Anomalous Quadrupole Topological Insulators in 2D Nonsymmorphic Sonic Crystals

TL;DR

This paper introduces a new class of anomalous quadrupole topological insulators in 2D nonsymmorphic sonic crystals, protected by glide symmetries, expanding the understanding of topological phases without flux.

Contribution

It presents a theoretical framework for anomalous quadrupole topological insulators in nonsymmorphic crystals, demonstrating their realization in sonic crystals with specific symmetry protections.

Findings

Protected by orthogonal glide symmetries in square or rectangular lattices

Features minimal four bands below the topological gap

Exhibits quantized Wannier band polarizations

Abstract

The discovery of quadrupole topology opens a new horizon in the study of topological phenomena. However, the existing experimental realizations of quadrupole topological insulators in symmorphic lattices with $\pi$-fluxes often break the protective mirror symmetry. Here, we present a theory for anomalous quadrupole topological insulators in nonsymmorphic crystals without flux, using 2D sonic crystals with $p4gm$ and $p2gg$ symmetry groups as concrete examples. We reveal that the anomalous quadrupole topology is protected by two orthogonal glide symmetries in square or rectangular lattices. The distinctive features of the anomalous quadrupole topological insulators include: (i) minimal four bands below the topological band gap, (ii) nondegenerate, gapped Wannier bands and special Wannier sectors with gapped composite Wannier bands, (iii) quantized Wannier band polarizations in these Wannier sectors. Remarkably, the protective glide symmetries are well-preserved in the sonic-crystal realizations where higher-order topological transitions can be triggered by symmetry or geometry engineering.

Equations

$$P_x=P_y=\frac{1}{4}(1-\xi ),\xi =\prod _n{\mathcal{I}}_n(\Gamma ){\mathcal{I}}_n(X),$$

$${\nu }_{x,n}(k_y)\stackrel{\underset{G_x}{}}{=}-{\nu }_{x,n^{\prime }}(k_y)+\frac{1}{2}\mathrm{mod}1,$$

$${\nu }_{x,n}(k_y)\stackrel{\underset{G_y}{}}{=}{\nu }_{x,n^{\prime }}(-k_y)+\frac{1}{2}\mathrm{mod}1,$$

$${\nu }_{x,n}(k_y)\stackrel{\underset{\mathcal{I}}{}}{=}-{\nu }_{x,n^{\prime }}(-k_y)\mathrm{mod}1,$$

$${\nu }_{x,n}(k_y)\stackrel{\underset{\mathcal{T}}{}}{=}{\nu }_{x,n}(-k_y)\mathrm{mod}1.$$

$$P_y^{{\nu }_x,I}\stackrel{\underset{G_x}{}}{=}{P}_y^{{\nu }_x,II}\mathrm{mod}1,$$

$$P_y^{{\nu }_x,I}\stackrel{\underset{G_y}{}}{=}-P_y^{{\nu }_x,I}\mathrm{mod}1,$$

$$P_y^{{\nu }_x,I}=P_y^{{\nu }_x,II}=0,\frac{1}{2}\mathrm{mod}1.$$

$$q_{xy}=2P_y^{{\nu }_x,I}{P}_x^{{\nu }_y,I}=\frac{1}{2}\mathrm{mod}1.$$

Full text

Topological insulators are unconventional materials which host robust edge states and quantized transport or electromagnetic properties as dictated by the topological invariants of the occupied bulk bands rev1 ; rev2 . These topological invariants result from nontrivial quantization of Berry’s phases arising from parallel transport in the Brillouin zone, which are essentially connected to dipole polarization and topological charge pumping in crystals pol1 . Though developed for electronic systems, topological band theory also applies for photonic ph1 ; ph2 ; ph3 ; ph4 ; ph5 ; ph6 ; ph7 ; ph8 ; ph9 ; ph10 ; ph11 ; ph12 ; ph13 and mechanical ac1 ; ac2 ; ac3 ; ac4 ; ac5 ; ac6 ; ac7 ; ac8 ; ac9 ; me1 ; me2 waves, yielding rich phenomena and applications in various disciplines of science.

