Non-conventional topological band properties and gapless helical edge states in elastic phononic waveguides with Kekul\'e distortion

TL;DR

This paper demonstrates that elastic phononic waveguides with Kekule9 distortion can host topologically protected gapless helical edge states, revealing novel pseudospin textures and Berry curvature patterns in a classical elastic system.

Contribution

It introduces a Kekule9-distorted elastic waveguide that exhibits topological properties similar to quantum spin Hall systems, with unique pseudospin textures and edge states.

Findings

Existence of nondegenerate pseudospin states

Presence of a 6-lobe pseudospin texture and Berry curvature pattern

Support for gapless helical edge states in elastic phononic systems

Abstract

This study investigates the topological behavior of a continuous elastic phononic structure characterized by a 3-term Kekul\'e distortion. The elastic waveguide consists of a hexagonal unit cell whose geometric dimensions are intentionally perturbed according to a generalized Kekul\'e 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. Using first-principles calculations, we show the existence of nondegenerate pseudospin states and a very peculiar 6-lobe pseudospin texture and Berry curvature pattern. Important insights are also provided concerning the topological states in the Kekul\'e lattice, so far considered indistinguishable, and their critical role in enabling unique gapless edge states, typically not achievable in phononic systems.

Equations

$$h_i$$

$${\delta }_i$$

$$|g\uparrow ⟩$$

$$|h\uparrow ⟩$$

$$H=\left(\begin{array}{cc}h(\mathbf{k})& 0\\ 0& h^{\ast }(-\mathbf{k})\end{array}\right),$$

Full text

††preprint: APS/123-QED

In the wake of the discovery of topological states of matter in quantum mechanical systems, researchers have explored the possibility to create analogue effect in classical waveguide systems including electromagnetic, acoustic, and elastic systems. In all these systems, the common interest lied in the possibility to develop scattering-free waveguides achieving, ideally, maximum transmission properties.

In acoustic and elastic systems, most of the work conducted in recent years has concentrated on creating mechanisms analogue to the Hall effect in its different forms. Systems preserving time-reversal (T-) symmetry have been of particular interest owing to their manufacturing and operating simplicity. Examples include the acoustic analogue of the quantum valley Hall effect (QVHE) Lu et al. (2016a, b); Pal and Ruzzene (2017); Vila et al. (2017); Liu and Semperlotti (2018, 2019); Zhu et al. (2018); Yan et al. (2018); Brendel et al. (2018); Ma et al. (2019) and of the quantum spin Hall effect (QSHE) Süsstrunk and Huber (2015); Mousavi et al. (2015); He et al. (2016); Deng et al. (2017a); Xia et al. (2017); Chaunsali et al. (2018a); Liu et al. (2017); Zhou et al. (2018); Chaunsali et al. (2018b); Xia et al. (2018); Miniaci et al. (2018); Zhang et al. (2017); Yves et al. (2017); Deng et al. (2017b). The later cases, also referred to as the acoustic topological insulator (TI), is achieved by leveraging pseudospins and degenerate double Dirac cones in mechanical lattices that allow mapping the effective Hamiltonian of the system to that of QSHE. A comparison between systems utilizing the two T-symmetry preserving mechanisms is investigated by Deng, et al. Deng et al. (2019). Moreover, Paulose et al. Paulose et al. (2015) showed that topological zero-frequency soft modes can exist near the local dislocated deformation of a Kagome or square lattice. We note that this study did not report any occurrence of topological edge states along the edge dislocation.

Recently Liu et al. Liu et al. (2017) and Zhou et al. Zhou et al. (2018) proposed discrete spring-mass models of phononic lattices in which the stiffness of the springs forming the unit cell was varied according to the Kekulé distortion pattern, as a way to devise QSHE analogue. In those studies, a perturbation approach was used to predict the formation of globally degenerate pseudospin modes. Note that these models can only capture the effective behavior in the vicinity of the degeneracy (i.e. the Dirac point) and are valid only for small values of the Kekulé’s distortion strength.

