Higher-Order Topological Insulators in Quasicrystals

TL;DR

This paper demonstrates that higher-order topological insulators can exist in quasicrystals, revealing new types of corner states and proposing experimental detection methods using electrical circuits.

Contribution

It introduces the realization of second-order topological insulators in quasicrystals, expanding HOTI concepts beyond crystalline materials with practical detection approaches.

Findings

Two types of quasicrystalline SOTIs constructed with different tiling patterns.

Identification of unique corner states in quasicrystalline SOTIs.

Electrical circuit simulation of quasicrystalline quadrupole insulator with measurable corner states.

Abstract

Current understanding of higher-order topological insulators (HOTIs) is based primarily on crystalline materials. Here, we propose that HOTIs can be realized in quasicrystals. Specifically, we show that two distinct types of second-order topological insulators (SOTIs) can be constructed on the quasicrystalline lattices (QLs) with different tiling patterns. One is derived by using a Wilson mass term to gap out the edge states of the quantum spin Hall insulator on QLs. The other is the quasicrystalline quadrupole insulator (QI) with a quantized quadrupole moment. We reveal some unusual features of the corner states (CSs) in the quasicrystalline SOTIs. We also show that the quasicrystalline QI can be simulated by a designed electrical circuit, where the CSs can be identified by measuring the impedance resonance peak. Our findings not only extend the concept of HOTIs into quasicrystals but also provide a feasible way to detect the topological property of quasicrystals in experiments.

Equations

$$H_{\text{QSH}}$$

$$H_{\text{m}}(\eta )=g\sum _{j\ne k}\frac{f(r_{jk})}{2}\cos \left(\eta {\varphi }_{jk}\right)c_j^{†}{s}_1{\tau }_1{c}_k,$$

$$H_{\text{QI}}=\gamma \sum _j{c}_j^{†}\left({\Gamma }_2+{\Gamma }_4\right)c_j+\frac{\lambda }{2}\sum _{j\ne k}f\left(r_{jk}\right)c_j^{†}T({\varphi }_{jk})c_k,$$

$$I_{pa}\left(\omega \right)=\sum _{q,b}{J}_{pa,qb}\left(\omega \right)V_{qb}\left(\omega \right),$$

$$H_{pa,qb}(\omega )=C_{pa,qb}-\frac{1}{{\omega }^2}{W}_{pa,qb}.$$

$$Z_{pa,qb}\left(\omega \right)=\sum _l\frac{{\left|{\psi }_l\left(pa\right)-{\psi }_l\left(qb\right)\right|}^2}{j_l\left(\omega \right)},$$

Full text

Introduction.—Since the discovery of topological insulators (TIs) Hasan and Kane (2010); Qi and Zhang (2011), tremendous effort has been devoted into the search for exotic topological phases (TPs) of matter. Recently, higher-order topological insulators (HOTIs) Benalcazar et al. (2017a) were proposed as an extension of TIs, which have been widely investigated in condensed matter as well as phononic, microwave, photonic and electrical circuit (EC) systems Langbehn et al. (2017); Benalcazar et al. (2017b); Song et al. (2017); Schindler et al. (2018a); Ezawa (2018a, b); Schindler et al. (2018b); Liu et al. (2019a); Serra-Garcia et al. (2018); Peterson et al. (2018); Noh et al. (2018); Imhof et al. (2018); Ni et al. (2018); Xue et al. (2018); van Miert and Ortix (2018); Ezawa (2018c, d); Xie et al. (2018); Kunst et al. (2018); Franca et al. (2018); Matsugatani and Watanabe (2018); Khalaf (2018); Geier et al. (2018); Zhang et al. (2019); Liu et al. (2019b); Trifunovic and Brouwer (2019); Xie et al. (2019); Chen et al. (2019); Serra-Garcia et al. (2019); Lee et al. (2019). Unlike conventional TIs in $d$ dimensions which have gapless states on the $d-1$-dimensional boundary, $n$th-order ($1<n\leq d$) TIs in $d$ dimensions have $(d-n)$-dimensional gapless boundary states Benalcazar et al. (2017a); Langbehn et al. (2017); Benalcazar et al. (2017b); Song et al. (2017); Schindler et al. (2018a). For instance, two-dimensional (2D) second-order topological insulators (SOTIs) display robust zero-energy modes (ZEMs) localized at their [math]-dimensional corners, dubbed corner states (CSs).

