Chern insulators in two and three dimensions: A global perspective

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.

💡 Research Summary

The paper presents a comprehensive second‑quantized field‑theoretic framework for Chern insulators in two and three dimensions that explicitly incorporates a static, lattice‑periodic vector potential ( \mathbf{a}{\text{static}}(\mathbf{x})). This vector potential breaks time‑reversal symmetry (TRS) spontaneously and generates a microscopic magnetic field ( \mathbf{b}{\text{static}}=\nabla\times\mathbf{a}_{\text{static}} ) consistent with the crystal’s periodicity. The authors argue that such a field can be interpreted as arising from local magnetic moments of one or more magnetic ions (e.g., Cr, V, Mn) residing in each unit cell, thereby providing a physically realistic source of TRS breaking in experimentally realized Chern insulators.

Model Construction
Starting from spin‑½ electron field operators (\hat\psi_{\sigma}(\mathbf{x},t)) obeying canonical anticommutation relations, the Hamiltonian density includes (i) the kinetic term with minimal coupling (\mathbf{p}=-i\hbar\nabla-e\mathbf{a}{\text{static}}), (ii) a static electrostatic potential (V{\Gamma}(\mathbf{x})) from the frozen ion lattice, (iii) a Zeeman term (-\frac{e\hbar}{2mc}\boldsymbol{\sigma}!\cdot!\mathbf{b}_{\text{static}}), and (iv) a spin‑orbit term (\frac{\hbar}{4m^{2}c^{2}}\boldsymbol{\sigma}!\cdot!