Topological Insulators and the Bulk–Boundary Correspondence

Presenter	Yiyang Jiang
Date	June 30, 2026
Venue	Topology Seminar
Notes	Typeset & handwritten versions below

These notes build the bulk–boundary correspondence of topological band insulators from the ground up — from the modern theory of polarization to the protected edge spectrum and the wider topological zoo. Both the typeset write-up and the original handwritten notes are attached below.

Abstract

We develop the bulk–boundary correspondence of topological band insulators starting from the modern theory of polarization and the position operator in the Bloch basis. The quantized Hall conductivity is obtained three equivalent ways — adiabatic pump, Kubo response, and Boltzmann anomalous velocity — and recast as a Berry-curvature / Chern-number topology via the Gauss map of the Brillouin zone onto the Bloch sphere. The integer quantum Hall system appears as the first topological insulator, with confinement-induced chiral edge modes and the velocity-cancellation argument that pins

$$ \sigma_{xy} = \frac{e^2}{h}\, C, \qquad C \in \mathbb{Z}. $$

The protected edge spectrum is then derived rigorously two ways — the Dirac (Jackiw–Rebbi) domain-wall construction and the Wilson-loop spectral flow — before surveying the broader topological zoo. All illustrations are redrawn in Python.

Topics covered

• Modern theory of polarization — position operator in the Bloch basis, $\hat{r} \leftrightarrow i\nabla_k$, Berry connection $A_n(k) = \langle u_n | i\nabla_k | u_n\rangle$.
• Quantized Hall conductivity, derived three equivalent ways (adiabatic charge pump, Kubo formula, Boltzmann anomalous velocity).
• Berry curvature & Chern number as the bulk topological invariant; the Gauss map of the BZ onto the Bloch sphere.
• Integer quantum Hall effect as the first topological insulator; chiral edge modes from confinement; the velocity-cancellation pinning of $\sigma_{xy}$.
• Bulk–boundary correspondence — Dirac / Jackiw–Rebbi domain walls and Wilson-loop spectral flow.
• The topological zoo — 2D/3D topological insulators, Weyl semimetals and Fermi arcs, higher-order topological phases.

Notes

The two PDFs cover the same material — the typeset version above, and the original handwritten lecture notes here: