with eigenvalue −E
When applying the chiral symmetry relatioin 3.10,
σzH(k)σz = 0 (3.17)
The above equation indicates dz = 0, so d(k) is restricted to lie on the dx − dy plane.
21

3.1.4 Higher-order topological insulators
There is a complete correspondence between d-dimensional bulk band structure and
topologically protected boundary states of dimension d−1 in topological phases only
with non-spatial symmetries [10]. This is called first-order topological insulator.
When the boundary dimension is less than d − 1, the material is called higherorder topological insulators (HOTI). The two dimensional SSH model which will be
introduced in chapter 4 is a good example of two dimensional HOTI with corner
modes characterizing the higher-order boundary. Three dimensional generalization
of two dimensional SSH model, the three dimensional SSH model, is also a secondorder topological phase.
3.2 Topological systems
The central concept of topological systems in condensed matter physics is the identification of topological invariants. These invariants are mathematical quantities that
characterize the topological properties of a material. Examples include the Chern
number, the Z2 topological invariant, and the winding number [11, 12]. The determination of these invariants often involves analyzing the electronic band structure
or other physical properties of the material. The aforementioned SSH model is a
great example for 1D case with non zero winding number. The system’s topology
relies heavily on its dimensionality and symmetries. 1D topological superconductors
exist only when the particle–hole symmetry applies. 3D topological insulators exist only when there is time-reversal symmetry. Quasiperiodic systems can also hold
22

�! �" �! �! �" �! �" �! �!
Molecule site Atom site Molecule site
Figure 3.3: Schematic of the Fibonacci chain model. Blue dots represent lattice
sites. Hopping amplitudes are tA (solid line) and tB (dashed line). Atom sites are
weaker hopping amplitude tA on both sides. Molecule sites have different hopping
amplitudes on left and right hand sides.
topological properties [13]. In Fibonacci chain, the gap label can serve as a topological invariant, allowing for the characterization of edge states [14]. We will focus on
Fibonacci chain in chapter 5. Fig. 3.3 shows the Fibonacci chain.
23

Chapter 4
Edge and corner superconductivity
in two-dimensional
Su-Schrieffer-Heeger model
4.1 Backgrounds
4.1.1 Two-dimensional Su-Schrieffer-Heeger model and higherorder topological insulators
The two-dimensional Su-Schrieffer-Heeger (2D SSH) model is an extension of the
well-known SSH chain [15, 16] to two dimensions, with alternating weak (tA) and
strong (tB) bonds along both spatial directions. It is defined for sites lying on
vertices of a square lattice of side a. Along the horizontal (x) direction, the sequence
24

of hopping amplitudes is alternating, just as in the parent 1D SSH model [15, 16].
The same is true for the hopping along vertical (y) direction, as shown in Fig. 4.1 (a)
and (b). We will assume tA ≤ tB and discuss properties of this model as a function of
the hopping ratio r = tA/tB ≤ 1. The hopping amplitudes are furthermore assumed
to be positive (since the sign changes can be gauged away). The non-interacting
Hamiltonian, similar to Equation 3.1 and discussed in detail in [17], is then
H0 = −
X
⟨i,j⟩
tij c
†
iσcjσ + h.c. (4.1)
where ⟨i, j⟩ denote nearest neighbor sites i and j, while σ denotes spin. The unit cell
consists of four sites. The infinite system or the finite system with periodic boundary
conditions (PBC) can be diagonalized by Fourier transforming. Note that the energy
spectrum can be obtained very easily since the problem is separable in x and y
variables, giving rise to two 1D SSH spectra. The 2D spectrum is just the direct sum
of the energy bands of the 1D SSH model, ϵ1D(kx) and ϵ1D(ky). The wave functions
are products of the 1D SSH wave functions, that is, ψ(kx, ky) = ψ1D(kx)ψ1D(ky).
The corresponding energy bands are given by
ϵ
(n)
2D(kx, ky) = ±ϵ1D(kx) ± ϵ1D(ky) (4.2)
ϵ1D(k) = tB
√
1 + r
2 + 2r cos 2ka (4.3)
where the wave vectors lie in the Brillouin zone, −π/2a ≤ kj ≤ π/2a (j = x, y).
See Appendix. 7.1 for the momentum-space Hamiltonian and the band structure.
The parameter determining the spectral properties is r, while tB serves only to
set the global scale of energy. The four choices of sign in the above expression
25

for ϵ2D (++, +−, −+, −−) correspond to four bands which we henceforth label by
n = 1, .., 4. The two overlapping central bands (n = 2, 3 corresponding to +−, −+)
intersect along the diagonals, where ϵ
(2)
2D = ϵ
(3)
2D = 0. They transform into each other
under the mirror symmetries that exchange kx ↔ ±ky. The energy spectrum is
particle-hole symmetric as the model is bipartite. When r < 1
2
, a gap separates the
lateral bands from the central bands. There are logarithmic van Hove singularities
at E = 0, and at the centers of the two lateral bands.
HOTI have been much studied recently. The formation of electric multipole
moments and charge pumping in such a lattice has been addressed [18, 19]. Photonic systems based on the 2D SSH model were investigated in [20]. A classification
scheme for topological superconductors and the bulk-boundary correspondence in
Bogoliubov-de Gennes type models has been discussed in [10, 21, 22]. In this dissertation we consider only the simplest possibility of s-wave pairing, however, we expect
that similarly interesting edge and corner phenomena should appear when the basic
model is extended to permit other types of superconducting pairing. The present
study constitutes, for example, a good starting point for investigations of corner and
edge Majorana fermions [23, 24].
While edge modes and resulting higher order topological superconducting phases
have been reported before in the literature, as on the honeycomb lattice [25–27],
the present model is of particular interest since it provides an analytically tractable
example with a tunable parameter, whose ground state and finite temperature properties can be described in detail. Furthermore, we observe an interesting interplay
between surface and bulk superconductivity which has not been reported in HOTI
26

structures so far.
4.1.2 Boundary states
With open boundary conditions (OBC), topologically protected states can arise at
the edges depending on the bond configuration. We use the term “weak edge” when
all sites lying on the edge are connected to the interior by weak bonds. This configuration results in the appearance of 1D edge modes, which have wave modulations
along the edge, but decay exponentially along the direction perpendicular to the
edge. Where two weak edges meet, there is an additional “zero-dimensional” corner
mode, which is exponentially decaying in both directions. The localization length ξ
is that of the edge states of the 1D SSH chain and depends on the hopping ratio,
ξ = 2a/| ln r|. A precursor of this 2D lattice, a ladder type system of 2 coupled
chains has been studied in [28, 29].
The total density of states (DOS) of the finite non-trivial and trivial 2D SSH
lattice is shown in Fig. 4.1 (c) and (d), respectively. Liu, et al. have proved this
nontrivial topological phases in 2D SSH model with non-zero 2D Zak phase [17]. The
non-trivial system has two supplementary bands corresponding to the quasi-1D edge
modes. The spectrum has four gaps for small r, which close when r = 1/3. If present,
each of the zero-dimensional modes localized on corners of the square contributes a
delta function δ(E) to the density of states. As for the trivial system (no weak
edges), we will not consider it any further, since the edge and corner phenomena
under discussion here are not present in this case.
In the interacting problem, an attractive onsite Hubbard term, Hint, is added to
27

(a) (b)
(c) (d)
Figure 4.1: Two finite systems showing a) a non-trivial case b) and the trivial case
(black dots represent lattice site, t1 is represented by blue thinner bond, t2 is green
thicker bond). (c) DOS for the OBC non-trivial lattice. Bands in this DOS plot
are labelled by B (bulk), E (edge) and the peak at E = 0 includes a contribution
C (corner). (d) DOS for the OBC trivial case. PBC lattice has the same DOS as
the OBC trivial lattice when the latter has enough unit cells. In these calculations,
r = 0.25.
28

