Delocalization and higher-order topology in a nonlinear elastic lattice

Topological elastic waves provide novel and robust ways for manipulating mechanical energy transfer and information transmission, with potential applications in vibration control, analog computation, and more. Recently discovered higher-order topological insulators (HOTIs) with multidimensional and hierarchical edge states can further expand the capabilities of topological elastic waves. However, the effects of nonlinearity on elastic HOTIs remain elusive. In this paper, we propose a nonlinear elastic higher-order topological Kagome lattice. After briefly reviewing its linear properties, we explore the effects of nonlinearity on the higher-order band topology and topological states. To do this, we have developed a method to calculate approximate nonlinear modes in order to identify the bulk polarization and probe the higher-order topological phase in the nonlinear lattice. We find that nonlinearity induces unusual delocalization of topological corner states, band crossing, and higher-order topological phase transition. The delocalization reveals that intracell hardening nonlinearity leads to direct delocalization of topological corner states while intracell softening nonlinearity first enhances and then reduces localization. The nonlinear higher-order topological phase is amplitude dependent, and we demonstrate a transition from a trivial to a non-trivial phase, enabling amplitude induced topological corner and edge states. Additionally, this phase transition corresponds to the closing and reopening of the bandgap, accompanied by an unusual band crossing. By examining the band topology before and after the band crossing, we find that the bulk polarization becomes quantized with respect to amplitude and can predict higher-order topological phases in nonlinear lattices. The obtained results are expected to be beneficial for the development of tunable and robust elastic wave devices.


Introduction
Topological insulators (TIs) can robustly conduct at edges, immune to defect perturbations, but insulate in the bulk, due to nontrivial band topology [1].This unique phenomenon can be dated back to the discovery of the quantum Hall effect, which opened up a new paradigm in condensed matter physics [2] beyond electronic systems, e.g.acoustical [3], photonic [4], and mechanical systems [5][6][7].Traditionally, a n-dimensional (nD) TI only supports (n-1)D edge states [1].Recently, higher-order TIs (HOTIs) that can support edge states lower than (n-1)D have been discovered [8,9], which not only expands the family of TIs, but also enables new potential applications, such as on-chip signal processing [10], directional emission [11], and topological laser [12].
Besides first-order elastic TIs [13][14][15][16][17][18], HOTIs have also been explored in elastic wave systems [8,[19][20][21][22][23][24] to manipulate mechanical energy transfer and information transmission [25,26].Typical examples include the development of new technologies and device for vibration control [21,27,28], information processing [29], and analog computation [30], among others.The multipole moment model [8] and the generalized Su-Schrieffer-Heeger (SSH) model [19] have been used to construct elastic HOTIs.Because the multipole moment model needs negative coupling and is difficult to realize [8], the generalized SSH model is often preferred.Based on the generalized SSH model, second-order topological corner states can be realized in structures with different crystalline symmetries [31], e.g.Kagome lattices [21], square lattices [32], and honeycomb lattices [33].These HOTIs have clear advantages in that their topological states are hierarchical and multi-dimensional.Consequently, their edge states and corner states can be selectively excited, with the corner states being capable of robust energy confinement with high-efficiency.
Following recent studies in nonlinear optical HOTIs [49][50][51], in this paper, we propose a nonlinear Kagome lattice based elastic HOTI and explore the nonlinear effects on higher-order band topology and the second-order topological corner states and the first-order topological edge states.To probe higher-order topological phases in the nonlinear regime, we use the harmonic balance method [52,53] (HBM) to calculate the approximate nonlinear modes to identify the nonlinear bulk polarization.The rest of the paper is organized as follows.The nonlinear elastic Kagome lattice is shown in section 2, and its linear topological properties are shown in section 3. Section 4 investigates nonlinear effects on band topology based on the HBM, and section 5 explores the nonlinear effects on topological states in nonlinear hardening and softening cases.Nonlinear higher-order topological phase transition is shown in section 6.Finally, conclusive remarks are given in section 7.

