Linyun Yang

Ying Wu

Kaiping Yu

Rui Zhao

Wei Wang

Bernard Bonello

Bahram Djafari-Rouhani

Robust Fano resonance between mechanical first-and second-order topological states

Keywords: Topological edge states, second order corner modes, Fano resonance, ultrahigh Q factor

The Fano resonance, as a phenomenon of wave scattering, is based on the interaction between a "bright mode" and a "dark mode" giving rise to an asymmetric and ultrasharp spectral peak.

However, the Fano resonant frequency is sensitive to structural imperfections such as defects or disorders, which will shift the resonant peak, or even damage the Fano line shape. Here, we harness the coupling between the first-order and the higher-order topological insulators to overcome this shortcoming. We construct a first-order topological edge state to serve as a bright mode, and a second-order topological corner state to be the dark mode using the same base configuration. As a result, a topological Fano resonance is expected to occur around the resonant frequency of the dark mode. The robustness of the Fano resonance is verified by deliberately introducing various types of imperfections into the system. Our findings may further enhance confidence in using the resonance such as acoustic switching, refractive index sensing, high quality factor filters, and accurate interferometers.

Introduction

Fano resonance, originally discovered in quantum systems, is a wave scattering phenomenon featuring asymmetric and ultra-sharp spectral peaks resulting from the interference between discrete and continuum scattering channels. It is named after Ugo Fano, for his contribution to the theoretical explanation for the scattering line shape of inelastic scattering between electrons and helium atoms [START_REF] Fano | Effects of configuration interaction on intensities and phase shifts[END_REF]. The wave interference is not exclusive of quantum mechanics, while it is a general wave phenomenon and the occurrence of Fano resonance has been evidenced in other fields such as photonic crystals [START_REF] Fan | Analysis of guided resonances in photonic crystal slabs[END_REF][START_REF] Rybin | Fano resonance between Mie and Bragg scattering in photonic crystals[END_REF], plasmonic nanostructures [START_REF] Luk'yanchuk | The Fano resonance in plasmonic nanostructures and metamaterials[END_REF][START_REF] Khanikaev | Fano-resonant metamaterials and their applications[END_REF] and phononic crystals [START_REF] Goffaux | Evidence of Fano-like interference phenomena in locally resonant materials[END_REF]. Two ingredients, including a broadband "bright" mode serving as a background state and a narrowband "dark" mode as an isolated state, are essential for the generation of Fano resonance in classical systems [START_REF] Joe | Classical analogy of Fano resonances[END_REF]. Around the dark mode resonant frequency, the bright mode magnitude varies slowly with frequency, while the dark mode changes abruptly both in magnitude and phase. When the driven force passes through the resonant frequency, both the in-phase and out-of-phase interactions between the two modes will take place within a very narrowband range. As a result, the constructive and destructive interferences enable the transmittance to reach its maximum from the minimum very quickly, which creates the asymmetric Fano-like profile.

Based on such a straightforward formulation, advances in theoretical and experimental demonstrations of Fano resonance have been achieved in mechanical and acoustic systems [START_REF] Goffaux | Evidence of Fano-like interference phenomena in locally resonant materials[END_REF][START_REF] Fellay | Scattering of vibrational waves in perturbed quasi-one-dimensional multichannel waveguides[END_REF][START_REF] Goffaux | Comparison of the sound attenuation efficiency of locally resonant materials and elastic band-gap structures[END_REF][START_REF] Kosevich | Fano-like resonance phenomena by flexural shell modes in sound transmission through two-dimensional periodic arrays of thin-walled hollow cylinders[END_REF][START_REF] Liu | Tunable transmission spectra of acoustic waves through double phononic crystal slabs[END_REF][START_REF] Pennec | Perpendicular transmission of acoustic waves between two substrates connected by sub-wavelength pillars[END_REF][START_REF] Amoudache | Optical and acoustic sensing using Fano-like resonances in dual phononic and photonic crystal plate[END_REF][START_REF] Jin | Tunable Fano resonances of Lamb modes in a pillared metasurface[END_REF]. Because of the asymmetric and ultra-sharp properties, Fano resonances in phononic crystals possess a variety of potential applications such as acoustic switching, refractive index sensing, high-quality factor filters, and highly accurate interferometers. While the high sensitivity of the Fano resonance requires precise control and accurate fabrication [START_REF] Zangeneh-Nejad | Topological Fano Resonances[END_REF]. Additionally, structural imperfections such as defects or disorder may shift the Fano resonant frequency, or even completely damage the Fano line shape. The developments of topological insulators (TIs) [START_REF] Hasan | Colloquium : Topological insulators[END_REF][START_REF] Moore | The birth of topological insulators[END_REF] and their classical analog in optical [START_REF] Haldane | Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry[END_REF][START_REF] Wang | Observation of unidirectional backscattering-immune topological electromagnetic states[END_REF][START_REF] Wu | Scheme for achieving a topological photonic crystal by using dielectric material[END_REF][START_REF] Gao | Topologically protected refraction of robust kink states in valley photonic crystals[END_REF], acoustical [START_REF] Xiao | Geometric phase and band inversion in periodic acoustic systems[END_REF][START_REF] Yang | Topological Acoustics[END_REF][START_REF] Khanikaev | Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice[END_REF][START_REF] He | Acoustic topological insulator and robust one-way sound transport[END_REF][START_REF] Lu | Observation of topological valley transport of sound in sonic crystals[END_REF][START_REF] Ding | Experimental Demonstration of Acoustic Chern Insulators[END_REF][START_REF] Gao | Sonic valley-Chern insulators[END_REF] and mechanical [START_REF] Torrent | Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates[END_REF][START_REF] Mousavi | Topologically protected elastic waves in phononic metamaterials[END_REF][START_REF] Chaunsali | Subwavelength and directional control of flexural waves in zone-folding induced topological plates[END_REF][START_REF] Miniaci | Experimental Observation of Topologically Protected Helical Edge Modes in Patterned Elastic Plates[END_REF][START_REF] Yan | On-chip valley topological materials for elastic wave manipulation[END_REF][START_REF] Chen | A study of topological effects in 1D and 2D mechanical lattices[END_REF][START_REF] Chen | Topological phase transition in mechanical honeycomb lattice[END_REF][START_REF] Zhou | Actively controllable topological phase transition in homogeneous piezoelectric rod system[END_REF][START_REF] Zhao | Non-reciprocal Rayleigh waves in elastic gyroscopic medium[END_REF][START_REF] Zheng | Tilted double Dirac cone and anisotropic quantum-spin-Hall topological insulator in mechanical granular graphene[END_REF][START_REF] Yang | Abnormal topological refraction into free medium at subwavelength scale in valley phononic crystal plates[END_REF][START_REF] Chen | Topology optimization of quantum spin Hall effect-based secondorder phononic topological insulator[END_REF] systems may provide an elegant solution to this problem. TIs support unique edge states protected by the intrinsic bulk bands topology, regardless of the local perturbations such as defects or disorders. Inspired by this preeminent feature, the ultra-sharp spectrum of Fano resonances can be guaranteed by topological protection in 1D acoustic/mechanical systems [START_REF] Zangeneh-Nejad | Topological Fano Resonances[END_REF][START_REF] Wang | Robust Fano resonance in a topological mechanical beam[END_REF]. In 2D systems, a topological waveguide and a trivial cavity are proposed to serve as the bright and dark modes, respectively [START_REF] Ji | Transport tuning of photonic topological edge states by optical cavities[END_REF][START_REF] Ji | Fragile topologically protected perfect reflection for acoustic waves[END_REF][START_REF] Ji | Robust Fano resonance in the photonic valley Hall states[END_REF]. By placing a care-

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Journal Pre-proof fully designed cavity besides the topological waveguide, a coupler can be constructed to support both constructive and destructive interferences between the two modes, which creates a topological Fano resonance. However, only the robustness of the bright mode is guaranteed by the topological protection, while the dark mode, i.e., the cavity mode, still remains trivial and less robust.

