Gauge-discontinuity contributions to Chern-Simons orbital magnetoelectric coupling

Abstract

We propose a new method for calculating the Chern-Simons orbital magnetoelectric coupling, conventionally parametrized in terms of a phase angle θ. According to previous theories, θ can be expressed as a 3D Brillouinzone 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 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.

Orbital Magnetization under an Electric Field and Orbital Magnetoelectric Polarizabilty for a Bilayer Chern System

“…where z 1 (z 2 ) is the z coordinate of the first (second) layer. It is known that with nonzero Chern number, the gauge ambiguity may lead to a z coordinate dependent result [27,41]. Therefore in order to further confirm this conclusion, we shift the origin of the z coordinate, so that z 1 = 0 and z 2 = 1 for two layers respectively.…”

“…However, the presence of an external electric field leads to some nontrivial problems which is closely related to the nontrivial nature of magnetoelectric coupling for topologically nontrivial materials [27,40,41]. Let us summarize the problems we are facing when calculating OM for a layered lattice from Eq.…”

“…Second, in the spatial tracing process, should the contribution from edges be counted in? Previous works show that it is impossible to calculate M CS in k space for Chern insulators due to its gauge de-pendence as mentioned above [26,27]. Is it possible to circumvent this problem in r space?…”

“…The Chern-Simons term is gauge dependent so its unambiguous definition needs a smooth gauge. While for a Chern insulator, it is simply impossible to construct a smooth gauge in the entire Brillouin zone [26,27].…”

“…Corresponding with the OM, OMP can also be written as the sum of similar three gauge invariant contributions originated from the OM formalism [26]. Contributions from the LC term and the IC term can be expressed as the standard linear response of the Bloch functions to the external electric field, denoted as the "Kubo term", and the contribution from the Chern-Simons term is called the Chern-Simons orbital magnetoelectric polarizablity (CSOMP) [27,[41][42][43]. CSOMP has attracted attention due to its relation with topological phases [44,45].…”