English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We propose a new method for calculating the Chern-Simons orbitalmagnetoelectric coupling, conventionally parametrized in terms of a phase angle$\theta$. According to previous theories, $\theta$ can be expressed as a 3DBrillouin-zone integral of the Chern-Simons 3-form defined in terms of theoccupied Bloch functions. Such an expression is valid only if a smooth andperiodic gauge has been chosen in the entire Brillouin zone, and even then,convergence with respect to the $\mathbf{k}$-space mesh density can bedifficult to obtain. In order to solve this problem, we propose to relax theperiodicity condition in one direction (say, the $k_z$ direction) so that agauge discontinuity is introduced on a 2D $\mathbf{k}$ plane normal to $k_z$.The total $\theta$ response then has contributions from both the integral ofthe Chern-Simons 3-form over the 3D bulk BZ and the gauge discontinuityexpressed as a 2D integral over the $\mathbf{k}$ plane. Sometimes the boundaryplane may be further divided into subregions by 1D "vortex loops" which make athird kind of contribution to the total $\theta$, expressed as a combination ofBerry phases around the vortex loops. The total $\theta$ thus consists of threeterms which can be expressed as integrals over 3D, 2D and 1D manifolds. Whentime-reversal symmetry is present and the gauge in the bulk BZ is chosen torespect this symmetry, both the 3D and 2D integrals vanish; the entirecontribution then comes from the vortex-loop integral, which is either 0 or$\pi$ corresponding to the $\mathbb{Z}_2$ classification of 3D time-reversalinvariant insulators. We demonstrate our method by applying it to theFu-Kane-Mele model with an applied staggered Zeeman field.

Gauge-discontinuity contributions to Chern-Simons orbital magnetoelectric coupling