Here, we harness the very recently discovered higher-order topological insulators (HOTIs) [START_REF] Benalcazar Wladimir | Quantized electric multipole insulators[END_REF][START_REF] Benalcazar | Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators[END_REF][START_REF] Serra-Garcia | Observation of a phononic quadrupole topological insulator[END_REF][START_REF] Peterson | A quantized microwave quadrupole insulator with topologically protected corner states[END_REF][START_REF] Mittal | Photonic quadrupole topological phases[END_REF][START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF][START_REF] Benalcazar | Quantization of fractional corner charge in $C n$symmetric higher-order topological crystalline insulators[END_REF][START_REF] Fan | Elastic Higher-Order Topological Insulator with Topologically Protected Corner States[END_REF][START_REF] Xue | Acoustic higher-order topological insulator on a kagome lattice[END_REF][START_REF] Yang | Helical Higher-Order Topological States in an Acoustic Crystalline Insulator[END_REF][START_REF] Xie | Higher-order quantum spin Hall effect in a photonic crystal[END_REF][START_REF] Wu | On-chip higher-order topological micromechanical metamaterials[END_REF][START_REF] Chen | Corner states in a secondorder mechanical topological insulator[END_REF][START_REF] An | Second-order elastic topological insulator with valley-selective corner states[END_REF][START_REF] Yang | Observation of Dirac Hierarchy in Three-Dimensional Acoustic Topological Insulators[END_REF] to achieve a robust dark mode. Generally speaking, an n-order topological insulator in a ddimensional system can support (dn) dimensional topological states, where the order n ≥ 2 indicates a higher-order topology, while n = 1 refers to the conventional TIs [START_REF] Xie | Higher-order band topology[END_REF]. Unlike the conventional TIs with edge states, which has one lower dimension than that of the bulk, the HOTIs host topological corner states whose dimension is at least two less than the bulk. Hence, we replace the trivial cavity by the second-order topological insulators (SOTIs), which supports topologically protected 0D corner states. By integrating the 0D topological corner state and the 1D edge state in a properly designed 2D phononic crystal structure, we realize a topological Fano resonance with both topologically protected bright and dark modes. Specifically, as conceptually depicted in Fig. 1, we construct a first-order topological edge state to serve as a bright mode [Fig. 1(a)], and a second-order topological corner state to be the dark mode [Fig. 1(b)]. The coupling of these two modes can be achieved by a proper arrangement of regions occupied by the topological insulator (TI) and the ordinary insulator (OI). As a result, a topological Fano resonance can be expected to occur around the resonant frequency of the dark mode, as shown in Fig. 1(c).

The arrangement of the rest of this paper is outlined as follows: In Sec. 2, we introduce a phononic crystal plate with breathing honeycomb unit cells, and show that the shrunken (type A) and expanded (type B) lattices belong to distinct topological phases. In Sec. 3, we demonstrate the existence of the first-order topological edge states confined at the domain wall formed by different topological structures. We then show that these two edge states are in fact not gapless, furthermore, the edge state gap can be tuned by alternating the geometrical parameters. This is followed by Sec. 4 in which we illustrate the second-order topological corner states that reside within the edge state gap. In Sec. 5, we propose a compound lattice structure with two types of domain walls that respectively supports topological edge states as well as the in-gap topological corner states at a common frequency. We show that within an extremely narrow range around this specific frequency, the topological edge and corner modes will undergo both constructive and destructive interferences, resulting to an asymmetric and ultra-sharp transmission line, i.e., a Fano-like resonance. We further illustrate in Sec. 6 that the topological Fano shape response can be preserved and the resonance frequency shift is very small at the presence of various geometrical imperfections, including random rotations or deformations. Finally, we summarize the main results of this paper in Sec. 7.

The bulk band structure and topology of the phononic plate

We consider the flexural wave propagation problem in a thin elastic plate attached by massspring oscillators with a honeycomb arrangement as depicted in Fig. 2 For a specific geometrical configuration that R = a/3 (denoted by R 0 ), the lattice sustains a fourfold degeneracy in the dispersion curves and a double Dirac cone occurs at the Brillouin zone center due to the zone-folding mechanism [START_REF] Wu | Scheme for achieving a topological photonic crystal by using dielectric material[END_REF]. By varying the ratio between R and a, the switching between distinct topological phases can be achieved. Briefly speaking, for a shrunken cell, i.e., R/a < 1/3, the corresponding band gap features a topologically trivial phase, while for an expanded cell that R/a > 1/3, it is topologically nontrivial [START_REF] Wu | Scheme for achieving a topological photonic crystal by using dielectric material[END_REF][START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF], as we will describe below later. It is worth noting that the above scheme to create topologically distinct phases was widely used to achieve TIs by mimicking the quantum spin Hall effect [START_REF] Wu | Scheme for achieving a topological photonic crystal by using dielectric material[END_REF][START_REF] He | Acoustic topological insulator and robust one-way sound transport[END_REF][START_REF] Chaunsali | Subwavelength and directional control of flexural waves in zone-folding induced topological plates[END_REF][START_REF] Yang | Topological spin-Hall edge states of flexural wave in perforated metamaterial plates[END_REF][START_REF] Yu | Elastic pseudospin transport for integratable topological phononic circuits[END_REF] or to realize recently proposed SOTIs [START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF][START_REF] Fan | Elastic Higher-Order Topological Insulator with Topologically Protected Corner States[END_REF][START_REF] Chen | Corner states in a secondorder mechanical topological insulator[END_REF][START_REF] Xie | Higher-order quantum spin Hall effect in a photonic crystal[END_REF] in classical wave systems. Here we adopt this mechanism to realize both the topological edge and corner states simultaneously at the same frequency, and investigate their coupling effects and the resultant topologically protected robust Fano resonance.

Weak form for elastic wave equations in periodic systems

Weak forms are powerful mathematical tools to solve partial differential equations (PDEs), especially in finite element formulations. To obtain the band structure and to discuss its bulk topology of the proposed phononic crystal plate, we first deduce the weak form formulation for three-dimensional elastodynamic equations in periodic systems. Its implementations to the flexural wave problems in phononic plate structures will be discussed in the next section.

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For elastic waves in homogeneous and linear elastic materials, the equations of motion without body forces expressed in terms of components with respect to a Cartesian coordinate system (x, y, z)

are σ i j, j -ρü i = 0 ( 1
)
where ρ is the mass density, σ i j and u i (i, j = x, y, z) denote respectively the Cauchy stress tensor and the displacement vector components, and üi represents the second derivative of u i with respect to time. Note that the comma in the subscript denotes partial differentiation, and that the summation convention over a repeated subscript is adopted.

The Cauchy stress tensor σ and the infinitesimal strain tensor ε are related by the constitutive equations, or known as Hooke's law,

σ i j = C i jkl ε kl (2)
where C i jkl is the four-order stiffness tensor, and the strain-displacement equations are

ε i j = 1 2 (u i, j + u j,i ) (3)
The weak form for Eq. ( 1) can be derived by multiplying itself by a test function using variation notation δu i , and then integrating the result over the solution domain Ω [START_REF] Zienkiewicz | The finite element method: its basis and fundamentals[END_REF]. The terms with derivatives of stresses are treated by integrating by parts using Green's theorem, which yields

Ω [ρδu i üi + δu i, j σ i j ]dV - ∂Ω δu i • σ i j n j dS = 0 (4)
where ∂Ω = ∂Ω u ∪ ∂Ω t denotes the boundary surface with the subscripts u and t refer to the boundaries with prescribed displacements and tractions respectively, n j is the j-th component of the normal vector. The boundary integration term can be simplified by removing the essential boundary condition part since δu i = 0 is valid on ∂Ω u , and preserving only the natural boundary condition part which is σ i j n j = f i with f i being the known traction on ∂Ω t .

According to the Bloch theorem, the displacement field u i = u i (x, y, z) in periodic structures with lattice vectors a i has the following form

u i = ûi exp(ik l x l ) (5)
where k i is the wave vector component, ûi is the cell-periodic counterpart of the Bloch wave u i , which has the same periodicity with the unit cell. We stress that the repeated subscript implies the J o u r n a l P r e -p r o o f

Journal Pre-proof summation convention. From Eq. ( 5), the derivatives of the displacement field can be expressed as

u i, j = (û i, j + ik j ûi ) exp(ik l x l ) (6)
Consequently, the variation term δu i in Eq. ( 4) should be replaced by its complex conjugation δu * i , since the integration represents an inner product operation. Inserting Eq. ( 6) back into Eqs. ( 2) -

(3), it can be seen that the stress and strain tensors are also Bloch functions, i.e., they have similar expressions to Eq. ( 5),

σ i j = σij exp(ik l x l ) ( 7a
)
ε i j = εij exp(ik l x l ) (7b)
where εij = 1 2 [(û i, j + ik j ûi ) + (û j,i + ik i ûj )], and σij = C i jkl εkl . Now inserting Eqs. ( 5) -( 7) back into Eq. ( 4) yields the weak form for wave equations in periodic structures as

Ω [ρδû * i üi + δ( ε * i j ) σij ]dV - ∂Ω t δû * i • fi dS = 0 (8)
in which the identity δu i, j σ i j = δ(ε i j )σ i j is used with the symmetry nature of the stress tensor being taken into consideration. We stress again that δu i and δu i, j have been replaced by their complex conjugations as required by the complex-valued inner product, and that an auxiliary function fi = f i exp(-ik l x l ) where f i = σ i j n j is introduced to make the expression of Eq. ( 8) remain compact.

