Nonlocal topological insulators: Deterministic aperiodic arrays supporting localized topological states protected by nonlocal symmetries

Significance A concept of nonlocal topological phases of DAAs introduced here establishes a different direction in topological physics and offers approaches to emulate higher-dimensional topology in lower-dimensional systems. Our study also unveils opportunities to engineer topologically protected states in aperiodic systems and paves the path to application of resonances associated with such states, whose robustness is ensured by nonlocal symmetries of DAAs. In particular, the possibility to engineer multiple localized resonances via dimensional reduction and their unique features, such as precise spectral properties stemming from their topological nature, offers remarkable opportunities for practical applications, from robust resonators to sensors and aperiodic topological lasers.

= ( , , , ), , , , = 0, , … , ( − 1) , each reciprocal direction of first Brillouin zone is discretized by steps, and = 2 / . | , ⟩ is the Wannier state, and , is the Wannier center of the ℎ occupied band with the Bloch eigenstate | , 〉, which describes the average position of the particle from the center of the unit cell in the direction at momentum vector . The polarization at a particular hinge is procured by integrating Wannier center over first Brillouin zone and summation over occupied bands, the alternative form in numerical calculation is, which is quantized as either 0 or 1/2 because of reflection symmetries. Similarly, polarizations in other directions can be also abstained based on the same procedures. Note to have a nonzero higher order topology in the system defined in the following text, the total polarization of a finite bulk is zero.
In higher-order topological insulator, the bulk states might not directly possess the topological phase characterized by the 1 st order Wannier bands, instead they inherit the higher order topological quantity from that of lower dimensional states. To determine the topological phase of boundary state which is one dimensional lower than its host bulk, 2 nd order Wilson loop (nested Wilson loop) has been invented recently (1,2) which is performed over the subspace of Wanniersector, for example, on Wannier-sector ± with 1 1 st order Wannier bands considered, and along in the Brillouin zone, , ± = ± (2 + , 2 + − ) … ± ( + 2δ , in which[ ± ( + δ , )] , ′ = 〈 ± , +δ | ± , ′ 〉, , ′ = 1,2, … , 1 , where the 1 st order Wannier state over Wannier-sector ± is defined as in which , ± , is the 2 nd order Wannier center of th (1 st order) Wannier band in the Wannier sector ± . The polarization over the Wannier sector ± is given by the equation, Note that index in Eq. (11) is redundant since , ± , is independent of it, however, we keep this index to have a traceable and compacted form of polarization which can be easily extended to the case in higher dimension, as we show in the following text. The quadrupole moment of a specific surface , is defined as = 2 ± ± , , = 1,2,3,4, ≠ , and it is quantized as either 1/2 or 0 because of the constraint by reflection symmetries. Continuing the above process, to get the topological invariant for the boundary states two dimension lower compared to the bulk, 3 rd order Wilson loop is constructed over the subspace of Wannier-sector ± ± with 2 2 nd order Wannier bands considered, which is defined along in the Brillouin zone, , ± ,± = ± ,± (2 + , 2 + − ) … ± ,± ( + 2 , + ) ± ,± ( + , ), (13) in which [ ± ,± ( + , )] , ′ = 〈 ± ,± , + | ± ,± , ′ 〉, , ′ = 1,2, … , 2 , where the 2 nd order Wannier state over Wannier-sector ± ± is defined as in which , ± ,± , is the 3 rd order Wannier center of th (2 nd order) Wannier band in the category of Wannier-sector ± ± . The polarization over the Wannier-sector ± ± is given by the equation, The octupole moment on an arbitrary surface can be obtained by which is quantized either to be 1/2 or 0 because of the conservation of reflection symmetries.

S3: Bulk localization of corner modes
The degree of localization can be quantified by defining the inverse participation ratio (IPR) of eigenstates (3) Since we introduced a tiny perturbation = 0.2 1 in (1) to shift one corner mode (| 1 >) from zero energy by in the 4d lattice model to guarantee this corner mode is orthogonal with other corner modes, in such way the corresponding projected state, according to the proof above, will be the edge mode (|̃1 >) of 1d system. ⟨ 1 | ⟩ = 0 for i = 2, 3 …16, after the Lanczos Transformation we have ⟨̃1|̃⟩ = 0 for i = 2, 3 …16, which means other zero energy modes will either localize at bulk of 1d system far away from the sites where |̃1 > is dominant, or at least orthogonal with |̃1 >.

S4: Amplitude Distribution
The Lanczos transformation maps four-dimensional (4d) localized modes to localized modes of the one-dimensional (1d) system and 4d delocalized modes to 1d delocalized modes. We plot 16 localized eigenstates of the 1d system in Fig.S4, these states are mapped from the 4d corner modes. There is one eigenstate localized at the edge while the 15 other eigenstates are localized in the bulk.