Nonlinear elastic Kagome lattice
We consider a nonlinear elastic Kagome lattice with a lattice constant L, as shown in figure 1.Each unit cell comprises three masses denoted by m with limited motion along the out-of-plane direction.The masses interact with each other by cubic nonlinear springs.The intracell spring k intra and intercell spring k inter follow the nonlinear force-displacement relations given by F = k intra (∆u) = k 1 ∆u + k c1 ∆u 3 and F = k inter (∆u) = k 2 ∆u + k c2 ∆u 3 , respectively, with k 1 and k 2 denoting the linear stiffness, and k c1 and k c2 the nonlinear stiffness coefficients.A positive nonlinear coefficient is referred to as hardening nonlinearity whilst a negative nonlinear stiffness coefficient denotes softening nonlinearity.An experiment could be that the masses are sliders mounted on a guide rail along the out-of-plane direction.The nonlinear springs, which can be either hardening or softening, can be realized through the use of mechanical metamaterials, allowing for a tailorable force-displacement relation [54].
The primitive vectors of the unit cell of the lattice with , where the superscript 'T' represents matrix transposition, and the masses in each cell are marked by {1, 2, 3}.The position of a cell, indexed by i and j, is r i,j = ia 1 + ja 2 .With the Hamiltonian (see appendix A), we can have the equations of motion of the nonlinear Kagome lattice as, where the over dot '•' represents differentiation with respect to time, and u i,j 3 represent the out-of-plane displacements.

Linear properties
Before investigating the nonlinear dynamics of the lattice, we first review the linear topological behavior of the Kagome lattice [55,56], in order for the content to be self-consistent.The inset in figure 1(b) shows the hexagonal and rhombic Brillouin zones (BZs) with the same origin, area and reciprocal primitive vectors Neglecting the cubic nonlinear terms in equation ( 1), we obtain a set of linear equations, in which Bloch's harmonic solution is assumed, i.e. where 2 is the Bloch wavenumber vector, ω is the angular frequency, t represents time, and T is the periodic part of the Bloch wave.Substituting equation ( 2) into (1), we have the following eigenvalue equation, where µ = mω 2 .The higher-order topological phase of the Kagome lattice is characterized by the bulk polarization P = (P 1 , P 2 ) [56,57], i.e.
in which ⟩ is the Berry connection, with U n (q) representing the wave amplitude polarization of the nth band below the concerned bandgap.Further, equation (4) can be written as [58] where θ 1 (q 2 ) = ´L(b1) Tr(A i (q))dq 1 (θ 2 (q 1 ) = ´L(b2) Tr(A i (q))dq 2 ) is the Berry phase along the loop q 1 (q 2 ) for a given q 2 (q 1 ), and L(b 1 ) (L(b 2 )) represents the projection length of the BZ along the b 1 (b 2 ) direction.Without losing generality, unit mass m = 1 and unit lattice constant L = 1 are assumed in this study.To show the higher-order topological phase transition of the Kagome lattice in the linear regime, we set the linear intercell stiffness to be k 2 = 1.When k 1 / k 2 = 0.75, the band structure of the lattice has a bandgap, as shown in figure 2(a), calculated using the hexagonal BZ for the convention of consistency.Increasing the linear intracell stiffness to k 1 / k 2 = 1, the bandgap closes, as shown in figure 2  phases.The Wannier band, i.e.Berry phase θ 2 , for the first acoustic band of the two lattices is shown in figures 2(d) and (e), calculated using the rhombic BZ due to the requirement of the Wilson loop approach [55].The bulk polarization P 2 is given by the mean value of the Wannier band, marked by the red dashed line in figures 2(d) and (e).Note that due to the three-fold symmetry of the lattice, we have P 1 = P 2 .It is observed that the lattice with k 1 / k 2 = 0.75 has a non-zero bulk polarization of −1/ 3 and the lattice with k 1 / k 2 = 1.25 has zero bulk polarization.These indicate that the lattice of k 1 / k 2 = 0.75 has a nontrivial higher-order topological phase, while the lattice of k 1 / k 2 = 1.25 is topologically trivial.The higher-order topological phase transition occurs at k 1 = k 2 , as the bulk polarization shown in figure 2 the lattice is a HOTI.
The eigenvalues and eigenmodes of a finite 21-cell lattice with various k 1 / k 2 are shown in figures 3(a) and (c)-(e), where k 2 = 1 and the fixed boundary condition is adopted.For the purpose of easy visualization, the eigenmodes in the form of displacement are represented by both the size and color of the solid circles in the figures.The grey area in figure 3(a) represents the eigenvalues of the Bloch band of corresponding infinite structure.When the size of the finite structure tends to infinity, the eigenvalues of the bulk modes of the finite structure would fill the eigenvalue area of Bloch bands.In the nontrivial topological phase, topological corner states and edge states are identified by observing the shape of modes.For example, there are 3 corner states and 15 edge states for the lattice with k 1 / k 2 = 0.25, as the eigenvalues shown in figure 3(b), in which the numbers 1, 4, and 7 indicate a corner mode, an edge mode and a bulk mode, respectively, see figures 3(c)-(e).It should be mentioned that when k 1 / k 2 varies from 0 to 1, corner states exist but gradually merge into bulk states, such as the corner state indicated by number 2 in figures 3(a) and (c).Moreover, large intracell linear stiffness k 1 appear to delocalize the states, as shown in figure 3(c).The delocalization of the edge states is similar to the corner states as k 1 / k 2 increases, see figure 3(d).