The weak form Eq. ( 8) is applicable for Bloch modes in general 3D periodic solid materials, which can be solved efficiently by using the weak form PDE module of COMSOL Multiphysics, combined with periodic boundary conditions imposed on ûi . Moreover, when considering the flexural waves in elastic plates whose thickness h is very small compared with the other two dimensions, additional assumptions regarding the behavior of such structures such as the Kirchhoff theory for thin plates or the Reissner-Mindlin theory for thick plates, can further reduce the formulation from complete three-to two-dimensional equations.

Weak form for flexural wave in phononic plates

In thin plate theory, the deformation is represented by a single variable w, the out-of-plane displacement of the mid-plane of the plate. This displacement is governed by a four-order PDE according to Kirchhoff's assumption, which introduces second derivatives of w in the expressions

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Journal Pre-proof of strains. However, the requirement that both w and its derivatives need to be continuous brings more difficulties in the finite element formulations. In contrast, it turns out that the thick plate theory is simpler to implement in the finite element analysis, since only up to the first derivatives of the dependent variables appear in the strain terms. According to the Reissner-Mindlin plate theory, in which the transverse shear and rotary inertia are taken into consideration, the displacement components for flexural motions are expressed in the forms [START_REF] Achenbach | Wave propagation in elastic solids[END_REF] u(x, y, z)

= -zθ x (x, y) (9a) v(x, y, z) = -zθ y (x, y) (9b) w(x, y, z) = w(x, y) (9c)
where θ x and θ y are the local rotations of the normal to the mid-plane of the plate, with respect to the x and y axis, respectively. The strains now can be separated into bending and transverse shear parts as follows

ε xx = -zθ x,x , ε yy = -zθ y,y , ε xy = - z 2 (θ x,y + θ y,x ) (10a)
ε xz = 1 2 (-θ x + w ,x ), ε yz = 1 2 (-θ y + w ,y ) (10b)
Note that the comma in the subscript implies partial differentiation. The flexural motions of the plate can be approximately regarded as a plane stress condition, thus the stresses corresponding to the strains above are

σ xx = E 1 -ν 2 (ε xx + νε yy ) (11a) σ yy = E 1 -ν 2 (ε yy + νε xx ) (11b) σ i j = µε i j (i j and i, j = x, y, z) (11c)
where E, µ and ν are the Young's modulus, shear modulus and Poisson's ratio of the isotropic plate material. Considering that the flexural wave propagates in the xy plane, the out-of-plane component of the wave vector should be zero, i.e., k z = 0, while the other two are retained. Therefore, according to Eq. ( 5) for the Bloch waves in three dimensions, the displacements in Eq. ( 9) can also be expressed in a similar manner, for example, u = -z θx exp ik x x + ik y y with θx (x, y) a periodic function. Finally, substituting Eqs. ( 9) -( 11) into Eq. ( 8), and analytically performing the

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Journal Pre-proof through-thickness integration yields

A 1 12 ρh 3 (δ θ * x θx + δ θ * y θy ) + ρhδ ŵ * ẅ + 1 12 h 3 δ( ε(0) * xx ) σ(0) xx + δ( ε(0) * yy ) σ(0) yy + 2δ( ε(0) * xy ) σ(0) xy + 2γh δ( ε(0) * xz ) σ(0) xz + δ( ε(0) * yz ) σ(0) yz -δ ŵ * fz dA = 0 ( 12
)
where A denotes the two-dimensional region of one unit cell, γ = π 2 /12 is the correction factor introduced to account for the fact that the shear stresses are not constant over the thickness [START_REF] Achenbach | Wave propagation in elastic solids[END_REF], h is the plate thickness, and the strain and stress terms with superscript • (0) represent their z-independent parts. The strain terms are given by ε(0

) xx = θx,x + ik x θx , ε(0) yy = θy,y + ik y θy , ε(0) xy = 1 2 ( θx,y + ik y θx + θy,x + ik x θy ) (13a) ε(0) xz = 1 2 (-θx + ŵ,x + ik x ŵ), ε(0) yz = 1 2 (-θy + ŵ,y + ik y ŵ) (13b)
and the stress terms can be subsequently obtained from Eq. ( 11).

The weak form formulation Eq. ( 12) is deduced from the weak form of 3D elastodynamic wave equations, with specific assumptions made on the displacement fields to describe the flexural motions of thick plates. Considering the periodicity of phononic crystal plates, the displacements are further regarded as Bloch waves, resulting in the emergence of their cell-periodic counterparts and the wave vector (k x , k y ) in the weak form expression. In this paper, Eq. ( 12) is solved for various wave vectors by using the weak form PDE module of the finite element software COMSOL Multiphysics with the eigenvalue solver in two dimensions. Then the dispersion curves can be obtained from the eigenfrequencies solutions. It is worth noting that the surface traction in the last term can be determined by the reaction forces from the oscillators as will be detailed below.

The band structure and the topological phase transition

Consider the phononic crystal plate shown in Fig. 2, with lattice constant a = 50 mm, thickness h = 0.5 mm and the material constants Young's modulus E = 70 GPa, Poisson's ration ν = 0.3 and mass density ρ = 2700 kg/m 3 . We assume that each oscillator has only one degree of freedom, the vertical displacement parallel to the z axis. The time-harmonic vibration of the resonators interacts with the elastic plate through the connecting springs, which gives the reaction force as [START_REF] Yang | Abnormal topological refraction into free medium at subwavelength scale in valley phononic crystal plates[END_REF] J o u r n a l P r e -p r o o f

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f z (r) = 6 α=1 m α ω 2 α ω 2 ω 2 α -ω 2 w(R α )δ(r -R α ) (14)
where m α , κ α and ω α = √ κ α /m α are respectively the mass, spring constant and the resonant frequency of oscillator α, which locates at R α . δ(r) denotes the Dirac delta function.

For simplification we also assume that the resonant frequencies of the oscillators are sufficiently higher than the working frequency we are interested in, with the limiting case being that the masses are rigidly connected to the plate surface. In other words, we let the spring constant κ α → +∞ and set the mass to be a finite value, for example m α = 5ρa 2 h. Consequently, the force term Eq. ( 14) can be further simplified as

f z (r) = 6 α m α ω 2 w(r α )δ(r -R α ) (15)
This treatment can keep our physical model rather simple without introducing additional resonant features, and in the meantime, it does not affect the demonstration of the topological Fano resonance phenomenon or the physical mechanism behind it.

We first consider the geometrical configuration for R/R 0 = 1 (notice that R 0 = a/3), which corresponds to a perfect honeycomb lattice, or a phononic analog of graphene. In this scenario, the primitive unit cell (i.e., the smallest one) consists of only two oscillators, while the unit cell shown in Fig. 2 to reconstruct the mode symmetries. We have also presented the phase map of these eigenmodes in For values of R R 0 , the translational symmetry of the above two-resonator "primitive cell" is broken, and thus the band folding mechanism is no longer applicable, which breaks the four-fold degeneracy and opens a complete band gap between the two-fold p modes and d modes. when reaching at the critical point R/R 0 = 1. If R/R 0 is further increased, the band gap will reopen again but with the eigenmodes switched their orders, i.e., the p modes correspond to the two bands above the gap while the d modes correspond to the two below. Such a band inversion is often accompanied by a topological transition from a trivial phase to a nontrivial one. In the case for time reversal symmetry preserved C 6v breathing lattices, for values of R/R 0 < 1 the band gap is topologically trivial and for those R/R 0 > 1 it is nontrivial [START_REF] Wu | Scheme for achieving a topological photonic crystal by using dielectric material[END_REF][START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF]. This can be explained by checking the (pseudo) spin Chern numbers based on an effective Hamiltonian obtained by k • p method (see Appendix B). To have a deeper understanding of these two topologically distinct phases, we plot the band structures of two example cases for R/R 0 = 0.9 (lattice A ) and R/R 0 = 1.1 (lattice B ) in Fig. 3(c) and(d). Good agreements between the k• p predictions and the first principle FEM results can be found around Γ point for bands 2 -5, indicating that we can use an effective Hamiltonian to describe the band structure and its topological properties. Besides, there are 3 bands below the complete band gap, and their corresponding eigenmodes at Γ and M points are displayed in the right panels of each figure. The sign + (or -) marked in the band diagrams indicates that the corresponding eigenmode has an even (or odd) C 2 rotation symmetry, and #Γ (2) (#M) is the number of eigenmodes that have C 2 rotation eigenvalue +1 at Γ (M) points for all the bands below the gap.

