Topological insulators and geometry of vector bundles

Dear Editor,

Please consider this re-submission of the manuscript "Topological insulators and geometry of vector bundles" for publication in the SciPost Physics Lecture Notes journal.

I am deeply grateful to the Referees for their numerous comments and recommendations, which showed many ways the notes can be improved. I apologize for the long time it took to prepare the new version. Most parts of the text have been heavily edited, some have been re-organized, and also there are several new topics (the additions mostly follow the suggestions). A detailed list of changes is given below.

One of the general changes aims to improve the connectivity and integrity of the text. A number of internal references have been added, both to the past and to the future sections. In many places, I have added or expanded introductory and concluding paragraphs, which explain motivation behind each section and its connection with other topics. Certain concepts and constructions are now listed as equations for easier referencing. After some exercises, there is a list of links to other exercises or sections that refer to this exercise.

Also, I have modified the choice of representative models to highlight the unity of the topological band theory. For example, a tight-binding model of a charge pump is introduced and extensively studied in Sec. 7. Then, this model is considered in the section on Chern insulators. Finally, a periodic evolution of the same system, which includes the closing of the bulk gap, gives a tight-binding model of a Weyl semimetal in Sec. 10.

Another major overall change is related to the context of discussion. I have added multiple paragraphs on the relevant experimental results and possible generalizations of the concepts discussed, as well as some historical remarks. Many of such paragraphs are collected in the new "Summary and outlook" sections. For example, these sections now contain all comments on the multi-band case, making the main text more focused on the two-band models.

Sec. 8 on Chern insulators and Sec. 9 on symmetry are re-organized around the concept of topological equivalence and the corresponding classification. This allows to put the material into the context of a general problem. Hopefully, this will give the reader a framework, which can help with navigating the original research articles. In particular, Sec. 9 starts with a formulation of the classification problem (9.2), which often underlies research works, but is rarely stated explicitly.

I have added numerous references to the literature, including:

• Key original works (or reviews by the same authors).

• Review articles, tutorials, and textbooks.

• Relevant experimental results.

• Historical studies and memoirs.

These references aim to guide an interested reader to additional information and provide further context.

There are also some new references to the sources of specific derivations. However, I have not inserted such references for all (sub)sections, as it was suggested by one of the Referees. In my view, the referencing in the lecture notes should follow the style adopted in textbooks, and not that of research articles. The material is often modified or does not follow any particular source, which makes such "local referencing" rather difficult. On the global level, the sources are listed in the "Sources and further reading" section in the Preface.

I decided against providing solutions of the exercises, for the following reasons. If the reader cannot solve an exercise immediately, it gives a good motivation to revisit the previous material or to consult other sources. A solution will provide an irresistible option of simply reading it and moving on without the firm understanding of the past material. On the other hand, the answers for the most important exercises are given, either implicitly or even explicitly, in the later sections.

I would also like to comment on Sec. 10.4 (former 9.3), which raised several concerns. This section provides a geometric interpretation of certain well-established results. I believe that it can serve as an interesting and simple illustration of the wide applicability of language of vector bundles, and so it is an appropriate part of these lecture notes. Unfortunately, the idea was poorly presented in the first version of the text. I hope that the new version gives a more clear and motivated exposition.

Please find the list of main changes below. Boldface numbers of subsections indicate substantial changes or new material. For the changes directly suggested in the reports, the number of the report is given in parentheses:

R1: Report of 2021-01-17

R2: Report of 2021-02-16

R3: Report of 2021-03-13

### Preface

Mostly rewritten.

### 1 Connection on a vector bundle

New figures: 1.2 (left (R3)), 1.5, 1.6 (left), 1.7

New exercises: 1.2, 1.3, 1.4

**1.1.1**: rewritten with an emphasis on tangent vectors. Added distinction between local and global sections (R1). Added the definition of ambient bundle (R3).