The study of TPs has been lately extended to aperiodic quasicrystalline systems Huang and Liu (2018a, b); Bandres et al. (2016); Tran et al. (2015); Fulga et al. (2016); Kraus et al. (2012), which lack translational symmetry and show forbidden symmetries in crystals such as the 5-fold and 8-fold rotation symmetries. The proposals for realizing conventional TIs Huang and Liu (2018a, b); Tran et al. (2015); Bandres et al. (2016) have been proposed in quasicrystalline systems. It is of interest to ask if it is possible to realize HOTIs in quasicrystals.

In this Letter, we propose SOTIs on 2D quasicrystalline lattices (QLs), which extend the concept of HOTIs to quasicrystalline systems. We demonstrate two general schemes to construct SOTIs in 2D quasicrystals. First, an SOTI can be realized by adding a proper mass term into the quantum spin Hall (QSH) insulator on a QL, where the edge states are gapped, and then topological CSs emerge. Second, the other type of SOTIs in quasicrystals originates from a quantized bulk quadrupole moment Benalcazar et al. (2017a, b), which is a quasicrystalline quadrupole insulator (QI). With the two mechanisms, the hallmark zero-energy CSs are found on the designed QLs. Our results reveal several distinguishing features of the quasicrystalline SOTIs: (1) the CSs strongly rely on the tiling pattern and boundary geometry of quasicrystals; (2) intriguing extended ZEMs exist in quasicrystals, quite unlike the normal CSs, they distribute along some segments of the edge; (3) 8-fold symmetric CSs are found in the Ammann-Beenker (AB) tiling quasicrystal, which are protected by a rotation symmetry $C_{8}$ forbidden in crystals, indicating that the quasicrystalline HOTI is actually distinct from that in crystalline systems. We also design an EC to simulate the quasicrystalline QI and show that the CSs can be measured as an impedance resonance peak at the EC corners Imhof et al. (2018). It may be the first feasible way to experimentally detect quasicrystalline HOTIs.

Mass term induced SOTIs in quasicrystals.—It was known that QSH states can be realized on 2D QLs Huang and Liu (2018a). The first scheme to make an SOTI in a 2D quasicrystal is to use an additional mass term to gap the topological edge states. Once this mass term results in a domain wall structure at two adjacent edges, a CS appears at the intersection of the two edges, i.e., an SOTI on the QL is achieved.

We first design a QSH insulator on a QL constructed according to the AB tiling with 8-fold rotation symmetry, where the plane is tiled using squares and rhombi as shown in Fig. 1. In our model, each lattice site has two orbitals and the Hamiltonian is

[TABLE]

where $c^{\dagger}_{j}=(c^{\dagger}_{j\alpha\uparrow},c^{\dagger}_{j\alpha\downarrow},c^{\dagger}_{j\beta\uparrow},c^{\dagger}_{j\beta\downarrow})$ are electron creation operators at site $j$. $\alpha$ and $\beta$ represent two orbitals at one site and spin degrees of freedom are considered. $M$ denotes the Dirac mass, $t_{1}$ and $t_{2}$ are hopping amplitudes. $s_{0}$ is the $2\times 2$ identity matrix, $s_{1,2,3}$ and $\tau_{1,2,3}$ are the Pauli matrices acting on the spin and orbital spaces, respectively. $\phi_{jk}$ is the polar angle of bond between site $j$ and $k$ with respect to the horizontal direction Fulga et al. (2016); Agarwala et al. (2019). $f\left(r_{jk}\right)=e^{1-r_{jk}/\xi}$ is the spatial decay factor of hopping amplitudes with the decay length $\xi$ and $r_{jk}=\left|\mathbf{r}_{j}-\mathbf{r}_{k}\right|$. $H_{\text{QSH}}$ preserves time-reversal $T$, particle-hole $C$ and chiral $S=TC$ symmetries, where $T=is_{2}\tau_{0}K$, and $C=s_{3}\tau_{1}K$ with complex conjugate $K$, therefore it belongs to the class DIII Schnyder et al. (2008); Kitaev (2009); Chiu et al. (2016). In some sense, this QSH model can be understood as a derivative of the Bernevig-Hughes-Zhang model Bernevig et al. (2006), which describes the QSH states in HgTe quantum wells. In the following calculations, we fix the side length of rhombi or squares $c=1$ and include hoppings up to the third nearest neighbors.

