Gauge-discontinuity contributions to Chern-Simons orbital magnetoelectric coupling

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We propose a new method for calculating the Chern-Simons orbital magnetoelectric coupling, conventionally parametrized in terms of a phase angle $\theta$. According to previous theories, $\theta$ can be expressed as a 3D Brillouin-zone 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 $\mathbf{k}$-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 $k_z$ direction) so that a gauge discontinuity is introduced on a 2D $\mathbf{k}$ plane normal to $k_z$. The total $\theta$ 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 $\mathbf{k}$ 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 $\theta$, expressed as a combination of Berry phases around the vortex loops. The total $\theta$ 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 $\pi$ corresponding to the $\mathbb{Z}_2$ 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.