Based on these two integer numbers, the C 2 topological invariant is defined as [START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF][START_REF] Benalcazar | Quantization of fractional corner charge in $C n$symmetric higher-order topological crystalline insulators[END_REF] [M] = #M -#Γ (2) (16

)
which is a measure of the difference between the subspaces spanned by the bands below the gap at M and Γ points in the C 2 representation. Besides, a C 3 topological invariant can also be defined in a similar manner as [K] = #K -#Γ (3) , where #K and Γ (3) are the number of eigenmodes below the gap that have C 3 rotation eigenvalue +1 for K and Γ points, respectively. Therefore, the bulk topology of the band gap can be characterized by [START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF][START_REF] Benalcazar | Quantization of fractional corner charge in $C n$symmetric higher-order topological crystalline insulators[END_REF]

χ = ([M], [K]) (17)
For example, a zero-valued χ implies that the k-dependent Hamiltonian can be continuously deformed along the path that joins the high symmetric points M (or K) and Γ without closing the band gap or breaking the symmetry, rendering the bulk topology trivial [START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF]. In contrast, χ (0, 0) reveals a nontrivial topology because the low-frequency subspaces below the gap at these points are R > R 0 . As for the C 3 invariant, [K] = 0 is valid for all ratios of R/R 0 in our system. This indicates that the configuration that R > R 0 is in a nontrivial phase with χ = (-2, 0) (0, 0) and consequently one can expect the observation of corner modes localized at the intersections of topological domain walls [START_REF] Noh | Topological protection of photonic mid-gap defect modes[END_REF].

The first-order topological edge states

According to the bulk-edge correspondence, it is known that there exist a pair of topological edge states confined at the domain wall formed by the two topologically distinct lattices. As shown in Fig. 4(a), we calculate the band structure of a supercell consisting of 16 layers of lattice B sandwiched in between 8 layers of lattice A (or A ) on each side, by applying periodic boundary conditions on both the left-right and top-bottom edge pairs. Note that the top and bottom termination can be viewed as they were sutured together due to the periodic boundary condition, which eliminates the boundary modes and allows us to focus on the edge states confined at the domain walls.

Let us first consider the edge states confined at A /B interface, as shown in Fig. 4(b). Clearly,

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Journal Pre-proof two branches of bands (red and blue solid lines) emerge in the common frequency range of the bulk band gaps of lattices A and B. Moreover, the corresponding eigenmodes of these two bands feature very localized amplitude-distributions near the topological domain walls (not shown). However, we would like to mention that these two edge states are not gapless, and that there exists a tiny gap between them. The reason is that the C 6 symmetry of the entire structure is not strictly preserved due to the deviations between lattices at different sides of the domain wall [START_REF] Chaunsali | Subwavelength and directional control of flexural waves in zone-folding induced topological plates[END_REF]. It can be expected that this tiny gap could even be extended to be sufficiently large if the configuration difference between the two lattices is further enlarged. This is indeed the case for the result shown in Fig. 4(c), where the lattice A , with geometrical configuration R/R 0 = 0.9, has been replaced by lattice A which has a smaller value of R/R 0 (= 0.7). Compared with Fig. 4(b), it is much more obvious that there exists a band gap in between the blue and red branches of edge states, making the topological edge states gapped themselves. One may notice that there is another band gap between the upper branch of topological edge state and the bulk states, but the second order topological corner modes are expected to reside inside the band gap of the first order topological edge states. We will show later that the existence of in-gap topological second-order corner states is guaranteed by the topological origin of this edge band gap.

The second-order topological corner states

It has been shown that inside the topologically nontrivial bulk band gaps there exist topological edge states along the domain walls. Similarly, if the edge band gap mentioned above also possesses nontrivial topological classification, a new type of topological states will emerge at the intersection of two domain walls, namely the second-order topological corner states. The corner modes are spatially very confined near the corners, and they are well isolated from the bulk and edge states in the frequency spectrum.

To illustrate this point, we consider a hexagon-shaped crystal stacked by type A unit cells (colored in blue) coated by a type B cells (red) which features a global C 6v symmetry that is compatible with the symmetry of the unit cell, as depicted in Fig. 5 shape lattice A is not considered here for two reasons, the first reason is the lattice termination or interfaces should be compatible with the unit cell symmetry [START_REF] Benalcazar | Quantization of fractional corner charge in $C n$symmetric higher-order topological crystalline insulators[END_REF], and the second reason is that the corner states at obtuse-angled corners are topologically protected while those at acute-angled corners are not [START_REF] Fan | Elastic Higher-Order Topological Insulator with Topologically Protected Corner States[END_REF]. Besides the perfect periodic structure, we also introduced three types of pertur- Ensured by their topological origin, the in-gap topological second-order corner states are not only frequency-isolated but also are robust against structure imperfections as long as the lattices on both sides of the interface hold the topological classification. In other words, the corner-localized modes will be preserved, and their corresponding eigenfrequencies won't deviate too much from the original values in the presence of various types of perturbations on the perfect lattice (without closing the bulk gap), such as removing resonators, randomly rotating or deforming several unit cells near the corners. The eigenfrequencies of the perturbed crystals for these three cases are listed in Fig. 5(d) -(f). Comparing them with Fig. 5(c) reflects two important facts: the first one is that the eigenfrequencies of the in-gap topological corner states almost remain unchanged, and the second one is that inside the trivial gap, the trivial corner modes (green dots), however, either appear or disappear under different circumstances. Even in the case of their presence, the trivial corner modes may emerge at different frequencies or even come into multiple groups, completely dependent on the specific perturbed crystal, which makes them much less useful when compared with the robust topological corner modes.

Occurrence of topological Fano resonance

We propose a three-component phononic plate structure, as shown in Fig. 6 dark mode has a phase shift of π near the resonant frequency, leading to a transition from an out-of-phase interference to an in-phase one, or vice versa. This abrupt change finally creates an asymmetric lineshape in the frequency response within a very narrow band, which is a Fano-like resonance with topological origins for both the bright and dark modes.

The FEM-based method in previous sections are suitable for band structure (eigenproblmes) calculations, while hereafter we turn to utilize the computationally efficient multiple scattering method (MST) [START_REF] Torrent | Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates[END_REF] (see Appendix E for details) to simulate the wave field distributions and frequency response of the proposed lattice under a specific source excitation. Note that even the MST based on the Kirchhoff-Love plate theory is valid only for low frequencies and long wavelength limit, the MST simulated results show good correspondence with the FEM calculated bandgaps (see Fig. and the minimum is 1/(3.557 × 10 -3 ) = 281. The Q-factor for the Fano resonance is given by the ratio between the average frequency and the frequency difference of the peak and dip value [START_REF] Sun | Q-factor enhancement of Fano resonance in all-dielectric metasurfaces by modulating meta-atom interactions[END_REF],

based on which we found the Fano resonance here has an ultrahigh value that

Q = f 0 /∆ f = 14582.
The out-of-plane displacement distributions exited by a unit-magnitude point source at f 1 and f 2 are shown in Fig. 6(c) and (d), respectively, which vividly demonstrate the forbidden and pass states due to the destructive and constructive interference between the bright and dark modes. This ultrasensitive switching phenomenon may provide potential applications in acoustic sensors or switches.

Moreover, because both the bright and dark modes, the fundamental ingredients to create the Fano resonance, have the topological protection features, this particular type of Fano resonance may stand out from its conventional counterparts as it is more robust against structural imperfections. It is also worth noting that for d = 2d 0 , the edge-corner distance is too close such that the response is affected strongly by the corner modes, and presents a different behavior with other cases. To qualitatively explain this, we simplify the topological lattice system into a 3 degree- of-freedom (DOF) system as shown in Fig. 7(f)-(g). Since the both corners states are isolated modes, thus they can be viewed as a 1-DOF oscillator with very small damping ratio, while the broad band topological edge states span over a broad spectrum, thus an oscillator with a large damping ratio is needed to exhibit a similar behavior (see the particular parameters we chose in Fig. 7). Next, we apply a force F(Ω) on m 1 and record its displacement response X(Ω) and

Robustness of topological Fano resonance

In this section, we investigate the robustness of the topological Fano resonance when various types of structural imperfections are deliberately introduced into the system. From the discussions in the previous sections, we know that the Fano resonance occurs around the resonant frequency of the dark mode, i.e., the topological second-order corner mode. So, it is natural to consider that the corner mode influences the Fano resonance more significantly, therefore we focus on the variations of the frequency response for the structural imperfections acting on the corners.

As a first example, we consider the rotational perturbations by assigning random rotational angles (schematically illustrated in Fig. 5 For each perturbative level, we generate 3 different groups of the random angles and repeat the calculation for each of them. From the results listed in Fig. 8, we can conclude that the Fano shape response is always preserved even though the resonant frequency may deviate slightly from the original point. As the magnitudes of the angles increase, the frequency shift of the Fano resonance tends to take a larger value. However, even for the case that |θ| < 10 • , we can see the largest relative frequency shift is approximately estimated as |333 -335|/335 ≈ 0.6%, which is very small. Then we investigate the influences of the deformations by assigning independent random resonator to centroid distances on the same seven cells considered above. Similar to the case of rotational perturbations, here we also consider different magnitudes of the distances in an increasing manner from |∆R/R 0 | < 5% to |∆R/R 0 | < 10%, and for each level we also generate 3 groups of geometrical configurations, upon which the calculation is repeated. The results shown in Fig. 9 are very similar to those shown in Fig. 8 for the rotational cases, from which we can conclude again that the Fano lineshape response is always preserved, and the frequency shift is comparatively small even we impose the maximum value of 10% geometrical perturbations.

