ABSTRACT

Title of Dissertation: LOW TEMPERATURE
SCANNING TUNNELING MICROSCOPY
OF TOPOLOGICAL MATERIALS
AND MAGNETIC STRUCTURES

Joseph Murray
Doctor of Philosophy, 2023

Dissertation Directed by: Professor Christopher Lobb
Department of Physics

Scanning tunneling microscopy (STM) provides an opportunity to study the physical and

electromagnetic properties of surfaces at the atomic scale. When performed at low temperatures,

in high magnetic fields, and with a variety of different probes, it offers a wide range of methods by

which novel materials of great practical and theoretical interest can be evaluated, characterized,

and even fabricated with atomic precision.

This thesis describes three independent STM studies performed at cryogenic temperatures.

In the first, I present an in-situ modification to our 4K STM which permits us to current-bias

our samples during STM operation. The modification can be used to study non-equilibrium

effects such as spin accumulation induced by a current through a spin Hall material and the spin-

momentum locking which is present at the surface of topological insulators.

Next, I examined oxygen-doped aluminum films with anomalously high kinetic inductance.

A suggested explanation was the migration of oxygen to the grain boundaries, forming a percola-



tion network separated by Josephson links. To determine the coupling between grains, I studied

the films using milliKelvin STM performed with a superconducting tip.

Finally, transport measurements performed by our collaborators indicated the possible pres-

ence of a topological Hall effect in thin films of Cr2Te3, induced by the presence of topologically

non-trivial magnetic textures called magnetic skyrmions. In order to provide more decisive evi-

dence, I studied the films using spin-polarized STM at 4K.

In addition to the experimental studies, this work explores the theoretical underpinnings of

the novel materials which constitute the frontier of our current understanding of condensed matter

physics. An emphatically pedagogical viewpoint is adopted throughout as part of a continuing

effort to bridge the gap between experiment and theory.



LOW TEMPERATURE SCANNING TUNNELING MICROSCOPY OF
TOPOLOGICAL MATERIALS AND MAGNETIC STRUCTURES

by

Joseph Edward Murray

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2023

Advisory Committee:
Professor Christopher Lobb, Chair/Advisor
Dr. Robert E. Butera, Co-Advisor
Dr. Michael Dreyer
Professor Johnpierre Paglione
Professor John Cumings



© Copyright by
Joseph Edward Murray

2023



Dedication

To my mother, whose unconditional love and support were the bedrock of my youth, and

whose memory will reverberate through the rest of my life. I love you.

ii



Acknowledgments

This work represents the end of a very long and winding road which I would never have

reached without the love and support of a vast community of mentors, friends and loved ones.

I owe an enormous debt of gratitude to my advisor Prof. Chris Lobb and my supervisor

Dr. Robert Butera, who took me on as a student and guided me to where I am today. Special

thanks are due to Dr. Michael Dreyer, who spent years teaching me the lion’s share of what I

know about experimental physics in general and scanning probe microscopy in particular. I am

also indebted to Profs. Johnpierre Paglione and John Cumings for adding their unique insights

and perspectives to my dissertation committee.

I would also like to thank my peers at the Laboratory for Physical Sciences. In par-

ticular, my heartfelt appreciation goes out to the LPS STM group which at various times in-

cluded Wan-Ting Liao, Kevin Dwyer, Sungha Baek, Azadeh Farzaneh, Jon Marbey, and Giovanni

Franco-Rivera; to Jimmy Kotsakidis for his experience, expertise, encouragement, and steadfast

Australian-ness; to Jennifer DeMell for her relentless support and friendship; and to the LPS ma-

chinists Don Crouse, Ruben Brun, and Paul Davis, without whom much of my work would have

been impossible.

I would not have made it through the turmoil of my first few years in graduate school with-

out the support of many dear friends I made at the University of Maryland, including Anastasia

Marchenkova, Gina Quan, Steve Ragole, Caroline Figgatt, Antony Speranza, David Somers, Matt

iii



Anthony, Connor Roncaioli, Pete Volpe, Min-A Cho Zeno, and Dalia Ornelas. But my support

network extended far beyond UMD - I have the greatest lifelong friends that could ever be asked

for (EEE), and could not begin to express the depth of my gratitude to each and every one of them.

Finally, I could not have made it to the end of this road without my father John, my brother

Steve, and my partner Elaine, who was a rock of support through good days and bad, and who

generously kept me alive during the long days and nights of writing.

iv



Table of Contents

Dedication ii

Acknowledgements iii

Table of Contents v

List of Tables viii

List of Figures ix

Chapter 1: Introduction 1

Chapter 2: A Brief Introduction to Topology 4
2.1 The Main Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Constructing Important Topologies . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Topological Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Topological Equivalence of Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Homotopy Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Homotopy Equivalence of Topological Spaces . . . . . . . . . . . . . . . 30

Chapter 3: Physical Applications of Topology I 37
3.1 Foundational Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 A Mathematical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 An Important Note Regarding Stability . . . . . . . . . . . . . . . . . . 41

3.2 Magnetic Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Homotopic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 2D Magnetic Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Skyrmion Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Abrikosov Vortices in Type-II Superconductors . . . . . . . . . . . . . . . . . . 54
3.3.1 Homotopic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 A Note on Magnetic Free Energy . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Ginzburg-Landau Theory of Superconductivity . . . . . . . . . . . . . . 59
3.3.4 Abrikosov Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Chapter 4: Elementary Band Theory 65
4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

v



4.2 Bloch Theorem in the Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Formulation of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Example Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Bloch Theorem on a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 Formulation of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Example Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Finite Systems and the Thermodynamic Limit . . . . . . . . . . . . . . . . . . . 79
4.5 The Emergence of Forbidden Bands . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 The Wannier Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Chapter 5: Physical Applications of Topology II: Topological Band Theory 91
5.1 The Bloch Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.1.1 The Nontrivial Topology of the Brillouin Zone . . . . . . . . . . . . . . 92
5.1.2 The Bundle Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.3 States in the Bloch Bundle . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1.4 The Valence Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1.5 Transport of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.6 Wilson Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1.7 The Role of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 103
5.1.8 Wilson Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.9 The Berry Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.10 The Abelian Simplification . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2 The Physicality of the Wilson Loop . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.1 The Zak Phase in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.2 The Bulk-Boundary Correspondence . . . . . . . . . . . . . . . . . . . . 111
5.2.3 The Surface Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3 Symmetry-Protected Topological Phases . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1 The Chern Insulator in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2 Inversion Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.3 Time Reversal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Chapter 6: Scanning Tunneling Microscopy in Theory and Practice 128
6.1 Principles of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.1.1 The Bardeen Tunneling Model . . . . . . . . . . . . . . . . . . . . . . . 129
6.1.2 Beyond Bardeen: The Tersoff-Hamman Formula . . . . . . . . . . . . . 138

6.2 Spin-Polarized STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.1 Polarized Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2.2 Differential Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2.3 Establishing a Spin-Polarized Tip . . . . . . . . . . . . . . . . . . . . . 142
6.2.4 Extracting a Spin-Polarized Signal . . . . . . . . . . . . . . . . . . . . . 145

6.3 STM at Cryogenic Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.1 Cooling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 7: Development of Cryogenic Current-Biased STM 154
7.1 Motivation for Current-biased STM . . . . . . . . . . . . . . . . . . . . . . . . 155

vi



7.1.1 Scanning Tunneling Potentiometry . . . . . . . . . . . . . . . . . . . . . 155
7.1.2 Local Sensing of Spin Transport Phenomena . . . . . . . . . . . . . . . . 159

7.2 Description of the LPS 4K STM . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.1 The UHV Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.2 The Cryostat and Magnet Dewar . . . . . . . . . . . . . . . . . . . . . . 163
7.2.3 The Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2.4 The Electronics and STM Control . . . . . . . . . . . . . . . . . . . . . 167

7.3 Modification Design and Technical Specifications . . . . . . . . . . . . . . . . . 167
7.3.1 The Sample Stud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.2 The Compensation Circuit . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.4 Experimental Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.4.1 Thermal Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.4.2 Validation via Potentiometry . . . . . . . . . . . . . . . . . . . . . . . . 173

7.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.5.1 Current Mapping and Potentiometry . . . . . . . . . . . . . . . . . . . . 175
7.5.2 Manipulation of Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Chapter 8: Experiment: Superconducting Gap Variation in Granular Aluminum Films 180
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.2 Experimental Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.2.1 The mK-STM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.2.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.3.1 Comparison with Previous Work . . . . . . . . . . . . . . . . . . . . . . 191
8.3.2 A Future Experiment: Vortex Matter in Granular Superconductors . . . . 193
8.3.3 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Chapter 9: Experiment: Magnetic Skyrmions in Cr2Te3 Thin Films 196
9.1 The Topological Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.1.1 2DEG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.1.2 An Alternative Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.2 Possible Topological Hall Effect in Cr2Te3 . . . . . . . . . . . . . . . . . . . . . 207
9.2.1 Structure and Properties of Cr2Te3 . . . . . . . . . . . . . . . . . . . . . 208
9.2.2 Experimental Description . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Bibliography 218

Bibliography 218

vii



List of Tables

3.1 Exact and approximate expressions for the various contributions to the total en-
ergy of an isolated, axially-symmetric magnetic skyrmion in the 2D limit d→ 0.
The I’s are numerical functions of the reduced skyrmion radius ρ ≡ R/∆, where
R is the radius of the mz = 0 contour and ∆ is the domain wall thickness. . . . . 51

viii



List of Figures

2.1 (a) A set is open if it constitutes a neighborhood for each point it contains. (b)
This set is not open because it contains points at the edge, and does not constitute
a neighborhood for those points. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Given an arbitrary set U , its interior U◦ is the largest open subset of U , the closure
is the smallest closed superset of U , and the boundary ∂U is the complement of
U◦ in U . U is open if and only if it does not contain any boundary points, i.e.
U ∩ ∂U = ∅. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 A set U ⊆ S1 is open in the subset topology inherited from R2 if there exists a set
V ⊆ R2 which is open in the topology on R2 such that U = V ∩ S1. . . . . . . . 11

2.4 A set U ⊆ S × T is open in the product topology, if for all p ∈ U , there exist
open sets A ⊆ S and B ⊆ T such that p ∈ A×B ⊆ U . . . . . . . . . . . . . . . 12

2.5 The set [0, 2π] is mapped to the quotient set [0, 2π]/ ∼ by the quotient map q.
A set U ⊆ [0, 2π]/ ∼ is open in the quotient topology if its preimage under q is
open in the topology on the original space [0, 2π]. . . . . . . . . . . . . . . . . . 14

2.6 A function f : A → B is said to be continuous at a ∈ A if its output can be
restricted to an arbitrarily small open neighborhood V ∋ f(a) by restricting its
input to a sufficiently small open neighborhood U ∋ a. . . . . . . . . . . . . . . 16

2.7 A sequence {fn} is said to converge to a limit L if every open set containing L
contains all but finitely-many points in the sequence. . . . . . . . . . . . . . . . 17