In this paper, we propose a continuous elastic phononic waveguide based on a 3-term Kekulé texture which introduces a component breaking space-inversion (or parity P-) symmetry. We first derive the effective Hamiltonian near the Dirac point by using low-order perturbation theory. The analysis of the Hamiltonian will show that such waveguide exhibits a dynamic behavior that is indeed the analogue of QSHE. Further, by using first-principle numerical calculations, we reveal finer structures of the phononic bands, namely: 1) the spurious degeneracy (predicted by low-order perturbation model) of the pseudospin states is lifted, and 2) the existence of an unusual six-lobe pseudospin texture and Berry curvature distributions around the $\Gamma$ point. These new fundamental findings could not be observed based on the conventional 4-level Hamiltonian model and provide a critical foundation for the analysis of the pseudospin-resolved Berry phase. In addition, some topological ambiguities, including the gauge dependence in mapping the system dynamics to the QSHE, are also revealed and extensively discussed. Then, we perform a thorough analysis of possible edge states supported by various domain wall configurations. These numerical results serve as a basis to clarify the nature of topological states in Kekulé lattices so far considered indistinguishable. This important insight allows elucidating the mechanism leading to the existence of gapless edge states. In fact, in contrast with existing acoustic topological materials based on zone-folding strategy Süsstrunk and Huber (2015); He et al. (2016); Deng et al. (2017a); Xia et al. (2017); Chaunsali et al. (2018a); Liu et al. (2017); Zhou et al. (2018); Chaunsali et al. (2018b); Xia et al. (2018); Zhang et al. (2017); Yves et al. (2017); Deng et al. (2017b), the generalized Kekulé distortion approach provides an additional degree of freedom that opens the way to realizing strictly gapless edge states (hence, further enhancing the robustness against back-scattering). Another very interesting and unique consequence of the above mentioned gauge dependence is the existence of edge states on edge dislocations of the Kekulé lattice.

The material system consists in a phononic waveguide made of a reticular plate having a hexagonal lattice structure (Fig. 1(a)) with constant $a$. Each slender beam in the reticular structure has a rectangular cross-section with width $b$ and varying height. The Kekulé distortion pattern is imprinted on the phononic lattice by controlling the height ($h_{i}$) of each connecting beam. Beams with three different heights are connected to a triangular prism having average height $h_{0}$ (Fig. 1(b)). Their height varies linearly in a small section $w$ connected to the prism (Fig. 1(a)) and their distribution is symmetric with respect to the mid-plane of the lattice.

The classic Kekulé pattern has two different bonds linking the atoms Kekulé (1865). Here a generalized Kekulé pattern is considered, where all three beams connected to the prism have different height expressed by

[TABLE]

The perturbation typical of the Kekulé pattern can be graphically interpreted as in Fig. 1(c) whereas the abscissa indicates the height $h$. The three parameters (average height $h_{0}$, distortion amplitude $\delta$, and initial phase $\phi$) can construct any set of three real numbers $h_{1,2,3}$. For the numerical simulations reported in this paper, we set $h_{0}=a_{0}/20$, $\delta=h_{0}/4$, $b=a_{0}/10$, $w=a/60$, and chose aluminum as base material. With these assumptions, the only free parameter is $\phi$. Given an input value of $\phi$, two types of lattices can be created based on different cyclic permutation orders of $h_{1,2,3}$ around a specific prism. The lattices are labeled as type-I and type-II, as illustrated in Fig. 1 (d,e). In type-I lattice, $h_{1,2,3}$ follows a counter-clockwise permutation around the $A$ prisms and clockwise around the $B$ prisms, and vice versa in type-II lattice.

Note that a substitution $\phi\rightarrow\phi+2\pi/3$ results in $h_{1,2,3}\rightarrow h_{2,3,1}$ which does not change the permutation order. This transformation produces a rigid translation $A_{3}\rightarrow A_{1}$ (or $\vec{T}=(0,a_{0})$) of the entire lattice therefore having no effect on the bulk properties. However, we anticipate that when finite boundaries or interfaces (such as domain walls) are considered, this translation can produce non negligible effects. More details will be provided below.