**1.1.2**: new section introducing bundle metric and standard basis choices (R1)

1.1.3: added the construction of constant section of $T\mathbb{R}^2$ based on linear structure (instead of that based on angular momentum conservation) (R1)

1.2: switched to covariant derivative along an arbitrary (non-coordinate) curve (R1), added a remark on covariance (R3)

1.2.2: added a remark on base space / fiber indices (R2)

1.2.3: added details on complex structure

1.3: added the geometric picture of the parallel transport ("no in-plane rotation")

1.3.1: removed discussion of conservation law of angular momentum and incorrect statement about constrained gyroscopes (R1)

1.3.1: added discussion of parallel transport in $TS^2$ along great circles

**1.3.2**: new section on the Foucault pendulum with a remark on inertial navigation systems based on constrained gyroscopes (R1, R2)

1.3.3: improved explanation of $2\pi n$ shift of $\Delta \alpha$ (R2)

1.4.1: extended discussion of $\omega_\tau$ as an angular velocity

Outlook: new comments on mathematical generalizations, added references (R1).

### 2 Electromagnetic field and curvature of connection

New figures: 2.7

New exercises: 2.3, 2.5

**2.1.1**: extended physical ("algebraic") discussion of gauge invariance, which is then contrasted with the geometric picture in 2.1.2 (R1)

2.1.2: improved description of the electromagnetic interpretation of plane bundles

2.2.1: added the definition of the curvature component (R1)

2.2.2: added a remark on covariantly constant vector fields on a sphere

2.2.3: improved description of the construction "vector field $\to$ map" (R2)

2.3.3: added a remark on covariantly constant vector fields on a cone

Outlook: non-Abelian gauge fields, formalism of differential forms (R3), history of gauge invariance.

### 3 Geometry of quantum states

New figure: 3.5

3: emphasized the difference between wave functions over the real space and eigenstate bundles over a parameter space (R3)

3.1.1: added a paragraph on persistent currents

3.1.1: extended physical and geometric description of particle on a ring (R1)

**3.1.2, 3.1.3**: rewritten description of geometric construction (R2)

3.1.3: added emphasis on the need for the vector bundle picture (R1)

3.2.1: added a reference for the calculation of the interference pattern (R1)

3.2.2: added a paragraph on Aharonov-Bohm experiments (R3), mentioned synthetic gauge fields

3.3.3: added details on the spectrum and eigenstates of a two-level system (R3)

3.3.5: added a reference for magnetic skyrmions

3.3.5: extended discussion of emergent electrodynamics with an emphasis on the mixture of two geometric pictures (R1, R2, R3)

Outlook: Aharonov-Anandan phase, Zak phase, Wilczek-Zee phase.

### 4 Topology of vector bundles

New figures: 4.4, 4.7

New exercises: 4.1

**4.1.1**: rewritten with an emphasis on continuous functions

4.2.1: mentioned that $c(M)$ is the first Chern number (R1)

**4.2.3**: new section discussing invariance of the Chern number under deformations

4.3.1: extended motivation and description of the pullback construction, added a figure (R2)

4.3.2: removed the mention of Whitney sum formula and a confusing footnote (R1, R2)

**4.4**: new section on the formalism of equivalence classes and topological classifications (preparation for charge pumps and Chern insulators)

Outlook: links to mathematical literature, Chern-Weil construction, idea of universal bundle, tautological bundles over projective spaces.

### 5 Tight-binding models and Bloch theory

This is a collection of several technical subsections, together with the discussion of graphene (all moved from other sections).

New figures: 5.2 (middle, right (R1))

New exercises: 5.1

5.1.3, 5.1.4: added remarks on the longer hopping amplitudes and the unit lattice constant (R3)

5.2.3: edited the proof of the Kramers theorem (R1)

### 6 Modern theory of electric polarization

New figures: 6.1

New exercises: 6.2

**6.1.1**: rewritten discussion of classical polarization, added a figure (R1)

6.1.2: replaced protocols in Fig. 6.3 with the full pumping cycles (R1)

6.2.1: added a remark on the gauge dependence of Wannier functions

6.2.2: expanded the derivation of the Zak phase (R2)

6.4.2: extended discussion of gauge-dependence (R1)

Outlook: Wannier functions and gauge freedom, multi-band case, multipole moments.