H0 in equation 4.1 with
Hint = −V
X
i
nˆiσnˆiσ¯ (4.4)
(V > 0), where ˆniσ = c
†
iσciσ is the number of electrons of spin σ on the site i and ¯σ
represents the opposite spin of σ. We assume that the instability of interest is the
s-wave superconducting instability. In particular, at half-filling, a competing charge
density wave instability could exist but be suppressed by doping or adding a small
next nearest neighbor hopping. To proceed, we use a standard mean field approximation introduced in section 2.4.2 to solve the superconducting related properties.
4.2 Numerical results
4.2.1 Order parameter results
The calculations have been done under a fixed bandwidth condition, that is, we
vary the hopping ratio r keeping the bandwidth W = 2(tA + tB) = 4t constant.
Results are reported in units of t, the average hopping amplitude. By applying the
Bogoliubov-de Gennes formalism on the finite 2D SSH tight-binding model with onsite attraction, we show that the superconducting pattern is inhomogeneous where
paring is site-dependent. In Fig. 4.2 we have illustrated the local OP distribution
for different band fillings, in a 20 × 20 lattice subjected to OBC, with r = 0.1 and
V = 1. N.b. in all the open systems considered, all four edges are taken to be
weak edges. Defining the total filling ⟨n⟩ =
1
Nsites
P
i
⟨c
†
i
ci⟩, panel (a) corresponds to
the edge band being half-filled (⟨n⟩ = 0.25, µ = 1.90t). Here the OP is largest on
29

(a) (b) (c)
Figure 4.2: Spatial variation of the local superconducting order parameters in a finite
2D Su-Schrieffer-Heeger lattice for different band fillings (a) 1/4-filling (in the edge
band), (b) 0.362-filling (in the central bulk band) and (c) 1/2-filling. Parameters
used: r = 0.1 and V = 1.
the edges and decays exponentially into the interior as we will show later. In panel
(b), the central bulk band is partially filled (⟨n⟩ = 0.362, µ = 0.19t). The OP is
accordingly largest in the bulk of the sample. In panel (c), the system is at halffilling (⟨n⟩ = 0.5, µ = 0). Here the OP is strongest on the four corner sites, followed
by the bulk, while the edges have negligibly small OP. The local densities of states
(LDOS) for these three band fillings on the corner, edge, and bulk sites are shown in
Fig. 4.3. These plots rationalize the site-dependent superconductivity pattern. The
gap in LDOS is seen only at the edge sites in (a), the bulk sites in (b), and both
corner and bulk sites in (c). The most interesting situations, corresponding to cases
a) and c), are discussed below.
Chemical potential at the band center (half-filling)
30

(a) (b) (c)
Figure 4.3: Plots of the LDOS of corner, edge, and bulk sites for different band
fillings. Parameters used: r = 0.1 and V = 1.
In Fig. 4.4, we plot T = 0 order parameters as a function of V for several values
of the hopping ratio r. The system sizes are large enough that the results have
converged, and the error bars are less than the size of the symbols used in the plots.
Panel (a) shows the quantity ∆bulk defined as the value of the OP at one of the
four equivalent central sites of the sample. For comparison, we also show values of
the order parameter computed for PBC, ∆, which of course is independent of the
position. At half-filling, we note that, as the hopping ratio r increases from 0 to 1,
∆bulk decreases. Panel (b) shows ∆corner defined as the value of the order parameter
at one of the four (equivalent) corner sites. Figs. 4.4 show that, when r = 0, all of
the order parameters are proportional to V . This is due to the form of the density
of states, which is sharply peaked around E = 0, as explained in section IV. In
contrast, for r approaching 1, the order parameters vary as exp−
√
cst/V , as obtained
31

(a) (b)
Figure 4.4: Plots of the T = 0 order parameters versus V with fitting curves. (a) ∆
for PBC (up-triangle with dash line), ∆bulk for OBC (down-triangle with solid line);
(b) ∆corner for OBC (circle with solid line). Points indicate numerical results, and
the continuous lines are fits to the analytical expressions (see text).
for the half-filled square lattice [30]. This type of scaling with V is expected when the
Fermi level is located at a logarithmic van Hove singularity [31, 32]. For intermediate
values of r the corner OP is well-fitted by an extrapolation between the linear and
exponential terms as follows
f(V ) = c1V + c2 exp−
√
c3/V (4.5)
Values of the fitted constants are given in Table. 4.1 for each of the OP. Note that c2
vanishes when r → 0, so that the variation is purely linear in V in this limit, while
for r = 1, the linear term vanishes. These behaviors will be explained in Sec. IV.
Chemical potential in the edge band
When the chemical potential lies within the edge band (i.e. just above 1
4
-filling),
32

(a) (b)
Figure 4.5: (a) Plot of the log of local order parameter ln(∆i) versus distance to the
edge under V = 1 and r = 0.25, showing the exponential decay and the odd-even
oscillation (see text). (b) Plot of localization length ξ as a function of 1/| ln r|. The
points show the values of ξ for different ratios of r, while the line shows the expected
theoretical dependence. r = 0.1, 0.15, 0.2, 0.25, 0.3, 0.333 from left to right.
Figure 4.6: PBC critical temperatures plotted vs. V for ratio values of 0.0, 0.1, 0.5,
0.9, 1.0.
33

Figure 4.7: Inset: T-dependence of ∆corner and ∆bulk for OBC and of ∆ for r = 0.5
and V = 1 lattice. The main figure shows ∆bulk versus T. The low temperature
region T < Tc,P BC is colored grey, and the tail of the bulk OP has been fitted to the
expression given in the text.
34

a superconducting gap is opened in the edge band, as can be seen in Fig. 4.3 (a). The
spatial dependence of the order parameter is governed by the spatial properties of
the 1D SSH edge modes, which are well-known. One thus observes, in addition to the
two-sublattice structure, an exponential decay of the order parameter as a function
of the distance from the edge. Fig. 4.5(a) presents a log-linear plot of the local order
parameter ∆i versus the distance from the edge. Fig. 4.5(b) shows the values of the
fitted localization length ξ (points) of the exponential decay of ∆i as a function of the
hopping ratio r, along with the expected dependence given by ξ = a/| ln(r)| (line).
We define ∆edge as the local order parameter of one of the two equivalent sites at
the center of one of the four equivalent edges. Finally, ∆edge has the standard BCS
dependence on V , namely ∆edge ∼ exp−cst/V
.
4.2.2 Finite temperature results
Fig. 4.6 shows results for the critical temperature of the periodic model at half-filling,
Tc,P BC, plotted against V , for different choices of the hopping ratio. Lines are fits to
the data using the form exp−
√
cst/V
.
For OBC, and a range of r-values, we find distinct transition temperatures for
the bulk and corner OP. This kind of two-step transition with corner (or surface)
superconductivity, followed by bulk superconductivity can be found more generally,
in systems with boundaries surface states, as shown in [33–35]. The temperaturedependence of the order parameter at bulk and corner sites, as well as for PBC, are
shown in Fig. 4.7 for r = 0.5 and V = 1. Fig. 4.7 shows that the corner and bulk
sites have different transition temperatures. The bulk OP which is expected to go
35

to 0 at the bulk transition temperature Tc,P BC, actually shows a non-zero tail above
this critical temperature. This tail arises from a proximity effect due to the corner
site where the OP is still non-zero, and it accordingly vanishes when T = Tc,corner.
One expects that the tail should be proportional to the corner OP, and also depend
on R, the distance of the midpoint from the corner as e
−R/λ(T)
. Here λ(T) is the
correlation length. Close to the bulk transition the correlation length should vary
as λ(T) = A/p
T − Tc,P BC in mean field theory. In this formula the prefactor A
depends on the hopping ratio, increasing monotonically as r → 1. Indeed, as shown
in Fig. 4.7, the tail is well fitted by this form. The tail is clearly visible only within
a range of r values. For very small hopping ratios, when the correlation length λ(T)
is too short for the tail to be observable. For larger r ∼ 1, the correlation length is
large and corner and bulk critical temperatures very close so that no tail is observed.
r 0.0 0.1 0.5 0.9 1.0
∆corner(0) c1 0.44 0.44 0.32 0.04 0
c2 0 0.02 4.96 23.32 26.27
c3 0 8.33 28.96 34.40 34.27
Table 4.1: Values of fitting parameters for zero-temperature corner site order parameters ∆corner(0) to the function f(V ) for different r.
36

