Quantum-spin-Hall topological insulator in a spring-mass system

Abstract

It is proposed that a lattice, with constituent masses and spring constants, may be considered as a model system for topological matter. For instance, a relative variation of the inter-and intra-unit cell spring constants can be used to create, tune, and invert band structure. Such an aspect is obtained while preserving time reversal symmetry, and consequently emulates the quantum spin Hall effect. The modal displacement fields of the mass-spring lattice were superposed so to yield pseudospin fields, with positive or negative group velocity. Considering that harmonic oscillators are the basis of classical and quantum excitations over a range of physical systems, the spring-mass system yields further insight into the constituents and possible utility of topological material.

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

“…We also computed the pseudoangular momentum based on the cell next to the domain wall, in order to illustrate the pseudospin polarization. In this specific configuration, we observe a gap between edge states indicating the coupling of the two counter-propagating pseudospin polarized modes [18,19]. In electronic TIs, the crossing of the edge state at Γ is protected by the degeneracy theorem of Kramers [28].…”

“…These numerical results also serve as basis to clarify the nature of seemingly topologically indistinguishable states in Kekulé lattices and their connection to the existence of gapless edge states. These key aspects were all unexplained in previous studies [17,18].…”

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.…”