Linear response of the Chern insulator MnBi$_2$Te$_4$: A Wannier function approach

Abstract

Recent work demonstrated that in the long wavelength limit the linear response of a Chern insulator to finite-frequency electric fields is the sum of two terms: A general frequency-dependent Kubo contribution that is present irrespective of band topology, and a topological Hall term that vanishes for topologically trivial insulators. Motivated by recent experiments and theoretical predictions, we use these expressions to calculate the optical conductivity and susceptibility of intrinsically magnetic MnBi$_2$Te$_4$ thin films with one, four, five, and eleven septuple layers by combining density functional theory with "single-shot" Wannier functions. To characterize the underlying topology of these systems, we compute the two-dimensional Chern number of these films using recently derived global expressions formulated in terms of Bloch energies and velocity matrix elements; the use of these expressions allows us to circumvent numerical issues at band crossings. Films with eleven septuple layers are of particular interest. We find that they have the same Chern number as five septuple layer films, in contrast to the reported "higher Chern-number phase" of these systems in other studies; we discuss a few possible reasons for the discrepancy. We also identify spin-orbit coupling-driven band inversions as a possible indicator of these topological phases.

Figures (5)

• Figure 1: (a) Simplified lattice structure of 5 SL MnBi$_2$Te$_4$, where blue circles represent the Mn atom with spin magnetic moments. (b) The ab initio and Wannier-interpolated band structure of 5 SL MnBi$_2$Te$_4$ along high symmetry path. The Fermi energy is centered at 0 eV and the horizontal dashed line at $1.54$ eV denotes the frozen energy window maximum used in the disentanglement step in the Wannier construction. (c) Density of states for 5 SL MnBi$_2$Te$_4$, grey dashed vertical lines mark the energy window shown in (b).
• Figure 2: Orbital resolved band structure identifying the contributions of Te p states in MnBi$_2$Te$_4$ thin films of (a) 1 SL, (b) 4 SL, (c) 5 SL, and (d) 11 SL. Band colors indicate Te p-orbital weight (scale shown in the upper-right panel of (d)). No topological band inversion occurs in 1 SL upon the inclusion of SOC, whereas it appears in 4, 5, and 11 SL systems. The corresponding Chern numbers are shown in the upper-right corner of each panel. Upper panels are calculated without SOC, while lower panels include SOC.
• Figure 3: Reactive and absorptive components of the two-dimensional Kubo susceptibility for thin films of (a) 1 SL, (b) 4 SL, (c) 5 SL, and (d) 11 SL MnBi$_2$Te$_4$. In the 4 SL system, the combination of parity-time symmetry and $C_{3z}$ rotational symmetry ensures that $\mathcal{R}(\chi^{xy})$ and $\mathcal{A}(\chi^{xy})$ vanish at all frequencies. The grey-dashed vertical line indicates the band gap energy. The real and imaginary parts of the tensor components that are not shown vanish at all frequencies.
• Figure 4: Ab initio and Wannier-interpolated band structures of 11 SL MnBi$_2$Te$_4$ plotted along high symmetry path. The Fermi energy is at 0 eV, and the dashed line at $1.54$ eV indicates the upper limit of the frozen energy window used in the Wannier disentanglement procedure.
• Figure 5: Kubo conductivity and joint density of states for thin films of (a) 1 SL, (b) 4 SL, (c) 5 SL, and (d) 11 SL MnBi$_2$Te$_4$. Similar to Fig. \ref{['fig:chi']}(b), the combination of parity-time symmetry in the 4 SL system, together with $C_{3z}$ rotational symmetry, enforces $\sigma_{\mathrm{K}}^{xy}(\omega)=0$ at all frequencies. Insets in the upper panels of (c) and (d) show Re$[\sigma_\mathrm{K}^{xx}]\approx$ Im$[\sigma_\mathrm{K}^{xy}]$ within a narrow infrared energy window, which is associated with nearly perfect magnetic circular dichroism. The grey-dashed vertical line indicates the band gap energy.