Note that if we remove one resonator around the corner as we did in Fig. 5(a), we can expect that the influence on the Fano resonance is also neglectable (not shown) since the eigenfrequency spectrum in Fig. 5(d) reveals that the topological corner modes are very robust. We compare the topological corner states and trivial defects with disorders in Appendix D, which indicates that the frequency of the topological corner states still remains in the gap of edge states and are only perturbed slightly, while the trivial point defect with disorders will suffer a huge shift across the whole bandgap of the bulk states. This unambiguously reflects the robustness of the topological

Discussions and Conclusions

In conclusion, we have investigated the topological Fano resonance induced by the interference between the first-order topological edge states and the second-order topological corner modes by taking an elastic phononic crystal plate as an example object. The breathing honeycomb lattice under consideration in this work is capable of hosting simultaneous topological edge states and corner states, which can be achieved simply by tuning the geometrical parameters. When both topological states are integrated into the same platform, they can have the possibility to couple with each other. Hence, the constructive and destructive interferences between them result in the ultra-sharp and asymmetric Fano resonance. Because both of the bright and dark modes are topologically protected, the resultant Fano resonance features a unique advantage of robustness against various kinds of geometrical imperfections, which is also quantitively confirmed by our numerical calculations.

We also demonstrated that the coupling strength between the topological edge and corner modes has a significant influence on the Fano-like spectrum. As the distance between the edge and corners increasing, the coupling strength decreases gradually, weakening the resultant topological Fano resonance. By qualitatively simplify the elastic continuum structure into a 3-DOF system, we found that if the two corner modes are not rigorously degenerate, there are in fact two Fano resonant profiles. The relative magnitudes of these two Fanos depends on the competition between the edge/corner and corner/corner couplings. In our case, only for d = 2d 0 could we observe two Fano resonance, while once d ≥ d 0 , one of them almost disappears and becomes not detectable.

Furthermore, we found that if two corners are placed with unequal distances from the topological edge, two nested ultra-sharp peak-dip pairs will also emerge.

It is worth noting that the lossless phononic plate attached with mass-spring resonators proposed here is a prototype design. For practical considerations, We discussed in Appendix G the effects of dissipation and concluded that the increasing of the loss coefficient will gradually flatten the ultrasharp Fano line-shape and the large peak-dip discrepancy. If the loss coefficient is large enough, the asymmetric Fano line will be eased out and the Q factor decreases quickly. given by

H i j = k • p i j (B.9)
Now let us take the symmetry of the eigenmodes into consideration to simplify the above expression. As demonstrated in Fig. 3, the p and d modes have different mirror symmetries along two coordinate axes. These four modes are arranged in the order of p x , p y , d x 2 -y 2 and d xy . Therefore, the calculation of p 11 require us to calculate p * x and i∇(∇ 2 p x ). To be brief, the parity of all the terms that are involved in the calculation of p 11 is listed in Table B.1, in which the sign + (or -) represent the certain term is even (or odd) with respective to the corresponding axis. For example, the signs of the first column in Table B.1 indicate that p x is odd and even with respect to the x and y axis, respectively. The rest entries can be understood in the same way.

〈

  The 1st-order topol。gical edge state

Figure 1 :

 1 Figure 1: Illustration of the occurence of toplogical Fano resonance. (a) The schematic view of the 1st-order topological edge state and its transmission spectrum, (b) The 2nd-order topological corner state and its transmission spectrum, (c) the robust topological Fano resonance and its ultra-sharp and antisymmetric transmission line resulted from the coupling between the topological edge and corner states.

Figure 2 :

 2 Figure 2: Schematic view of the phononic plate. (a) Top view of the phononic plate in the xy plane, each white circle represents one oscillator. (b) The perspective view of one unit cell. (c) The first Brillouin zone of the honeycomb lattice.

  (b) is in fact a supercell. This specific selection of an enlarged unit cell allows the formation of a double Dirac cone at Γ point due to a band folding mechanism. By solving Eq. (12), the gapless band diagram for R = R 0 shown in Fig. 3(a) unambiguously reveals the occurrence of a double Dirac cone at 331.7 Hz, which is again confirmed by the theoretically predicted results by k • p method (blue solid lines) which show a very good correspondence with the finite element method (FEM) results (white open circles). For the details of the implementation of k • p method, please refer to Appendix B. The right panel shows the field distributions of the four-fold degenerate modes, which are denoted by p x , p y , d x 2 -y 2 and d xy according to their symmetries. For example, the dipole-like modes p x(y) are antisymmetric about the x(y) axis, but symmetric about the y(x) axis. While the quadruple-like modes d x 2 -y 2 (d xy ) are symmetric (antisymmetric) about both x and y axes.Besides the mirror symmetries, under a r2 operation (180 • rotation), the p-modes will flip their sign while the d-modes remain unchanged. It is worth noting that these four degenerate modes are generally hybridized in numerical calculations and therefore the aforementioned symmetries J o u r n a l P r e -p r o o f Journal Pre-proof are usually broken and thus may need to be constructed by a combination of the degenerate eigenmodes. To overcome this shortcoming, we present a scheme based on projection (see Appendix A)

Fig. A. 10

 10 Fig. A.10 for a better visualization of the rotational eigenvalue of them.

Fig. 3

 3 (b) presents the varying of the upper and lower gap limits when R/R 0 takes different values. Starting from R/R 0 = 0.7, as we increase the value of R/R 0 , the band gap decreases gradually and closes

Figure 3 :

 3 Figure 3: Bulk band structures and the topological transition. The band diagram for the critical configuration when R 1 = R 0 , where a double Dirac cone is formed. The open circles refer to the FEM simulations whereas blue curves are the results of k • p theory (see Appendix B). The eigenvectors at the Γ point are shown on the right. (b) the topological phase transition as R 1 varies revealed by the lifting of the Dirac cone degeneracy. (c) and (d) the band structures and eigenmodes at Γ and M point for configuration A and B , respectively.

Figure 4 :

 4 Figure 4: The first-order topological edge states. (a) Schematic view of a ribbon consisting of 16 layers of lattice B sandwiched in between 8 layers of lattice A (or A ) on both sides. The band structures for (b) A /B and (c) A/B interfaces along the k x direction. The inset in (b) shows a zoom in view near the tiny topological edge band gap.

Figure 5 :

 5 Figure 5: The second-order topological corner states. (a) The schematic view of the hexagonal structure and various lattice imperfections. (b) The normalized out-of-plane displacement distribution of the corner state. (c)-(f) The eigenfrequency spectra of the lattice structure under various geometrical perturbations: (c) without any perturbations, (d) with a cavity, or disorders introduced by randomly (e) rotating or (f) deforming several cells near each corner.

  bations on the crystal to demonstrate the robustness of the corner states with topological protection, including a cavity by removing one resonator near the corner, or disorders induced by randomly rotating or deforming seven-unit cells close to the corner, schematically shown in the right panels in Fig.5(a). The calculated eigenfrequency spectra for the perfect and perturbed crystals are shown in Fig.5(c) -(f), and a typical mode shape corresponding to one of the six corner states highlighted by red dots in Fig.5(c) for the perfect lattice is plotted in Fig.5(b). The result in Fig.5(c) again illustrates the existences of topological edge states (blue dots), and more importantlyJ o u r n a l P r e -p r o o fJournal Pre-proof the in-gap topological second-order corner states (red dots, around 335 Hz), which are not directly predictable from the edge-projected band diagrams. The spatial localization of the corner states is unambiguously demonstrated by the eigenmode distribution, in which only a small region near the corners has nonzero out-of-plane displacements.

Figure 6 :

 6 Figure 6: The occurrence of topological Fano resonance. (a) schematic view of a sandwiched lattice with A /B/A configuration that supports the coupling between topological edge states and second-order corner modes. (b) The normalized frequency response of the topological edge state. (c) and (d) the wave fields magnitudes |w(x, y)| at the dipand peak-frequency, respectively.