When $\delta=0$, hence in the absence of Kekulé distortion, the lattice has $C_{6v}$ symmetry and the unit cell is a $\sqrt{3}$-sized superlattice of its primitive cell with a corresponding folding of the original Brillouin zone. Following this folding, the Dirac cones at the original valleys $K_{0}$ and $K^{\prime}_{0}$ overlap at the $\Gamma$ point (see Fig. 2 (c)), forming a double cone having 4-fold degeneracy (Fig. 2 (a)). When the distortion is introduced, it breaks the original lattice periodicity (having lattice constant $a_{0}$ shown in Fig. 1 (a)) and lowers the symmetry to $C_{3v}$. Note that, differently from all previously considered acoustic zone-folding QSHE implementations Xia et al. (2017); Chaunsali et al. (2018a, b); He et al. (2016), the Kekulé pattern breaks P-symmetry. Once the Kekulé distortion is applied, the original 4-fold degeneracy at $\Gamma$ is lifted (due to mixing of the two valley modes) and a bandgap opens between two 2-fold degenerate bands, as shown in Fig. 2 (b). This specific example considers $\phi=30^{\circ}$ which yields $h_{1}>h_{3}>h_{2}$.

In order to understand the general characteristics that could be expected in such bulk medium, we derive the effective Hamiltonian which describes the eigenstates near the $\Gamma$ point of the distorted lattice. We assume that these states can be expanded using the 4-fold degenerate modes of the undistorted lattice as basis. Since the states of the distorted and undistorted lattices are defined in different spaces (different lattice geometry), a homeomorphic transformation is required to map the eigenstates of the distorted lattice back to the undistorted one. Once the eigenstates are expressed under the same frame, their inner product can be readily performed and used to the determine the expansion coefficients. The 4-fold degenerate basis contains two $g$- and two $h-$ orbit-like (with eight and ten angular nodal planes) states, respectively. A pseudospin basis can be constructed by generating linear combinations of symmetry adapted states as follows

[TABLE]

(See Sup for a video of the eigenstates). With the basis in the order $(\ket{g\uparrow},\ket{h\uparrow},\ket{g\downarrow},\ket{h\downarrow})$, the $4\times 4$ effective Hamiltonian (complete derivation in Sup ) assume a form similar to the Bernevig-Hughes-Zhang model Bernevig et al. (2006)

[TABLE]

in which $h(\mathbf{k})=(C-Dk^{2})\sigma_{0}+\mathcal{A}(k_{x}\sigma_{x}-k_{y}\sigma_{y})+(M+Bk^{2})\sigma_{z}$, where $\sigma_{x,y,z}$ are Pauli matrices, and $\sigma_{0}$ is the identity matrix. $M$ indicates the coupling of $\ket{g}$ and $\ket{h}$ states yielding the bandgap. Also, recalling that the spin Chern number $C_{S}=\pm(sgn(M)+sgn(B))/2$, the same sign of $M$ and $B$ will yield $C_{S}=\pm 1$ hence suggesting the existence of edge states at the interface of topologically distinct materials.

This perturbation approach yields two 2-fold degenerate upper (lower) pseudospin bands, with isotropic circular cone-like dispersion. However, this formulation is only valid in the neighborhood of the $\Gamma$ point and for small levels of distortion. In practice, the degeneracy only occurs at the $\Gamma$ point while the lattice symmetry only guarantees $C_{6v}$ symmetric dispersion. If the two pseudospin states are not degenerate, how are they distributed? It is certainly not possible for each band to represent a pseudospin, since it would imply that each band has a (pseudospin-resolved) Berry curvature that peaks at $\Gamma$, hence inconsistent with the premise of intact T-symmetry.

To answer this question, we no longer use the 4-fold degenerate states as the basis, but instead extract pseudo parameters from the numerical solution. First, we calculate the pseudoangular momentum of each eigenstate in $\mathbf{k}$-space which allows tracking the pseudospin states. We define a pseudoangular momentum similarly to the expected value of $z$-angular momentum in the quantum mechanical case, $L_{z}=-i\braket{\mathbf{u}}{\rho\partial_{\theta}}{\mathbf{u}}$ Sup , where $\mathbf{u}=(u_{x},u_{y},u_{z})$ is the eigenstate displacement field, $\rho$ is the mass density, and $\theta$ is the angular coordinate of the cylindrical reference frame with the longitudinal axis parallel to $z$-axis and centered at the unit cell center (see Fig. 1 (a)).

The resulting pseudoangular momentum is plotted in color on the corresponding dispersion curves over a square domain that is approximately $0.4|\Gamma K|$ in size (Fig.3 (a)). Fig.3 (b) show the pseudoangular momentum of the fourth band as an example. The lower inset in Fig.3 (a) shows the frequency difference for the lower two states. This plot highlights that the two bands are detached everywhere except at the $\Gamma$ point.