Nonlinear higher-order band topology
To investigate the nonlinear effect, we employ a special version of the Galerkin method-HBM-to calculate the band structure of the lattice.It has been shown that HBM is effective in studying 1D nonlinear elastic TIs [39,45], even for elastic wave propagation in strongly nonlinear periodic structures [52].Based on HBM, we develop a method to characterize the higher-order band topology in the nonlinear regime.
In the presence of nonlinearity, we assume that a Bloch wave is approximately valid [52] and therefore use the ansatz real solution, in which ξ = qr i,j − ωt, and A 1 , A 2 , A 3 , B 1 , B 2 , and B 3 are real harmonic coefficients.Note that although the solution is real, it contains enough information to recover a complex mode.We will show in the following that it allows us to calculate the nonlinear bulk polarization.A Galerkin procedure is adopted to numerically solve equation ( 1) in the frequency domain.Substituting equation ( 6) into (1), and then multiplying both sides by cos(ξ ), and separately sin(ξ ), and finally integrating with respect to ξ from 0 to 2π , we can transform equation ( 1) into a set of algebraic equations, see equation (B.1) in appendix B. It is noticed that there are 6 unknown wave amplitude coefficients and one unknown frequency with only 6 equations in total in equation (B.1).Therefore, we impose an additional amplitude condition [39] to render the problem solvable, i.e.
where A is a given total wave amplitude.Giving a wavenumber q, and solving equations (B.1) and ( 7) iteratively, the solution (6) can be obtained for a given amplitude A. By doing so, we can calculate nonlinear bulk modes to probe the effects of nonlinearity on bulk band topology.Recent works [47,59] showed that the relative phase between the masses in nonlinear bulk modes plays a key role in band topology.Therefore, we use the harmonic amplitudes in the solution (6) to recover phase information and construct an nonlinear complex bulk mode, and then utilize bulk polarization, i.e. equation (5), to approximately characterize the higher-order band topology of the lattice in the nonlinear regime.In the weakly nonlinear region, equation ( 5) is also valid [49].So, we rewrite equation (6) as Since the phase φ l ranges from 0 to 2π , it is sufficient to represent the wave amplitude and phase by a complex value.Thus, we have Substituting the complex bulk mode into equation ( 4) yields a nonlinear bulk polarization.Next, we will numerically demonstrate that not only this complex nonlinear bulk mode gives quantized bulk polarizations, but also the changes in the bulk polarization correspond to the closing and opening of the bandgap with an unusual band crossing.
At first, we focus on a lattice with a hardening nonlinearity of k 1 = 0.25, k 2 = 1, k c1 = 0.1, and k c2 = 0, meaning that the intracell springs are nonlinear while the intercell springs are linear.The corresponding linear lattice by neglecting the nonlinear term is topologically nontrivial.The nonlinear band structures of the lattice are calculated by iteratively solving equations (B.1) and ( 7) using the fsolve function in the software MATLAB based on the trust-region-dogleg algorithm, the linear eigenmode with a total amplitude of 1 × 10 −5 and the linear eigenvalue as initial values.Then, using equations ( 4), ( 8) and ( 9), we can obtain the nonlinear Wannier band of the acoustic band.The results of the linear and nonlinear cases with the amplitudes A = 0.5, 0.8 are shown in figure 4. It is observed that the hardening nonlinearity causes the frequency to shift to a higher frequency as the wave amplitude increases, and the bandgap becomes narrower.However, due to the intracell nonlinearity, the effect is different for different bands.For example, the nonlinear effect on the acoustic band is significant in the frequency region near the bandgap, while it is limited for the second band in the region near the upper boundary of the bandgap, see figure 4(a).As shown for the bulk polarization, the nonlinear effect is hard to be observed directly.Importantly, the nonlinear bulk polarization remains almost −1/ 3, the same as the linear bulk polarization, indicating that the nonlinear topological phase remains nontrivial and nonlinear topological corner states and edge states exist.
The softening nonlinearity effects on the band structure and the bulk polarization are shown in figure 5, with k 1 = 0.25, k 2 = 1, k c1 = −0.1, and k c2 = 0.It is observed that the bandgap becomes wider as the amplitude increases, since the softening nonlinearity induces that the acoustic band shifts to low frequency, while the second band at the high symmetry K point remains unchanged.Note that in the linear case a wider bandgap results in greater wave attenuation and increases the localization of topological corner states.In this regard, the increment of nonlinear wave amplitude may lead to a higher localization of the corner states.However, we will show later that this is not always true.Same as the hardening case, the bulk polarization remains at an unchanged finite value when the amplitude increases.If we carefully check the Wannier bands, we can find that the Wannier band gets flatter with the amplitude increasing.

