Gauge-discontinuity contributions to Chern-Simons orbital magnetoelectric coupling

We propose a method for calculating Chern-Simons orbital magnetoelectric coupling, conventionally parametrized in terms of a phase angle θ. According to previous theories, θ can be expressed as a three-dimensional (3D) Brillouin-zone (BZ) integral of the Chern-Simons 3-form defined in terms of the occupied Bloch functions. Such an expression is valid only if a smooth and periodic gauge has been chosen in the entire Brillouin zone, and even then, convergence with respect to the k-space mesh density can be difficult to obtain. In order to solve this problem, we propose to relax the periodicity condition in one direction (say, the kz direction) so that a gauge discontinuity is introduced on a two-dimensional (2D) k plane normal to kz. The total θ response then has contributions from both the integral of the Chern-Simons 3-form over the 3D bulk BZ and the gauge discontinuity expressed as a 2D integral over the k plane. Sometimes, the boundary plane may be further divided into subregions by 1D "vortex loops" which make a third kind of contribution to the total θ, expressed as a combination of Berry phases around the vortex loops. The total θ thus consists of three terms which can be expressed as integrals over 3D, 2D, and 1D manifolds. When time-reversal symmetry is present and the gauge in the bulk BZ is chosen to respect this symmetry, both the 3D and 2D integrals vanish; the entire contribution then comes from the vortex-loop integral, which is either 0 or π corresponding to the Z2 classification of 3D time-reversal-invariant insulators. We demonstrate our method by applying it to the Fu-Kane-Mele model with an applied staggered Zeeman field.