  4 and Fig. C.11 for comparison). Fig. 6(b) shows the MST simulated frequency response with respect to its maximum. Here the frequency response is defined as the ratio between J o u r n a l P r e -p r o o f Journal Pre-proof the detected displacement signal and the excited source signal. In the calculation, we have set d = 3d 0 where d 0 = √ 3a/2 is the height of each unit cell, and the wave field is excited by a unit-magnitude point source. Around the resonant frequency of corner modes (∼ 335 Hz), we can clearly see the resultant asymmetric and ultra-sharp Fano line. It reaches the maximum very quickly at f 2 = 335.3870 Hz from the minimum at f 1 = 335.3640 Hz, and the ratio between the maximum

  The lattice configurations for other values of d will affect the peak-and dip-value of the Fano resonance, but the resonant frequency, as well as the ultra-sharp and asymmetric features of the response curve will be preserved. The frequency responses of the topological edge states coupled with the corner modes at different values of d are shown in Fig.7(b)-(e). As a reference, the edge states frequency response without edge-corner couplings are also displayed in Fig.7(a), which is in fact a zoom-in view of Fig.C.11(d). The main profile of the frequency response in Fig.7(a), especially the two large broad peaks around 333 and 335 Hz, are maintained even in the presence of the edge-corner couplings. The significant result of the edge-corner coupling is the emergence of the Fano resonance. In addition, by comparing Fig.7(c)-(e), it can be concluded that larger values of d means weaker coupling strengths, which will decrease the ratio between the peak-and dip-values, making the Fano resonance less observable. Note that around the Fano resonance, the response changes abruptly, thus a very fine frequency sweep are needed, as shown in the insets.

Figure 7 :

 7 Figure 7: Fano lineshapes in the MST simulated frequency responses for various edge/corner distances. (a) The frequency response for the A /B edge states. (b)-(d) correspond to d = 2d 0 to 5d 0 with d 0 the height of one unit cell. The insets show sufficient fine sweep of the response spectra near the Fano resonant frequency in the numerical calculations. (f) The schematic view of the lattice configuration that supports edge-corner couplings. (g) The simplified discrete 3 degree-of-freedom (DOF) mass-spring system. (h)-(l) The frequency response for the 3-DOF system in (g) for various values of coupling strengths ξ. The dimensionless parameters are k 1 = 1, m 1 = 1, c 1 = 0.3, k 2 = 4, m 2 = 1, c 2 = 0.01, k 3 = 1.1k 2 , m 3 = m 2 , c 3 = c 2 , η = 0.5, and ξ is the tuning parameter.

  obtain the frequency response function H(Ω) = X(Ω)/F(Ω) where Ω is the dimensionless angular frequency. The amplitudes of H(Ω) for various values of ξ are plotted in Fig.7(h)-(l). Here ξ denotes the coupling strength between m 1 and m 2,3 , directly mimicking the couplings between the topological edge and corners. If we ignore the first peak in Fig.7(a)-(e), it can be seen thatJ o u r n a l P r e -p r o o fJournal Pre-proof our simplified 3-DOF system matches well with the real topological edge/corner coupling lattice by comparing the frequency responses of both systems. Note that the two corner modes are not completely degenerate, thus we have let k 3 slightly deviates from k 2 . Through the calculation on the 3-DOF system, we found that there are in fact always two Fano resonant peaks [see Fig.7(i)],corresponding the two corners. The competition between edge-corner coupling (ξ) and the cornercorner coupling (η) determines which one of the two Fano profiles is dominant. For ξ = 4η, both Fano profiles are observable, which explains Fig.7(a). As the coupling strength ξ decreases shown in Fig.7(j)-(l), both Fano resonances decrease quickly and only one of them is detectable, which explains Fig.7(c)-(e).

  (a)) independently for each of the seven cells around the corners. For simplification, both corners are set to be under exactly the same perturbation. The random angles are given with an increasing of the angle magnitudes, from |θ| < 5 • to |θ| < 10 • .

Figure 8 :

 8 Figure 8: The robustness of the topological Fano resonance when different random rotational angles are introduced for (a) |∆θ| < 5 • , (b) |∆θ| < 8 • and (c) |∆θ| < 10 • when d = 3d 0 .

Figure 9 :

 9 Figure 9: The robustness of the topological Fano resonance when different random resonator-centroid distances are introduced for (a) |∆R/R 0 | < 5%, (b) |∆R/R 0 | < 8% and (c) |∆R/R 0 | < 10% when d = 3d 0 .

Figure A. 10 :

 10 Figure A.10: The reconstruction of the symmetric eigenmodes from the hybridized modes. The phases of these modes are also shown to illustrate their rational eigenvalues.

Table B .

 B 1: The parity of each term needed in the calculation of p 11p x i∂ x (∇ 2 p x ) i∂ y (∇ 2 p x ) p * x • i∂ x (∇ 2 p x ) p * x • i∂ y (∇ 2 p x )According to the last two columns of TableB.1, one can conclude that

	x -y +	+ +	--	-+	+ -
		cell	p * x • i∂ x (∇ 2 p x )dr = 0	(B.10a)
		cell	p * x • i∂ y (∇ 2 p x )dr = 0	(B.10b)
	Hence, p 11 = (0, 0).

Acknowledgments Y. W. acknowledges the support from China Postdoctoral Science Foundation (No. 2020M672615). R. Z.acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 12102103).

Journal Pre-proof on-chip micromechanical phononic plates with pillars etched on silicon chips [START_REF] Yan | On-chip valley topological materials for elastic wave manipulation[END_REF][START_REF] Wu | On-chip higher-order topological micromechanical metamaterials[END_REF], or similar arrangements on the macroscale phononic crystals by using 3D printing [START_REF] Miranda | Wave attenuation in elastic metamaterial thick plates: Analytical, numerical and experimental investigations[END_REF] or other techniques [START_REF] Chen | Corner states in a secondorder mechanical topological insulator[END_REF][START_REF] Chaunsali | Experimental demonstration of topological waveguiding in elastic plates with local resonators[END_REF]. Our findings may find applications in highly sensitive and accurate filters, sensors, and switches.

During the submission of this paper, we became aware of Ref. [START_REF] Kim | Ultrahigh-Q Fano resonance using topological corner modes in secondorder pseudospin-hall photonic systems[END_REF] which also utilized the SO-TIs to realize the robust dark modes to construct a topological Fano resonance in photonic crystals.

We would like to note that Ref. [START_REF] Kim | Ultrahigh-Q Fano resonance using topological corner modes in secondorder pseudospin-hall photonic systems[END_REF] adopted both armchair and zigzag topological interfaces to realize gapless topological edge states and in-gap corner modes with common frequencies to further guarantee the Fano interaction, while in our work it is simply realized by independently tuning the geometrical configurations of unit cells in different regions of the phononic crystal, and only one type of topological interface is needed.

Appendix A. Symmetry reconstruction of degenerate eigenmodes

The Bloch modes usually exhibit certain symmetries at high-symmetric point in the reciprocal space. But for degenerate modes, these symmetries might be hidden because of mode hybridization, which is very common in numerical simulations. In this section we demonstrate how to recover the symmetry of the modes from the hybridized ones, which is very useful for topological index analysis and effective Hamiltonian deduction. Journal Pre-proof which unambiguously reveals that r2 ψ i is also an eigenvector of H, corresponding to the same eigenvalue λ. Therefore, r2 ψ i must lies in the subspace spanned by ψ i (i = 1, 2, 3, 4),

As shown in

in which S i j represent some expansion coefficients that can be identified as

Once the 4 × 4 matrix S is obtained, we consider the following eigenvalue problem

in which η and v are the eigenvalue and eigenvector of S . Now let v be the coordinate of some vector in span{ψ i }, denoted as ϕ, which means

Therefore, r2 ϕ can be evaluated as follows,

We find that ϕ is an eigenvector of r2 , with the corresponding eigenvalue being η. This reveals that ϕ defined in Eq. (A.7) is symmetric under the operation r2 .

The aforementioned procedure indeed helps us to transform ψ i to ϕ i , as shown in the second row of In this section we give the explicit form of the reduced effective Hamiltonian of the continuum elastic system by using the k • p perturbation method. We start from a group of eigenfrequencies ω nk 0 and their corresponding eigenmodes w nk 0 at k 0 point, which can be numerically obtained by the method illustrated in Sec. 2 and Appendix A. With these eigen-solutions at hand, the eigenmodes at k 0 + ∆k are supposed to be the linear superposition of w nk 0 , and are inserted back into the J o u r n a l P r e -p r o o f

Journal Pre-proof

After analyzing all the term of p i j in the similar way, the matrix form of p is collected as follows

in which B is a pure imaginary number which need to be further numerically calculated. Substituting Eq. (B.11) back into Eq. (B.7), we obtain the following eigenvalue problem,

, we arrive at the final eigenvalue problem

in which the eigenvalue is λ = ω 2ω 2 0 , with the corresponding eigenvector Φ to be the linear superposition of the expanding coefficients A according to the above base-transformation, and the effective Hamiltonian is given by

where k = |k|, k ± = k x ± ik y , and α = |B| 2 /2∆. The eigenvalues of H are analytically obtained as

which can well describe the real behavior of the band structures in the vicinity of Γ point, as shown in Fig. 3(a), (c) and (d).