4.3 Theoretical results
4.3.1 Calculation for periodic boundary conditions
We begin by considering a translationally invariant 2D SSH square of side L with
N = (L/2a)
2 unit cells, subject to periodic boundary conditions. Due to the lattice
symmetries, all sites have the same order parameter ∆i = ∆. We outline the gap
equation that is obeyed in the particle-hole symmetric half-filled lattice, µ = 0. We
will consider the weak coupling limit, V small, and define the expectation values b⃗k
by
b
(n)
⃗k
= ⟨η
(n)
⃗k↓
η
(n)
−⃗k↑
⟩ (4.6)
with all other expectation values of two annihilation operators assumed to vanish,
by the symmetries of the problem. η
(n)
⃗kσ
are eigenmodes in Fourier representation.
The order parameter ∆ can be written in terms of a sum over bands using the
transformation to the diagonal basis. One has ∆ = P
n ∆(n) where
∆
(n) =
V
4N
X
⃗k
b
(n)
⃗k
(4.7)
We can additionally simplify by assuming that only the two central bands contribute,
and fix the band index at n = 2 (it suffices to keep only one of the two central bands
in the sums over ⃗k, by virtue of their symmetry under exchange of kx and ky). The
37

BdG equations for different ⃗k decouple, giving rise to 2 by 2 matrices of the form
H⃗k =


ϵ2D(
⃗k) ∆
∆ −ϵ2D(
⃗k)

 (4.8)
where ϵ2D(
⃗k) is given by Eq. 4.2. Diagonalization yields quasiparticle energies of the
form E(
⃗k) = q
ϵ
2
2D(
⃗k) + ∆2
. As in the standard case, the gap equation is obtained
from the self-consistency condition, which reads
∆(T) = V
2
Z
dϵ ρ(ϵ)
∆(T)
E
th(βE/2) (4.9)
where β = 1/kBT is the inverse temperature and ϵ = ϵ2D is the single particle energy.
This gap equation predicts that, for fixed V , the order parameter ∆(0) decreases as
a function of r. In particular, for small r, perturbative expansion predicts a decrease
of the OP proportional to r
2
. This is indeed seen in Fig.4.4.
In the limit r → 0, the gap equation can be solved to obtain the T = 0 order
parameter ∆(0) as a function of V . As the width of the central band tends to zero,
the DOS can be approximately replaced by a delta function Aδ(E) where A ≈ 0.5 is
the fraction of states lying within this band (neglecting correction of order 1/L). The
gap equation yields ∆(0) ∼ V . The critical temperature can be determined from the
gap equation from the requirement that ∆(Tc) = 0. In the limit of small r, Tc scales
similarly as the OP, that is, Tc ∝ V .
For non-zero r the integral in Eq.4.9 is determined by the logarithmic van Hove
singularity at E = 0. Instead of the standard BCS form, Tc ∝ exp−1/N0V
, that is
expected for a regular density of states (where N0 is the DOS at the Fermi level),
38

the critical temperature here has a V -dependence given by Tc ∼ exp−
√
cst/V [31, 32].
These behaviors are confirmed by the numerical calculations, as shown in Fig. 4.6.
4.3.2 Calculation for open boundary conditions
Consider an open square sample of side L, with two weak edges which meet at the
corner situated at the origin. Thus, two perpendicular sets of 1D edge modes and
one 0D corner mode are present, in addition to the bulk modes. The extension
to situations with more than one corner mode is straightforward. To simplify the
analyses, we will assume that the sample is large so that the number of bulk modes
is much larger than the number of edge modes which is smaller by a factor 1/
√
N.
For convenience, we assign the site index O to the corner site on the top left, the
site index B to the site in the middle of the 2D sample, and the site index E to
the central site of one of the two equivalent weak edges (the upper edge and the
left edge). We will consider the superconducting OP at three specific locations as
follows:
– corner site OP (site index i = O): ∆corner = V ⟨cO↓cO↑⟩
– the OP at the bulk site(s) of the sample (site index i = B): ∆bulk = V ⟨cB↓cB↑⟩
– the OP for a site at the midpoint(s) of a weak edge (site index i = E): ∆edge =
V ⟨cE↓cE↑⟩
Let ϵν be the eigenvalues of the noninteracting Hamiltonian Eq. 4.1 and {ην} the
eigenmodes. We will suppose that they are ordered such that the first index ν = 1
denotes the corner mode, followed by the edge modes denoted by ν = 2, . . . , 2L − 1,
and finally the 2D bulk modes (ν = 2L, . . . , L2
). By diagonalization one obtains the
39

transformation U which relates one from the real space basis set {ci} to the expasion
in a new basis {ην}, i.e.
cj =
X
ν
Uiνην ην =
X
i
U
−1
νi ci (4.10)
where U
−1 = U
T
, the matrix U being real. The absence of translational symmetry makes it difficult to solve the coupled gap equations for OBC. However, with
simplifications, some limiting cases are solvable, as shown below.
System at Half-filling
In terms of the U transformation matrix, one can write the order parameter for
the midpoint at T = 0, ∆bulk, as follows:
∆bulk = V
X
ν∈bulk
U
2
Bνbνν (4.11)
where contributions of expectation values bµν = ⟨ηµ↓ην↑⟩ for µ ̸= ν are neglected.
In the small r limit, the corner mode contribution and edge mode contributions can
be dropped – the former decays very fast and therefore is zero in the center of the
sample, and the latter is very small because the edge band is far from the Fermi
level. One has then
∆bulk ≈
V
4N
X
ν∈bulk
bνν (4.12)
where the sum is over the bulk modes ν ≥ 2L. In the equation above, we have
simplified by replacing the coefficients U
2
M ν by their average value U
2
= 1/4N. To
compute the bνν, we assume that the interaction term can be decomposed into 2 by 2
40

blocks Hν in the space {cν↓, c
†
ν↑
}, as in the periodic case. In the case of bulk modes,
the energies ϵν are essentially the same as the energies ϵ2D in Eq. 4.2. As a result,
one obtains the same gap equation as in Eq. 4.9. In conclusion, ∆bulk ≈ ∆ and the
bulk OP is essentially the same as the order parameter found for PBC.
One can proceed in a similar way for the corner site OP ∆corner. One finds
∆corner = V U2
11b11 + V
X
ν∈bulk
U
2
Oνbνν + ...
≈ a1V + a2∆ (4.13)
where the numerical index of the corner site is 1 and ∆ represents the bulk OP
given by Eq. 4.9. In the second line, the coefficients U
2
Oν have been replaced by their
average value, written as a2/N with a2 < 1 a constant of order 1. In addition, we
used the result of the BdG hamiltonian for the η0 mode, which gives b11 = 1. The
constant a1 = U
2
11. Both the coefficients a1 and a2 depend on the hopping ratio.
When r → 0, a1 → 1
4
and a2 → 0. Then ∆corner = V/4. Similarly, in this limit,
the critical temperature for the transition at the corner can be shown to scale as
Tc,corner ∼ V .
When r ∼ 1, all coefficients of the U matrix are of the same order of magnitude,
O(1/
√
N). In this case, bulk modes contribute to leading order to all ∆i
, while
the corner and edge modes can be neglected. This results in bulk OP and corner
OP of the same order of magnitude, and both are similar to ∆ computed for the
periodic case. This explains the results shown in Fig. 4.4 for the corner and bulk
order parameters as a function of V for different hopping ratios r.
Chemical potential in an edge band.
41