2.8 Given a parent space (M,OM) and two subsets A and B, the subspace (S, OS)
(where S = A ∪ B and OS is the subset topology) is disconnected if and only if
A ∩B = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.9 (a) In T0 spaces, no two distinct points s ̸= t are in exactly the same open sets, but
it’s possible that every open set containing s also contains t (but not the reverse).
(b) In T1 spaces, we can always find open sets containing s but not t and t but
not s, but it’s possible that such sets necessarily intersect somewhere. (c) In T2
spaces, we can always find open sets around any two distinct points which don’t
intersect at all.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.10 If M is a bounded subset of Rn, then its closure M equipped with the subset
topology is a compact space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.11 f and g are homotopic if they can be continuously deformed into one another.
The curves in (a) are homotopic (with varying stages of the homotopy shown
with dashed lines), but the curves in (b) are non-homotopic because they may not
cross over the puncture in the center of the disk. . . . . . . . . . . . . . . . . . . 24

ix



2.12 (a) A contractible space is homotopy-equivalent to a point. (b) The punctured
disk is not contractible, but it is homotopy-equivalent to S1 (i.e. its boundary). (c)
The surface of a hollow cube is homotopy equivalent to S2 (the surface of a ball). 32

2.13 The punctured diskD⋆ has a non-trivial fundamental group given by π1(D⋆; d0) ≃
Z for arbitrary d0. Curve f belongs to the equivalence class with winding number
+1.g1, g2, g3 are all homotopic and belong to the class with winding number 0.
h belongs to the class with winding number −2, with the minus sign originating
from the clockwise orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 The homotopic equivalence of n-dimensional loops is preserved under the ac-
tion of a continuous state ψ : R → M, which motivates the definition of the
maps ψ⋆n : πn(R) → πn(M) which map families of equivalent n-loops in R to
corresponding families inM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 An example of three states ψ1, ψ2, ψ3 : R → M and their corresponding topo-
logical indices (ν+, ν−). Under our classification scheme, ψ1 has indices (0, 0),
ψ2 has indices (1, 1), and ψ3 has indicies (1, 0). . . . . . . . . . . . . . . . . . . 40

3.3 The punctured space R3 − 0⃗ is homotopic to the 2-sphere S2. On this domain,
point defects take the form of monopole configurations, and our classification
scheme essentially counts the number of monopoles which are present at the lo-
cation of the puncture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 The stereographic projection map Π provides a continuous, one-to-one corre-
spondence between points on S2 and points in R2 ∪ {∞}, where the so-called
point at infinity is associated to the north pole of the sphere. . . . . . . . . . . . . 44

3.5 An example of a magnetic texture with skyrmion number Σ = +1. . . . . . . . . 46
3.6 The profile of a magnetic skyrmion in the 360◦ domain wall model with ρ0 =

0.25. The color scale denotes the polar angle θ, with the asymptotic magnetiza-
tion pointing down. The inner circle is the parameter ρ0, while the outer circle is
the more intuitive ρ ≡ sinh−1

(
cosh(ρ0)

)
. . . . . . . . . . . . . . . . . . . . . . 49

3.7 A U(1)-valued order parameter exhibits non-trivial states only if π1(R) ̸= {0}.
This applies to point defects in 2D (a) and line defects in 3D (b), but not point
defects in 3D (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.8 Abrikosov vortices and their associated magnetic fields for (a) κ = 0.2 and (b)
κ = 10. The density ρ is reflected in the surface shape, while the color reflects
the magnitude of the magnetic field. Note that in the κ ≪ 1 limit, the flux is
restricted to the very center of the core, whereas for κ≫ 1 the flux extends deep
into the bulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 The Wigner-Seitz unit cell of a 2D triangular lattice, consisting of all of the points
nearer to the origin than to any other lattice site. . . . . . . . . . . . . . . . . . . 67

4.2 The free electron dispersion relation for the n = 0 and n = ±1 bands. The
degenerate points where the bands intersect are denoted by red dots. . . . . . . . 70

x



4.3 Illustrations of the allowed energies in the Kronig-Penney model for several val-
ues of the barrier strength. The allowed energies E ≡ q2ℏ2/2m are those for
which cos(ka) = cos(qa) + α sin(qa)/qa ∈ [−1, 1]. When α = 3 (Figs. a and
b), the δ-barriers are fairly weak and the forbidden gaps are small. When α = 10
(Figs. c and d), the δ-barriers are strong and the gaps are correspondingly large
compared to the width of the bands. . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 The dispersion relation for the bipartite chain with on-site energy±ϵ and intracell
and intercell hopping amplitudes v and w. Allowed energy ranges are highlighted
in green. In (a), (ϵ, v, w) = (0.5, 1, 1) and the system is gapped. In (b), (ϵ, v, w) =
(0, 1, 1) and the gap closes, making the system a semimetal. . . . . . . . . . . . . 78

4.5 The origin of the band gap in nearly free electron models at the edge of the Bril-
louin zone. States at the edge of BZ can be written as (co)sinusoidal waves whose
corresponding probability densities are maximized (red) or minimized (blue) on
the lattice sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6 A cartoon illustrating the breaking of the isolated atom energy degeneracy by
inclusion of nearest-neighbor interactions. The dashed lines correspond to the
unperturbed wavefunctions centered at lattice sites n and n + 1. The symmetric
linear combination (red) is larger than the antisymmetric combination (green) in
between the lattice sites, and therefore feels the potential due to the neighboring
sites more strongly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7 A representation of the magnitude of the conduction band Wannier functionW1,0

for the bipartite lattice (see Sec. 4.3.2.2), where the intercell hopping amplitudew
is half the value of the intracell hopping amplitude v. Whereas the Bloch waves
unk have uniform probability density on the A and B sublattices, the Wannier
representation clearly shows a strong intra-cell bond. . . . . . . . . . . . . . . . 88

5.1 A cartoon schematic of the Bloch bundle. The Brillouin zone BZ is called the
base space, and the copies of L2(C) which are attached to each point of BZ are
called the fibers. This type of structure is called a fiber bundle; because L2(C) is
a vector space, the Bloch bundle is a special type of fiber bundle called a vector
bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 A genuine function f : BZ → L2(C) (left) vs. a section σ of the Bloch bundle
(right). f sends all points in BZ to the same vector space, while σ does not.
This has critical implications for differentiation, which fundamentally relies on
expressions like f(k+ dk)− f(k) which don’t make any sense if f(k+ dk) and
f(k) live in different vector spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 A generic band structure before (a) and after (b) subtracting the center of the gap
at each point in the Brillouin zone. This can always be done to a gapped system
without changing its topology, so we will always assume that the gap is centered
at 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 The transport procedure defined by the projectors Pk. As the initial vector is
moved in infinitesimal steps along the curve, it generically leaves the subspace of
occupied states, but the relevant projection operator acts to project it back. In the
limit as |δki| → 0, this process preserves the normalization of the initial vector,
and the operator W [γ] will be seen to be unitary. . . . . . . . . . . . . . . . . . . 100

xi



5.5 The periodic evolution of the Wannier charge center ∆y as kx is swept from −π
to π. The trivial phase (a) can be continuously deformed to ∆y = 0, but the
topological phase (b) cannot, as it undergoes non-zero winding. . . . . . . . . . . 112

5.6 (a) In the trivial case, the spectrum of Z is periodic in k∥ and exhibits no winding,
so when it is squashed into a compact interval (b) there is no guarantee that it will
cross the Fermi level. (c) In the trivial case, the spectrum of Z exhibits winding
as k∥ varies across the BZ, and so the squashed edge spectrum (d) must intersect
the Fermi level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.7 As the flat-band Hamiltonian is deformed back into a realistic one, the edge state
spectrum should change continuously, in particular retaining its topological prop-
erties (e.g. whether or not it exhibits winding). . . . . . . . . . . . . . . . . . . . 116

5.8 The Wannier charge center evolution (a), the spectrum of the projected position
operator along y (b), the top edge state dispersion (c), and the helical edge mode
(d) of a 2D Chern insulator with Chern number Ch = +1. In (c), the bulk bands
are shown in gray, and the Fermi level is denoted by a wavy line. . . . . . . . . . 119

5.9 Several possible Wilson loop spectra which obey the constraints of time-reversal
symmetry - namely, symmetry under kx 7→ −kx and double degeneracy at kx =
0, π. The trivial phase (a) can be continuously deformed to ∆y = 0, but the non-
trivial phases (b) and (c) cannot. Note that the non-trivial phases can nonetheless
be deformed into one another, which corresponds to the Fu-Kane invariant being
an element of Z2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.10 The Wannier charge center evolution (a), the spectrum of the projected position
operator along y (b), the top edge state dispersion (c), and the chiral edge modes
(d) of a 2D time-reversal symmetric topological insulator. The bulk states in (c)
are shown in gray, and the Fermi level is denoted by a wavy line. Solid and
dashed curves reflect Kramers partners with opposite spin. . . . . . . . . . . . . 126

6.1 A simple model of tunneling due to Bardeen, in which we treat the two electrodes
as quasi-independent systems weakly coupled by a barrier potential. . . . . . . . 129

6.2 The decomposition of the potential V (x) as described in Eq. 6.1. Note that the
potentials do not sum to V (x), but HA +HB +∆ does sum to H . . . . . . . . . 131

6.3 Examples of spin-polarized STM. (a) is a cartoon depiction of SPSTM with an
in-plane polarized tip on a body centered tetragonal (bct) Mn(001) film. The
step-wise antiferromagnetism of bct-Mn(001) results in an enhanced tunneling
current when the tip and sample magnetizations are parallel and a suppression
when they are antiparallel. (b) shows the topography (left) and differential con-
ductance (right) of a real scan performed with a Fe-coated tungsten tip, taken at
+0.2 V. The spin-polarization is not immediately obvious in topography, but is
manifested as a very clear pattern of alternating bright and dark steps in dI/dV .
(c) and (d) are the topography and differential conductance of a scan I performed
with a bulk Cr tip on a Cr(001) single crystal, in which the characteristic, step-
wise pattern is visible in dI/dV . (a) and (b) are reused with permission from [1]. 143

xii



6.4 A photograph and schematic diagram of a liquid helium transfer dewar. Helium
is transferred in and out of the dewar using an insulated hose which is inserted
into a fill port at the top of the dewar. A vent port enables the user to control the
pressure of the helium atmosphere which exists above the liquid. A safety valve
(or more typically, several of them) release excessive pressure if needed. The
crosshatched region is typically vacuum space and multilayered radiation shielding.148

6.5 A phase diagram for 3He-4He mixtures. Published by Mets501 as Helium phase
diagram and licensed under CC BY-SA 3.0. . . . . . . . . . . . . . . . . . . . . 150