Nonlinear higher-order topological states
Non-zero bulk polarization indicates nontrivial higher-order topology and the existence of higher-order topological states.Here the nonlinear effect on higher-order topological states in finite lattice is also explored using the HBM [45].We consider a N-cell finite triangle Kagome lattice.The nodal displacements of the lattice masses are Substituting equation ( 10) into (1), following the above Galerkin method with the following equation, and using equation ( 9), we can have the normalized nonlinear modes of the finite lattice, i.e.
The effect of the hardening nonlinearity of k c1 = 0.1 on a 28-cell lattice with k 1 / k 2 = 0.25, k 2 = 1 and k c2 = 0 is shown in figure 6.The eigenvalue µ = mω 2 of the modes of the nonlinear finite lattice is calculated and plotted in figure 6(a), and the eigenvalue of the Bloch bulk bands are also calculated and plotted, based on equation (7), as the grey areas in figure 6(a).The bulk modes of the finite lattice are almost completely within the Bloch bands, and just one mode detaches from the low-frequency Bloch band, which could develop into a localized mode [45].
Importantly, we can see that the frequency of the topological corner and edge states is amplitude-dependent, shifting to higher frequencies as the amplitude increases.The evolution of a corner mode, an edge mode and a bulk mode corresponding to the marked numbers in figure 6(a) are shown in figures 6(c)-(e), respectively.These correspond to the band structures shown in figure 4. The bulk mode located at the upper boundary of the bandgap is consistent with the prediction from the band structure that the nonlinear effect on such bulk mode is negligible, see figure 6(e).However, the nonlinear effect on the observed edge mode is significant, see figure 6(d).Inspired by [45], we use the inverse participation ratio (IPR), to quantize the nonlinear effect on the localization of the mode.For the corner mode, we here use the IPR to quantize localization of corner modes, as shown in figure 6(b), where a large IPR represents a high degree of localization.It shows that increasing the amplitude delocalizes the corner state, which can also be seen by directly comparing the modes at A = 0.1 and A = 0.8 in figure 6(c).The disorder effect on nonlinear topological states is also explored, see appendix C. The effect of softening nonlinearity with k 1 / k 2 = 0.25, k 2 = 1, k c1 = −0.1, and k c2 = 0 is shown in figure 7. Overall, the frequency of these modes becomes lower as the amplitude increases, see figure 7(a).But the bulk modes near the upper boundary of the bandgap are immune to the intracell nonlinearity, see figure 7(e).For the edge states, the softening nonlinearity also affects the mode shape, as shown in figure 7(d).As to the corner state, in contrast to the hardening case, the IPR of the corner mode first increases, reaches a maximum, and then decreases as the amplitude increases, in which the maximum approaches the limit value of 1/ 3 of a corner mode, according to equation (13).The maximum IPR corresponds to the mode shown in the middle in figure 7(c) with approximate zero bulk components hard to be observed.Recalling the nonlinearity effect on the band structure shown in figure 5, we can explain that the increment of the IPR results from the bandgap widening induced by increasing the amplitude.Subsequently, the decreasing IPR may be due to the delocalization nature of nonlinearity.For example, Vila et al [39] experimentally demonstrated that the mechanical softening nonlinearity eventually causes a first-order topological interface state to merge into the bulk.Similarly, in nonlinear optical experiments [50,51], it was shown that moderate nonlinearity could delocalize a topological corner state before the formation of soliton, although these nonlinearities had different origins.Here, our results reveal that the delocalization process for the topological corner state is not simple.