When the Fermi level lies within an edge band, the superconducting gap opens
within this band. The edge mode contributions are the most important and the OP
is the largest on the edges. Writing out the expansion of ∆edge, one has
∆edge = V
X
ν∈edge
U
2
Eν⟨cν↓cν↑⟩ + ... (4.14)
where i is the index of the midpoint of a weak edge, and the sum runs over the indices
µ = 2, 2L−1. As before, we approximate the coefficients U
2
Eν by their average values.
The equation can then be simplified to give the self-consistent equation for this OP
at T = 0 as follows
∆edge ≈
V
4
√
N
X
ν
q
∆edge
(ϵν − uHF ν − µ)
2 + ∆2
edge
(4.15)
In this expression, for small r the single particle energies ϵν are essentially the 1 dimensional energies ϵ1D written in Eq. 4.2. Qualitatively, the above equation predicts
that when the chemical potential lies within this band, the solution for the OP is
expected to have the usual BCS form 1
. The numerical results described in section
III are in good accord with the analysis given here.
1Note however that when the chemical potential is in the region of the square root van Hove
singularities, the dependence of the OP on V should in principle be modified, with ∆edge ∼
√
V .
However, the degeneracy of the edge bands (a factor 1/L smaller than that of the bulk) is too small
for this effect to be of practical significance.
42

4.4 Related works and applications
An interesting direction for future work consists of extending the length of the unit
cell of the 1D chains used in defining the 2D model. One can get edge and corner
modes in 2D systems by considering chains of period 3 – following a {t1t2t1} sequence
in x and y directions. This is a member of a set of finite sequences which in the
infinite limit give rise to the Fibonacci quasicrystal, known to host topological edge
modes [36, 37]. It is not difficult to generalize our model to 3D, by taking a direct
product of three orthogonal SSH chains, in which case, vertex edge, surface and
bulk modes should appear. Variants of the 2D SSH model in the presence of nextnearest-neighbor hoppings have recently been considered experimentally [38–40] and
theoretically [20, 41]. As we discuss in the Appendix. 7.2, this term, when small
enough, does not result in qualitative changes of the above findings.
The s-wave superconducting phases we considered here are topologically trivial.
Introducing gauge fields or spin-orbit interactions are some means to induce topologically non-trivial superconducting phases, as in [42]. Adding spin-orbit interactions
should lead to new interesting edge phenomena and competition between order parameters of different pairing symmetries close to the edges, as discussed in [43].
A recent paper has introduced a route towards a 2D topological superconductivity
starting from SSH chains [44]. In this work, however, boundary phenomena were
not discussed and it would be interesting to study their model predictions for finite
samples. Last but not least, Floquet topological systems such as those discussed in
[45, 46] constitute another class of systems likely to host interesting edge and corner
43

superconducting states.
Future work will involve going beyond mean field theory to investigate the stability of the low dimensional phases described in this work. It will be interesting
to see how the results are modified when fluctuations are included. It would be an
interesting experimental challenge to realize edge and corner superconductivity. 2D
SSH lattices could be realized by the bottom-up assembly of atoms, a method which
has been to fabricate “designer” structures, as described in [47]. Obtaining a 2D SSH
lattice in the non-interacting limit should be possible, with the attractive Hubbard
Hamiltonian harder to obtain. If such a system is realized, electronic properties of
the edge and corner superconducting phases could be probed by scanning tunnelling
spectroscopy (STS).
44

Chapter 5
Superconductivity in the Fibonacci
chain
5.1 Backgruonds
Superconductivity in a quasicrystal was first reliably reported in an Al-Zn-Mg alloy
in 2018 [48], with a critical temperature of 0.05 K. More recently, superconductivity
has been reported in van der Waals layered quasicrystal of T a1.6T e with bulk critical
temperature ∼ 1K in 2023 [49]. This has led to many theoretical questions regarding
the nature of the superconducting instability, and the structure of Cooper pairs. The
standard BCS theory does not, of course, apply for these systems, due to the absence
of translation invariance. While attempts have been made to employ the superposition of nearly degenerate eigenfunctions for constructing extended quasiperiodic
Bloch wavefunctions in momentum space [50], the task of providing an appropriate
45

theoretical framework for describing interacting quasicrystals remains challenging.
There have been a number of previous theoretical studies for two-dimensional models. A real-space dynamical mean field theory treatment of the negative-U Hubbard
model on the Penrose vertex model was used to study the spatial pattern of the superconducting order parameter [51]. Further studies have been carried out using the
Bogoliubov de Gennes mean field approach for models on the Penrose tiling [51–55]
and the Ammann-Beenker tiling [54, 56, 57]. Other numerical works on 2D models
have studied the superconducting state in the presence of a magnetic field [58] and
topological superconductivity [59–61]. The results from these studies typically show
that the spatial variations of the superconducting order parameter and of the local
density of states are exceedingly complex functions of the local environment and the
band filling.
To understand the systematics of spatial variations in quasiperiodic systems, we
consider a simple one-dimensional system, the Fibonacci chain. We describe in detail
the dependence of the order parameter on the local environment, and explain it in
terms of approximate analytical solutions in the perturbative limit. The 1D model
allows to give the dependence of the transition temperature on the parameters of the
Hamiltonian, namely hopping ratio, chemical potential and interaction strength U.
In our study we use the Bogoliubov-de Gennes framework to treat the negativeU Hubbard model on the Fibonacci chain, assuming an inhomogeneous BCS s-wave
superconducting order. To justify our mean field approach, we argue that our 1D
results can be carried over to a 3D extension of the Fibonacci chain, in which periodic
2D lattices are stacked in the third direction in a quasiperiodic way. Such Fibonacci
46

