# How to choose half-Brillouin-zone (HBZ) in Fukui & Hatsugai's numerical scheme for the Z2 invariant?

EDIT: Please see my first comment on this question first.

In this paper, Fukui and Hatsugai present a numerical scheme for the calculation of the $$\mathbb{Z}_2$$ index that uses the following definition: $$\tag{1} \mathbb{Z}_2 = \frac{1}{2\pi}\left[ \oint_{\partial HBZ} A dk - \int_{HBZ} \Omega_z d^2k \right] (\text{mod } 2).$$ In Figure 1 of the paper (screenshot below), they identify three types of $$k$$-points in a discretized Brillouin zone lattice. These three sets are denoted by $$\mathcal{B}_s^\pm$$ and $$\mathcal{B}_s^0$$. They define what each of these points are:

• $$\mathcal{B}_s^0$$ are time-reversal $$\mathcal{T}$$-invariant sites: $$\mathcal{T}H(k_l)\mathcal{T}^{-1}=H(k_l)$$.
• As a $$\mathcal{T}$$ constraint, states at $$-k_l\in\mathcal{B}_s^+$$ are Kramers doublets of states at $$k_l \in \mathcal{B}_s^-$$.
• If the spectrum at each $$k_l$$ is arranged as $$\epsilon_n (k_l)\leq\epsilon_{n+1}(k_l)$$, the states at $$-k_l$$ can be constrained as: $$|n(-k_l)\rangle=\mathcal{T}|n(k_l)\rangle,$$ for $$k_l\in\mathcal{B}_s^-$$.
• Both Kramers doublets are included in $$\mathcal{B}_s^0$$. Arranging the spectrum here as $$\epsilon_{2n-1}(k)= \epsilon_{2n}(k)\leq \epsilon_{2n+1}(k)$$, they impose a final constraint: $$|2n(k_l)\rangle=\mathcal{T}|n(k_l)\rangle$$, for $$k_l\in\mathcal{B}_s^-$$.

They use these constrained states to define a link variable per the usual Fukui-Hatsugai-Susuki scheme (I think). I have implemented this common scheme in Python and MATLAB with no issue, but I am having trouble imposing the required constraints and properly choosing one half-Brillouin-zone for eq. $$(1)$$ above. I can see it being easy to do with some hard-coding in a low-resolution lattice, but I was wondering whether someone give me insight on translating the above constraints to code, and properly choosing the three different types of $$k$$-points ($$\mathcal{B}_s^i$$) automatically without hard-coding. Thanks.

• Actually, after typing all this out, I realized that this might be very easy. Do I just have to pick the required area by eliminating certain points from the area of integration, using the points $(i,j)$ with $i,j \in \{\pm \pi, 0\}$ as reference? I don't think I have to care about 'imposing' those conditions as long as I know my eigenvalues are ordered. Is this correct? Jul 2 at 16:38
• They seem to exemplify this in pg 14 of t-ozaki.issp.u-tokyo.ac.jp/meeting16/OMX-Sawahata-2016Nov.pdf Jul 5 at 18:36

I think I figured this out. For most practical purposes, I think it is fine to just choose $$-\pi\leq k_x\leq \pi$$ and $$0\leq k_y\leq \pi$$ (half-BZ, with exact ranges depending on the model). I think I over-complicated the authors' work. It is unlikely that the vortices/singularities will occur near the boundary of the chosen region. So, I can just take the difference between the Berry connection and curvature integrals. If the singularities do occur near the boundary, I think it is okay to offset the region of integration in a consistent manner to avoid having to choose the region of integration so carefully as in Figure 1. Someone please correct me if I am wrong. Thanks.