Exceptional second-order topological insulators

TL;DR

Non-Hermitian point-gap topology is extended to second-order phases protected by reflection and pseudo-reflection symmetry. The authors classify these phases in 2D and 3D by Hermitizing the non-Hermitian Hamiltonians and identify conditions for second-order boundary states, including both second-order skin effects and exceptional second-order topological insulators. They present a 2D example in class D with corner exceptional points protected by pseudo-reflection and a 3D example in class AIII with hinge exceptional points, constructed via a layer approach from 2D layers. The work adds a new family of point-gap topological phases and suggests experimental routes in metamaterials and photonics, highlighting the rich boundary physics enabled by crystalline symmetry in non-Hermitian systems.

Abstract

Point-gap topological phases of non-Hermitian systems exhibit exotic boundary states that have no counterparts in Hermitian systems. Here, we develop classification of second-order point-gap topological phases protected by reflection symmetry. Based on this classification, we propose exceptional second-order topological insulators, exhibiting second-order boundary states stabilized by point-gap topology. As an illustrative example, we uncover a two-dimensional exceptional second-order topological insulator with point-gapless corner states. Furthermore, we identify a three-dimensional exceptional second-order topological insulator that features hinge states with isolated exceptional points, representing second-order topological phases intrinsic to non-Hermitian systems. Our work enlarges the family of point-gap topological phases in non-Hermitian systems.

Figures (13)

• Figure 1: (a-c) Complex energy spectra of the Hamiltonian (\ref{['eq:classD_2D_pseud']}) under the (a) full periodic boundary conditions (PBC), (b) open boundary conditions (OBC) in the $x$ direction and PBC in the $y$ direction, and (c) full OBC for a square crystal with the edges perpendicular to the $(1,1)$ and $(1, -{1})$ directions. (d) Real-space distribution of one of the right eigenstates encircled by the circle in (c). The parameters are $m_1=0.1$ and $m_2=0.3$. The system size is 51 in the $x$ direction with the momentum resolution $\Delta k_y=2\pi/10000$ in (b). The system size is 61 in the $x$ and $y$ directions in (c) and (d).
• Figure 2: (a) Layer construction for realizing a three-dimensional (3D) exceptional second-order topological insulator with reflection symmetry. The colors of the layers indicate the sign $(\pm)$ of $H^{(\pm)}_{\rm 2D}(\boldsymbol{k})$. The dotted box indicates the unit cell. (b) 3D reflection-symmetric exceptional second-order topological insulator with hinge states by the layer construction.
• Figure 3: (a-c) Complex energy spectra of the Hamiltonian (\ref{['eq:intrinsc3d']}) under the (a) full periodic boundary conditions (PBC), (b) open boundary conditions (OBC) in the $z$ direction and PBC in the $x$ and $y$ directions, and (c) OBC in the $x$ and $z$ directions and PBC in the $y$ direction. (d) Real-space distribution of one of the right eigenstates encircled by the circle in (c). (e) Energy dispersion of the hinge states. The parameters are $t_1=0.4$ and $t_2=0.3$. The system size in the $x$ direction is $41$ in (c) and (d), and the number of layers is $81$ in (b-d). The momentum resolutions in the $k_i$ ($i=x,y$) direction are $\Delta k_i=2\pi/80$ in the whole Brillouin zone in (b) and (c). For $-\pi/1000 \leq k_y \leq \pi/1000$, the momentum resolution is set to $\Delta k_y=2\pi/40000$ to obtain the energy spectra of the hinge states.
• Figure 4: (a) Complex energy spectrum of the Hamiltonian $H_{\tilde{\mathcal{M}}}(\boldsymbol{k})$ in Eq. (\ref{['eq:classD_2D_pseud']}) of Sec. \ref{['sec:2DETI']} with $m_1=0.1$ and $m_2=0.3$ under the open boundary conditions (OBC) in the $y$ direction. (b) Energy dispersion of $H_{\tilde{\mathcal{M}}}(\boldsymbol{k})$. The blue points indicate the exceptional points. The system size is 51 in the $y$ direction with the momentum resolution $\Delta k_x=2\pi/10000$.
• Figure 5: Complex energy spectra of the Hamiltonian $H'_{\tilde{\mathcal{M}}}(\boldsymbol{k})$ in Eq. (\ref{['eq:2DclassD_line']}) with $m_1=0.1$, $m_2=0.3$, (a,b) $m_3=m_4=0$, (c) $m_3=m_4=0.01$, and (d) $m_3=m_4=0.5$ under (a) the full periodic boundary conditions (PBC) and (b-d) the open boundary conditions (OBC) in the $y$ direction and PBC in the $x$ direction. The system size is $51$ in the $y$ direction with the momentum resolution $\Delta k_x=2\pi/2000$ in (b-d).