This result can also be interpreted from a different perspective. One pseudospin state has a $C_{3v}$ symmetric dispersion with constant $L_{z}$ while the other pseudospin (that is the T-counter part) has a dispersion surface that is $180^{\circ}$-reversed and displays opposite $L_{z}$. The coupling between the two-fold degenerate bands having opposite pseudospins results in band repulsion (along $\Gamma$-M directions) which leads to nondegeneracy except at $\Gamma$. It is found that the repulsion strength also depends on parameter $\phi$ Sup . It is also worth noting that the value of $L_{z}$ is gauge dependent: unit cells with $\phi=\phi_{0}+n(2\pi/3),\forall n\in\mathbb{N}$ all represent the same bulk lattice but $L_{z}$ varies depending on the number $n\mod 3$. This gauge transformation is equivalent to shifting the position of the longitudinal ($z$) axis of the cylindrical coordinates when calculating $L_{z}$. Nevertheless, with any fixed gauge choice, $L_{z}$ of inverted lattices (types-I/II obtained by applying the inversion operator about the unit cell center) always show opposite signs, therefore confirming that band inversion occurs during continuous morphing from one lattice to the other.

Fig. 3 (c) shows the Berry curvature of the lowest band in a squared area ($\sim 0.25|\Gamma K|$) around the $\Gamma$ point Sup . The Berry curvature concentrates near $\Gamma$ and shows a 6-lobe pattern as observed for $L_{z}$. A similar behavior is found for the remaining bands (i.e., both lower and upper bands) provided a change in signs. Clearly, $\Omega(\mathbf{-k})=-\Omega(\mathbf{k})$ of each band is a consequence of T-symmetry and it results in a vanishing accumulating Berry phase over the entire Brillouin zone. However, the pseudospin-resolved Berry phase is nonzero. We also note that, the Berry curvature of each band of type-II lattice, is identical to that of type-I but with opposite signs. Numerical integration of the Berry curvature shows that the pseudospin Chern numbers are $C_{\uparrow/\downarrow}=\pm 1$, hence leading to nontrivial $\mathbb{Z}_{2}$ invariant $\frac{C_{\uparrow}-C_{\downarrow}}{2}\mbox{ mod }2=1$ Kane (2013). These topological invariants show that the bulk lattices have a non-trivial topological significance.

Based on the information from both the asymptotic effective Hamiltonian and the first principle numerical calculations, topologically protected edge states should be expected at the interface between type-I and II lattices. Given that the two lattices are mirrored images to each other, a symmetric domain wall (DW) can be constructed, as shown in Fig. 4 (a).

In electronic TIs, the crossing of the edge state at $\Gamma$ is protected by the degeneracy theorem of Kramers Hasan and Kane (2010). In the acoustic case, it is no longer guaranteed given phonons are bosons. It follows that gaps are typically found in the edge states of most acoustic TIs based on zone-folding strategy Süsstrunk and Huber (2015); He et al. (2016); Deng et al. (2017a); Xia et al. (2017); Chaunsali et al. (2018a); Liu et al. (2017); Zhou et al. (2018); Chaunsali et al. (2018b); Xia et al. (2018); Zhang et al. (2017); Yves et al. (2017); Deng et al. (2019, 2017b). In stark contrast with these cases, the generalized Kekulé distortion (Eq.1) offers an additional degree of freedom that allows control on the dispersion of edge states. This degree of freedom is directly connected to the parameter $\phi$ that controls the breaking of P-symmetry, which is intact when $\phi=n\pi/3$ and broken otherwise. The proper selection of this parameter yields gapless edge states by purposely exploiting the use of accidental degeneracies, which allow complete decoupling of counter-propagating (i.e., helical) edge states. Note that topological materials relying solely on P-breaking in honeycomb lattices belong to a different topological phase typically related to analogue QVHE Lu et al. (2016a, b); Pal and Ruzzene (2017); Vila et al. (2017); Liu and Semperlotti (2018, 2019); Zhu et al. (2018); Yan et al. (2018); Brendel et al. (2018); Ma et al. (2019).