S5: Lanczos Tridiagonalization of Chiral Symmetry and Reflection Symmetries
The emergence of a quantized multipole moment and corner states is deeply related to the symmetries of the system, i.e., the presence of anti-commuting reflection symmetries in the 4D h-HOTI. Similarly, the "zero-energy" of the states is ensured by chiral symmetry, which stems from the fact that in the 4D h-HOTI the sites that belong to the same sublattice do not couple with each other. In addition, the Hamiltonian (1) possesses reflection symmetries that anti-commutate with each other because of the flux of in each plaquette of the hypercubic lattice. The chiral symmetry of the system, on the other hand, with matrix representation Γ = 3 ⊗ 3 ⊗ 3 ⊗ 0 , is expressed as ΓĤ 4D Γ −1 = −Ĥ 4 , which ensures the overall symmetry of the spectrum and "zero-energy" of the corner states. The fact that this property is retained during the dimensional reduction implies that the chiral symmetries is preserved in new form and can be expressed as ̂Γ̂− 1 . Indeed, due to its local character, the chiral symmetry can be written as a symmetry operator for the finite h-HOTI as Γ = × ⊗ Γ, where × is the by identity matrix and is the number of degrees of freedom (unit cells), and thus is (block)-diagonal. The Lanczos transformation changes the local character of the chiral symmetry by mixing different sites (except the anchor site) Γ 1 = Γ̂− 1 , thus inducing the new non-local form of chiral symmetry Γ 1 Ĥ 1 Γ 1 −1 = −Ĥ 1 of the effective 1D Hamiltonian. This effective chiral symmetry of the 1D Hamiltonian plays the same role as the original chiral symmetry for 4D h-HOTI and it ensures spectral stability of the modes of aperiodic 1D array. However, the non-local character of the symmetry operator Γ 1 in 1D is reflected in the presence of correlations of parameters in different part of array, hopping amplitudes and on-site energies, within the 1D system, responsible for the "zero-energy" of the localized states. The reflection symmetries and the resultant quantized multipole moment can be similarly analyzed in the dimensionally reduced system, and the quantized multipole moment of the bulk bands in the finite array can be extracted from the respective wave functions. Thus, despite its low-dimensional character, the effective 1D system inherits the properties of higher-dimensional h-HOTI. Therefore, the projected corner states, localized either in the bulk or on the edge of the 1D array, are induced and protected by the symmetries of the original 4D system, thus ensuring their very existence and stability specific to topological systems.
The finite 4D lattice model satisfy chiral symmetry Γ , The chiral symmetry in the finite 4D lattice model is local operator. However, The LTD transformation maps the local operator to non-local operator (as shown in Fig.S4) and the effective 1D lattice Hamiltonian satisfies the same anti-commutation relation with the non-local operator, For the 4D lattice with 3 4 sites, the reflection symmetry ̂, ] and the resultant quantized multipole moment, can be similarly analyzed in the dimensionally reduces system, and the quantized multipole moment of the bulk bands in the finite array can be extracted from the respective wave functions. Thus, despite its low-dimensional character, the effective 1D system indeed inherits all the characteristics of higher-dimensional h-HOTI. Therefore, the projected corner states, localized either in the bulk or on the edge of the 1D array, are induced and protected by the symmetries of the original 4D system, thus ensuring their very existence and stability that are specific to topological systems.
LTD map reflection symmetry ̂ to ̂̂̂− 1 = ̂1 , the reflection symmetry loss its locality to be a global operator as shown in Fig.S5.

S6: Topological invariant (ℎ ) of deterministic aperiodic 1D array.
We use the real space formula of hexadecapole moment(4) wherẽ= { | 1 >, | 2 >, … |̃> } and ̃= ̂̂̂+ . We argue the ℎ is a reasonable quantity to capture the topological transition of the 1D array since it depends only on the occupied eigenstates (|̃>) of the effective 1D array. We show ℎ and ℎ in Fig. S6, two quantities have the same topological transition region, which means the topological signatures of the 4D h-HOTI are captured by the corresponding 1D array. As shown in Fig.S5, we calculated the band spectra and intensity distribution of the effective 1D model truncated after the 30 th site, finding that the zero energy and the corresponding intensity distributions of localized topological states are not affected, which confirm the feasibility of the dimensional reduction.
By using the above equation, we can map the coupling amplitude of our effective 1D chain to the distance of our experiment. The experiment data is shown in Table.1. In this paper, we focus on the first 20 sites of 1D chain, and each resonator has the same onsite energy. Therefore, these resonators can be set at the same height 0 .

S9: Quality factor
Two major loss mechanisms in our experiment are the absorption in the resin used in 3D printing and the leakage through the probe holes, which are deliberately introduced in the design of the individual resonators to allow excitation and probing of the acoustic field. Where Ω is the resonance frequency and Γ is the linewidth of the resonance. We measure the frequency distribution with different heights of the resonator (See Fig. S9). By using equation (27) and quality factor = Ω Γ , we get the quality factor within the range of 50 to 60, and therefore, the corresponding finite lifetimes of the modes of the lattice are long enough not to alter their topological nature, making them clearly observable in our 1D array.