Symmetry-Indicated Time-Reversal-Doubled Axion Insulators

Abstract

The axion insulator exhibits a topological magnetoelectric effect characterized by an axion angle $θ=π$, while the time-reversal-doubled axion insulator (T-DAXI) can be viewed as two copies of an axion insulator related by time-reversal symmetry. In this work, we show that a topological crystalline insulator with nonsymmorphic glide or screw symmetry hosts the T-DAXI phase. The spin-resolved topology of the T-DAXI phase is guaranteed by the nonsymmorphic symmetry invariant $δ_g=1$ or $δ_s=1$ in certain spin directions. In this phase, the partial axion angles are quantized to $π$, and the gapped surfaces realize half-quantized quantum spin Hall states. By applying an external magnetic field along the $z$ direction, electrons with opposite spins accumulate on opposite $(001)$ surfaces, producing a topological spin polarization in real space. When the magnetic field is time-periodic, this leads to an alternating spin current detectable in experiment. Using $\mathrm{\textit{ab initio}}$ calculations, we demonstrate that mixed bismuth monohalides Bi4Br3I and Bi4BrI3 realize the nonsymmorphic T-DAXI with $δ_g=δ_s=1$. Our findings not only reveal the symmetry-enforced T-DAXIs in nonsymmorphic topological crystalline insulators, but also introduce the spin magnetoelectric effect as a novel topological spin response.

Figures (5)

• Figure 1: (Color online) (a) A $T$-invariant gTCI in the layer construction scenario, where two glide-related QSHIs are stacked in the $z$-direction. (b) Spin-resolved topology in the $s_y$ spin direction for a gTCI of (a). The two QSHIs carry opposite spin Chern numbers due to the glide symmetry. The system therefore realizes a $T$-DAXI with partial axion angles quantized by glide. (c) Schematic plots of the helical hinge states (dark green lines) for a gTCI in a prismatic geometry, with an even number of QSHI layers. The hinge states on the top and bottom surfaces lie on the same side of the glide plane (light-green plane). (d) Hinge-state configuration with an odd number of QSHI layers, obtained by removing the topmost QSHI layer. The hinge states on the top surface shift to the opposite side. In both cases, the helical hinge states propagate around the entire sample.
• Figure 2: (Color online) (a) Bulk energy bands of the gTCI model in Eq. (\ref{['Hamiltonian']}). The parameters are $-m=t=2$, $v_x=v_y=A_x=A_y=2$, $t_1=t_4=0.5$, $t_5=t_2=2$ and $t_6=-t_3=1.5$. (b) WCCs $\gamma_1$ of the $k_x$-directed Wilson loop for the four occupied bands. The zigzag pattern of hourglass Wilson bands characteristic of the gTCI phase is clearly observed. (c) Adapted from Ref. Aris-PRX: schematic illustration of the Wilson loop for a gTCI with hourglass-type connection. (d) Schematic plot of a rod geometry used in (f), where the (1$\bar{1}$0) and (110) directions are open while the (001) direction is periodic. (e) Gapped surface energy bands of the (110) surface, which are the same as those of the (1$\bar{1}$0) surface due to $g_y$. (f) Energy bands for the rod geometry shown in (d). The helical hinge states localized at the interface of the (1$\bar{1}$0) and (110) surfaces are clearly visible.
• Figure 3: (Color online) (a) Partial WCCs $z^{+}_{n{\vb* k}}$ for the $s_y$ upper spin bands of a 4-unit slab, where 8 upper spin Wannier bands are located on 8 layers (two QSHI layers per unit cell). (b) Evolution of the slab partial axion terms $\theta^\pm_{slabL}$ in the $s_y$ resolution as a function of $1/L$. $\theta^{\pm}_{slabL}$ are quantized to $\pm\pi$ as $L\rightarrow\infty$, indicating a $T$-DAXI phase. (c) Upper panel: layer-dependent spin Chern number $C_{s}(l)$ for a 15-unit-cell slab. $C_{s}(l)$ is mainly confined to the surfaces and oscillates around zero in the bulk. Lower panel: accumulated spin Chern number $C_{s,\mathrm{int}}(l)$ up to depth $l$, which yields a vanishing total spin Chern number but exhibits a clear half-quantized plateau, with half-quantized spin Chern number $C_{s,\text{bot}}=-C_{s,\text{top}}=1/2$ on the bottom and top surfaces (yellow shaded regions). (d) Layer-dependent spin Chern number and its accumulation for a modified slab after removing the topmost QSHI layer. The half-quantized plateau persists, while the sign of $C_{s,\text{top}}$ flips. The total spin Chern number thus becomes $C_s=1$.
• Figure 4: (Color online) (a) Topological charge polarization induced by an external magnetic field in a magnetic AXI. (b) Topological spin polarization induced by an external magnetic field in a $T$-DAXI. (c) For $s_y$-conserved parameters by setting $v_x=A_y=t_3=t_6=0$ and $t_1=t_4=1.5$, the $s_y$ spin distributions in each unit cell are shown along the $z$ direction under a $z$-directed magnetic field. Red and blue dotted lines indicate $s_y$ up and $s_y$ down, respectively. The total spin polarization reaches the quantized value $P_{s}=eB/8\pi$. (d) AC spin current in a gTCI tunneling junction induced by a time-periodic external magnetic field.
• Figure 5: (Color online) (a) Crystal structure of Bi$_4$Br$_3$I. (b) Bulk Brillouin zone. (c) Electronic band structure (including SOC). (d) $k_z$-directed Wilson loop. (e) $k_1$-directed nested Wilson loop $\gamma_2$ for the WCCs around $\gamma_1=0$ inside the blue frame, which shows a helical winding pattern. The WCCs around $\gamma_1=0.5$ should show the same nested Wilson loop pattern due to $g_y$ or $s_2$. These form a QSHI layer construction on the $(001;0)$ and $(001;\frac{{\vb* c}}{2})$ planes, indicating a nonsymmorphic TCI with $\delta_g=\delta_s=1$ and $(\nu_0;\nu_1\nu_2\nu_3)=(0;000)$. (f) Spin bands formed by eigenvalues of ${\vb* S}_y({\vb* k})$, where a spin-band gap is clearly observed.