Appendix C. The frequency responses for topological edge states

To provide more insights on the topological edge states and the aforementioned tiny gap, we present the frequency response spectrum for the phononic crystal constructed from topologically distinct lattices in Therefore, it can be concluded that the topological corner modes are more robust than the trivial cavity modes.

Appendix E. The multiple scattering method for flexural wave in thin plates

In this section we briefly introduce the MST method following Ref. [START_REF] Torrent | Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates[END_REF] for the total wave field simulations under arbitrary incident wave fields, which is used in this paper for the calculation of wave field distributions and the spectrum response curve. We can rewrite Eq. (B.3) as

where κ = (ρhω 2 /D) 1/4 is the wave number in the homogeneous plate, α is a label of the resonators, and t α measures the resonant strength, which is given by

For truncated finite clusters of resonators, the total wave field distribution under an external incident wave can be semi-analytically solved using MST method: each resonator acts like a point source and radiates a cylindrical wave. The total field is thus the superposition of all the scattered fields and external incident field,

where w inc (r) is the incident field which is known beforehand, and G(r; R α ) = i/(8κ 2 )[H (1) 0 (κ|r -R α |) + 2i/πK 0 (κ|r -R α |)] is the Green's function which satisfies ∇ 4 w(r)κ 4 w(r) = δ(r -R α ) [START_REF] Torrent | Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates[END_REF].

J o u r n a l P r e -p r o o f

Journal Pre-proof Considering the total field at an arbitrary resonator's position r = R β , from Eq.(E.3) we have the linear system

from which we can derive the displacement of each resonator. The total field can then be constructed by using Eq.(E.3). This semi-analytical method by using Green's function constitutes an efficient approach for the calculations of the total wave field response under any given incident wave.

Appendix F. Double asymmetric Fano profiles by two different waveguide-corner coupling strengths

In this section we consider the case when the distances between the edge and the two corners are different, in which the frequency spectrum exhibits a double Fano-like ultrasharp profiles. Note that we have demonstrated the influence of different values of d on the Fano-line profile in Fig. 7.

Since the Fano profile is almost not noticeable for d ≥ 5d 0 , here we study the effects of different We have also shown the wave field distributions for the dip and peak frequencies for both Fano Journal Pre-proof only one corner mode has a large magnitude. As for the peak of Fano 2, we can see topological edge state and both corner modes are excited. In the case for Fano 2, both at the dip and peak frequencies the two corner modes are excited, and the edge mode discrepancy between the dip and peak is not as evident as that for Fano 1.

Appendix G. The loss effects on the Fano resonance

In practical situations, the loss effects of the dissipation should be taken into consideration.

We consider again the three-component phononic crystal illustrated in Fig. 6(a). The geometrical parameters remain the same as the lossless case, and the only difference is that we introduced a complex-valued Young's modulus such that E = E 0 (1 + iη) where E 0 is the modulus without loss and the parameter η is a coefficient to account for the loss effects. A harmonic point source is applied on the topological waveguide near the left port, and the simulated out-of-plane displacement on the waveguide near the right port is recorded. The magnitude of the frequency response, i.e., the ratio between the detected signal and the source signal, is plotted below for various choices of η. Note that the lossless case [Fig. 6(b)] is also shown by dashed blue line for reference. We can clearly see that as the loss coefficient η increases, the ultrasharp Fano line-shape and the large peak-dip are flattened gradually. If the loss coefficient is large enough, the asymmetric Fano line will be eased out and the Q factor decreases quickly. The influence of the loss effects on the Fano line-shape spectrum response. The loss is introduced by using a complex-valued Young's modulus E = E 0 (1+iη) for the plate material. The calculated displacement responses for all cases are normalized with respect to the same factor, i.e., the maximum of the lossless response spectrum.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:


Mathieu Acher
email: mathieu.acher@irisa.fr

Luc Lesoil
email: luc.lesoil@irisa.fr

Georges Aaron Randrianaina
email: georges-aaron.randrianaina@irisa.fr

Xhevahire Tërnava
email: xhevahire.ternava@irisa.fr

Olivier Zendra
email: olivier.zendra@inria.fr

A Call for Removing Variability

Keywords: software variability, removing variability, software bloat

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

WHY REMOVING VARIABILITY?

To support and deliver values to a wide spectrum of users, today's software systems include an abundance of features that tend to expand over time in real-world industrial contexts [START_REF] Holzmann | Code inflation[END_REF][START_REF] Tërnava | On the Interaction of Feature Toggles[END_REF]. For instance, a highly complex system is the Linux kernel with more than 25 M lines of code and 20 K configuration options, and it keeps growing [START_REF] Martin | Transfer learning across variants and versions: The case of linux kernel size[END_REF][START_REF] Passos | Coevolution of variability models and related software artifacts[END_REF]. Current software development approaches also strive for increasing variability in software systems. To enable expanding features, significant progress is being made in the forward and reverse engineering of software variability, which is frequently mentioned in research on software reuse, software families, or software product lines (SPL) (e.g., [START_REF] Capilla | Systems and software variability management[END_REF][START_REF] Capilla | Opportunities for software reuse in an uncertain world: From past to emerging trends[END_REF][START_REF] Clements | Software product lines[END_REF][START_REF] Mcilroy | Mass-Produced Software Components[END_REF]19]). But, in the frame of variability management and evolution, the need for a systematic removal of superfluous or underutilized variability has received far less to no attention from the community in software variability.

Since more support for usage context in a software system is desirable, the removal of variability might look counter-intuitive. Hence, the question is: how reducing variability can be a good idea?

The ultra-high amount of variability in software systems is exceeding human and even machine limits to deal with it, namely, to manage, test, comprehend, or even be able to use every feature or option ever [START_REF] Martin | Transfer learning across variants and versions: The case of linux kernel size[END_REF][START_REF] Passos | Coevolution of variability models and related software artifacts[END_REF]. Xu et al. [START_REF] Xu | Hey, you have given me too many knobs!: Understanding and dealing with over-designed configuration in system software[END_REF] show that up to 54.1% of configuration options in a system are rarely set by any user. Some systems are also completely avoided by users, who are overwhelmed by * This is the corresponding author. The authors' names are in alphabetical order. the too many choices available [START_REF] Noyes | Does Linux Offer Too Much Choice?[END_REF][START_REF] Torvalds | Fragmentation is Why Linux Hasn't Succeeded on Desktop[END_REF]. More often, end-users lack the expertise and time to configure the system to get the right functional and non-functional properties for their usage context.

Yet, Soto-Valero et al. [START_REF] Soto-Valero | A comprehensive study of bloated dependencies in the maven ecosystem[END_REF] show that up to 75.1% of software libraries that exist in a software system are not needed to compile and run it. Still, they are packed in the system binaries. Quite similarly, 75% of feature toggles, a temporary form of variability, become unused after 49 weeks [START_REF] Hoyos | On the Removal of Feature Toggles[END_REF][START_REF] Murali | Piranha: Reducing feature flag debt at Uber[END_REF]. It's normal to imagine an underutilized feature or library could be useful in the future, yet the opposite is true. However, 89.2% of libraries as direct dependencies that are unused today in a software system will likely never be used [START_REF] Soto-Valero | A comprehensive study of bloated dependencies in the maven ecosystem[END_REF]. These and similar findings show that today's software systems, ranging from small-scale size utilities in Unix [START_REF] Holzmann | Code inflation[END_REF][START_REF] Pike | Program design in the UNIX environment[END_REF] to large-scale size web browsers, such as Chromium [START_REF] Qian | Slimium: debloating the chromium browser with feature subsetting[END_REF], are bloated.

Discovering the unneeded variability in software systems is not trivial and is a complex socio-technical task. So, why should someone bother and remove variability from a codebase?

Removing unneeded or unused variability from software systems is very important first because it can negatively impact the system security: the disabled or never used features in a software system may contain security vulnerabilities that could be exploited, threatening the whole system and its users. On the other hand, the attacker can for example chain small code sequences called gadgets and threaten the security of the system [START_REF] Qian | Slimium: debloating the chromium browser with feature subsetting[END_REF][START_REF] Sharif | TRIMMER: Application specialization for code debloating[END_REF]. The larger the binary size of a software system, the larger its attack surface [START_REF] Michael | Is less really more? towards better metrics for measuring security improvements realized through software debloating[END_REF].

Another reason lies in the system's reliability. Studies show that many software failures arise from the misconfiguration of software systems [START_REF] Sayagh | Software configuration engineering in practice interviews, survey, and systematic literature review[END_REF][START_REF] Yin | An empirical study on configuration errors in commercial and open source systems[END_REF]. In such cases finding the cause of a failure among large sets of options is difficult. Moreover, unused variability can introduce technical debt, for example, when a feature toggle has served its purpose and needs to be removed from the codebase [START_REF] Murali | Piranha: Reducing feature flag debt at Uber[END_REF].