Amplitude induced higher-order topological phase transition
In this section, we demonstrate an amplitude induced higher-order topological phase transition and topological states in our nonlinear lattice.First, we show that the amplitude can induce bandgap closing and reopening, which can be described by the HBM, but this process has an unusual band crossing.Importantly, before the closing and after the reopening, the nonlinear lattice has quantized nonlinear bulk polarizations.Then, we observe the steady-state response of the lattice to explore the bulk-edge correspondence in the nonlinear case.
More generally, we consider the lattice with intercell hardening nonlinearity and intracell softening nonlinearity, i.e.
In the linear case, the lattice has a nontrivial higher-order topology, and the band structure of the lattice with a small bandgap is shown in figure 8(a).As the amplitude increases, the bandgap closes, and an unusual band crossing occurs at K point, see figure 8(b) with A = 0.45.Furthering increasing the amplitude reopens the bandgap, see figure 8(c).The evolution of eigenvalue of the band structure with respect to amplitude is shown in figure 8(d).We can identify that the onset of the band crossing, i.e. the bandgap closing point, occurs at A = 0.36, and the bandgap reopening point is at A = 0.51.This unique nonlinear band crossing is different with the linear case shown in figure 3(a).The amplitude can thus be considered as an active knob for controlling the bandgap.
In the linear Kagome lattice, the closing and reopening of the bandgap indicate a higher-order topological phase transition, as shown in figure 2. Here, we show that the amplitude induced band crossing corresponds to a nonlinear higher-order topological phase transition.The nonlinear bulk polarization and Wannier band, which vary with the amplitude, are calculated according to the nonlinear mode, equation ( 9), as shown in figure 9(a).Remarkably, the bulk polarization is quantized, 0 or −1/ 3, out of the band crossing region.Before the bandgap closing, the bulk polarization is 0, and after the bandgap reopening, the bulk polarization is −1/ 3. Thus, the topological phase transition occurs in the band crossing region.Note that the band topology is ill defined when bandgap closes and is thus not shown in the band crossing region, i.e. the white area in figure 9. From the Wannier bands, we can find that the Berry phase θ 2 undergoes a dramatic change near q 1 = ±1/ 3 × 4 √ 3π / 3L, corresponding to the K and K' points in the reciprocal space.This change is more drastic for amplitudes closer to the amplitude region of the phase transition.Surprisingly, this dramatic change can also be captured by the method we propose.
The bulk-edge correspondence is the foundation of TIs.Now, we inspect it in the nonlinear case.The above topological phase transition from a topological trivial phase to a nontrivial phase indicates the existence of topological corner modes and edge modes for large amplitudes.We calculate the nonlinear eigenvalues and modes of a corresponding finite lattice consisting of 28 cells, and the results are shown in figure 10.By directly observing the nonlinear large-amplitude mode shape, we can distinguish corner modes, edge modes, and bulk modes.The evolution of the corresponding eigenvalues is highlighted in figure 10(a).The evolutions of the modes corresponding to the linear eigenvalues µ = 4.0681, 4.0683, and 1.6000, at different amplitudes A = 0.1, 0.4, 1, 1.5, and 1.8 marked in figure 10(a) by the light blue dashed lines, are shown in figures 10(c)-(e), respectively.All these modes at low amplitude are bulk modes, see the modes at A = 0.1 and 0.4 in figures 10(c)-(e).However, although the linear modes of µ = 4.0681 and 4.0683 evolving into the low amplitude region remain as bulk modes numbered {1, 2} and {6, 7}, respectively, they finally develop into a corner mode and an edge mode in the large amplitude, numbered {3, 4, 5} and {8, 9, 10}.That is, the amplitude induced bulk-to-corner and bulk-to-edge transitions occur.According to the bulk-edge/corner correspondence, there is indeed a higher-order topological phase transition induced by the amplitude, consistent with the prediction from the nonlinear bulk polarization, see figure 9.The physical mechanism of this phase transition is that the intrinsic nonlinear response is amplitude dependent.As the amplitude increases, the bulk modes in the high-frequency band move to lower frequencies and the bulk modes in the low-frequency band move to higher frequencies, as shown in figure 8(d), leading to band inversion, i.e. the topological phase transition.In addition, the IPR of the corner state corresponding to figure 10(c) is shown in figure 10(b).It is observed that the IPR also has a maximum, and finally decreases due to strong nonlinearity, indicating that a localized corner mode may emerge into the bulk by further   increasing the amplitude.It is noted that the IPR drops and changes dramatically at small amplitudes due to the lack of localization of the bulk mode.
We must emphasize that the nonlinear higher-order topological states are amplitude-dependent.We demonstrate such a property in the time domain.The time-domain evolution of the nonlinear modes is calculated by directly solving equation ( 1) using the Runge-Kutta method.We observe three corner modes with A = 1.5, in which the non-degenerate one shown in figure 10(c