6.6 A 1K pot, capable of reaching temperatures of ∼ 1.5 K. Liquid helium is drawn
into the pot by a pump, where it evaporates and provides cooling power. The
helium level is controlled by a needle valve, allowing the user to ensure that the
helium level is neither above nor below the pot itself. . . . . . . . . . . . . . . . 151

6.7 A schematic diagram of a dilution refrigerator. Reused with permission from [2]. 152

7.1 Simulated IV curves and their corresponding standard deviations. (a) simulates
curves taken at five different values of the tunneling resistance RT , resulting in a
sharper peak in (c). (b) simulates five curves taken at the same value of RT , and
differ due to thermal noise which is minimized at the true zero of the tip-sample
voltage. As shown in (d), this method results in a less precise measurement of the
current and voltage offsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2 A schematic drawing of a sample mounted for scanning tunneling potentiometry.
Current is injected and extracted through the gold contacts. . . . . . . . . . . . . 158

7.3 A schematic drawing of the 4K STM system, reused with permission from [3].
Samples are loaded into the preparation chamber via a load lock, then into the
transfer chamber via a horizontal magnetic transfer rod. A vertical transfer rod
picks up the sample and transfers it into the cryostat which is housed within the
LHe dewar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.4 A schematic drawing of the sample holder and transfer rod, reused with permis-
sion from [3]. The sample holder (1) is outfitted with a spring plate (2) which
holds the sample stud (3) in place. Once inserted into the holder through the key-
hole in the back, it is rotated to lock it in place. The sample holder is manipulated
in the UHV subsystem via the tab at the top. The sample stud is extracted from
the sample holder and moved into the cryostat via a transfer rod with a leaf spring
friction coupling (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.5 The 4K cryostat probe. The vacuum can which houses the cryostat is screwed
in to the top flange (1). Copper heat sinks (2) keep the microscope wiring cool,
while thin metallic shutters (3) block room-temperature electromagnetic radia-
tion. A keyhole (4) ensures that the transfer rod remains aligned so it can reach the
microscope stage (5) and microscope (6). Figure reused with permission from [3]. 165

7.6 The locking mechanism at the top of the 4K microscope. The sample stud is
inserted sample-side down into a keyhole and then twisted to lock it in place. The
annular bias plate (shown in gold) provides electrical contact to the sample as
well as mechanical stability. In this figure, the bias plate has been split in half to
allow for current-biasing of the sample. . . . . . . . . . . . . . . . . . . . . . . 166

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https://commons.wikimedia.org/wiki/File:Helium_phase_diagram.svg
https://creativecommons.org/licenses/by-sa/3.0/legalcode


7.7 The original (a) and redesigned (b) sample studs. (1) The body of the stud is made
from SS, to which metallic straps (2) are spot welded. A sample (3) is loaded into
the trench before being pressed upward into the straps by set screws (4). In the
redesigned stud with copper body, a ceramic insert (5) insulates the sample from
the stud body. The straps are fixed to the insert by M1 screws (6). A second
ceramic insert (7) isolates a contact on one of the wings (8), which is connected
to the adjacent strap via an enameled wire which sits in a groove (9). The other
strap is connected to the sample body. . . . . . . . . . . . . . . . . . . . . . . . 168

7.8 The voltage compensation circuit which allows for a dynamic modulation of the
bias current. A fraction ξ of the potential difference across the film is subtracted
from the nominal bias voltage. When ξ is properly calibrated to the tip location,
the local tip-sample voltage remains at V0 while the bias current and ∆V are
allowed to vary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.9 Equilibriation times for the z position of the tip (a) and the thermometers on the
microscope body and sample stage (b). The change in vertical tip position is well
within the dynamic range of the z piezo, and can easily be accommodated by the
feedback controls. The far longer equilibriation time of the thermometers is due
to the large thermal masses of the microscope body and sample stage. . . . . . . 173

7.10 Scanning tunneling potentiometry of a NbTiN thin film, with Ibias = −3 mA
(left) and +3 mA (right). The average potential difference across the scan window
agrees with the expected value (assuming a homogeneous film) to within 5%. . . 174

8.1 The mK-STM system at the Laboratory for Physical Sciences, consisting of a
room temperature UHV subsystem for sample preparation and transport and a
cryostat oufitted with a 1K pot and dilution refrigerator. Figure published in [4]. . 182

8.2 A close-up photograph of the outer and inner tip configuration (a) and an over-
all schematic (b) of the dual-tip mK STM. The outer (1) and inner (2) tips are
mounted on separate outer (3) and inner (4) piezo tubes for lateral scanning con-
trol. Coarse motion for the outer assembly is provided by external piezo stacks
(5) which move an outer sapphire prism (6). The inner tip can be independently
controlled via an inner prism (7) and piezo stacks (8), while also riding along
with the outer assembly (9). Figure published in [4]. . . . . . . . . . . . . . . . . 185

8.3 AFM topography and histograms of grain sizes for films grown with varying
pressures of O2 in the deposition chamber. The scan windows are all 500 × 500
nm2, with the exception of the 9 µTorr scan which was truncated (but remains to
scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.4 (a) The empirical fit function I(V ) given in 8.2 and its derivative. (b) An example
of the fit being applied to one of the measured spectra. Despite its simplicity, it fits
the data well and serves a straightforward way to extract the gap with substantial
computational efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.5 The topography (a) and measured gap widths (b) of a granular Al film, taken over
a 500× 500 nm2 scan area. The scale on the topography map is in nm, while the
scale on the gap map is in mV. Both images have a lateral resolution of 128×128
pixels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

xiv



8.6 Histogram of the measured values for the superconducting map shown in Fig.
8.5. The mean gap is µ∆ = 1.75 mV and the standard deviation is σ∆ = 257 µV. . 189

8.7 The deviation of ∆ from its mean value, shown as a spatial plot and as a his-
togram. The color map has been superimposed on top of a grayscale topography
image for visual clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.8 The topography (a) and ∆ map (b) of a 50 nm thick granular TiN film, taken with
permission from [5]. The anticorrelation between the grain topography and ∆ is
qualitatively similar to that shown in Fig. 8.5. . . . . . . . . . . . . . . . . . . . 192

9.1 A magnetic phase diagram for MnSi, feautring a Skyrmion lattice phase identified
via the Hall resistivity ρxy (color scale). Figure published in [6] and reused under
a Creative Commons license. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.2 A typical presentation of the topological Hall effect, observed in a Bi2Te3/CrTe2
heterostructure. The the sign of the anomalous Hall resistivity changes at T = 40
K, but the topological hall effect is not dramatically changed. Figure published
with permission from [7], Copyright 2021 American Chemical Society. . . . . . 204

9.3 A cartoon illustration of a two-component AHE, in which the dark curve is the
sum of the two lighter curves. The superposition of two ordinary AHE signals
with opposite signs and different coercive fields gives rise to localized features
which mimic the characteristic sign of a true (skyrmionic) topological Hall effect. 206

9.4 Imaging of magnetic skyrmions by Lorentz transmission electron microscopy (a)
and out-of-plane and in-plane spin-polarized STM (b,c). Lorentz TEM has the
advantage of not relying on an atomically sharp tunnel junction, but SPSTM
offers the ability to resolve both in-plane and out-of-plane magnetization. The
LTEM image is reproduced with permission from [8], while the SPSTM image is
reproduced with permission from [9]. . . . . . . . . . . . . . . . . . . . . . . . . 206

9.5 Hall measurements performed on a quasi-2D Cr2Te3 film with a thickness of 6
unit cells. The emergence of pronounced, localized humps in the vicinity of the
coercive field suggests either the presence of a topological Hall effect or of a
multi-domain anomalous Hall effect. Figure published with permission from [10]. 207

9.6 The crystal structures of CrTe2 (left) and Cr2Te3 (right). CrTe2 has a trigonal unit
cell (α = β = 90◦, γ = 120◦) with lattice constants a = b = 3.86 Å, c = 6.2
Å. Cr2Te3 also has a trigonal unit cell, but the lattice constants are enlarged to
a = b = 6.9 Å, c = 12.5 Å, as the larger unit cell is required by the intercalation
layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.7 Surface morphology of a Cr2Te3 film at (a) 2 µm and (b) 250 nm scales with an
electrochemically etched bulk Cr tip. (c) shows the height profile of the line cut
shown in (b). (d) is a filtered atomic resolution image of one of the island features
and was taken with an electrochemically etched tungsten tip. All images taken at
It = 100 pA and V = 100 mV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.8 (a) Topography and (b) differential conductance maps of a Cr2Te3 film for µHz =
+2 T, 0 T, −0.5 T, and −0.9 T, taken with an in-plane polarized STM tip. dI/dV
images were processed via Fourier filtering and decorrelation from topography.
Possible stripe domains emerge in the −0.5 T and −0.9 T scans, but no signs of
a skyrmionic phase are visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

xv



9.9 (a) Topography and (b) differential conductance maps of a Cr2Te3 film for µHz =
−2 T, 0 T, 0.5 T, and 0.9 T, taken with an in-plane polarized STM tip. A mask
was applied to emphasize the terrace under examination. dI/dV images were
processed via Fourier filtering and decorrelation from topography. Possible mag-
netic skyrmions emerge at µ0Hz = 0.9 T before disappearing at higher fields. . . 215

9.10 A comparison of transport measurements (a) vs. the emergence of possibly
skyrmionic structures at µ0Hz = 0.9 T. Based on the hysteresis curves, this
is in the immediate vicinity of the coercive field of the film. (c) shows a line
cut through the differential conductance map which exhibits symmetric positive
and negative variation across the structure, as one would expect from a magnetic
skyrmion. Figure (a) published with permission from [10]. . . . . . . . . . . . . 216

xvi



Chapter 1: Introduction

In 1981, Binning and Rohrer [11] developed the scanning tunneling microscope (STM) -

an instrument capable of measuring the topography and electrical characteristics of surfaces with

sub-nanometer precision without direct physical contact with the sample. If a small voltage is

applied between a sharp metallic probe and a sample which are separated by a few angstroms, the

quantum tunneling effect allows electrons to traverse the classically-forbidden vacuum gap and

establish a small but measurable current. In the tunneling regime, this current varies exponentially

with the tip-sample distance; if the latter decreases by 1 Å, then the former typically increases by

roughly a factor of 10. By slowly scanning the probe across the surface and adjusting its position

to maintain a constant tunneling current, the surface topography (as well as its electromagnetic

properties) can be measured with sub-nanometer precision.

The STM has revolutionized the way that we study and manipulate physical systems on the

smallest of scales. Its sub-nanometer spatial resolution has enabled the characterization of the

topography and physical structure of surfaces, atomic features, and composite heterostructures

with unprecedented detail, while its wealth of variations has provided a window into the elec-

tromagnetic and chemical properties of materials and how they vary over scales from microns to

angstroms. The rise of STM-based atomic precision fabrication [12, 13] has even allowed us the

ability to build our own nanostructures atom-by-atom. It has become one of the most essential

1



and direct tools in the arsenal of experimental solid-state physics.