### 7 Charge pumping and topology

New figures: 7.3, 7.5, 7.6, 7.7, 7.8, 7.9, 7.10

New exercises: 7.1, 7.2

7.1.1: altered the continuous pumping protocol

**7.1.2**: added discussion of the charge quantization and of the choice of connection (R2)

7.2.1: added discussion of avoided level crossing

**7.2.2:** rewritten

* new introduction with an emphasis on what happens with the Bloch theory description once periodicity is broken

* added numerical spectrum for the continuous pump (Fig. 7.6)

* added an illustration of the accuracy of the ansatz (Fig. 7.7)

* added a plot of average positions of states illustrating localization of the end states (Fig. 7.8)

7.3.1: extended discussion of stability of end branches

**7.3.2**: new discussion of classification of pumps in periodic and finite settings

7.4: extended caption and discussion of Fig. 7.11 (R3)

Outlook: multi-band Chern number, Floquet theory (R3), experiments with cold atoms.

### 8 Chern insulators

This section is re-organized around the concept of topological equivalence of two-band Hamiltonians, which is studied in one, two, and three spatial dimensions. Qualitative discussion of the multi-band case is removed (R2).

New figures: 8.1, 8.3, 8.5, 8.7, 8.8

New exercises: 8.1, 8.2, 8.3, 8.4, 8.5

**8.1**: new section introducing the topological equivalence and corresponding classification problem (R3)

8.2.1: added the Bloch Hamiltonian of a Chern insulator derived from the continuous charge pump introduced in Sec. 7, and a plot of the corresponding vector field

8.2.2.: added a reference for the exponential localization of Wannier functions (R2)

8.2.3: specified an edge in the definition of $n_c$ (R1); mentioned local Chern marker

Ex. 8.2: new exercise about a boundary between two Chern insulators (R1)

**8.2.4**: rewritten the section on the topological phase transition (as a preparation for Weyl semimetals)

8.3: moved Haldane model here from the section on semimetals (R1)

**8.3.1**: added plots and discussion for edge states and Zak phase in graphene (R1), boron nitride, and Haldane model

8.3.2: edited edge spectra in Fig. 8.7 (R2)

**8.4**: new section introducing 3D Chern insulators and Hopf insulators (R2)

Outlook: multiple bands, Hopf insulator vs. K-theory; continuum limit; equatorial waves; experiments with amorphous lattices of gyroscopes; QAH experiments, including twisted bilayer graphene.

### 9 Role of symmetry

This section mostly consists of new material.

New figures: 9.2

New exercises: 9.1, 9.2, 9.3

9.1: added a remark on the other type of inversion, which preserves sublattices (R3)

**9.1.2**: new discussion of the interplay between gauge freedom and inversion symmetry

**9.1.3**: new section on symmetry-adapted Wannier functions

**9.2.1**: new derivation of the quantization of the Zak phase without using reflection symmetry (R2). Added discussion of the Zak phase quantization in graphene

9.2.2: extended discussion of polarization in periodic vs. finite crystals (R1), added a reference for the filling anomaly (R2)

**9.3**: new section on topological quantum chemistry (R1, R2)

**9.4**: edited to make better contact with the previous discussion (R1). Clarified the role of inversion symmetry. Added notes on experiments. Reduced discussion of 3DTI to a single paragraph.

Outlook: chiral symmetry (R3), comparison of inversion-symmetric chain with SSH model; periodic table for internal symmetries (R2); many bands in TQC, fragile topology (R1).

### 10 Topological semimetals

New figures: 10.2, 10.3 (left)

New exercises: 10.1, 10.2

**10.2**: new section on nodal line semimetals and drumhead surface states (R3)

**10.3.1**: new discussion of Weyl semimetals including

* a tight-binding model derived from the model for Chern insulator used before (R2)

* linearization of the Hamiltonian near a Weyl point (R1)

* Nielsen-Ninomiya theorem and symmetry constraints on Weyl points (R3)

* examples of inversion- and time-reversal-symmetric Weyl semimetals

10.3.2: added a link to the surface states of 3D Chern insulators

10.3.2: added discussion of experiments

**10.4**: extended introduction and motivation (R1, R2, R3)

10.4.2: corrected the definition of non-symmorphic symmetry group (R1)

Outlook: dimensional hierarchy; Berry phase detecting the nodal points; topology of real vector bundles; Dirac semimetals (R3); chiral anomaly in Weyl semimetals.