Nonconventional topological band properties and gapless helical edge states in elastic phononic waveguides with Kekulé distortion

Abstract

This study investigates the topological behavior of an elastic phononic structure characterized by Kekulé distortion. The truss-like elastic waveguide consists of a hexagonal unit cell whose geometric dimensions are intentionally perturbed according to a Kekulé scheme. The resulting structure exhibits an effective Hamiltonian that resembles a quantum spin Hall system, hence suggesting that the waveguide can support helical topological edge states. This study also reveals the existence of finer structures including frequency splitting of pseudospin states in Kekulé distorted lattices as well as their 6-lobe alternating pattern. Important insights are also provided concerning the, so far considered, topologically indistinguishable states in Kekulé lattices and their connection to the existence of gapless edge states.

Synthetic Kramers Pair in Phononic Elastic Plates and Helical Edge States on a Dislocation Interface

“…Nevertheless, it was reported that phononic systems exploiting this mechanism could give rise to gapped edge states at zero momentum where the ω– dispersion curves of the counter‐propagating edge states repel each other, due to coupling between them. [ 21–31 ] These results showed that the edge states are not a Kramers pair and do not have a continuous spectrum across the bulk band gap. In addition, although the zone‐folding approach is already widely adopted, previous studies concentrated on mapping the system back to the electronic counterpart but usually omitted explaining some discrepancies between the synthetic phononic pseudospins and the electron's intrinsic spin; hence, leaving behind some obscure points such as the indeterminate pseudospin states, and the seemingly indistinguishable topological phases.…”

“…However, these systems required the breaking of time reversal symmetry (TRS), which imposes significant practical complexities due to the need for either special magneto‐optic and elastic materials, or for carefully controlled external input. [ 9–18 ] More recently, mechanisms analog to TRS‐preserved quantum spin Hall effect (QSHE) [ 19–34 ] and quantum valley Hall effect [ 31,35–46 ] were also proposed. These systems could be built based on ordinary dielectric or linearly elastic materials, and only required the breaking of spatial symmetry, which was a considerably more practical approach.…”

“…Researchers have proposed different ideas to synthesize properties analog to electron spins, often referred to as “pseudospins”. In phononic elastic systems, examples include mixing of symmetric and antisymmetric Lamb modes in waveguides, [ 33,34 ] and a “zone‐folding” method [ 21–32 ] to attain a doubly degenerate Dirac cone at the center of the Brillouin zone that acts as a degree of freedom that resembles electron spins. Nevertheless, it was reported that phononic systems exploiting this mechanism could give rise to gapped edge states at zero momentum where the ω– dispersion curves of the counter‐propagating edge states repel each other, due to coupling between them.…”

“…This procedure leads to a description of the system in the form of a 4 × 4 block diagonal Hamiltonian that is quadratic in . After the application of a proper change in basis, [ 26,30 ] this Hamiltonian can be mapped to the Bernevig–Hughes–Zhang (BHZ) model of topological insulators. This simplified Hamiltonian yields four frequency bands that are quadratic in with the two upper and the two lower bands being doubly degenerate, respectively.…”

“…[ 22 ] However, it can be seen (details provided in the Supporting Information) that in a continuous and periodic medium, the parity of the wavefunctions used in classifying p‐ or d‐orbitals depends on the reference frame. It is also reported that the mapping to the BHZ model actually depends on the gauge choice or the unit cell selection, [ 26,30 ] therefore the possible “nontrivial” character of the lattice is indeterminate. On the other hand, unlike nontrivial 2D topological insulators that support gapless helical edge states on the boundary, the phononic lattices classified as “nontrivial” still do not support edge states on its boundary with either vacuum or air.…”