Over roughly the same period of time, topology has emerged as an indispensable tool for

our theoretical understanding of physical systems. In 1980, Klaus von Klitzing discovered the

quantum Hall effect [14], in which a material exhibits a quantized Hall conductance of such high

quality that it initially led to a suspicion of the laboratory equipment. The incredible robustness

of this quantization in the presence of even quite strong disorder suggested an underlying mech-

anism which in some sense operated globally, completely glossing over the localized messiness

of an imperfectly clean system. Since then, the mathematical tools and concepts from topology

have permeated essentially every subfield of condensed matter physics and beyond, informing a

complementary picture of global and local influences.

This work reports several novel studies performed using STM as the principal experimental

tool, while simultaneously developing a unified theoretical understanding of the phenomena un-

der investigation through the lens of topology. Chapter 2 serves as a self-contained introduction to

the basic ideas of topology from a purely mathematical viewpoint before Chapter 3 applies those

ideas to the study and characterization of defects in ordered media such as magnetic materials

and superconductors. Chapter 4 reviews the fundamentals of the electronic structure of solids

before Chapter 5 develops an understanding of the novel topological phases which can emerge in

quantum matter and of the surprising correspondence between topological properties of the bulk

of a material and the unique and robust electronic states which are localized to its boundaries.

Having built this broad theoretical base, Chapter 6 reviews the subject of STM, the princi-

ples on which it operates, and the practical side of operating an STM under cryogenic conditions

before the remaining chapters report some of the experimental studies performed over the last

several years in the STM group at the Laboratory for Physical Sciences. Chapter 7 describes an

2



in-situ modification which I implemented on our low-temperature STM to permit the study of

non-equilibrium effects which arise only in the presence of a transverse electrical current. Chap-

ter 8 is a report on a study I performed on superconducting granular aluminum at mK tempera-

tures to study the coupling between grains and the somewhat unconventional nature of granular

superconductivity in general. Finally, Chapter 9 describes my search for topological magnetic

structures as part of a larger, collaborative effort to explain the unexpected electromagnetic be-

havior of the quasi-2D layered ferromagnet Cr2Te3.

3



Chapter 2: A Brief Introduction to Topology

This chapter constitutes a purely mathematical but hopefully fairly gentle introduction to

topology for someone who had never seen it before. It is my belief that it is helpful to be willing to

set physics aside for a moment and wrestle with some mathematical ideas without being burdened

by the necessity of practical application. On the other hand, being a physicist, I want to get to the

good stuff eventually. The best resources I’ve found which manage to bridge this divide are the

books by Nakahara [15] and Frankel [16]. Additionally, I must credit the lecture course by Dr.

Frederic Schuller entitled The Geometrical Anatomy of Theoretical Physics [17] as perhaps the

most self-contained introduction to mathematics for physicists that I’ve ever seen, which spurred

me in this direction some years ago. At the time of writing, it is freely available on YouTube and

from FAU’s video archives.

2.1 The Main Ideas

The most fundamental concept in topology is that of a neighborhood. Given a set S and a

point p ∈ S, a topology on S amounts to a definition of what constitutes a neighborhood of p.

The following constraints are imposed on all such definitions:

1. If N is a neighborhood of p, then p ∈ N

2. If N is a neighborhood of p and N ⊆M , then M is also a neighborhood of p

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https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic
https://www.fau.tv/course/id/242.html


Figure 2.1: (a) A set is open if it constitutes a neighborhood for each point it contains. (b) This
set is not open because it contains points at the edge, and does not constitute a neighborhood for
those points.

3. If N and M are neighborhoods of p, then their intersection N ∩M is a neighborhood of p

4. If N is a neighborhood of p, then there exists some neighborhood M of p such that N is a

neighborhood of every point in M .

Given a notion of neighborhood as defined above, we define a set U to be open if it is a

neighborhood of every point p ∈ U which it contains. All topological properties of a space can

be framed in terms of these open sets. Given a set S, the set of all open sets U ⊆ S - which we

denote by O - is called the topology on S. The pair (S,O) is then called a topological space.

These requirements do not uniquely associate a topology to any given set. In fact, it’s

quite the opposite - for a generic set, there are a vast number of topologies which satisfy these

conditions.

5



2.1.0.1 A Few Simple Topologies

Let S be any set. The following definitions of neighborhoods are consistent with the re-

quirements of a topology.

• The Discrete Topology Every set containing a point p ∈ S is a neighborhood of p.

• The Indiscrete Topology The only neighborhood of a point p ∈ S is S itself.

• The Particular Point Topology Let q ∈ S be an arbitrary point. A set N containing p ∈ S

is a neighborhood of p if and only if N also contains q.

• The Cofinite Topology We call a set N ⊆ S cofinite if its complement of N in S contains

an infinite number of points. A set N ⊆ S is a neighborhood of p in the cofinite topology

if p ∈ N and N is cofinite.

The discrete and indiscrete topologies are not particularly interesting because of how in-

clusive and exclusive they are, respectively. The particular point and cofinite topologies is less

trivial, but are also of limited importance for our purposes. However, it is easy to show that all

four satisfy the requirements of a topology, so it is a good exercise to do so.

It is worth emphasizing that given a set S, whether a subset U ⊆ S is open clearly depends

on the topology we choose for S. For example, if we equip S with the discrete topology defined

above, then every possible U ⊆ S is open. On the other hand, if we equip S with the indiscrete

topology, then no subsets U ⊆ S are open, with the exception of ∅ and S itself. Therefore, when

there is a possibility for ambiguity it is important to specify that a set is open in a particular

topology. Since a topology O formally consists of the collection of all sets which are to be

considered open, the statement that a set U is open in that topology can also be written as U ∈ O.

6



2.1.0.2 Terminology

It’s worth taking a moment to explicitly mention some important terminology. Let (S,OS)

be a topological space. As mentioned above, a subset U ⊆ S is called open if it is an element of

OS . If the complement of U is open, then U is called closed. Confusingly, observe that openness

and closedness are neither exhaustive nor mutually exclusive; a set could be either open or closed,

both1 open and closed, or neither open nor closed.

Given a set U ⊆ S, the interior of U , denoted U◦, is the collection of points p ∈ U such

that there exists a neighborhood N of p which lies entirely within U (note that if U is open, then

U◦ = U ). Equivalently, U◦ is the union of all open sets X ⊆ U . The closure of U , denoted U , is

the intersection of all closed sets X ⊇ U ; as one might expect, U is a closed set (in some sense,

it is the smallest closed set containing U ). The boundary of U , denoted ∂U , is the set of points

in U which are not elements of U◦.

In more intuitive terms, a set U is open if it does not contain any of its boundary points. For

the topologies of general interest to physics, this means that open sets have the property that their

elements remain elements when they are shifted or changed by small amounts. In contrast, the

boundary points are teetering on the edge, such that any arbitrarily small shift may be enough to

move it outside. Put differently, if U is open then membership in U is a property which is robust

to tiny perturbations.

1Note that for any topological space (S,OS), both S and the empty set ∅ are always both open and closed.

7



Figure 2.2: Given an arbitrary set U , its interior U◦ is the largest open subset of U , the closure is
the smallest closed superset of U , and the boundary ∂U is the complement of U◦ in U . U is open
if and only if it does not contain any boundary points, i.e. U ∩ ∂U = ∅.

8



2.1.1 Constructing Important Topologies

The topologies given above are straightforward to define, and serve as a good warm-up to

test our understanding of important ideas. However, in physics we are interested in more complex

topologies. The most important topology for our purposes is the so-called standard topologyORd

on Rd. Once we’ve defined this fundamental topology, we can inherit a large number of additional

topologies from it. For example, any subset M ⊆ Rd can be endowed with the subset topology

inherited from ORd .

2.1.1.1 The Standard Topology

To define this topology, we introduce the concept of the open ball of radius r > 0 centered

at a point x ∈ Rd, denoted Br(x); it is defined as

Br(x) :=

{
y ∈ Rd

∣∣∣∣ d∑
i=1

(yi − xi)2 < r2

}

That is, Br(x) is the set of points within a distance r of x, where the notion of distance is defined

in the preceding expression. Having made this definition, we say that a set N is a neighborhood

of a point p ∈ Rd if there exists some r > 0 such that Br(p) ⊆ N , and that a set U ⊆ Rd is

open (in the standard topology) if it is a neighborhood of every point p ∈ U . See Fig. 2.1 for an

illustration of this topology.

This construction provides a topology on Rd, and whenever we speak of Rd in the future

we will assume it to be equipped with this topology unless otherwise specified. However, any set

which can be expressed as a subset of Rd can inherit a so-called subset topology from its parent,

9



as we see from the following construction.

2.1.1.2 The Subset Topology

Let (S,OS) be a topological space, and let M ⊆ S. We can define a topology OM on M

called the subset topology inherited from OS as follows. A set U ⊆ M is an element of OM if

there exists some V ∈ OS such that U = V ∩M . In words, U is open in the subset topology if

it can be expressed as the intersection of the subset M with a set V which is open in the parent

topology.

As an example, let S = R2 and OS be the standard topology on R2, and let M = S1 where

S1 :=

{
(a, b) ∈ R2

∣∣∣∣ a2 + b2 = 1

}

is the unit circle (also called the 1-sphere). Let U = {(a, b) ∈ S1 | a > 0} be the right semicircle

of S1. It’s easy to see that U is not an element ofOS (see the definition above), but it is an element

of the subset topology OM . This can be seen by letting V = {(a, b) ∈ R2 | a > 0} and noting

that U = V ∩ S1.

The subset topology allows us to start from the standard topology on Rd and construct a

topology for any of its subsets. A very large number of topological spaces2 can be expressed

in this way, making this a very powerful construction. The product topology construction adds

another crucial tool to our toolbox, which provides a topology to the Cartesian product of spaces.

2The n-spheres, the n-tori, and spaces of n× n matrices (interpreted as elements of Rn2

) are classic examples.

10

https://en.wikipedia.org/wiki/N-sphere
https://en.wikipedia.org/wiki/Torus#n-dimensional_torus


Figure 2.3: A set U ⊆ S1 is open in the subset topology inherited from R2 if there exists a set
V ⊆ R2 which is open in the topology on R2 such that U = V ∩ S1.

2.1.1.3 The Product Topology

Let (S,OS) and (T,OT ) be topological spaces, and let S × T be the Cartesian product set

whose elements are the ordered pairs (s, t) where s ∈ S and t ∈ T . A subset U ∈ S × T is

open in the product topology if for all p ∈ U there exists some A ∈ OS and B ∈ OT such that

p ∈ A×B and A×B ⊆ U .

The final method for constructing a topology which we will consider here is the quotient

topology construction, which is particularly useful for constructing topologies on spaces which

are obtained by identifying elements of other spaces as equivalent. For example, the circle S1 can

be obtained by starting from an interval [0, 2π] and then identifying the points 0 and 2π - hence

”gluing” the ends of the interval together.