With the designed QL above, we get a QSH state as shown in Figs. 1(a) and 1(b). We set $\xi=1$, $M=-1$ and $t_{1}=t_{2}=1$, and diagonalize $H_{\text{QSH}}$ under a periodic boundary condition (PBC) and an open boundary condition (OBC). The energy spectrum in Fig. 1(a) exhibits an energy gap for the PBC, while for the OBC, gapless in-gap states occupy the bulk gap. Figure 1(b) displays the wavefunction probability of a typical in-gap state, which suggests a localized edge state. Due to time-reversal symmetry (TRS), each in-gap energyvalue is doubly degenerate, corresponding to two counter-propagating edge states, which characterize the QSH state. Note that the QLs, which can host QSH states, are not unique. Recently, the QSH insulator on the Penrose tiling QL was also proposed Huang and Liu (2018a, b).

To transform the QSH insulator on the QL to an SOTI, we induce a TRS breaking Wilson mass term, which reads

[TABLE]

where $g$ and $\eta$ describe the magnitude and the varying period of the Wilson mass, respectively. Thus, the total Hamiltonian of the SOTI is $H_{\text{I}}=H_{\text{QSH}}+H_{\text{m}}$. In the following, we fix $g=1$ unless otherwise specified.

When turning on $H_{\text{m}}$, the edge states are gapped out and then the hallmark CSs of HOTIs emerge. Figure 1(c) gives the energy spectrum of the QL with a square boundary when $\eta=2$, we can see that four ZEMs emerge inside the edge gap and are symmetrically localized at the four corners of the QL [Fig. 1(d)]. Essentially, the location of CSs can be explained by the Jackiw-Rebbi mechanism Jackiw and Rebbi (1976) that a topological ZEM appears when a mass domain wall forms. Note that the Wilson mass term in Eq. (2) depends on the polar angle of the bond $\phi_{jk}$. However, due to the lack of translation symmetry in quasicrystals, we can not give an analytic expression of the effective Wilson mass for edge states. A rough approximation is to consider an edge of the sample boundary as a long “bond”, so that the sign of the effective Wilson mass for the edge state depends on the orientation of the edge (or the polar angle of the edge $\theta_{\text{edge}}$). Surprisingly, this approximation works quite well compared with the numerical results. It can even be used as an intuitive rule to determine the appearance of CSs in any quasicrystal polygons.

In Fig. 2, we consider quasicrystal pentagon and octagon, which are constructed in the same way as Fig. 1 but of different boundary shapes. We give the results of the quasicrystal pentagon with $\eta=2$ in Figs. 2(a), 2(e) and 2(i). At all the corners of the pentagon, in-gap ZEMs are found except the top one, and the distribution of the ZEMs become asymmetric [see Figs. 2(e) and 2(i)]. Generally speaking, the factor $\cos{\eta\theta_{\text{edge}}}$ in the Wilson mass will distinguish two different regions of the edge orientation $\theta_{\text{edge}}$, which have opposite sign of the Wilson mass. For example, the two regions for $\eta=2$ are illustrated in Figs. 2(a) and 2(b), where the red region is determined by $\theta_{\text{edge}}\in(-\frac{\pi}{4},\frac{\pi}{4})\cup(\frac{3\pi}{4},\frac{5\pi}{4})$, and the blue region is $\theta_{\text{edge}}\in(-\frac{\pi}{4},-\frac{3\pi}{4})\cup(\frac{\pi}{4},\frac{3\pi}{4})$. For the top corner of the pentagon, the two edges are both in red regions [Fig. 2(a)], which means that their masses are of the same sign. This is the reason why there is no ZEM at this corner. Meanwhile, for the other four corners, the two edges of each corner lie in two different regions, so that a ZEM appears at each corner. The case of the quasicrystal octagon [Figs. 2(b), 2(f), and 2(j)] is special. Note that the Wilson mass at four edges of the octagon is zero because the polar angles of the four edges are $\theta_{\text{edge}}=\pm\frac{\pi}{4},\pm\frac{3\pi}{4}$. At the same time, the remaining four edges of the octagon have a finite Wilson mass. In this situation, we find four in-gap ZEMs distributing along the four edges with zero Wilson mass [Figs. 2(f) and 2(j)], quite unlike the normal CSs. To the best of our knowledge, these extended ZEMs in the quasicrystal haven’t been reported in other HOTIs.