Fig. 4 (b) and (c) plot the eigenfrequency for both symmetric and antisymmetric modes of the supercell at $k_{x}=0$ with $\phi$ $\in[0,2\pi]$. The blue shaded zone indicates the bulk bands, while the points in the bandgap are the edge states. The width of the gaps of both the bulk and the edge states are also shown. Recall that $\phi+2n\pi/3$ yields the same bulk lattice of $\phi$, so the bulk band pattern repeats every $2\pi/3$. However, the DW configuration changes with different $n$, so the edge states have no repeating pattern. The red arrows in Fig. 4 (b) and (c) indicate examples where the gap of the edge states closes, while Fig. 4 (d) and (e) plot the corresponding supercell dispersion. In these two cases we obtain helical edge states. We also computed the pseudoangular momentum based on the cell next to the domain wall which is indicated in color to illustrated the pseudospin polarization.

However for the two lattices supporting edge states in between, it is possible to find a path in the parameter space $(\delta_{1},\delta_{2},\delta_{3})$ linking the two configurations such that when the lattice evolves adiabatically along it, the bulk gap never closes, hence implying no topological phase transition. And both bulk lattices can have nontrivial $\mathbb{Z}_{2}=1$. This hold true for different (e.g., assymetric) types of DW assemblies Sup . These considerations indicate that, while these bulk lattices are seemingly indistinguishable from a topological perspective, they can still yield non-trivial edge states following translations perpendicular to the domain wall. This observation is consistent with the fact that the existence of edge states on DWs between Kekulé lattices depends not only on the bulk properties of the two individual lattices but also on the same reference gauge choice. This result is a direct consequence of the gauge-dependence of both the $L_{z}$ calculation and the mapping to the BHZ model. Based on this idea, one can even devise helical edge states along an edge dislocation of a Kekulé lattice (see supplemental material Sup for examples). Another related work recently reported the possibility to create DW between nontrivial phases controlled by different analogue quantum mechanisms Miniaci et al. (2019).

Finally we performed full-field numerical simulations to demonstrate the robustness of the edge states. Symmetric DW configuration with $\phi=229.9^{\circ}$ (see Fig. 4 (b)) is selected in this case. Fig. 5 (a) and (b) show both the configuration and the steady-state response of a Z-shaped DW under a harmonic excitation applied at one end (red dot). The edge state concentrates along the Z-path with uniform amplitude indicating no significant reflections at the corners. The second geometric configuration shows the case of an intersection between two DWs with excitation at one of the four terminals (Fig.5 (c)).

In this case, the $+x$ propagating state is a pseudospin up. After the intersection, only the oblique branches support pseudospin-up out-going modes while the remaining wall along the $+x$ axis is associated with pseudospin down. The result is that the edge state continues propagating only along the two oblique walls. The different amplitude of the mode along the two DW branches is caused by the different configuration of the walls. The lower branch has the same configuration as the input one, while the upper one is equivalent to a configuration with $\phi-2\pi/3$ (see Fig. 4 (a)) which is not optimized to supports gapless edge states at the selected frequency. This is an interesting feature of this DW and can be used to create complex patterns of propagation.

In summary, we proposed a continuous elastic phononic structure composed of slender beams and exploiting Kekulé distortion to achieve an elastic analogue of the spin Hall effect for flexural waves. We derived the effective Hamiltonian based on conventional perturbation analysis and pseudospin bases and showed that it maps to the BHZ model for QSH materials. Furthermore, first-principles calculations revealed the finer structure of the Kekulé lattices that further the understanding of these material phases beyond the low-energy Hamiltonian approximation available in the literature. The finer structure includes a peculiar 6-lobe alternating pattern of both the pseudospin texture and the Berry curvature. We also investigated the unique gauge dependence of the topological states and its connection to the occurrence of helical edge modes. A remarkable consequence of this finding is the possible existence of topological edge states on edge dislocations of a lattice. We show, also for the first time, that the gapless edge states in an elastic Kekulé lattice (associated with a degeneracy at the $\Gamma$ point) can always be achieved by combining lattice symmetry and accidental degeneracies. Numerical simulations confirmed that the counter-propagating pseudospin polarized edge states are decoupled and robust to disorder.

I Acknowledgments

The authors gratefully acknowledge the financial support of the National Science Foundation under grant MOMS #1761423.