Conclusions
In this paper, we propose a nonlinear elastic higher-order topological Kagome lattice and a HBM based method to calculate the approximate nonlinear modes to identify higher-order topology of the nonlinear lattice.The effects of cubic nonlinearity, both hardening and softening, on the higher-order topological state of the lattice are explored.Unusual delocalization of topological corner states induced by nonlinearity is revealed.Intracell hardening nonlinearity can directly delocalize topological corner states, while intracell softening nonlinearity first enhances localization and then delocalizes topological corner states.From the higher-order topology in the nonlinear lattice, quantized nonlinear bulk polarization with respect to amplitude is obtained, which can characterize the higher-order topological phase transition.It is revealed that such nonlinear phase transition is accompanied by an unusual band crossing.We also demonstrate amplitude induced bulk-to-edge and bulk-to-corner transitions in a finite nonlinear lattice, which can be predicted from the bulk polarization calculated by our method.We show that such amplitude dependent corner states can persist for a long time in the time domain, while such corner states with small amplitudes immediately merge into the bulk.The findings can inspire the studies of nonlinear HOTIs in other branches of physics, such as topological circuits and topological acoustics and may also enlighten new device designs for elastic wave control, vibration control, and mechanical computing [60][61][62].
Appendix B. Nonlinear algebraic equations obtained from Galerkin procedure

Appendix C. Disorder effect on nonlinear higher-order topological states
For exploring the robustness of nonlinear higher-order topological states, we consider a 28-cell hardening-nonlinearity finite lattice with mass disorder of 5%, and k 1 = 0.

Figure 1 .
Figure 1.Nonlinear elastic Kagome lattice.(a) Designed model.(b) Schematic of the lattice.a1 and a2 are the primitive vectors of the lattice.The inset in (b) represents the hexagonal and rhombic Brillouin zones.b1 and b2 are reciprocal primitive vectors.

Figure 3 .
Figure 3.The eigenvalues and eigenmodes of the finite lattice with fixed k2 = 1.(a) The eigenvalues of the lattice as a function of the normalized linear stiffness ratio k1/ k2.The grey area in the background represents the eigenvalues of band structure of the infinite lattice with varying k1/ k2.(b) The eigenvalues of the lattice with k1/k2 = 0.25.(c), (d), (e) The eigenmodes corresponding to the eigenvalues marked in (a) or (b).

Figure 4 .
Figure 4.The effects of hardening nonlinearity on band structure and band topology.(a) Amplitude-dependent Band structures.(b) Linear and (c), (d) nonlinear Wannier bands of the acoustic band with different amplitudes.The blue dotted line in (b)-(d) represents the Wannier band, and the red dashed line represents the mean value of the Wannier band.