11



Figure 2.4: A set U ⊆ S × T is open in the product topology, if for all p ∈ U , there exist open
sets A ⊆ S and B ⊆ T such that p ∈ A×B ⊆ U .

12



2.1.1.4 The Quotient Topology

An equivalence relation on a set S, denoted by∼, is a relation which satisfies the following

conditions:

• s ∼ s for all s ∈ S (reflexivity)

• s ∼ t ⇐⇒ t ∼ s for all s, t ∈ S (symmetry)

• s ∼ t and t ∼ u implies that s ∼ u for all s, t, u ∈ S (transitivity)

If s ∼ t, then we say that s and t are equivalent under ∼.

Every equivalence relation on S partitions S into so-called equivalence classes. Given an

element s ∈ S, the equivalence class [s] is the subset of all elements of S which are equivalent

to s. It’s easy to show that each element of S corresponds to one and only one equivalence class.

The set of all equivalence classes is denoted S/ ∼ and is called the quotient space of S under ∼.

Finally, the quotient map q : S → S/ ∼ is the map s 7→ [s] which takes each element of S to the

corresponding equivalence class.

Having made these definitions, let (S,OS) be a topological space and let ∼ be an equiva-

lence relation on S. A subset U ⊆ S/ ∼ is open in the quotient topology3 if its preimage4 under

the quotient map q is open in the parent topology OS .

To flesh out the example mentioned previously, let S = [0, 2π] and OS be the topology

inherited from the standard topology on R. Define the equivalence relation ∼ on S such that for

3This construction actually works for an arbitrary function f : A → B, and is more generally called the final
topology on B inherited from f (where ”final” refers to the fact that f takes points in A to some destination in
B). The quotient topology is then the final topology applied to a quotient map which arises due to an equivalence
relation.

4Given a map f : A→ B, the preimage of a set U ⊆ B, denoted preimf (U) or f−1(U), is the set of all a ∈ A
such that f(a) ∈ U .

13



Figure 2.5: The set [0, 2π] is mapped to the quotient set [0, 2π]/ ∼ by the quotient map q. A set
U ⊆ [0, 2π]/ ∼ is open in the quotient topology if its preimage under q is open in the topology
on the original space [0, 2π].

all s ∈ S, s ∼ s and 0 ∼ 2π. Consider the set of points

U =

{
[s] ∈ S/ ∼

∣∣∣∣ s ∈ [0, π/2) ∪ (3π/2, 2π]

}

The set preimq(U) = [0, π/2) ∪ (3π/2, 2π] is open in the parent topology OS , and so U is open

in the quotient topology.

We can find compatibility with our earlier constructions by noting that if we embed S/ ∼

into R2 with the map [s] 7→
(
cos(s), sin(s)

)
, then we obtain the familiar circle S1, where the

quotient topology on S/ ∼ coincides with the subset topology on S1 inherited from the standard

topology on R2.

14



2.1.2 Topological Ideas

Armed with the ability to endow a large variety of different spaces with topologies, we can

now explore some of the ideas which a topology allows us to define.

2.1.2.1 Continuity

One of the most important ideas which arises in topology is that of continuity. A function

f between topological spaces is continuous if small changes in the input yield correspondingly

small changes in the output. Formally, let (A,OA) and (B,OB) be topological spaces, and let

f be a map from A → B. We say that f is continuous at a point a ∈ A if, for any open set

V ∋ f(a), there exists an open set U ∋ a such that f(U) ⊆ V . If f is continuous at each point in

its domain, we simply say that f is a continuous function.

Intuitively, f : A → B is continuous at a if we may restrict the output to an arbitrarily

small neighborhood of f(a) ∈ B by restricting the input to an sufficiently small neighborhood of

a ∈ A. If we cannot do this, then that is an indication that the function makes a ”jump” at a.

In practice, an extremely useful alternative definition of continuity goes as follows. Given

topological spaces (A,OA) and (B,OB), a function f : A → B is continuous if the preimage

f−1(U) of all open sets U ⊆ B are open - that is, U ∈ OB =⇒ f−1(U) ∈ OA. It is an excellent

exercise to demonstrate that this definition is equivalent to the more intuitive one given above.

2.1.2.2 Convergence

Another crucial idea which emerges from topology is that of convergence. A sequence

f : N → S is formally a map from the natural numbers (0, 1, 2, . . .) to some other set S. By

15



Figure 2.6: A function f : A→ B is said to be continuous at a ∈ A if its output can be restricted
to an arbitrarily small open neighborhood V ∋ f(a) by restricting its input to a sufficiently small
open neighborhood U ∋ a.

convention, we often write f(n) ≡ fn.

If (S,OS) is a topological space, then we can talk about a sequence converging to a limit L.

The formal definition goes as follows. A sequence f : N → S converges to a limit L ∈ S in the

topology OS if every open set containing L contains all but finitely-many points in the sequence.

Alternatively, we say f converges to L if, given any open set U ∋ L, there exist some N ∈ N

such that fn ∈ U for all n > N . Proving that these definitions are equivalent is a useful exercise.

Intuitively, for {fn} to converge to L we want its elements to get arbitrarily close to L

(where we define arbitrary close as belonging to an arbitrary open neighborhood of L) and stay

there (which we ensure by demanding that past a certain N , all points fn with n > N remain in

that neighborhood).

16



Figure 2.7: A sequence {fn} is said to converge to a limit L if every open set containing L
contains all but finitely-many points in the sequence.

17



2.2 Topological Equivalence of Spaces

If (A,OA) and (B,OB) are topological spaces, it is worth asking whether there is a sense

in which they are the same space.

For example, we first defined the circle S1 as the topological space whose underlying set

is the unit circle in R2 and whose topology is inherited by the subset topology construction.

However, when discussing the quotient topology we started with the interval [0, 2π], identified

the endpoints as equivalent via an equivalence relation ∼, and then equipped [0, 2π]/ ∼ with the

quotient topology. We then claimed that this was an equivalent construction of the circle.

Obviously the underlying sets of the two constructions are totally different - the former

consists of ordered pairs of R-numbers, while the latter consists of equivalent classes of elements

of the interval [0, 2π]. However, from a topological perspective, all of their properties are exactly

the same. The reason for this is the existence of the function ϕ : [s] 7→
(
cos(s), sin(s)

)
which was

mentioned during the discussion of the quotient topology. This function is invertible, and both ϕ

and ϕ−1 are continuous; as a result, the open sets in the domain are in one-to-one correspondence

with the open sets in the codomain. Any property which can be phrased entirely in terms of the

open sets of a topological space must therefore be shared between the two.

Formally, let (A,OA) and (B,OB) be topological spaces and let ϕ : A → B be a map

between them. If ϕ is invertible and ϕ and ϕ−1 are both continuous, then ϕ is called a homeomor-

phism (or topological isomorphism). If there exists a homeomorphism between two topological

spaces, then the spaces are called homeomorphic and they are equivalent from a topological

standpoint.

The following are several examples of topological properties, their definitions in terms of

18



Figure 2.8: Given a parent space (M,OM) and two subsetsA andB, the subspace (S, OS) (where
S = A ∪B and OS is the subset topology) is disconnected if and only if A ∩B = 0.

open sets, and an explicit demonstration that they are preserved under homeomorphisms:

2.2.0.1 Connectedness

A topological space (S,OS) is called disconnected if it can be written as the union of

two disjoint, non-empty open subsets. Otherwise, it is called connected. This definition can be

motivated by the observation that if we start with an arbitrary space (M,OM), take two non-

empty subsets U, V ⊂ M , and let S = U ∪ V and OS be the subset topology, then (S,OS) is

disconnected if and only if U and V are disjoint. That is, if (S,OS) arises as a subspace of a

larger space (M,OM), then S is disconnected precisely if it can be written as disjoint pieces of

M . The non-intuitive definition is given because it does not require any reference to a ”parent”

space.

Let (A,OA) and (B,OB) be topological spaces and let f : A → B be a homeomorphism.

19



If B = U ∪ V for non-empty U, V ∈ OB, then

A = f−1(B) = f−1(U ∪ V ) = f−1(U) ∪ f−1(V ) (2.1)

where f−1(U) and f−1(V ) are non-empty (because f is surjective) and open (because f is con-

tinuous). As a result, if B is (dis)connected then A is (dis)connected, and vice-versa.

2.2.0.2 Separation (T0, T1, T2)

A topological space (S,OS) is called:

• T0 if for any two distinct points s ̸= t ∈ S, there exists an open set U which contains s

but not t or t but not s. This property means that any two points in s are topologically

distinguishable; otherwise, the topology is too coarse to tell them apart. For an example

of a topological space for which this fails, consider any set equipped with the indiscrete

topology.

• T1 if for any two distinct points s ̸= t ∈ S, there exists an open set U containing s which

does not contain t. Note the subtle distinction between this and T0 - the latter admits the

possibility that there are some points t such that the only open set containing t is the entire

set S, while the latter does not. For an example of a space which is T0 but not T1, consider

any set equipped with the particular point topology.

• T2 if for any two distinct points s ̸= t ∈ S, there exist open sets Us ∋ s and Ut ∋ t such that

Us ∩ Ut = ∅. In such spaces, any two distinct points can be respectively enclosed within

their own open sets which do not intersect. This is crucially important for convergence

20



Figure 2.9: (a) In T0 spaces, no two distinct points s ̸= t are in exactly the same open sets, but it’s
possible that every open set containing s also contains t (but not the reverse). (b) In T1 spaces,
we can always find open sets containing s but not t and t but not s, but it’s possible that such sets
necessarily intersect somewhere. (c) In T2 spaces, we can always find open sets around any two
distinct points which don’t intersect at all..

because it implies and is required by the uniqueness of limits - a convergent sequence has

a unique limit if and only if the space in question is T2. All topologies we will work with

are T2, but for an example of a topology which is T1 but not T2, consider any set equipped

with the cofinite topology.

To prove that these properties are preserved under homeomorphism, let (A,OA) and (B,OB)

be topological spaces and let f : A→ B be a homeomorphism between them. Assuming that B

has any of the properties listed above, simply replace sets and points with their preimages under

f to prove that A has the same properties.

21



Figure 2.10: IfM is a bounded subset of Rn, then its closureM equipped with the subset topology
is a compact space.

2.2.0.3 Compactness

An open cover C of a topological space (S,OS) is a collection of open sets {U1, U2, . . .}

whose union is equal to S. A subcover C ′ of a cover C if C ′ ⊆ C and C ′ is also a cover -

intuitively we obtain C ′ from C by throwing away some of the U ’s.

We say that (S,OS) is non-compact if there exists an open cover C = {Ui} which has no

finite subcover - that is, there is no finite subset of these U ’s which covers the space. Otherwise,

we say that the space is compact. Some intuition for compactness is obtained by considering

subsets of Rn. A subset M ⊂ Rn is called bounded if there exists some r > 0 such that

M ⊆ Br(0), i.e. M can be contained inside a ball of finite radius centered at the origin. If M

is bounded then (M,O) is a compact space, where M is the closure of M and O is the induced

subset topology.