2 0 2
E/t
0.00
0.25
0.50
0.75
1.00
DOS
(b) U=0
2 0 2
E/t
(c) U=1.3t
A A B A B A A B A B A A B A B A A B A B
(a)
Figure 5.1: (a) cut-and-project method of generating the Fibonacci chain: Projecting
selected sites of the 2D square lattice (within the blue strip) on to a line with slope
equal to the golden ratio generates the Fibonacci chain. The chain linked by ‘A’s
and ‘B’s is shown near the bottom of this figure. Atom sites (defined in Sec. 5.2)
are marked by green in the Fibonacci chain. Global density of states in a 610-
site approximant of the Fibonacci tight-binding chain without (b) and with (c) the
attractive Hubbard U. When an attractive Hubbard interaction is introduced, a
superconducting gap appears in the density of states at the Fermi level, indicated by
the arrows in (b). The modulation strength w = 0.2 in (b) and (c).
47

superlattices, which have been studied both theoretically and experimentally, provide a justification for the use of mean field theory which would otherwise not be
meaningful for a single chain. When the in-plane periodic hopping amplitudes are
much weaker than the aperiodic hopping amplitudes in the out-of-plane direction,
one finds a close similarity between the results for the 1D and 3D systems.
5.2 Model
We consider the off-diagonal Fibonacci tight-binding model. This is defined by a
binary hopping sequence generated by the infinite limit of the substitution rule tA →
tAtB, tB → tA on an initial sequence consisting of a single hopping, tA. The nearestneighbor hopping integrals, ti
, take one of two values, tA or tB, according to the
Fibonacci sequence, where i is the position index. In numerical calculations, we
use finite hopping sequences generated by applying the substitution rule a finite
number of times denoted by n. These are the approximants of the Fibonacci chain—
finite structures that locally retain quasiperiodic character. The nth generation
approximant contains a number of atoms equal to the Fibonacci number, Fn =
Fn−1 + Fn−2. We parametrize the strength of the quasiperiodic potential by the
modulation strength w = tB −tA, with tB > tA. We fix the total bandwidth with the
constraint that the average hopping t =
Fn−1tA+Fn−2tB
Fn
= 1. With this constraint, tA
and tB are fully specified by w. The hopping ratio is given by ρ = tA/tB = 1 −
wτn
τn+w
,
where τn =
Fn
Fn−1
. The τn are called convergents of the golden ratio τ , and τn → τ in
the limit n → ∞.
48

0 250 500
Site Number
0.075
0.100
0.125
(a)
0 250 500
Conumber
0.075
0.100
0.125 (b)
molecule
atom
molecule
2 0 2
E/t
LDOS
(c)
0.075 0.100 0.125
Coherence
Peak Height
(d)
0.0 2.5 5.0
w2
0.1
0.2
0.3
kB
Tc
/
t
(e)
0.04 0.09 0.16
0.05
0.07
Figure 5.2: (a) Superconducting order parameter in real space and (b) in perpendicular space for a chain with 610 sites at half-filling. Panels (a)-(d) share all the
same calculation parameters, chain length is 610, modulation strength w = 0.2 and
BCS pairing strength U = 1.3t. (c) Local density of states (LDOS) at the four different sites marked in (b) with different values of the order parameter. The width
of the central gap is the same in the LDOS for all sites. (d) coherence peak height
in LDOS vs. the order parameter at the corresponding site. The order parameter
at a given site is positively correlated with the height of the coherence peak in the
local density of states for that same site. (e) Dependence of the critical temperature
on the square of the modulation strength for a chain with 610 sites and BCS pairing strength U = 1.3t. The data points within the red frame in the main figure is
zoomed in in the inset, which shows a clear linear scaling between critical temperature and the square of the modulation strength. In the inset of panel (e), w values
are 0.0,0.1,0.2,0.3,0.4 from left to right.
49

The Fibonacci chain has been well studied as a prototypical quasicrystal due to
its simple structure and special physical properties, chief among which is its fractal
eigenvalue spectrum. In the approximant structure, there are Fn distinct bands, and
Fn − 1 gaps. As n → ∞, the gaps become dense and the spectrum becomes singular
continuous. In fig. 5.1 (b), we present the global density of states of a non-interacting
610-site approximant of Fibonacci chain. Several of the larger gaps are visible, while
the smaller gaps are smoothed over by the broadening. Each gap can be assigned a
unique integer q in accordance with the gap-labelling theorem [62, 63].
It is often useful to index the sites of the Fibonacci approximant by their conumbers rather than their positions. The conumber indexing is derived by considering
the projection on to the perpendicular space in the cut-and-project scheme or model
set method [64] of generating the Fibonacci chain. The conumber c(i) ∈ [0, Fn−1] of
a site with position index i can be defined for a given Fibonacci approximant by [65,
66]
c(i) = Fn−1i mod Fn. (5.1)
The conumbering index classifies sites according to their local environment—sites
with similar local environments are closer to each other. In particular, the conumber
indexing partitions sites into atom or molecule sites. There are Fn−3 sites that have a
weak hopping tA in both directions. These are called atom sites and their conumber
falls in the central window of the conumber sequence, c(i) ∈ [Fn−2, Fn−1]. In Fig. 5.1
(a), atom sites are marked by green dots. The 2Fn−2 molecule sites have a strong
hopping tB in either the left or the right direction, and accordingly, their conumber
lies in the right (c(i) > Fn−1) or the left (c(i) < Fn−2)window.
50

A local attraction of the form Hˆ
I =
P
i −
U
2
c
†
i↑
c
†
i↓
ci↓ci↑ causes electrons to pair
and form a superconducting condensate. We apply the Bogoliubov-de Gennes meanfield approximation to this model with the introduction of two mean-field terms—
the Hartree Fock shift ϵ
HF
i = −U⟨c
†
i↑
ci↑⟩ and superconducting pairing amplitude
∆i = U⟨ci↑ci↓⟩. The Hamiltonian is shown below:
Hˆ = −
X
iσ
ti(c
†
i+1σ
ciσ + c
†
iσci+1σ)
+
X
iσ
(ϵi + ϵ
HF
i − µ)c
†
iσciσ
− U
X
i

∆ic
†
i↑
c
†
i↓ + ∆∗
i
ci↑ci↓

. (5.2)
where ciσ (c
†
iσ) is the electron annihilation (creation) operator with spin σ at site
number i, ϵi
is an on-site potential, and µ is the chemical potential. The second and
third line is the result of a mean-field treatment of the negative-U Hubbard term.
The two mean-field terms must be computed self-consistently,
∆i = U
X
n
v
∗
inuin (1 − 2f(En, T)) (5.3)
ϵ
HF
i = U
X
n
|uin|
2
f(En, T) + |vin|
2
(1 − f(En, T)) (5.4)
where En are the positive eigenvalues, and (uin, vin) are the eigenvectors of the
Bogoliubov-de Gennes pseudo Hamiltonian operator corresponding to (5.2). All the
following results are reported in units of t, the average hopping amplitude.
51

5.3 Results
5.3.1 Order Parameter and critical temperature at half-filling
At half-filling, an attractive Hubbard term opens a superconducting gap at the Fermi
level. Fig. 5.1(b) and (c) shows the density of states of the half-filled Fibonacci chain
without (b) and with (c) a non-zero Hubbard attraction. Note the opening of the superconducting gap at the Fermi level in (c) with coherence peaks appearing on either
side. This gap is in addition to the several intrinsic gaps of the non-interacting Fibonacci chain which can also be seen in (b). The opening of the superconducting gap
corresponds directly with a non-zero superconducting order parameter ∆i shown in
Fig. 5.2(a). Because the quasicrystal does not have translational symmetry, the order
parameter is position dependent. Its fluctuations follow the underlying quasiperiodic
modulation. The complex real-space pattern in Fig. 5.2(a) is considerably simplified
by transforming to conumber space following eq. (5.1), where sites with similar local
environments are closer to each other. Fig. 5.2(b) shows the same ∆i
in (a) but with
the indices permuted according to the conumber transform. The site-by-site fluctuations seen in real space take a relatively smoother layered structure that looks like a
sequence of plateaux. Note that this curve is not smooth in the technical sense, but
discontinuous everywhere, and one can see a self-similar fractal pattern emerging.
The conumber indexing allows us to connect the magnitude of the order parameter
with the local neighborhood of the sites of the Fibonacci chain. At half-filling, we
find that generically the atom sites (with a weak bond to either side) have a higher
52

