Two-dimensional weak-type topological insulators in inversion symmetric crystals

Abstract

The Su-Schrieffer-Heeger (SSH) chain is an one-dimensional lattice that comprises two dimerized sublattices. Recently, Zhu, Prodan, and Ahn (ZPA) proposed in [L. Zhu, E. Prodan, and K. H. Ahn, Phys. Rev. B 99, 041117 (2019)] that one-dimensional flat bands can occur at topological domain walls of a two-dimensional array of the SSH chains. Here, we newly suggest a two-dimensional topological insulator that is protected by inversion and time-reversal symmetries without spin-orbit coupling. It is shown that the two-dimensional SSH chains realize the proposed topological insulator. Utilizing the first Stiefel-Whitney numbers, a weak type of Z2 topological indices are developed, which identify the proposed topological insulator, dubbed a two-dimensional Stiefel-Whitney insulator (2DSWI). The ZPA model is employed to study the topological phase diagrams and topological phase transitions. It is found that the phase transition occurs via the formation of the massless Dirac points that wind the entire Brillouin zone. We argue that this unconventional topological phase transition is a characteristic feature of the 2DSWI, manifested as the one-dimensional domain wall states. The new insight from our work could help efforts to realize topological flat bands in solid-state systems.

Experimental realization of two-dimensional weak topological insulators

“…The capacitor between two nodes imitates the hopping term in tight-binding model. 22 Parameters e (a uniform square-to-rectangle distortion) and d = (d x , d y ) (a staggered distortion) are used to describe the deformation of the crystal structure (see Sec. I in Supporting Information for details), as depicted in Figure 1b.…”

“…In such a case, the Z 2 -quantized first Stiefel-Whitney number ν c = 1 π P dk • A(k) can be used to distinguish the topological phases with A(k) the Berry connection and P being the path-ordering operator. 22,55 In the presence of the inversion symmetry: H(−k) = PH(k)P −1 , one can calculate the Z 2 topological index by evaluating the parity of the occupied band eigenstates at the four timereversal invariant momenta. Besides, H(ω 0 ) hosts the chiral symmetry due to ΓHΓ −1 = −H with Γ = σ z , so a nontrivial bulk topology will yield the flat edge bands pinned at zero admittance.…”

“…One can also explore the properties of the circuit in higher dimensions for realizing other fascinating topological states, such as the nodal-ring and Weyl semimetals. [69][70][71] In Jeon and Kim's proposal, 22 it requires centrosymmetric deformations of the 2D square lattice to realize the 2D WTI, which is extremely challenging from the experimental aspect. In our circuit model, by introducing inductors and capacitors, one can successfully simulate the lattice deformations demanded in the tight-binding modelling.…”

“…[18][19][20][21] Quantum spin Hall insulator, with helical edge modes spreading over all boundaries, is regarded as a strong TI in 2D systems. It is therefore intriguing to ask if there exist 2D WTIs with robust edge states emerging only along certain boundaries 22 and particularly how to realize them in experiments.…”