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 $\theta$. According to previous theories, $\theta$ 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 $\mathbf{k}$-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 two-dimensional (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.

Figure 1

A planar gauge discontinuity $\mathcal{S}$ in the 3D BZ can be expanded into a fictitious slab whose thickness dimension is described by a parameter $\lambda \in [0,1]$ that interpolates smoothly between the gauge just below $\left(\lambda =0\right)$ and just above $\left(\lambda =1\right)$ the plane $\mathcal{S}$.

Figure 2

Schematic illustration of a 3D BZ containing a 2D plane of gauge discontinuity that is divided into two patches ${\mathcal{S}}_{\mathrm{GD}}$ and ${\overline{\mathcal{S}}}_{\mathrm{GD}}$ by a vortex loop (red line).

Figure 3

3D plots of the two eigenvalues (colored red and cyan) of $B(k_x,k_y)=ilnU(k_x,k_y)$ for the Fu-Kane-Mele model at half-filling: (a) when the system is a TI, i.e., $\beta =\pi$, with the branch choice taken as $(-7\pi /4,\pi /4]$; (b) when $\beta =0.95\pi$, with the branch choice $(-7\pi /4,\pi /4]$; (c) when $\beta =\pi$, with the branch choice $(-5\pi /4,3\pi /4]$; and (d) when $\beta =0.95\pi$, with the branch choice $(-5\pi /4,3\pi /4]$. The wave vectors $k_x$ and $k_y$ are reduced such that $k_j\in [0,1]$ generates the 2D BZ.

Figure 4

Axion response $\theta$ for the Fu-Kane-Mele model. Blue circles denote the total response. Red diamonds indicate the contribution from the gauge discontinuity, including both the 2D surface integral ${\theta }_{\mathrm{GD}}$ and the 1D vortex-loop integral ${\theta }_{\mathrm{VL}}$. Black crosses represent the contribution from the bulk integral without enforcing periodicity in the $k_z$ direction.

Figure 5

Difference between the $\theta$ values calculated with two different branch choices (see text) for the Fu-Kane-Mele model. The blue diamonds, black pluses, and red circles denote the differences for ${\theta }_{\mathrm{GD}},{\theta }_{\mathrm{VL}}$, and ${\theta }_{\mathrm{VL}}+{\theta }_{\mathrm{GD}}$, respectively.