local order parameter than the molecule sites (with a strong bond to one or the other
side). This is a consequence of the fact that the eigenstates at half-filling have higher
spectral weight on the atom sites than on the molecule sites [67, 68].
Even though the local order parameter is site dependent, the superconducting gap
in the local density of states is the same in all sites. This is shown by example of four
sites with distinct neighbourhoods in Fig. 5.2 (c). The vertical dashed line marks the
smallest energy with a non-zero value of the local density of states. This is a corollary
of the fact that there is only one critical temperature. The entire Fibonacci chain
goes superconducting at the same temperature. This is consistent with the findings
of [56] where the same is found to be true in the Ammann-Beenker tiling, and in
contrast with the disordered case, where strong disorder leads to the formation of
superconducting islands [69, 70]. The magnitude of the order parameter is instead
reflected in the height of the coherence peaks. The positive correlation between
the coherence peak height and the magnitude of the order parameter is shown in the
scatter plot in Fig. 5.2 (d). Consistent with BCS theory, the average order parameter
scales as √
Tc − T close to and below the critical temperature (see Appendix 7.5).
Fig. 5.2 (e) shows the critical temperature plotted against the square of the modulation strength w
2
. It shows that the critical temperature increases monotonically
with the modulation strength. This is a consequence of the fact that with increasing w, the intrinsic gaps of the Fibonacci chain increase, occupying an ever higher
proportion of the total bandwidth. Since we keep the bandwidth constant in our calculation, the density of states in the ungapped regions must become higher as more
states get squeezed into smaller energy intervals. This ultimately leads to a higher
53

number of states being available to form the superconducting condensate around
the Fermi level. The functional form of the the critical temperature as a function
of the modulation strength can be understood in two opposite limits. In the weak
modulation limit, ρ → 1, w → 0, the critical temperature goes quadratically in w
(see inset of Fig. 5.2). This follows by treating the quasiperiodic modulation as a
perturbation to the periodic system (following from [71], where one finds that the
gap width grows linearly in w [37]). We show in Appendix 7.4 that this results in
a BCS order parameter that scales quadratically with w. In the strong modulation
limit, ρ → 0, w →∼ 2.6, Tc becomes essentially independent of w. In this limit,
the Fibonacci chain decouples into a series of disconnected atoms (monomers) and
molecules (dimers). The density of states at half-filling takes the form of a delta
function localized entirely on atom sites. In this limit, the critical temperature saturates to the value expected for a single-site Bogoliubov-de Gennes calculation, see
Appendix 7.7 for analytical expression.
5.3.2 Scaling of critical temperature with U
Multifractality of spectrum and states plays a crucial role in controlling the superconducting order parameter and critical temperature. The singularities of the spectrum
offer a means to control these properties – for example, to enhance the value of Tc
by tuning µ.
In this section we show results obtained for Tc as a function of U for different
quasiperiodic modulation strengths. In an infinite chain, the density of states is
expected to have power law singularities throughout the spectrum. The curves for
54

Tc(U) are shown for different values of w in Fig. 5.4 (a). The critical temperatures
and gap width ∆g are proportional in all cases as shown in Fig. 7.5. The coefficient
of proportionality which is predicted to be 3.52 in BCS theory increases with w
and ranges from 3.46 for w = 0.2 to 3.98 for w = 2.5. This coefficient is slightly
lower than the BCS value and then become larger than 3.52, which may indicate
different pairing mechanisms under weak and strong modulaiton limits. Because of
this proportionality, the gap width also follows a power-law scaling with respect to
U, shown in Fig. 5.4 (b). This study at half-filling shows that the quasiperiodic
modulation is a sensitive tuning parameter to change the critical temperature and
spectral gap width for a given value of interaction strength U. Other band fillings
are expected to have similar behavior, with different values of the power law.
5.3.3 The order parameter as a function of band filling
In this section we study the effects of varying the chemical potential µ in (5.2). In
Fig. 5.5(a), the color map shows the amplitude of the local order parameter ∆i
in
conumber space for a range of electron densities (parametrized by µ along the yaxis). In our convention, µ = 0 represents the half-filled chain. If the Fermi level is
tuned to a gapless part of the non-interacting spectrum, (or, anticipating the results
of the following section, if the gap is not too large), the system is superconducting
below a Tc and ∆i
is non-zero everywhere and has a complex self-similar plateauxlike structure as in the trace in Fig. 5.2(a), except that the relative values of the
plateaux depends on the filling. On the coarse level, the relative magnitude of the
local order parameter at a given site depends on the immediate local environment of
55

0.035 0.040 0.045
kBT
0.000
0.025
0.050
0.075
0.100
(a)
w=0.2
0.24 0.26 0.28
kBT
0.0
0.1
0.2
0.3
0.4 (b)
w=1.5
Figure 5.3: Order parameter vs. temperature for a chain of length 610 and attraction
U = 1.3t at two modulation strengths (a) w = 0.2 and (b) w = 1.5. Each column
of data at a specific temperature represents all the order parameter values of the
Fibonacci chain.
56

2 1 0
ln U
4
3
2
1
0
ln
g
(b)
w=0.2
w=0.4
w=0.6
w=0.8
w=1.0
w=1.5
w=2.0
w=2.5
2 1 0
ln U
6
5
4
3
2
1
ln
kB
Tc
(a)
Figure 5.4: (a) Plots of ln kBTc versus ln U for varying values of w. Calculations are
done for a chain of 610 sites. This indicates a power law dependence between the
critical temperature and attraction strength. (b) Plots of gap width ln ∆g vs. ln U
showing that they are proportional, with a w-dependent coefficient of proportionality.
57

the site. When the system is close to half-filling, atom sites (the sites between two
red lines) have higher OP values; away from half filling, molecules sites have higher
OP values. The finer structure in Fig. 5.5 is governed directly by the electron density
around the Fermi level. Note the qualitative similarity between Fig. 5.5(a) and the
local electron density map shown in Fig. 7.4 of [72] (reproduced in Appendix 7.6).
When tuning the chemical potential µ and attraction U at the same time, we got
the average OP distribution pattern (Fig. 5.5 (b)) closely related to the spectrum of
Fibonacci approximants. There are two distinct phases that can be accessed in this
way, depending on whether the Fermi level lies in a region of finite density of states,
or an intrinsic gap.
5.3.4 filling-U phase diagram
If the Fermi level is tuned to a gap, the system may be insulating or superconducting,
based on a competition between the strength of the attraction U and the width of
the gap. A minimal model with this phenomenology is the BCS model for the
semiconductor [73, 74]—a two band model with an attractive BCS term. In this
model, when the Fermi level lies within a gap, there is a sharp transition from the
superconducting to the insulating state as the Cooper pair binding energy crosses
below the gap width. An analogous phenomenology is reflected in the Fibonacci
chain, where the non-interacting spectrum contains a heirarchy of gaps. When the
Fermi level is close to or in a particular gap, the low-energy physics can, at least
heuristically, be approximated by the two-band model, as the fine structure in the
density of states further away from the Fermi level is less relevant. Fig. 5.5(b) is
58

1000 2000
conumber
2.0
1.0
0.0
-1.0
-2.0
/
t
(a)
0.000
0.025
0.050
0.075
0.100
0.4 0.8 1.2 1.6 2.0
U/t
0.0
-0.5
-1.0
-1.5
/
t
(b)
0.00
0.05
0.10
0.15
0.20
Figure 5.5: (a) Local order parameter distribution in conumber scheme for BCS
attraction U=0.9t at various chemical potentials for a chain of 2584 sites. (b) Average
order parameter as a function of the BCS attraction U and the chemical potential µ.
a color map of the average order parameter in the chain as a function of µ and
U. We see in Fig. 5.5(b), that as the BCS attraction is reduced, a hierarchy of
insulating regions (with ∆average = 0) start to appear. Each of the insulating regions
is associated with an intrinsic gap of the Fibonacci chain. This gives rise to a
complex phase diagram where at smaller values of U, a series of superconductorinsulator transitions occur as the chemical potential is raised or lowered.
59