To further confirm the topological origin of CSs, we first examine robustness of the CSs by applying various symmetry-breaking perturbations Sup . It’s found that the CSs in the quasicrystal square (Fig. 1) are protected by the combined symmetry $C_{4}m_{z}$ and $C$ symmetry, where $C_{4}=e^{-i\frac{\pi}{4}s_{3}\tau_{3}}\mathcal{R}_{4}$ is a 4-fold rotation symmetry and $m_{z}=s_{3}\tau_{0}$ is the mirror symmetry about the $x-y$ plane. Moreover, the CSs in the AB tiling quasicrystalline HOTI can be characterized by a $\mathbb{Z}_{2}$ topological invariant, which is determined by the product of Pfaffians at high-symmetry momenta in a 4D momentum hyperspace Varjas et al. (2019). $\mathbb{Z}_{2}$ phase diagram of the SOTI model is shown in Ref. Sup . Note that the symmetry of a quasicrystal sample also depends on its boundary shape. Thus, the CSs of the quasicrystal pentagon in Fig. 2 (i) aren’t protected by the above spatial symmetry. In this case, we actually realize an “extrinsic” HOTI Geier et al. (2018), which hosts termination-dependent corner states instead of spatial-symmetry-protected ones.

We can also change the angular dependence of the Wilson mass by choosing a different $\eta$. With $\eta=4$, We calculate the quasicrystal octagon [Figs. 2(c), 2(g), and 2(k)] and pentagon [Figs. 2(d), 2(h), and 2(l)]. Now the red and blue regions for the edge orientation to determine the sign of Wilson mass have been changed [Figs. 2(c) and 2(d)]. The red region becomes $\theta_{\text{edge}}\in\cup(-\frac{\pi}{8}+\frac{n\pi}{4},\frac{\pi}{8}+\frac{n\pi}{4})$, where $n=0,1,2,3$. In a similar way, we can predict that the quasicrystal octagon has eight CSs and the quasicrystal pentagon has two corners. It’s in good agreement with the numerical results as shown in Figs. 2 (g), 2(k), 2(h), and 2(l). For $\eta=4$, the CSs of quasicrystal octagon are protected by the combined symmetry $C_{8}m_{z}$ and $C$ symmetry. Similarly, we can also define a $\mathbb{Z}_{2}$ topological invariant, and the $\mathbb{Z}_{2}$ phase diagram for $\eta=4$ is given in Ref. Sup . We emphasize that the 8-fold symmetric CSs here are forbidden in crystals, as crystals don’t have a $C_{8}$ symmetry. This implies that the quasicrystalline HOTI is beyond the framework of crystalline HOTIs.

The results above indicate that, with an angular dependent Wilson mass term, SOTIs can be achieved on the constructed QLs, and CSs can be manipulated by choosing proper boundaries (See Ref. Sup for other geometries). It should be emphasized that we can also use other kinds of tilings, e.g., the Penrose tiling Sup , to construct a quasicrystalline SOTI.