Figure 5 .
Figure 5.The effects of softening nonlinearity on band structure and band topology.(a) Amplitude-dependent Band structures.(b) Linear and (c), (d) nonlinear Wannier bands of the acoustic band with different amplitudes.The blue dotted line in (b)-(d) represents the Wannier band, and the red dashed line represents the mean value of the Wannier band.

Figure 6 .
Figure 6.Effects of hardening nonlinearity on topological states.(a) The amplitude-dependent eigenvalues.(b) The IPR of one corner mode.The amplitude-dependent (c) corner states, (d) edge states, and (e) bulk states, corresponding to the eigenvalues marked in (a).The grey areas in (a) represent the eigenvalues of the corresponding Bloch bands.

Figure 7 .
Figure 7. Effects of softening nonlinearity on topological states.(a) The amplitude-dependent eigenvalues.(b) The IPR of one corner mode.The amplitude-dependent (c) corner states, (d) edge states, and (e) bulk states, corresponding to the eigenvalues marked in (a).The grey areas in (a) represent the eigenvalues of the corresponding Bloch bands.

Figure 9 .
Figure 9. Amplitude induced higher-order topological phase transition.(a) The bulk polarizations.The white area corresponds to the band crossing region.(b)-(i) Wannier bands for various amplitudes.

Figure 10 .
Figure 10.Amplitude induced topological states.(a) Eigenvalues.(b) The IPR of a corner mode shown in (c).(c) Amplitude induced corner mode from the linear mode of µ = 4.0681.(d) Amplitude induced edge mode from the linear mode of µ = 4.0683.(e) Nonlinear bulk modes.The grey areas in (a) represent the eigenvalues of the corresponding Bloch bands.
) has µ = 4.468 and the other two are degenerate with µ = 4.556.Using the three corner modes as initial conditions, the responses within 50 time periods are shown in figures 11(a), (c) and (e).The results show that these corner modes persist for a long time.However, when the amplitude of these modes is reduced to a small amplitude, they do not persist and become bulk modes and lose localization largely, see figures 11(b), (d) and (f).For the time evolution of the nonlinear corner mode with A = 1.5 and µ = 4.468, i.e. the mode in figure11(a), the continuous time signals of a corner mass, an edge mass and a bulk mass and their Fourier transforms are presented in figure12.The maximum of the Fourier spectra is at µ = 4.468, agreeing closely with the eigenvalue of the nonlinear mode calculated from the HBM.This further confirms the existence of the nonlinear topological corner states and the validity of our method.

Figure 11 .
Figure 11.The evolution of the nonlinear corner states varying amplitude in the time domain.(a), (b) The non-degenerate corner mode with A = 1.5 and µ = 4.468.(c), (d) The first one and (e), (f) the second one of the two degenerated corner modes with A = 1.5 and µ = 4.556.

Figure 12 .
Figure 12.The Fourier transforms of the time signals of the nonlinear corner mode with A = 1.5 and µ = 4.468.(a) The locations of masses for extracting time signals.The time signal and the Fourier transform of (b) the corner mass, (c) the edge mass and (d) the bulk mass.

25 , k 2 = 1 ,
k c1 = 0.1, and k c2 = 0.The masses in each cell remain the same, but different cells have different mass densities.The mass distribution is a uniform random distribution between [0.975, 1.025].The nonlinear eigenvalues and eigenmodes of the disorder finite lattice with varying amplitude are shown in figureC1.The results show that the nonlinear topological corner and edge states are robust and can be immune to a certain degree of disorder.

Figure C1 .
Figure C1.Mass disorder effect on nonlinear topological states in the finite lattice with k1 = 0.25, k2 = 1, kc1 = 0.1, and kc2 = 0. (a) Linear eigenvalues.(b) The amplitude-dependent eigenvalues.The amplitude-dependent (c) corner states, (d) edge states, and (e) bulk states, corresponding to the eigenvalues marked in (b).The grey area in the background in (b) represents the eigenvalues of the band structure of the corresponding infinite lattice without mass disorder with varying amplitude.