Phonon-Induced Modulation of the Axion Angle in MnBi₂Te₄

The Chern-Simons axion coupling θ characterizes the topological contribution to the magnetoelectric response of three-dimensional insulators. In systems preserving time-reversal or inversion symmetry, θ is quantized to 0 or π (mod 2π); in more general cases, it can take continuous values. MnBi₂Te₄ (MBT) is a magnetic topological insulator whose inversion symmetry and the combined symmetry S = Θτ_1/2 (time-reversal followed by a half-translation) enforces a quantized θ = π. In the presence of perturbations that break all of the θ-quantizing symmetries, θ can vary continuously away from π. In our work, we consider phonon modes that break the inversion and S symmetry in MBT and linearly couple to θ, allowing it to be modulated away from θ = π. We identify the relevant modes at the T-point in the hexagonal Brillouin zone of the rhombohedral space group R-3m in the non-magnetic setting. Starting from the ab initio band structure, we construct the Wannierized tight-binding Hamiltonian and evaluate the first-order phonon-induced (λ) response of θ using two approaches: a gauge-invariant four-curvature formulation where ∂_λθ is computed from the second Chern character, and a gauge-dependent formulation where θ is computed from the Chern-Simons 3-form. In the latter case, we fix a smooth gauge using the projection method and derive a Kubo-like expression for θ, eliminating the need for finite difference evaluations in k.