QIs in quasicrystals.—The other feasible scheme to construct an SOTI on a QL is to consider the QI with a quantized quadrupole moment. Referring to the 2D square lattice model of QI in Refs. Benalcazar et al. (2017a, b), we design a 2D QL as shown in Fig. 3(a). Here, four sites form a cell, and we use the cells to construct the AB tiling QL. The Hamiltonian is

[TABLE]

with $T(\phi_{jk})=\left|\cos\phi_{jk}\right|\Gamma_{4}-i\cos\phi_{jk}\Gamma_{3}+\left|\sin\phi_{jk}\right|\Gamma_{2}-i\sin\phi_{jk}\Gamma_{1}.$ Here, $c^{\dagger}_{j}=(c^{\dagger}_{j1},c^{\dagger}_{j2},c^{\dagger}_{j3},c^{\dagger}_{j4})$ is the electron creation operator in cell $j$. The first and second terms in Eq. (3) are the intra-cell and inter-cell hoppings with amplitudes $\gamma$ and $\lambda$. $\Gamma_{4}=\tau_{1}\tau_{0}$ and $\Gamma_{\nu}=-\tau_{2}\tau_{\nu}$ with $\nu=1,2,3$. $\tau_{1,2,3}$ are the Pauli matrices representing the sites in one cell, and $\tau_{0}$ is the identity matrix.

We consider the AB tiling QL with a square boundary and include the hoppings to the third neighbors. By numerically diagonalizing $H_{\text{QI}}$ with $\gamma=0.1\lambda$, we find four zero-energy CSs on the designed QL in Fig. 3(b). This system has two mirror symmetries $m_{x}$ and $m_{y}$ as well as a rotation symmetry $C_{4}$, resulting in a quantized quadrupole moment $Q_{xy}=0,e/2$. Thus, $Q_{xy}$ is a natural topological invariant and can be calculated in real space Wheeler et al. (2018); Kang et al. (2018). Numerical calculations show that corner states appear when $Q_{xy}=e/2$ Sup . These results clearly indicate that Eq. (3) on the designed QL can produce a QI. The results of the Penrose tiling QL are presented in Ref. Sup .

Electrical-circuit realization.—Here, we show that the QL hosting the QI we proposed can be mapped onto an EC lattice, and the zero-energy CSs will result in an impedance resonance peak at the EC corners.

In a recent work Imhof et al. (2018), 2D crystalline QI has been simulated by an EC, and the CSs have been observed in experiments. The case of the quasicrystalline QI is rather similar. The designed EC is shown in Figs. 4(a) and 4(b). Each node in the EC corresponds to a QL site. Similar as the lattice configuration in Fig. 3, we use four nodes to form a cell [Fig. 4(b)], and the cells are connected to produce an AB tiling EC [Fig. 4(a)]. The inter-cell and intra-cell connections between the nodes with capacitors or inductors are given in Fig. 4(b), and more details are presented in Sup . Here, we make a simplification about the inter-cell hopping on the QL and set $T\left(\phi_{jk}\right)=\left(1-n\right)\left[\Gamma_{4}-i\left(-1\right)^{m}\Gamma_{3}\right]+n\left[\Gamma_{2}-i\left(-1\right)^{m}\Gamma_{1}\right]$ for $\phi_{jk}\in\left(-\frac{\pi}{4}+n\frac{\pi}{2}+m\pi,\frac{\pi}{4}+n\frac{\pi}{2}+m\pi\right)$, where $n=0,1$ and $m=0,1$. With this simplification, the QL still hosts a QI Sup .

According to the Kirchhoff’s law Imhof et al. (2018); Lee et al. (2018); Hofmann et al. (2019); Ezawa (2018e); Albert et al. (2015); Luo et al. (2018a, b), we have

[TABLE]

where $I_{pa}(t)=I_{pa}\left(\omega\right)e^{i\omega t}$ [$V_{qb}(t)=V_{qb}\left(\omega\right)e^{i\omega t}]$ is the current (voltage) at node $a$ (b) in cell $p$ ($q$), and $\omega$ is the frequency of circuit. The circuit Laplacian is $J_{pa,qb}\left(\omega\right)=i\omega H_{pa,qb}(\omega)$, where

[TABLE]