Recently, topological band theory was generalized from conventional (dipole) topology to quadrupole and octopole band topology qti1 ; qti2 ; exp1 ; exp2 ; exp3 ; exp4 , opening a pathway toward higher-order topology hoti1 ; hoti2 ; hoti3 ; hoti4 ; hoti5 ; new1 ; new2 ; new3 ; new4 ; new5 ; new6 ; new7 ; aqti1 . For instance, a 2D quadrupole topological insulator (QTI) has gapped edge states and topologically-protected corner states. QTI was proposed theoretically using a $\pi$-flux tight-binding model with noncommutative mirror symmetries qti1 ; qti2 and was later observed in mechanical exp1 and microwave metamaterials exp2 , $LC$-circuits exp3 , and coupled optical waveguides exp4 . However, the existing material realizations of QTIs often break the mirror symmetry that protects the quadrupole topology exp1 ; exp2 ; exp3 .

In this work, we present a theory of anomalous QTI (AQTI) in nonsymmorphic crystals without gauge flux. Using 2D square-lattice sonic crystals (SCs) as material realizations, we find that the AQTI is a topological state protected by two orthogonal glide symmetries. Due to these glide symmetries, the AQTI phase has several features that are distinct from the QTI model proposed in Refs. qti1 ; qti2 . These features include: (i) minimal four bands below the topological band gap which originates from the band-sticking effect and the parity inversions, (ii) nondegenerate, gapped Wannier bands and special Wannier sectors with gapped composite Wannier bands, (iii) quantized Wannier band polarizations in the special Wannier sectors. Using SCs, we demonstrate that higher-order topological transitions can be triggered by symmetry or geometry engineering. We further present a full symmetry analysis for square and rectangular wallpaper crystals in the pursuit of quadrupole topology in flux-threading-free systems. Our discovery can be directly generalized to other classical-wave systems, e.g., photonic crystals, mechanical metacrystals. An experimental demonstration of the $p4gm$ SC as the acoustic AQTI has been demonstrated by some of us in Ref. ourexp based on the theory in this work.

Nonsymmorphic $p4gm$ SCs.—We start with the AQTI phase in the $p4gm$ SCs [see Fig. 1(a)]. In each unit-cell, four identical arch-shaped epoxy blocks, with width $w$, height $h$ and length $l$, serve as the scatterers for acoustic waves. The scatterers are arranged such that the SC has $p4gm$ nonsymmorphic symmetry group. The four key symmetries are: two orthogonal glide symmetries, $G_{x}=\{m_{x}|{\bf\tau}_{y}\}$ and $G_{y}=\{m_{y}|{\bf\tau}_{x}\}$ where $m_{x}:=x\to\frac{a}{2}-x$, $m_{y}:=y\to\frac{a}{2}-y$, ${\bf\tau}_{y}:=y\to y+\frac{a}{2}$ and ${\bf\tau}_{x}:=x\to x+\frac{a}{2}$ ($a$ is the lattice constant, and the origin of the coordinate is at the center of the unit-cell), the inversion symmetry ${\cal I}$ and the $C_{4}$ rotation symmetry.

The acoustic band structure, as obtained from COMSOL Multiphysics, is shown in Fig. 1(b). The band gap of concern is that between the fourth and the fifth bands. The dipole polarization of the bands below the gap is given by qti2 ,

[TABLE]

where ${\cal I}_{n}(\Gamma)$ and ${\cal I}_{n}(X)$ are the parity eigenvalues of the $n^{th}$-band ($n=1,2,3,4$) at the $\Gamma$ and X points, respectively. From the parity eigenvalues in Fig. 1(b), the first and second (third and fourth) bands have a dipole polarization of ${\vec{P}}=(\frac{1}{2},\frac{1}{2})$. Therefore, the band gap of concern has a vanishing dipole polarization, which is necessary for the emergence of the quadrupole topology qti1 .