Poor configuration choices in a software system may also result in bad or even its worst possible performance (see e.g., Hadoop [START_REF] Herodotou | Starfish: A Self-tuning System for Big Data Analytics[END_REF]). Assuming that the unused features may be among the poor choices and will further degrade system performance is reasonable. Moreover, software systems and their variability grow whether or not there is a rational need for them [START_REF] Holzmann | Code inflation[END_REF], and successful and widely used software systems tend to become encrusted with dubious features [START_REF] Pike | Program design in the UNIX environment[END_REF]. Some unnecessarily added features can even change the primary purpose of a system and threaten its reliability. Last but not least, attempting to use stale variability within a system has the power to bankrupt the entire company, as in the often-cited case of the Knight Capital Group. It unintentionally did a poor use of a stale feature toggle, causing it to lose more than 460 M dollars in 45 minutes and go bankrupt [START_REF]SEC Charges Knight Capital With Violations of Market Access Rule[END_REF][START_REF] Tabbaa | The Rise and Fall of Knight Capital -Buy High, Sell Low. Rinse and Repeat[END_REF].

Thus, exploring methods and tools for removing underutilized variability from software systems is vital and a realistic way to reduce system complexity and testing burden, improving its comprehension, reliability, security, maintainability, and performance.

HOW TO REMOVE VARIABILITY

Why is removing variability not yet a major trend?

First, because removing code in general, including variability, is not a rewarding activity for developers and product (line) managers, since it does not bring any new functionality or feature.

The main challenge to stakeholders in removing variability is the lack of automated or integrated technologies. In general, there are limited studies, understanding, and expertise on removing variability, specifically how stakeholders (may) operate over software artefacts in modern development workflows. To confirm our assumption, we investigated all publications in the past 10 years in the VaMoS 1 and SPLC 2 venues. Specifically, we searched the presence of "remove", "reduce", "debloat", and "delete" keywords, including their variations, in their title. It resulted that only 3 papers contain the word "reducing", but they are about reducing feature models or energy consumption, and not variability 3 . The existing technologies for removing code, such as dead code, technical debt, or software bloat, perhaps can be adapted to a certain extent. But, our call is for removing variability, which is different from disabling it, debloating system binaries [START_REF] Soto-Valero | Software debloating papers[END_REF][START_REF] Qi Xin | Studying and Understanding the Tradeoffs Between Generality and Reduction in Software Debloating[END_REF], or removing an unreachable piece of code.

Thus this section elaborates on possible research directions to support variability removal, discussing early attempts in literature.

Debloating variability. Software debloating has been recently explored to reduce the size of deployed containers [START_REF] Rastogi | Cimplifier: Automatically debloating containers[END_REF], or reduce the attack surface of specific programs (e.g., [START_REF] Heo | Effective program debloating via reinforcement learning[END_REF][START_REF] Koo | Configuration-driven software debloating[END_REF][START_REF] Kuo | Set the configuration for the heart of the OS: On the practicality of operating system kernel debloating[END_REF][START_REF] Quach | Debloating software through piece-wise compilation and loading[END_REF][START_REF] Sharif | TRIMMER: Application specialization for code debloating[END_REF][START_REF] Soto-Valero | Software debloating papers[END_REF][START_REF] Qi Xin | Program debloating via stochastic optimization[END_REF][START_REF] Qi Xin | Subdomainbased generality-aware debloating[END_REF]). Often, proposed approaches debloat a system compiled binaries [START_REF] Qian | Slimium: debloating the chromium browser with feature subsetting[END_REF][START_REF] Qi Xin | Studying and Understanding the Tradeoffs Between Generality and Reduction in Software Debloating[END_REF], remove its unused libraries [START_REF] Soto-Valero | A comprehensive study of bloated dependencies in the maven ecosystem[END_REF], and rarely configuration options [START_REF] Kuo | Set the configuration for the heart of the OS: On the practicality of operating system kernel debloating[END_REF]. Existing approaches are all heirs of existing works in program specialization [START_REF] Mcnamee | Specialization tools and techniques for systematic optimization of system software[END_REF], so the idea of removing code is not completely new. Yet, the proposed techniques do not cover the plethora of variability units and mechanisms that can be subject to removal and debloating, namely features, command-line options, feature toggles, design patterns, and configuration files. For instance, removing run-time options within the source code still is an investigation direction with many challenges: How to trace and locate run-time options within the source code? How to remove a subset of run-time options without breaking other options and core functionality? How to guarantee that the removal is safe, that is, the remaining functionalities are unaffected?

Revisiting the reverse engineering of variability. Numerous works have been conducted on the reverse engineering of variability and to specifically create variability representations of artefacts in other 1 VaMoS proceedings: https://dl.acm.org/conference/vamos/proceedings 2 SPLC proceedings: https://dl.acm.org/conference/splc/proceedings 3 The list of resulting papers: https://doi.org/10.5281/zenodo.7360593 forms or at higher levels of abstraction. Feature extraction and location have for example been subject to intensive research [START_REF] Wesley Kg Assunção | Reengineering legacy applications into software product lines: a systematic mapping[END_REF]. In our case, removing unused variability in existing systems requires locating and tracing variability that can cross-cut numerous functions, classes, modules, or files. Reverse engineering approaches have been proposed to support the maintenance and re-engineering of legacy variability-rich systems; most of the literature aims to add features and create new variants with richer functionalities. We make the point that reverse engineering techniques should be revisited in light of a new objective, that is, of removing the unneeded variability from legacy software systems while expecting novel and more precise variability location methods and metrics.

Designing for variability removal. We argue that, when developing software variability, developers should also pursue the objective of easing the removal of variability. Developers should thus find it gratifying to remove features and quantify its benefits. Any newly devised approach should support the removal of different variability units, be they features, configuration options, settings, or feature toggles. To this end, we see two complementary paths. First, the development of automated tools and methods to address a legacy, existing codebase, so that with limited effort developers can pilot tools to remove specific variability in a software system. A second and unexplored direction is to design, by construction, variabilityrich systems that are "variability removal friendly". Extensibility and modularity, two key principles of software engineering, have been subject to intensive inquiry; we argue that similar research should be conducted now to ease variability removal. In both cases, the technical process should also include the removal of all other artefacts bound to a variability unit, such as tests or comments. Ideally, the remained system after removing some variability should not only be compilable and functional but also well formatted.

Developers' workflow for removing variability. Currently, there are very few proposed methodologies to support variability removal and quantifying its benefits. A notable case is the semi-automated approach of Piranha to remove stale feature toggles in Uber applications realized in Java, Swift, and Object-C [START_REF] Murali | Piranha: Reducing feature flag debt at Uber[END_REF]. In general, we are missing studies that observe how developers remove code and specifically variability-related code. We ignore what are the difficulties of removing variability. Characterizing and understanding such difficulties could help to design tools supporting their activities. There also exists little quantification of the human cost of removing variability. Another open question is how to integrate variability removal as part of modern development workflows (i.e., code reviews, continuous integration, issues, ticketing systems, etc.).

Removing variability: application or domain engineering? SPL engineering usually distinguishes two complementary phases: domain engineering and application engineering [19]. Removing variability can occur within application engineering when concrete products are derived using the common and reusable artefacts developed in domain engineering. The issue is that tailoring a product to remove unneeded variability can be very specific (i.e., to the product) and not reusable for other products. In a sense, the effort of removing variability would be one-shot and hard to replay in the long run, for instance, when software evolves. Hence, another possibility is to lift the removal of variability as part of domain engineering. It would allow developers to then systematically configure what variability can be easily removed in the future for any product. If not designed for removal, the counterpart is that this lifting has arguably a development cost. Overall, it is an open question of where to position the removal of variability as part of SPL engineering.

Scenarios for removing variability. One possible scenario is to remove all variability (a.k.a., based on a usage profile), for example, fixing a configuration once and for all and forbidding any reconfiguration at run-time. This scenario is extreme and not applicable in many contexts, since users would lack the needed flexibility for their systems [START_REF] Qi Xin | Studying and Understanding the Tradeoffs Between Generality and Reduction in Software Debloating[END_REF]. Another scenario is to find the unused variability units (e.g., stale feature toggles) and remove them. In a sense, the variability space is specialized and gradually reduced. Ideally, developers should have the power to control what variability can be removed and kept.

CONCLUSION

As we are already aware, we are surrounded by variability in our software world. The ability of a software system to be efficiently extended, changed, and customized for use in a particular context is precisely what makes software "soft" and adaptable. Hence, although not new in the software world, the idea of removing software variability might sound counter-intuitive at first.

But there is evidence that software systems are bloated with variability units (i.e., features and options). Removing variability is a missing piece in the research landscape in software variability. In this paper, we call for developing approaches and tools by our community which will facilitate the removal of variability and bloat from software systems, as a way to improve their security, performance, reliability, comprehension, maintainability, and testing.