Unconventional hybrid-order topological insulators

Abstract

Exploring novel topological matters with exotic quantum states has always been a core issue in the field of condensed matter physics, which can update the understanding of topological phases and broaden the classification of topological materials. Here, we report a class of unconventional hybrid-order topological insulators (HyOTIs), which simultaneously host various different higher-order topological states in a single band gap. Such topological states exhibit a unique bulk-boundary correspondence that is different from the well-known first-order topological states, higher-order topological states, and the coexistence of both. Particularly, we develop a generic surface theory to precisely capture them and discover a three-dimensional unconventional HyOTI protected by inversion symmetry, which renders both helical and corner topological states and exhibits an unprecedented bulk-edge-corner correspondence. By adjusting the parameters of the system, we also observe the nontrivial phase transitions between the inversion-symmetric HyOTI and other conventional phases. We further propose a circuit-based experimental scheme to detect these interesting results. Remarkably, we demonstrate that a modified tight-binding model of bismuth can support the unconventional HyOTI, suggesting a possible route for its material realization. This work shall significantly advance the research of hybrid topological states in both theory and experiment.

Figures (8)

• Figure 1: (a) The generation mechanism of a 3D inversion-symmetric unconventional HyOTI in a cubic crystal. (b) The distribution of unit mass fields enclosing 8 corners of (a). The bottom right and top left corners host the mass defects, giving the nonzero $w^{(1)}_1$ and driving the corner states. The front and back edges host the mass defects, giving the nonzero $C^{(1),(2)}_0$ and driving the helical states. (c)-(d) The OBC energy spectrum gives four zero-energy states located in corners and hinges. The lattice size is $10\times 10\times 10$. The other parameters are $m_0=t_0$ and $B_0=0.35t_0$.
• Figure 2: Topological phase diagram of the model \ref{['Hamiltonian_HOHTP']}, including HyOTI-1, HyOTI-2, SOTI-1, SOTI-2, WS, and trivial phases. The OBC energy spectra are shown in (b) HyOTI-2, (c) WS, and (d) trivial phase, SOTI-1, and SOTI-2. The insets give the distribution of zero-energy states. Here the parameters are $m_0=5t_0$ and $B_0=0$ for (b), $m_0=3t_0$ and $B_0=t_0$ for (c), $m_0=3t_0$ and $B_0=3t_0$, $0.35t_0$, and $0$ for (d).
• Figure 3: (a) General circuit models for realizing distinct interactions. Here, INIC unit acts as a positive (negative) capacitor/resistor from right to left (left to right). (b) Circuit implementation of the magnetic field terms (left), hopping term $h_4$ (middle), and on-site potentials (right). (c)-(e) Other hopping terms along the $x$, $y$, and $z$ directions for first node. (f) Numerical impedance of a finite-size circuit model.
• Figure 4: The OBC energy spectrum of the tight-binding model (a) without and (b) with additional mass terms on a hexagonal structure. The 100 eigenstates near the Fermi level are plotted, where two insets illustrate the real-space contributions of the states with red color and the signs of effective surface masses, respectively. The system size is $20\times 20\times 20$.
• Figure S1: Schematic diagram of a $4$D HyOTI with surface, hinge, and corner states, which is characterized by the nonzero $C^{(1),(2),(3)}_0$, $w^{(1),(2),(3)}_1$, and $C^{(1)}_1$, respectively.