Compact spaces have a number of very useful properties. For example, any R-valued

22



continuous functions on a metric space are bounded, and achieve both minimum and maximum

values. More generally, the image of a compact space under a continuous function is also com-

pact. More sophisticated operations such as integration are often better-behaved on compact

spaces, and a number of important theorems such as the Chern-Gauss-Bonnet theorem hold only

for compact spaces.

To see that this property is preserved under homeomorphism, let (A,OA) and (B,OB) be

topological spaces and f : A → B be a homeomorphism. Assume that (A,OA) is compact.

Then any open cover {Ui} of B can be mapped to an open cover {f−1(Ui)} of A. Because

A is compact, this implies that this cover admits a finite subcover, which can be mapped via

f = (f−1)−1 to a finite open cover of B.

2.3 Homotopy

Sometimes we are interested not in the equivalence of different spaces, but rather in the

equivalence of maps between spaces. For example, a global map of the magnetic field orientation

could be understood as a function from S2 (whose points represent positions on the surface of the

Earth) to S2 (whose points represent unit vectors which specify the direction of the magnetic field

at the Earth’s surface). Two different configurations can be viewed as topologically equivalent if

it is possible to continuously deform one into the other (in a sense to be defined momentarily). If

this is not possible, it suggests that the two configurations are distinct in some topological sense,

and that a transition between the two is possible only by introducing a discontinuity in the map.

The concept at play here is called homotopy. Let (A,OA) and (B,OB) be topological

spaces and let f, g : A → B be two continuous functions between them. If there exists a

23

https://en.wikipedia.org/wiki/Chern%E2%80%93Gauss%E2%80%93Bonnet_theorem


Figure 2.11: f and g are homotopic if they can be continuously deformed into one another. The
curves in (a) are homotopic (with varying stages of the homotopy shown with dashed lines), but
the curves in (b) are non-homotopic because they may not cross over the puncture in the center
of the disk.

continuous map h : [0, 1]×A→ B such that h(0, a) = f(a) and h(1, a) = g(a), then it is called

a homotopy between f and g, and f and g are called homotopic. The intuition for this is that the

parameter λ ∈ [0, 1] in the expression h(λ, a) parameterizes the deformation of f into g; as λ

varies from 0 to 1, h(λ, a) varies in a continuous way between f(a) and g(a).

As a trivial example, consider two functions f, g : R→ R2 such that

f(t) = (t, 0) g(t) = (t, t2)

We can easily find a homotopy h between these maps:

h(λ, t) = (t, λt2) (2.2)

Note that h can be written as h(λ, t) = (1 − λ)f(t) + λg(t). This raises the question as to why

24



we couldn’t always do this - why can’t we always find a homotopy between any two functions in

a similar way?

The answer is that when we multiplied f and g by (1− λ) and λ and again when we added

the results together, we made implicit use of the canonical vector space structure on R2, where

λ(a, b) = (λa, λb) and (a, b) + (a′, b′) ≡ (a + a′, b + b′). If a space does have such a vector

space structure, then indeed all continuous functions are homotopic to one another. However,

this assumption fails for almost all spaces. Consider points on the unit circle - what does it mean

to add them together, or to multiply them by scalars?

For a less trivial example, consider functions f, g : [0, 2π]→ S1 such that

f(t) =
(
cos(t), sin(t)

)
g(t) =

(
cos(2t), sin(2t)

)

These functions are also homotopic via the homotopy

h(λ, t) =
(
cos([1 + λ]t), sin([1 + λ]t)

)

Though this homotopy is slightly less trivial, it may not yet be clear why it is impossible to con-

struct a similar homotopy for arbitrary functions. Functions from [0, 2π] → S1 can be imagined

as continuous arcs on the unit circle (which may double back on themselves), and intuitively any

such arc should be continuously deformable into any other. And indeed, this is also true, a fact

which can be traced back to the triviality of the domain [0, 2π].

To see an example of when it is not true, consider functions f, g : S1 → S1. For ease of

notation, we will identify S1 with points z on the unit circle in the complex plane, so z ∈ C and

25



|z| = 1. Let

f(z) = 1 g(z) = z2 (2.3)

I claim that it is not possible to produce a homotopy h : [0, 1] × S1 → S1 between these two

functions. Intuitively, the image of f is a single point while the image of g wraps around S1, and

it is not possible to continuously expand f to g (or shrink g to f ) without leaving the unit circle or

having an intermediate stage in which the loop is ”cut” - but a continuous function whose domain

is S1 must form a closed loop at all times.

2.3.1 Homotopy Invariants

The intuitive argument is convincing, but it would be nice to have a rigorous proof which

extends to more general functions and spaces. The problem lies in the difficulty of proving that

it’s impossible to construct something. The general way to do this is to define some quantity

which is homotopy-invariant, meaning that if two functions are homotopic then the quantity

must be the same for both of them. Once we have this, it follows that if two functions differ in

the value of this quantity, then they must not be homotopic to one another (and so a homotopy

cannot possibly be constructed).

2.3.1.1 The Winding Number

An extremely useful homotopy invariant for maps from S1 to S1 is given by the winding

number ν, which counts the number of times the output of the function winds around the circle

when the input winds around once. A complex number u with modulus 1 can be assigned an

angle in the complex plane given by θ(u) = log(u)/i; along a contour γ, the total change in that

26



angle is given by

∆θ =

∫
γ

dθ =
1

i

∫
γ

du

u
(2.4)

Since the angle changes by 2π for each winding, we divide by 2π to yield the winding number of

the curve γ. If u = f(z) is the image of a point z ∈ S1 under the differentiable mapping f , then

du = f ′(z)dz and it follows that the winding number associated to the function f is

ν[f ] =
1

2πi

∮
|z|=1

f ′(z)

f(z)
dz (2.5)

To show that this is homotopy-invariant, consider two functions f, g and a homotopy h

between them. We may write

ν[h, λ] =
1

2πi

∮
|z|=1

d
dz
h(z, λ)

h(z, λ)
dz

Because h is a continuous function of λ, ν must also be a continuous function of λ. However,

we know that ν takes only integer values, while λ varies continuously from 0 to 1. The crucial

insight5 is that the only way to have a continuous function of λ take only discrete values is for it

to be constant - otherwise it will have a discontinuity somewhere. As a result, ν[f ] = ν[h, 0] =

ν[h, 1] = ν[g].

Applying this to our previous example 2.3, we see immediately that ν[f ] = 0 and ν[g] = 1,

and so the two functions are not homotopic.

5Proving this is a nice exercise!

27



2.3.1.2 The Wrapping Number

The concept of a winding number can be extended to higher dimensions. Consider a con-

tinuous map f : S2 → S2, where S2 is the 2-sphere defined as

S2 :=

{
(a, b, c) ∈ R3

∣∣∣∣ a2 + b2 + c2 = 1

}
(2.6)

By way of analogy with the winding number, we begin by writing a normalized integral

over the area of S2:

1

4π

∮
sin(α)dαdβ (2.7)

where α ∈ [0, π] and β ∈ [0, 2π] are the polar and azimuthal angles, respectively. Letting

α = α(θ, ϕ) and β = β(θ, ϕ), we find the wrapping number w of the map f to be

w[n] =
1

4π

∮
sin(α)dαdβ

=
1

4π

∮
sin
(
α(θ, ϕ)

)(∂α
∂θ

∂β

∂ϕ
− ∂β

∂θ

∂α

∂ϕ

)
dθdϕ

(2.8)

where ∂α
∂θ

∂β
∂ϕ
− ∂β

∂θ
∂α
∂ϕ

is the Jacobian of the coordinate transformation (α, β) 7→ (θ, ϕ).

From an intuitive standpoint, one can imagine a region R ⊂ S2 with boundary ∂R and

image f(R). The image of the boundary f(∂R) traces out a loop in the target space of the func-

tion. As R grows to cover the entire domain, this loop shrinks down to a point and disappears. It

could do this by wrapping around the target space once, or by wrapping partway around the target

28



space and doubling back on itself (think of a deflated balloon being draped over a basketball), or

by wrapping itself around the unit sphere any integer number of times (with negative wrapping

number corresponding to a reversal of orientation, where the image is turned ”inside out”).

2.3.1.3 The Degree of A Continuous Map

The construction we’ve used so far seems readily generalizable, which turns out to be

true. Let M and N be smooth manifolds, which for the moment we will loosely describe as

topological spaces which can be smoothly and consistently coordinatized6. The dimension of a

smooth manifold is equal to the number of coordinates necessary to specify a point.

If M and N are compact, boundary-less smooth manifolds of the same dimension, then

every continuous function f : M → N can be characterized by an integer homotopy invariant

called its degree, which can be obtained by first writing down the natural volume7 integral over

N (suitably normalized) and then taking its coordinates to be the images of points in M under

the map f . If the volume of the space N is given by

vol(N) =

∮
ω(y)dy1 . . . dyd (2.9)

where ω is the so-called volume form8, then the degree of a function f : x 7→ y is given by

deg[f ] =
1

vol(N)

∮
M

ω
(
y(x)

)
det

(
∂yi

∂xj

)
dx1 . . . dxd (2.10)

6A given coordinate system may only be valid on a particular patch of the manifold, but by coordinatizing each
patch and smoothly transitioning between coordinate systems in the overlap region we can cover the entire manifold.

7Here we mean volume in a generalized sense - for a one- or two-dimensional manifold, we mean length and
area, respectively.

8For our parameterization of S1, the volume form was 1; for S2, it was sin(α).

29



For maps from S1 → S1 and S2 → S2, the degree of the map is given by the winding

and wrapping numbers, respectively. More generally, one might consider the degree of a map

between n-spheres, n-tori, and other similar spaces.

2.3.2 Homotopy Equivalence of Topological Spaces

The concept underlying homotopy is the continuous deformation of continuous maps into

one another. Using this concept, one can speak of a kind of equivalence between the topological

spaces themselves.

Let (A,OA) and (B,OB) be topological spaces, and let f : A → B and g : B → A be

continuous maps. If g ◦ f is homotopic to idA
9 and f ◦ g is homotopic to idB, then the spaces are

said to be homotopy equivalent.

Homotopy equivalence is a strictly weaker notion than that of topological equivalence (via

a homeomorphism). For homotopy equivalence, we must be able to find some f and g such that

that g ◦ f and f ◦ g are homotopic to idA and idB, respectively; if A and B are homeomorphic, it

is possible to find f and g such that g ◦ f and f ◦ g are actually equal to idA and idB (in which

case g = f−1). Therefore, if two topological spaces are homeomorphic then they are necessarily

homotopy equivalent, but the reverse is not true. With that being said, homotopy equivalence is

often easier to work with in practice, and many topological invariants are also homotopy invariant.

