Chern insulators in two and three dimensions: A global perspective

TL;DR

The paper develops a continuum, second-quantized model for Chern insulators in which a lattice-periodic, time-reversal-symmetry-breaking vector potential yields a microscopic magnetic field. It derives globally defined expressions for topological invariants—the Chern number $C_{ ext{V}}$ in two dimensions and the Chern vector $m{C}_{ ext{V}}$ in three dimensions—built from velocity matrix elements and the full band structure, and connects these invariants to linear and optical responses through a Kubo-type conductivity and an effective dielectric tensor that exhibits optical activity. The formalism succeeds in linking bulk topology to measurable responses (quantum anomalous Hall effect in 2D and related phenomena in 3D), while remaining gauge-invariant and robust to degeneracies via the occupation-difference factors $f_{nm}$. It also provides a practical route to compute these invariants for real materials using Bloch- or quasi-Bloch-frame methods and paves the way for higher-order and nonlinear responses in Chern insulators.

Abstract

We introduce a second-quantized field theory for Chern insulators in which the Hamiltonian features a static vector potential that has the periodicity of the crystal's lattice and spontaneously breaks time-reversal symmetry in the system's ground state. Such a vector potential generates a magnetic field at the microscopic level that may be thought of as arising from local moments associated with one or more magnetic ions in each unit cell. Considering spinor electrons, we study the Chern invariants characterizing the topology of the occupied valence bands of Chern insulators in both two and three dimensions - the Chern number and the Chern vector, respectively - and we derive novel expressions for these topological invariants that are globally defined across the Brillouin zone and involve the full band structure of the system. We also study the long-wavelength response of a Chern insulator to electromagnetic fields at finite frequency, generalizing the quantum anomalous Hall effect in the static limit to the optical regime.