The glide symmetries lead to the double degeneracy for all Bloch states on the XM and YM lines. This double degeneracy can be understood by constructing the anti-unitary operators $\Theta_{j}=G_{j}*{\cal T}$ ($j=x,y$) where ${\cal T}$ is the time-reversal operator (i.e., complex conjugation). It is straightforward to obtain that $\Theta_{y}^{2}\psi_{n,{\vec{k}}}=e^{ik_{x}a}\psi_{n,{\vec{k}}}$. Therefore, $\Theta_{y}^{2}\psi_{n,{\vec{k}}}=-\psi_{n,{\vec{k}}}$ at the XM line, i.e., $k_{x}=\frac{\pi}{a}$. As an analog to the Kramers theorem for fermions, such a relation results in the double degeneracy for all Bloch states on the XM line ph10 . According to the $C_{4}$ symmetry, the YM line has the same property.

The parity operator ${\cal I}$ can be expressed using the glide operators, ${\cal I}={\bf\tau}_{y}^{2}G_{x}G_{y}={\bf\tau}_{x}^{2}G_{y}G_{x}$. Obviously, the two glide operators do not commute. Besides, we find that the double degeneracy at the $X$ point consists of Bloch states with opposite parities, since $\Theta_{y}{\cal I}=-{\cal I}\Theta_{y}$. This property constraints the parity-inversion between the $\Gamma$ and $X$ points and hence the dipole polarization. In this way, the quadrupole topology, since it requires a vanishing dipole polarization, emerge in our system when there are four bands below the band gap.

Wannier bands and nested Wannier bands.—The quadrupole topology is characterized by two features: (i) gapped Wannier bands for both $\nu_{x}$ and $\nu_{y}$ qti1 ; qti2 ; (ii) quantized Wannier band polarizations. The second feature requires the mirror or glide reflection symmetry with respect to both $x$ and $y$. The first condition, however, cannot be fulfilled in the presense of commutative mirror symmetries, as shown in Ref. qti2 .

The Wannier bands are calculated from the Wilson-loop approach hx . For instance, $\exp[i2\pi\nu_{x}(k_{y})]$, is obtained from the eigenvalues of the Wilson-loop operator $\hat{W}^{x}_{\vec{k}}={\cal T}_{P}\exp[i\oint\hat{A}^{x}dk_{x}]$ where the superscript $x$ and the subscript ${\vec{k}}$ denote the direction and starting point of the Wilson-loop, respectively. $\hat{A}^{x}$ is the non-Abelian (matrix) Berry connection for the first four bands. The matrix element is $A^{x}_{nm}=\langle u_{n}({\vec{k}})|i\partial_{k_{x}}|u_{m}({\vec{k}})\rangle$ for $n,m=1,...,4$. Here, $u_{n}({\vec{k}})$ is the periodic part of the Bloch wavefunction, and ${\cal T}_{P}$ is the path ordering operator. The results in Fig. 1(c) show that there are four nondegenerate, gapped Wannier bands.

The symmetry constraints on the Wannier bands are (see Supplemental Material for detailed derivations),

[TABLE]

Here, $n^{\prime}\neq n$ are Wannier band indices. We notice that ${\cal I}$ transforms the first (second) Wannier band to the fourth (third) Wannier band. Besides, $G_{x}$ transforms the first (third) Wannier band into the second (fourth) Wannier band, while $G_{y}$ transforms the first (second) Wannier band to the third (fourth) Wannier band. Thus, the four Wannier bands are intrinsically connected to each other.

Following Refs. qti1 ; qti2 , for the description of the quadrupole topology, it is important to introduce a division of the Wannier bands below and above $\nu_{y}=0$. For the situations with more than two Wannier bands, such a division can have more possibilities. From the symmetry analysis above, we find that the division “1+2” and “3+4” always yield gapless composite Wannier bands, i.e., $\nu_{x,1}+\nu_{x,2}=\nu_{x,3}+\nu_{x,4}=\frac{1}{2}~{}{\rm mod}~{}1$ [see Fig. 1(c)]. Only the division “1+3” and “2+4” leads to gapped composite Wannier bands (“gapped” meaning that the Wannier bands are away from the special values of 0 and $\pm\frac{1}{2}$) [see Fig. 1(c)]. We denote the Wannier sectors, “1+3” and “2+4”, as I and II, respectively. The Wannier sector I (II) has a Wannier center below (above) the unit-cell center.