5.4 Conclusion
In this work, we explore both local and global superconducting features of Fibonacci
approximant chain. These features engage in mutual interactions. The order parameter magnitude is modulated by local hopping environments and the average electron
density of the chain. All the sites share the same critical temperature is related to
the uniform superconducting gap width across the chain. The self-similar pattern
of order parameter distribution exists in both real space and perpendicular space
with tuning the chemical potential in a wide range. Upon simultaneous adjustment
of chemical potential µ and attraction U at the same time, the resulting average
OP distribution displays a strong resemblance to the spectrum of Fibonacci aprroximant. The multifractal characteristics of the Fibonacci chain lead to a significant
alteration in the scaling exponent between the critical temperature (average order
parameter) and attraction strength. Focusing solely on the half-filling scenario, we’ve
observed substantial variations in the scaling exponent from weak to strong modulation limits. This distinctive trait provides abundant opportunities for the controlled
manipulation of associated physical properties.
60

Chapter 6
Conclusions
Exploring superconductivity using a mean-field way provides us with the opportunity to unravel the complex interactions within many-body systems. Conventional
superconducting properties at zero and finite temperatures have been investigated
in a periodic system, the 2D SSH model, and a quasiperiodic system, the Fibonacci
approximant chain. In the 2D SSH model, we focused on the edge modes and corner
modes. The superconducting order parameter on surface sites scales differently with
attraction magnitude and temperature when compared with those of the bulk sites.
This feature provides a way of controlling the superconductivity. When it’s close
to the critical temperature, the corner site superconductivity penetrates into the
bulk sites next to them due to the proximity effect, which leads to inhomogeneous
critical temperatures in the lattice. By tuning the hopping sequence in the lattice,
the superconducting pattern can be adjusted. This model serves as a prototype for
forming various superconducting arrays which can be promising for quantum com61

puting related applications. In the Fibonacci approximant chain, we studied the
superconducting order parameter distribution along the Fibonacci chain in both real
space and perpendicular space. The distribution in both these two spaces inherit
the self-similar features of Fibonacci hopping. The superconducting order parameter is determined both by local environment and electron filling. Even though the
superconducting order parameters are fluctuating along the chain, all the sites share
the same superconducting gap width and critical temperature, which is totally different from disordered systems. Critical temperature and spectral gap width both
increases with the attraction value following a power law. The FC approximant
chain holds a rich phase diagram. Compared with the 2D SSH model, Fibonacci
approximant’s multifractal characteristics in their energy spectrum enable it much
more possibilities by tuning the electron filling to regulate the critical temperature
and superconducting properties. Incorporating multifractality into the study of superconductivity expands our comprehension of superconductivity. The Fibonacci
chain is a good platform to study the interplay between disorder and topological
properties, as they are intermediate between periodic and disordered systems.
These investigations serve as a reference for understanding the superconducting
characteristics of low-dimensional topological materials. Furthermore, these fundamental models extend their applicability beyond actual materials, as they can be
replicated in optical lattices and with electromagnetic waves. [75]. These provide
abundant experimental opportunities to investigate these systems. The considered
s-wave superconducting phases are topologically trivial. Introducing gauge fields or
spin-orbit interactions offers a way to induce topologically non-trivial superconduct62

ing phases, resulting in intriguing edge phenomena, greater potential to control the
superconducting properties in these systems, and richer phase diagram. Future research will move beyond mean-field theory to explore the stability of low-dimensional
phases. Adding disorder to these low-dimensional systems is also a promising research
direction.
63

Chapter 7
Appendix
7.1 Model analysis in momentum space
In each unit cell of the periodic lattice, we label the top left, top right, bottom left
and bottom right sites with a numerical index starting from 1. In momentum space,
the non-interacting 4 × 4 matrix Hmomentum is given by
Hmomentum =


0 α(kx) β(ky) 0
α
∗
(kx) 0 0 β(ky)
β
∗
(ky) 0 0 α(kx)
0 β
∗
(ky) α
∗
(kx) 0


. (7.1)
Here, the matrix elements are
α(kx) = H12 = H34 = tAe
ikxa + tBe
−ikxa
, (7.2)
β(ky) = H13 = H24 = tAe
ikya + tBe
−ikya
. (7.3)
64

(a)
(b)
Figure 7.1: (a) Band structure of the periodic 2D SSH model on a square lattice.
Here, the high symmetry points are M(
π
2a
,
π
2a
), X(
π
2a
, 0.0), and Γ(0.0, 0.0). (b) Band
structure of the SSH model on a ribbon with periodic boundary conditions in one
direction and open boundaries in the other direction. Here, the path in momentum
space is one-dimensional, and the high symmetry points are Γ(0.0) and M(
π
2a
). We
use r = 0.25.
In Fig. 7.1 (a), the two orange dispersion curves correspond to the central bulk
band in Fig. 4.1, whereas the blue dispersion curves belong to the two lateral bulk
bands. When open boundary conditions are applied in one direction, as shown in
panel (b), two additional edge bands emerge, which are represented by the two green
curves in Fig. 7.1. Furthermore, the flat purple band at energy 0 stems from the
corner sites.
65

(a)
(b)
(c)
Figure 7.2: (a) Illustration of the 2D SSH model, including intra-cellular nextnearest-neighbor hoppings. t1 is represented by the thin blue bonds, whereas t2
depicted by the thick green bonds. The intra-cellular next-nearest-neighbor (NNN)
hopping, tNNN is represented by the orange bonds. (b) Band structure of the periodic
2D SSH model with NNN hoppings. Here, the high symmetry points are M(
π
2a
,
π
2a
),
X(
π
2a
, 0.0), and Γ(0.0, 0.0). (c) Band structure of the 2D SSH model on a ribbon
structure with NNN hopping. Here, we used t1 = 0.4, t2 = 1.6, and tNNN = 0.2.
66

7.2 Effects of intra-cellular next-nearest-neighbor
hopping
In order to make a comparison with recent experimental and theoretical work on
related systems that include longer-range hopping [olekhno2021higher, 20, 39–
41], here we consider the effects of two additional intra-cellular next-nearest-neighbor
hopping matrix elements, tNNN . Comparing the band structures of the lattices in
Fig. 7.1 (a) and Fig. 7.2 (b), the only qualitative difference is an asymmetry with
respect to the dispersions of the top and the bottom curves, because next nearest
neighbor hopping breaks the chiral symmetry and the particle-hole symmetry is lost.
Furthermore, in Fig. 7.1 (b) and Fig. 7.2 (c), the two green edge bands and purple
near-flat corner band still exist when introducing tNNN . These features remain the
same also when the two diagonal tNNN ’s are not equal to each other.
7.3 Properties of the convergents of the golden
ratio
The golden ratio is the positive root of of the polynomial x
2−x−1. This immediately
implies two useful identities:
τ
2 = 1 + τ (7.4)
τ = 1 +
1
τ
(7.5)
67

The analogous identities for the convergents are as follows:
τnτn−1 = 1 + τn−1 (7.6)
τn = 1 +
1
τn−1
(7.7)
The application of these identities allows us to compactly express the relation between
the modulation strength w, the hoppings tA and tB, and the hopping ration ρ =
tA/tB.
tA = 1 −
w
τnτn−1
(7.8)
= 1 −
w
1 + τn−1
(7.9)
tB = 1 +
w
τn
(7.10)
= 1 + w(τn+1 − 1) (7.11)
ρ = 1 −
wτn
τn + w
(7.12)
For the full Fibonacci chain in the limit n → ∞, the convergents τn can be replaced
by the golden ratio τ in the above expressions.
68