Here, $H$ is just the Hamiltonian of the tight-binding (TB) model of the EC. $C_{pa,qb}$ is the capacitance between two nodes, and $W_{pa,qb}=L^{-1}_{pa,qb}$ is the inverse inducitivity between two nodes. For the diagonal components with $pa=qb$, we have $C_{pa,pa}=-C_{pa,g}-\sum_{q^{\prime}b^{\prime}}C_{pa,q^{\prime}b^{\prime}}$, and $W_{pa,pa}=-L^{-1}_{pa,g}-\sum_{q^{\prime}b^{\prime}}L^{-1}_{pa,q^{\prime}b^{\prime}}$. Subscript $g$ means the ground. In principle, the EC Hamiltonian in Eq. (5) can be considered as the TB Hamiltonian of QL, expect for $D(\omega)$ i.e., $H(\omega)=H_{\text{QL}}(\omega)+D(\omega)$, and $D(\omega)$ is a diagonal matrix collecting all the diagonal elements of $H(\omega)$.

Now we discuss how to measure the CSs in the EC. As illustrated in Ref. Imhof et al. (2018), the impedance between two nodes $Z_{pa,qb}(\omega)\equiv(V_{pa}-V_{qb})/I_{0}$ can be directly measured in experiments. Diagonalizing the $J$ matrix, we get $J_{pa,qb}(\omega)=\sum_{l}j_{l}(\omega)|\psi_{l}(pa)\rangle\langle\psi_{l}(qb)|$, where $j_{l}$ is the $l^{th}$ eigenvalue and $|\psi_{l}\rangle$ is the corresponding eigenvector. Then,

[TABLE]

and $Z_{pa,qb}$ diverges whenever $j_{l}\left(\omega\right)=0$. The topological CS of $H_{\text{QL}}$ gives rise to $j_{l_{0}}=0$ when $D(\omega)=0$ at the resonance frequency $\omega_{0}$, so that $Z(\omega_{0})$ shows a peak. $D(\omega_{0})=0$ is guaranteed by a suitable choice of grounding Sup . Meanwhile, we choose pairs of capacitors and inductors ($C_{0,1,2,3}$ and $L_{0,1,2,3}$) in the EC [Fig. 4(b)] to satisfy $\omega_{0}=1/\sqrt{C_{0}L_{0}}=1/\sqrt{C_{1}L_{1}}=1/\sqrt{C_{2}L_{2}}=1/\sqrt{C_{3}L_{3}}$. Of course, the CS induced resonance peak can only be observed when the nodes are at the corner.

Figure 4(c) shows the spectrum of the circuit Laplacian as a function of the normalized frequency $\omega/\omega_{0}$. Four degenerate isolated ZEMs inside the bulk spectral gap at $\omega_{0}$. In Fig. 4(d), we plot the impedance between two nearest-neighbor nodes in one circuit cell located at the corner, the edge and in the bulk, respectively. We can see that the corner impedance shows an obvious resonance peak at $\omega_{0}$, indicating a quasicrystalline QI.

Conclusion.—In summary, we have constructed quasicrystalline SOTIs by two distinct ways. One is the edge gapped quasicrystalline QSH insulator, and the other is the quasicrystalline QI. The quasicrystalline SOTIs exhibit some unusual characteristics. We also propose that the quasicrystalline QI can be simulated by an EC. Our work indicates that the concept of HOTIs is also valid in quasicrystalline systems, which enriches the HOTI family.

Acknowledgments.—We would like to thank Xin Liu, Jin-Hua Sun, Yi Zhou, Rui Yu and Zhi-Hong Hang for helpful discussions. R.C. and D.-H.X. were supported by the NSFC (Grant No. 11704106). J.H.G. was supported by the NSFC (Grant No. 11534001 and 11874160). C.Z.C. was funded by the NSFC (Grant No. 11974256) and the NSF of Jiangsu Province (Grant No. BK20190813). D.-H.X. also acknowledges the financial support of the Chutian Scholars Program in Hubei Province.

Note added.—Recently, we became aware of a complementary study, which focuses on realizing a higher-order topological superconductor phase on an 8-fold symmetric patch of the AB tiling Varjas et al. (2019).