For Wannier sectors I and II, the Wannier band polarizations are calculated as $P^{\nu_{x},I}_{y}=\frac{1}{2\pi}\int dk_{x}p^{\nu_{x}}_{y,1}(k_{x})+p^{\nu_{x}}_{y,3}(k_{x})$ and $P^{\nu_{x},II}_{y}=\frac{1}{2\pi}\int dk_{x}p^{\nu_{x}}_{y,2}(k_{x})+p^{\nu_{x}}_{y,4}(k_{x})$, respectively. The nested Wannier bands, $p^{\nu_{x}}_{y,n}(k_{x})$ ($n=1,2,3,4$), are calculated through the nested Wilson-loop approach (see Supplemental Material for details). The symmetry constraints on the Wannier band polarizations are (see Supplemental Material for proofs),

[TABLE]

The Wannier band polarizations is thus quantized as

[TABLE]

Similarly, the polarizations are quantized for the Wannier bands $\nu_{y}(k_{x})$, leading to $P^{\nu_{y},I}_{x}=P^{\nu_{y},II}_{x}=0,\frac{1}{2}\ {\rm mod}\ 1$. These constraints are confirmed by numerical calculations. In the $p4gm$ SC, $P^{\nu_{x},I}_{y}=P^{\nu_{x},II}_{y}=\frac{1}{2}\ {\rm mod}\ 1$ [see Fig. 1(d)], leading to nontrivial quadrupole topological index

[TABLE]

Symmetry considerations.—The above results, in particular, the quantization of the bulk quadrupole moment $q_{xy}$ is determined by the glide symmetries, $G_{x}$ and $G_{y}$. Therefore, these results are also applicable to $p2gg$ SCs ($p2gg$ is similar to $p4gm$ but without the $C_{4}$ rotation symmetry), as shown in next section. We now consider possible emergence of the AQTI phase in square/rectangular groups within the 17 wallpaper groups. Fig. 2(a) illustrates the structures of four types of SCs with $p4gm$, $p2gg$, $p2mg$ and $p4mm$ symmetries, separately. Analysis on other symmetry groups are presented in the Supplemental Material. We remark that although we often use square-lattices as examples, the analysis can be directly generalized to rectangular lattices. For the sake of continuity, the $p2gg$ SC is constructed from the $p4gm$ SC by truncating the gray regions as depicted in the figure. The $p2mg$ SC has a glide symmetry, $G_{y}$, and a mirror symmetry, $M_{x}:=(x,y)\to(-x,y)$. The $p4mm$ SC has two mirror symmetries, $M_{x}$ and $M_{y}:=(x,y)\to(x,-y)$. The unit-cells of the $p2mg$ and $p4mm$ SCs are doubled to enable comparison and analysis of the band structures and the Wannier bands for all SCs. The acoustic band structures for the $p2gg$ and $p4mm$ SCs are presented in Figs. 2(b) and 2(c), respectively. From the parity eigenvalues at the high-symmetry points of the Brillouin zone, we conclude that, for both SCs, the second band gap carries vanishing dipole polarization. However, the $p2gg$ SCs can have nontrivial quadrupole topology, whereas the $p4mm$ SCs cannot. The underlying reason is that the commutative mirror symmetries, $M_{x}$ and $M_{y}$, in the $p4mm$ SCs lead to degenerate Wannier bands and gapless composite Wannier bands. Similar results were found in the $p2mm$ and $c2mm$ wallpaper crystals. A special case is $p2mg$ crystals where the commutation between the mirror and glide symmetry at special high-symmetry points yields gapped Wannier bands for $\nu_{y}$, but gapless Wannier bands for $\nu_{x}$. These wallpaper crystals cannot support quadrupole topology. We find that to have nondegenerate, gapped Wannier bands, both glide symmetries $G_{x}$ and $G_{y}$ are needed, or if the system has neither glide nor mirror symmetry in the $x$ and/or $y$ direction. In the latter case (including $p2ll$, $p1ml$, $p1gl$, $c1ml$, $p4$ wallpaper crystals), however, the Wannier band polarizations are not quantized (at least for one of them, $\nu_{x}$ or $\nu_{y}$), because the quantization of the Wannier band polarizations are dictated by the glide or mirror symmetry. Therefore, only for the $p4gm$ and $p2gg$ SCs, the AQTI phase can emerge.