2.3.2.1 Contractibility

An important notion for topological spaces is that of contractibility. A contractible space

is a topological space which is homotopy-equivalent to a set with a single element, which means

9Given a set S, the function idS : S → S is the identity map which maps each s ∈ S to itself.

30



that it can be continuously shrunk down into a point. The interval [−1, 1] ⊂ R is contractible, but

[−1, 0) ∪ (0, 1] is not. Similarly, the unit disk D = {(a, b) ∈ R2 | a2 + b2 < 1} is contractible,

but the punctured disk D⋆ = {(a, b) ∈ R2 | 0 < a2 + b2 < 1} is not.

More generally, let (S,OS) be a topological space and P = {0} be a topological space

with one element10. The only map from S → P is f : s 7→ 0, which is easily seen to be

continuous. Let g : P → S be a map defined by g(0) = s0 for some arbitrarily chosen point

s0 ∈ S, which is also easily shown to be continuous. The map g ◦ f is the constant map which

takes s 7→ s0, while the map f ◦g is the identity map which takes 0 7→ 0. Therefore, by definition

(S,OS) is homotopy equivalent to P (and is therefore contractible) if and only if the identity map

idS : s 7→ s is homotopic to any constant map s 7→ s0. A map which is homotopic to a constant

map is called null-homotopic.

As a simple example, let S = [−1, 1] ⊂ R and s0 ∈ S. The constant map idS is homotopic

to f : s 7→ s0 via the following homotopy:

h(λ, s) = λs0 + (1− λ)s (2.11)

As a result, [−1, 1] is contractible.

To see the importance of contractibility, consider two spaces (A,OA) and (B,OB) and a

continuous map f : A → B between them. If (A,OA) is contractible, then there exists some

hA : [0, 1] × A → A such that hA(0, a) = a and hA(1, a) = a0 for some constant a0 ∈ A.

Therefore, the map f ◦ hA : [0, 1] × A → B has the property that
(
f ◦ hA

)
(0, a) = f(a) and(

f ◦hA
)
(1, a) = f(a0), meaning that f is homotopic to a constant map (i.e. it is null-homotopic).

10Up to homeomorphism, there is only one topological space with one element, with the obvious (discrete) topol-
ogy.

31



Figure 2.12: (a) A contractible space is homotopy-equivalent to a point. (b) The punctured disk
is not contractible, but it is homotopy-equivalent to S1 (i.e. its boundary). (c) The surface of a
hollow cube is homotopy equivalent to S2 (the surface of a ball).

If this is the case, then all of its homotopy-invariant properties (e.g. its wrapping number or, more

generally, topological degree) must be trivial. Similarly, if (B,OB) is contractible then there

exists some hB : [0, 1] × B → B such that hB(0, b) = b and hB(1, b) = b0 for some b0 ∈ B.

From there, hB ◦ f constitutes a homotopy between f and a constant map on b, and once again

we find that f must be null-homotopic.

The conclusion, then, is that for a continuous map f : A→ B to have non-trivial homotopy-

invariant properties, both (A,OA) and (B,OB) must be non-contractible. In essence, contractible

spaces are too trivial for maps to or from them to have interesting properties (at least through the

lens of homotopy).

2.3.2.2 Homotopy Groups

If a space is not contractible, then it is because there is some kind of obstruction which

prevents us from continuously shrinking it down to a point. The examples [−1, 0) ∪ (0, 1] ⊂ R

32



and the punctured disk cannot be shrunk down because they have holes in them. The same is

true of the circle S1, the 2-sphere S2, and more generally the n-spheres11. The homotopic non-

triviality of a space can, in essence, be associated to the presence and nature of these holes.

Consider a topological space (A,OA). We can define a loop in A to be a continuous map

f : S1 → A. It is convenient to take S1 to be constructed from interval [0, 2π] with its endpoints

glued together, in which case we can think of a loop as a continuous map f : [0, 2π] → A with

f(0) = f(2π). We call the point f(0) ≡ a ∈ A the base point of the loop.

Two loops f and g which share the same base point can be concatenated to form a third

loop f ∗ g, which follows f and then subsequently g. Concretely, we define

(
f ∗ g

)
(t) =


f(2t) 0 ≤ t < π

g(2(t− π)) π ≤ 0 ≤ 2π

(2.12)

Let La be the set of all loops with base point a. We define the equivalence relation ∼

on La such that f ∼ g ⇐⇒ f and g are homotopic. Therefore, the equivalence classes [f ]

consist of families of loops which can be continuously deformed into one another. We can define

a kind of composition operation ◦ on La/ ∼ by letting [f ] ◦ [g] := [f ∗ g]; it can be shown

straightforwardly that this respects the equivalence relation. Equipped with this operation, the

quotient space La/ ∼ has the structure of a group which we call π1(A; a) - the first homotopy

group of A based at a.

The unit disk D ⊂ R2 is contractible, and so every loop based at any d0 ∈ D is homotopic

to the constant loop f : t 7→ d0. This implies that every loop based at d0 is homotopic to every

11The n-sphere Sn is given by the set {(x1, x2, . . . , xn+1) ∈ Rn+1 |
∑n+1

i=1 x
2
i = 1} equipped with the topology

inherited from Rn+1.

33



other, and therefore equivalent to every other under the equivalence relation ∼. As a result,

π1(D; d0) is the trivial group with a single element, which we may write as π1(D; d0) ≃ {0}.

Clearly this is true for any contractible space. In general, if π1 is trivial then we call the space

simply connected12.

On the other hand, the punctured disk D⋆ ⊂ R2 is not contractible. The constant loop t 7→

d0 ∈ D⋆ is not homotopic to a loop which encircles the origin, which in turn is not homotopic to a

loop which encircles the origin twice. It turns out that the equivalence classes [f ] ∈ π1(D⋆; d0) ≃

Z, where the association [f ] ↔ n ∈ Z means that every loop in [f ] winds13 around the origin n

times.

The general homotopy groups πn(A; a) are constructed analogously to π1(A; a), with the

only difference being that the continuous maps in question are from Sn → A. The 0-sphere S0 is

the set of points x ∈ R such that x2 = 1 - that is, the set {−1, 1}. A map f : {−1, 1} → A with

base point a must have f(1) = a, and so is completely characterized by f(−1). Two such maps

f, g are homotopic if there exists a homotopy h such that h(0,−1) = f(−1) and h(1,−1) =

g(−1) - that is, if there exists a continuous path p : [0, 1] → A such that p(0) = f(−1) and

p(1) = g(−1). This property is called path-connectedness; a generic space can be decomposed

into pieces which are each path-connected, and π0(A; a) counts the number of such pieces.

One can show that if A is path-connected, then the homotopy groups πn(A; a) are indepen-

dent of a, and in such cases the groups are often written as πn(A). However, in general this is not

possible, and the homotopy groups of a generic space do depend on the chosen base point.

12Observe that contractibility implies simple-connectedness, but simple-connectedness does not imply con-
tractibility

13We can define a generalized winding number about the origin (i.e. the puncture). Because the winding number
is a homotopy invariant, f ∼ g =⇒ ν[f ] = ν[g], and in this particular example the reverse is also true. As a result,
the equivalence classes in π1(D⋆; d0) can be precisely identified by their winding numbers.

34



Figure 2.13: The punctured diskD⋆ has a non-trivial fundamental group given by π1(D⋆; d0) ≃ Z
for arbitrary d0. Curve f belongs to the equivalence class with winding number +1.g1, g2, g3 are
all homotopic and belong to the class with winding number 0. h belongs to the class with winding
number −2, with the minus sign originating from the clockwise orientation.

35



2.3.2.3 Interplay between Homotopy Groups and Invariants

As one might be able to suspect by now, the homotopy groups πn of a given space (and

base point) are intimately related to the types of homotopy-invariant quantities which that space

may support. As an example, π1 (also called the fundamental group) describes the homotopy-

equivalent families of loops which can exist in the space, and therefore constrais the types of

winding numbers which can be defined. π1(D⋆) ≃ π1(S
1) ≃ Z, which means that these spaces

support integer-valued winding numbers. For the 2-torus14,π2(T2) ≃ Z × Z (loops may wind

an integer number of times around the major (exterior) radius and an integer number of times

around the minor (interior) radius, and so it supports two integer-valued winding numbers. The

somewhat exotic space RP3 - which can mysteriously be associated with 3-dimensional rotations

- has fundamental group π1(RP3) ≃ Z/2Z (the group of integers modulo 2), meaning that there

are only two distinct classes of loops on this space. This is intimately related to the fact that there

are only two types of fundamental particle (fermionic and bosonic) in three spatial dimensions.

14The 2-torus T2 is the surface of a doughnut.

36



Chapter 3: Physical Applications of Topology I

Much of the content of this chapter can be learned from the excellent overview of the

topologically ordered media by Mermin [18]. His approach is somewhat more sophisticated and

powerful, but it comes at the cost of working with quotient groups from the start. It would be an

excellent idea to study his review if the material in this chapter is deemed interesting.

3.1 Foundational Ideas

An immediate application of topology arises when we seek to classify ordered media -

physical systems occupying some region R ⊆ Rd which can be described by a continuous,

spatially-varying quantity called the order parameter. The order parameter associates to each

point r ∈ R a value which lies in some parameter spaceM. The archetypal example of an order

parameter arises in a magnetic material, in which each spatial point x is associated to the local

magnetization m(x) ∈ R3; other examples include the translations, rotations, and distortions of

infinitesimal elements of a crystalline solid, and the orientations of molecules in a liquid crystal.

The topological classification of various configurations of the order parameter (which we

will refer to as states) is built on the idea that not all continuous states can be continuously

deformed into one another; in the language of topology, not all continuous states (which are

ultimately functions from R →M) are homotopic. Given a choice of domain R and parameter

37



Figure 3.1: The homotopic equivalence of n-dimensional loops is preserved under the action of a
continuous state ψ : R→M, which motivates the definition of the maps ψ⋆n : πn(R)→ πn(M)
which map families of equivalent n-loops in R to corresponding families inM.

space M , we would like to decompose the space of all continuous states1 into homotopically

equivalent families.

Given a state ψ : R → M, any n-dimensional loop γ : Sn → R in the domain can be

mapped to an n-dimensional loop ψ ◦ γ : Sn → M in the parameter space, as in Fig. 3.1.

It’s not difficult to show that if two loops γ, δ in R are homotopic, then the loops ψ ◦ γ and

ψ ◦ δ inM are homotopic. As a result, for each non-negative integer n, a state ψ induces a map

ψ⋆n : πn(R) → πn(M) which acts as follows. Given any equivalence class C ∈ πn(R), choose

a representative loop γ ∈ C and map it toM via ψ. The output of function is the corresponding

equivalence class in πn(M). Symbolically, let [γ] be the equivalence class to which the loop γ

1Note that we may consider only those states which are compatible with some prescribed boundary conditions.

