Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Euler band topology and multiple hinge modes in three-dimensional insulators

Yutaro Tanaka, Shingo Kobayashi

Abstract

In two-dimensional systems with space-time inversion symmetry, such as $C_{2z}T$, the reality condition on wave functions gives rise to real band topology characterized by the Euler class, a $\mathbb{Z}$-valued topological invariant for a pair of real bands in the Brillouin zone. In this paper, we study three-dimensional $C_{2z}T$-symmetric insulators characterized by $\bar{e}_2$, defined as the difference in the Euler classes between two $C_{2z}T$-invariant planes in the three-dimensional Brillouin zone. By deriving effective surface Hamiltonians from generic low-energy continuum Hamiltonians characterized by the topological invariant $\bar{e}_2$, we reveal that multiple gapless boundary states exist at the domain walls of the surface mass, which give rise to the multiple chiral hinge modes. We also show that three-dimensional insulators characterized by $\bar{e}_2=N$ support $N$ chiral hinge modes. Notably, due to the constraint of two occupied bands in our system, these phases are distinct from stacked Chern insulators composed of $N$ layers. Furthermore, we construct tight-binding models for $\bar{e}_2=2$ and $3$ and numerically demonstrate the emergence of two and three chiral hinge modes, respectively. These results are consistent with those obtained from the surface theory.

Euler band topology and multiple hinge modes in three-dimensional insulators

Abstract

In two-dimensional systems with space-time inversion symmetry, such as , the reality condition on wave functions gives rise to real band topology characterized by the Euler class, a -valued topological invariant for a pair of real bands in the Brillouin zone. In this paper, we study three-dimensional -symmetric insulators characterized by , defined as the difference in the Euler classes between two -invariant planes in the three-dimensional Brillouin zone. By deriving effective surface Hamiltonians from generic low-energy continuum Hamiltonians characterized by the topological invariant , we reveal that multiple gapless boundary states exist at the domain walls of the surface mass, which give rise to the multiple chiral hinge modes. We also show that three-dimensional insulators characterized by support chiral hinge modes. Notably, due to the constraint of two occupied bands in our system, these phases are distinct from stacked Chern insulators composed of layers. Furthermore, we construct tight-binding models for and and numerically demonstrate the emergence of two and three chiral hinge modes, respectively. These results are consistent with those obtained from the surface theory.

Paper Structure

This paper contains 18 sections, 73 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Euler band topology in 3D insulators
  3. Euler class $\bar{e}_{2}=1$
  4. 3D Euler insulators with $\bar{e}_{2}=2$
  5. Continuum theory
  6. Tight-binding model
  7. 3D Euler insulators with $\bar{e}_{2}=3$
  8. Continuum theory
  9. Tight-binding model
  10. 3D Euler insulators with $\bar{e}_2=N$
  11. even $N$
  12. odd $N$
  13. Conclusions and discussion
  14. Derivation of the surface Hamiltonian from the lattice model
  15. The number of gapless boundary states governed by $m$-th order spatial derivative
  16. ...and 3 more sections

Figures (6)

  • Figure 1: Three-dimensional Euler insulators with multiple hinge modes. The left panels show schematics of the Euler class $e_2(k_z)$ in the $k_z=0$ and $k_z=\pi$ planes within the Brillouin zone for (a) $\bar{e}_2:=e_2(0)-e_2(\pi)=1$, (b) $\bar{e}_2=2$, and (c) $\bar{e}_2=N$. The right panels depict the real-space configurations of the corresponding Euler insulators with multiple hinge modes.
  • Figure 2: (a) The spatial profile of $\lambda_x$ around $x=0$ for the ($100$) surface. (b) The surface mass terms on the ($100$), ($\bar{1}00$), ($010$), and ($0\bar{1}0$) surfaces.
  • Figure 3: (a) The Brillouin zone and the high-symmetry points of the model $\mathcal{H}^{(\bar{e}_{2}=2)}_{\boldsymbol{k}}$ [Eq. \ref{['eq:lattice_model']}]. (b) The bulk band structure along the high-symmetry lines. (c,d) The spectra of the $x$-directed Wilson loop matrix at (c) $k_z=0$ and (d) $k_z = \pi$ [$\lambda=1$, $v_1=0.5$, $v_2=2$, $v_{z}=1$, $v_{xz}=v_{yz}=0.5$, $B_1=B_2=0.1$, $\Delta=0.6$].
  • Figure 4: (a-c) Band structures of the model $\mathcal{H}^{(\bar{e}_{2}=2)}_{\boldsymbol{k}}$ [Eq. \ref{['eq:lattice_model']}] along the $k_z$ direction under (a) the periodic boundary conditions (PBCs) both in the $x$ and $z$ directions and the open boundary condition (OBC) in the $y$ direction, (b) the PBCs both in the $y$ and $z$ directions and the OBC in the $x$ direction, and (c) the PBC in the $z$ direction and the OBCs both in the $x$ and $y$ directions. (d) The real-space distributions of the boundary states colored in (c) at $k_z=-0.168\pi$. The parameter values are the same as those in Fig. \ref{['fig:bulk_EI']}. The system sizes in the $x$ and $y$ directions are $L_x=30$ and $L_y=30$, respectively.
  • Figure 5: (a) A three-dimensional stacked triangular lattice of the model $\mathcal{H}^{(\bar{e}_{2}=3)}_{\boldsymbol{k}}$ [Eq. \ref{['eq:model_e2=3']}]. (b) The Brillouin zone and the high-symmetry points for the model. (c) The bulk band structure of the model. (d,e) The spectra of the Wilson loop operator at (c) $k_z=0$ and (d) $k_z=\pi$ [$\lambda=1$, $v_1=0.4$, $v_2=0.5$, $v_z=0.5$, $v_{xz}=4$, $v_{yz}=6$, $B_1=0.15$, $B_2=0.1$, $\Delta=0.3$].
  • ...and 1 more figures