7.4 Quadratic scaling of the superconducting gap
with the modulation strength
When the modulation strength w = 0, eigenstates are doubly degenerate. ϵ
+
n = ϵ
−
n =
ϵn. Turning on w, the degeneracy splits linearly in w [raiBulkTopologicalSignatures2021],
ϵ
±
n = ϵn ± αnw, (7.13)
where 2αn is the gap width corresponding to the pair of states at ϵn. The contribution of a state at energy ϵ
±
n
to the density of states at energy ϵ
′
, DOS(ϵ
′
), is a
monotonically decreasing function of ∆ϵ
±
n = |ϵ
±
n − ϵ
′
|, which we denote as f(∆ϵ
±
n
).
Expanding f(∆ϵ
±
n
) to second order in w,
DOS(ϵ
′
) =X
n
f(∆ϵ
+
n
) + f(∆ϵ
−
n
) (7.14)
=
X
n
[2f(∆ϵn) + f
′
(∆ϵn)(αnw − αnw)
+2f
′′(∆ϵn)(αnw)
2

(7.15)
=
X
n
2f(∆ϵn) + w
2X
n
2f
′′(∆ϵn)α
2
n
(7.16)
=DOSw=0(ϵ
′
) + w
2F(ϵ
′
), (7.17)
where we let F(ϵ
′
) = P
n
2f
′′(∆ϵn)α
2
n
. Thus, the density of states of a given energy
is proportional to the square of modulation strength.
In BCS theory, ∆ ∝ e
− 1
N(ϵf
)U
, where N(ϵ) is the density of states of the noninteracting system without any quasiperiodic modulation. we Taylor expand this to
69

first order in w
2
,
∆(w) ∝e
− 1
U(N(ϵf
)+w2F (ϵf
))
(7.18)
=e
− 1
UN(ϵf
) +
w2F (ϵf
)
UN(ϵf
)2 +O(w4
)
(7.19)
=e
− 1
UN(ϵf
)

1 +
w
2F(ϵf )
UN(ϵf )
2
+ O(w
4
)

(7.20)
=∆(ω = 0)
1 +
w
2β(ϵf )
UN(ϵf )

+ O(w
4
) (7.21)
where β(ϵf ) = F(ϵf )/N(ϵf ).
7.5 Average order parameter as a function of temperature
In Fig. 7.3, the average order parameter magnitude is plotted vs. temperature near
the critical temperature. These curves follow the √
Tc − T relation in BCS theory.
The average order parameter magnitudes for both atom sites and molecule sites also
fit the same √
Tc − T relation just by multiplying different constant coefficients.
70

0.040 0.045
T
0.0000
0.0005
0.0010
0.0015
0.0020
2
a
v
g
(a) w=0.2
0.120 0.125 0.130
T
0.0000
0.0005
0.0010
0.0015
0.0020
2
a
v
g
(b) w=0.6
0.275 0.280 0.285
T
0.0000
0.0005
0.0010
0.0015
2
a
v
g
(c) w=1.5
0.315 0.320 0.325
T
0.0000
0.0005
0.0010
2
a
v
g
(d) w=2.5
Figure 7.3: (a)-(d) The square of average order parameter at different temperatures
near its corresponding Tc value under different modulation strengths. The dots
represent the data point. The curve is the fitting curve following the formula ∆2
avg =
const. × (Tc − T) (which is the same as ∆avg = const. ×
√
Tc − T). The Fibonacci
approximant length is 610 and the attraction value is 1.3t.
71

Figure 7.4: LDOS intensity distribution as a function of energy and conumber indexing. The darler color represents the higher LDOS intensity. Reproduced with
permission from [72].
7.6 Intensity distribution of local density of states
in perpendicular space and energy space under strong modulation limit
In Mac´e et al. ’s work, they display the LDOS intensity in the energy and perpendicular space, which looks similar to our Fig. 5.5 (a). The electron density distribution
inherits the self-similar feature in perpendicular space. These two plots prove that
the superconducting order parameter distribution follows the electron density distribution.
72

7.7 Critical temperature at strong modulation limit
When w → 2.6, only the atom sites contribute to the pairing if the system is at
half-filling. It’s only necessary to consider the atom sites in Hamiltonian.
Hatom =


0 U∆
′
U(∆′
)
∗ 0

 . (7.22)
where ∆′ =
P
n
v
∗
nun at zero temperature and n represents the different eigen energy
states. ∆′
is ∆ without multiplying by U. After the first iteration of diagonalizing
the Hamiltonian above, the eigenstate corresponding to the only positive eigenvalue
U
p
|∆′
|
2
is:
1
2


1
1

 . (7.23)
Then (∆′
)
(1) =
1
2
. Plug the current (∆′
)
(1) back into the Hamiltonian Hatom and
diagonalize again, it finally results in a self-consistent ∆′
value 1
2
. So, the order
parameter magnitude at zero temperature ∆(0) = U
2
.
Based on the conventional BCS theory, Tc =
∆
1.76 . Substitute our ∆(0) into the
above equation,
Tc =
U
2 × 1.76
=
U
3.52
(7.24)
When U = 1.3t, this results in Tc ≈ 0.369. The numerically calculated Tc value is
0.326.
As shown in Fig. 7.5, ∆g
kBTc
= 3.98 when w = 2.5. If we plug ∆(0) = U
2
into it, Tc
73

0.0 0.1 0.2 0.3
kBTc
0.0
0.5
1.0
1.5
g
w=0.2
w=0.4
w=0.6
w=0.8
w=1.0
w=1.5
w=2.0
w=2.5
Figure 7.5: Plots of gap width ∆g vs. kBTc showing that they are proportional, with
a w-dependent coefficient of proportionality.
in the strong modulation limit is 0.327, which is almost the same as our numerical
Tc value 0.326.
74

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Abstract (if available)

Abstract Fundamental research in low-dimensional topological systems is an exciting and rapidly evolving field, offering numerous avenues for exploration. We focus on the superconducting properties at zero and finite temperatures in basic low-dimensional topological models, such as the two dimensional Su-Schrieffer-Heeger (SSH) model and the Fibonacci chain. We show analytically and by numerical diagonalization that the superconductivity on surface sites can be tuned by the electron filling. The properties of the system are controlled by a central parameter which enters the Hamiltonian, the hopping ratio. The scaling relations involving superconducting observables--such as the superconducting order parameter, the spectral gap width, and the critical temperature are investigated. These scaling relations are determined by the singularity strength of the electronic spectrum. The critical temperature can be uniform or site-dependent, depending on the specific model. When the critical temperature distribution is not homogeneous, a novel proximity effect arises when the temperature is in between the higher and the lower critical temperature, here the sites with larger superconducting order parameter induce a nonzero tail of the superconducting order parameter in nearby sites. In contrast, the uniform spectral gap width in the quasiperiodic Fibonacci chain is a corollary of the fact that there is only one critical temperature in this system.

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Creator Wang, Ying (author)

Core Title Superconductivity in low-dimensional topological systems

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Degree Conferral Date 2023-12

Publication Date 11/17/2023

Defense Date 11/02/2023

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

Tag critical temperature,OAI-PMH Harvest,superconducting order parameter,superconductivity,topological models

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Language English

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Advisor Haas, Stephan (committee member), Mak, Chi (committee member), Nakano, Aiichiro (committee member), Takahashi, Susumu (committee member)

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