38



belongs; then we may write

ψ⋆n :πn(R)→ πn(M)

[γ] 7→ [ψ ◦ γ]
(3.1)

If two states ψ and ϕ are homotopic, then it is straightforward to show that ψ⋆n and ϕ⋆n

are the same maps for all n. This implies that if ψ⋆n and ϕ⋆n are different (for any n) then ψ

and ϕ cannot be homotopic. This provides the foundation for our classification scheme. Once

we have specified R andM, we may systematically ask what kinds of maps we can have from

πn(R)→ πn(M) for each n, and distinguish different states based on the results.

In particular, note that if πn(R) = {0} or πn(M) = {0}, then the only possible map from

πn(R)→ πn(M) is the zero map. In order to have a meaningful classification scheme for a given

n, it is necessary (and, as it turns out, sufficient) that πn(R) and πn(M) be non-trivial. In the

following, we restrict our attention to n ≤ 2, while remarking that there is room for some very

interesting research surrounding the application of higher homotopy groups to the classification

of physical systems.

3.1.1 A Mathematical Example

To make this rather abstract discussion concrete, we will consider a visual (but still purely

mathematical) example before moving on to a more in-depth discussion of physical phenomena.

Let R = D⋆ be the punctured unit disk andM = C⋆⋆ be the doubly-punctured complex plane,

defined as

C⋆⋆ :=
{
z ∈ C

∣∣ z ̸= ±1}

39



Figure 3.2: An example of three states ψ1, ψ2, ψ3 : R→M and their corresponding topological
indices (ν+, ν−). Under our classification scheme, ψ1 has indices (0, 0), ψ2 has indices (1, 1),
and ψ3 has indicies (1, 0).

as shown in Fig. 3.2. Note that π1(D⋆) ≃ Z and π1(C⋆⋆) ≃ Z × Z, so for n = 1 we have a

nontrivial classification scheme.

Loops in D⋆ are classified by an integer winding number which describes how many times

they encircle the central puncture (with positive n denoting a counterclockwise loop). Loops in

C⋆⋆ are classified by a pair of integer winding numbers which describe how many times they

encircle each of the two punctures. Therefore, we may classify a state ψ by choosing a loop γ

in D⋆ with winding number 1 (e.g. the positively-oriented unit circle) and assigning it a pair of

integers (ν+, ν−) which denote the respective winding numbers around the punctures at +1 and

−1 inM. It is an instructive exercise to work out that those winding numbers are given by

ν± =
1

2π

∫ 2π

0

dθ
d

dθ
Im

[
log

(
ψ(r, θ)∓ 1

|ψ(r, θ)∓ 1|

)]
(3.2)

where (r, θ) are the coordinates on D⋆ and log is the natural logarithm. To see this classification

40



scheme at work, consider the following three states:

ψ1(r, θ) = 0

ψ2(r, θ) = (1 + r)eiθ

ψ3(r, θ) = 1 + e−reiθ

(3.3)

It is straightforward to verify that our classification scheme labels ψ1 with (0, 0), ψ2 with (1, 1),

and ψ3 with (1, 0), which matches the picture shown in Fig. 3.2.

3.1.2 An Important Note Regarding Stability

Before studying physical examples, it’s important to note that topological classification can

have implications about the dynamical stability of various structures in ordered media, but these

implications are often subtle. Topological stability and dynamical stability are often improperly

conflated, and so disentangling these notions is crucially important to physical applications.

It is always worth remembering that modeling states of an ordered system as continuous

maps is always, on some level, an approximation. Sometimes the approximation arises from

some form of coarse-graining procedure, in which e.g. spins on a lattice may be approximated as

a continuous function on a line in the suitable thermodynamic limit. Other times, the approxima-

tion arises when we assert that a many-particle system can be described as a spatially coherent

condensate, as in superfluids or superconductivity.

In a generic ordered medium, there is an energy cost associated with spatial variation in the

order parameter, such that the introduction of a discontinuity would require a formally infinite

41



amount of energy. Since transitions between states which are topologically inequivalent requires

such a discontinuity, one might think that such transitions are energetically forbidden, but this

reasoning holds only as long as our underlying assumptions do.

Once the introduction of discontinuities in an order parameter becomes physically allowed

due to the breakdown of those assumptions, any topologically non-trivial state gains a viable

pathway by which it can be annihilated to a trivial state. That pathway generally takes the form

of a continuous deformation to a regime in which the assumption of continuity becomes question-

able, followed by a transition to an energetically favorable trivial state. Therefore, the stability

of a topological state depends on the presence of an energy barrier which prevents this pathway

from being followed. With that in mind, the relationship between topological non-triviality and

dynamical stability is something which must be examined on a case-by-case basis, as we will do

in the following two examples.

3.2 Magnetic Skyrmions

3.2.1 Homotopic Preliminaries

If we model a d-dimensional magnetic medium with an order parameter m : Rd → R3,

then because M = R3 is contractible, there are no topologically distinct configurations of m.

However, if we require that the magnetization be everywhere non-vanishing, then the parameter

space becomesM = R3 − {⃗0}, which is homotopic to S2 (see Fig. 3.3). In less technical terms,

if m(x) ̸= 0 for all x, then we may consider the alternative order parameter n : Rd → S2 where

n(x) := m(x)/|m(x)| is a unit vector defined by a point on the unit sphere.

The first two homotopy groups of the 2-sphere are well-known and can be understood

42



Figure 3.3: The punctured space R3 − 0⃗ is homotopic to the 2-sphere S2. On this domain,
point defects take the form of monopole configurations, and our classification scheme essentially
counts the number of monopoles which are present at the location of the puncture.

intuitively:

• S2 is simply-connected, which means that its fundamental group is π1(S2) = {0}, and all

closed loops on S2 can be deformed into one another2.

• The second homotopy group is given by π2(S2) = Z, suggesting that the wrapping number

may be a viable classification tool.

This suggests that if we restrict our attention to materials with a non-vanishing local mag-

netization, then defects may be characterized by some kind of wrapping number. One example is

point singularities in three dimensional systems; taking the singularity to occur at the coordinate

origin, the region R on which the magnetization m is continuous and non-vanishing is given by

R = R3−{0}, again homotopic to S2. This suggests surrounding the singularity with a spherical

surface, defining a mapN : S2 → S2 to be the restriction of n to the aforementioned surface, and

computing the degree of N . Physically, deg(N ) would be the normalized flux of m through the

2This is often intuitively phrased as ”you cannot lasso a basketball.”

43



Figure 3.4: The stereographic projection map Π provides a continuous, one-to-one correspon-
dence between points on S2 and points in R2 ∪ {∞}, where the so-called point at infinity is
associated to the north pole of the sphere.

surface, and could be called the monopole number.

3.2.2 2D Magnetic Skyrmions

Alternatively, consider a 2D magnetized film which is asymptotically uniform (lim|r|→∞ n(r)

is well-defined and unique along all directions) along ẑ, and whose magnetization direction n is

a continuously-varying function from R2 to S2. As shown in Fig. 3.4, we may define the stere-

ographic projection Π : S2 → R2 ∪ {∞} where the north pole of the sphere is mapped to the

so-called ”point at infinity”3 From there, the function N := n ◦ Π is a map from S2 → S2; note

that n(∞) is well-defined because of the asymptotically uniform magnetization.

As per our classification scheme, the only non-trivial map between homotopy groups is

π2(R)→ π2(M). A continuous state n : R2 ∪ {∞} → S2 can then be classified precisely by the

3Because Π provides a homeomorphism between S2 and R2 ∪ {∞}, the two spaces must share all topological
properties - in particular, compactness. For this reason, R2 ∪ {∞} is called a (one-point) compactification of R2.

44



degree of N ≡ n ◦ Π, given explicitly by

deg(N ) :=
1

4π

∫∫
R2

dxdy n · (∂xn× ∂yn) (3.4)

In the present context, we refer to this integer Σ ≡ deg(N ) as the skyrmion number4 of the

magnetic texture. As long as Σ is well-defined, it cannot be changed by continuous variation in

N . That is, we might parameterize the magnetization with a continuous variable λ such that for

each λ ∈ R, Nλ corresponds to a different function. If Nλ is a continuous function of λ, then we

have that

d

dλ
Σλ ≡

d

dλ
deg
(
Nλ
)
= 0 (3.5)

Physically, this implies that magnetic textures with different skyrmion numbers cannot be de-

formed into one another without introducing a discontinuity in the magnetization direction. As

such, Σ constitutes a topological invariant, and so textures with Σ = 0 are referred to as (topo-

logically) trivial while textures with Σ ̸= 0 are referred to as (topologically) non-trivial. The

topologically non-trivial magnetization textures obtained here are called magnetic skyrmions.

3.2.3 Skyrmion Stability

As mentioned in Sec. 3.1.2, fact that magnetic skyrmions cannot be created or deleted

without introducing a discontinuity into the magnetization is often referred to as topological

protection, and taken as an indication that these configurations are stable against perturbations

and thermal noise. However, it should be noted that stability is really a statement about energetics,

and topology alone is insufficient to determine whether a particular configuration is energetically

4The name is in deference to Tony Skyrme, who developed the relevant ideas in the context of nuclear physics.

45

https://en.wikipedia.org/wiki/Tony_Skyrme


Figure 3.5: An example of a magnetic texture with skyrmion number Σ = +1.

stable.

3.2.3.1 Energy of Magnetic Structures

An axisymmetric magnetization texture m can be described using cylindrical coordinates

(r, φ, z) as

m(r, φ, z) = m0


sin
(
θ(r)

)
cos(φ+ ψ)

sin
(
θ(r)

)
sin(φ+ ψ)

cos
(
θ(r)

)

 (3.6)

where θ(r) is the polar angle (measured from ẑ) of the magnetization direction and ψ is the so-

called domain wall angle which quantifies the direction of the in-plane component of m with

respect to r̂. For ψ = 0 (resp. π) m points radially outward (resp. inward), and the skyrmion is

called Néel-type; if ψ = π/2 (resp. −π/2) m winds around counterclockwise (resp. clockwise)

as observed from above and the skyrmion is called Bloch-type. In general, a skyrmion may have

any ψ, but the energetics of a particular material may dictate that one of these two extremes is

46



preferred.

The energy of such a configuration can be expressed as the sum of several distinct contri-

butions:

• The demagnetization energy Ed quantifies the interaction of the magnetization m with

the demagnetizing field H which it produces, which typically acts to reduce the overall

magnetic moment of the film. In the absence of other interactions, it favors the in-plane

magnetization of a ferromagnetic film.

• The anisotropy energy Ek quantifies the preference toward out-of-plane magnetization for

2D films, and competes directly with the demagnetization energy. Electrons at interfaces

(and in particular, in quasi-2D materials) can experience strong spin-orbit interactions

which tend to align their spins along the out-of-plane direction.

• The exchange energy EA quantifies the tendency o