Edge and corner states.—The nontrivial quadrupole topology leads to gapped edge states and in-gap corner states. To demonstrate these topological boundary states, we study a finite SC with $8\times 8$ unit-cells cladded by hard-wall boundaries. A finite-width ($0.28a$) air channel is introduced between the hard-wall boundaries and the SC to tune the edge band gap to the middle of the bulk band gap. Such a narrow air channel cannot introduce waveguide modes in the bulk band gap, since the waveguide modes can appear only at much higher frequencies ($>30$ kHz). In accordance with our theory, the calculated spectrum [see Fig. 3(a)] shows the emergence of four corner states in the common spectral gap of the edge and bulk. The acoustic wavefunctions of the corner and edge states are shown in Figs. 3(b) and 3(c), respectively. The four degenerate corner states exhibit the quadrupole symmetry.

Geometry engineering in the $p2gg$ group.—Fig. 3(d) presents the evolution of the bulk, edge and corner states as a function of the geometry parameter $\delta$. For $\delta=0$, the SC has $C_{4}$ symmetry and belongs to the $p4gm$ group where all four corner states are degenerate. As $\delta$ increases, the degeneracy is lifted and split into two doublets: the left-upper and right-lower corner states form a doublet, while the right-upper and left-lower corner states form another doublet. At large $\delta$, the edge band gap is closed ($\delta=0.233a$), while the corner states merge into the edge, i.e., their wavefunctions gradually transform from fully localized to extensive along the edges. The closure of the edge band gap signals the transition into a quadrupole trivial phase. Further increase of $\delta$ leads to bulk band gap closing at $\delta=0.3a$ where the edge states merge into the bulk (see Supplemental Material for details).

Symmetry engineering.—Starting from the $p4gm$ SC, one can keep the $C_{4}$ symmetry and gradually transform into the $p4mm$ SC, by reducing the geometry parameter $h$. We study a continuous transformation process with first (i) $h$ going from $0.25a$ to 0, and then (ii) $l$ going from $0.5a$ to $0.35a$, and lastly (iii) $w$ going from $0.05a$ to $0.25a$. The phase diagram for the corner, edge and bulk spectra are shown in Fig. 4(a) which indicates a topological transition taking place exactly at the geometry transition point, i.e., $h=0$. While the SC is transformed from the $p4gm$ group to the $p4mm$ group, the bulk band gap closes and the corner states merge into the bulk. The edge states remain gapped and gradually merge into the bulk bands before the bulk band gap closing. The bulk band gap remains closed in stage (ii). When the band gap is reopened in stage (iii), the corner states disappear in the bulk band gap. The phase diagram in Fig. 4(a) thus manifests directly the bulk-corner correspondence due to the quadrupole topology.

Such a topological transition is also manifested in the Wannier bands. As shown in Fig. 4(b), the Wannier band gap for the “1+3” and “2+4” sectors is closed when the SC expriences the symmetry transition from the $p4gm$ group to the $p4mm$ group. We find that the $p4gm$ SCs have nontrivial quadrupole topology, i.e., $q_{xy}=\frac{1}{2}$ (see Fig. 4(c); This is confirmed in a recent experiment ourexp ). In contrast, when the band gap is reopened in the $p4mm$ SC, the Wannier bands are degenerate which cannot support the quadrupole topology.

Finally, we remark that the acoustic topological band gaps studied here do not have the chiral symmetry. Consequently, the corner states are not fixed at the mid-gap frequency. However, as shown in Ref. bic , this property can be used to move the corner states into the bulk continuum where the corner states still survive.

Note added: At the final stage of this work, we became aware of recent works on boundary obstructed topological phases obstructed (including some quadrupole topological insulators) and quadrupole topological insulators in magnetized systems mQTI1 ; mQTI2 .

Acknowledgments.— J.H.J, Z.K.L, H.X.W, and Z.X acknowledge supports from the Jiangsu specially-appointed professor funding and the National Natural Science Foundation of China (Grant No. 11675116). M.H.L thanks the support from the National Key R&D Program of China (2017YFA0303702 and 2018YFA0306200). J.H.J thanks Prof. Gil Young Cho for illuminating discussions.