In-Gap Edge and Domain-Wall States in Largely Perturbed Phononic Su-Schrieffer-Heeger Lattices

1. Introduction

Topological insulators (TIs) are intrinsically electrical insulators with conducting surfaces/edges/corners when interfaced with trivial insulators, including vacuum. These surface/edge/corner states result from TI bulk properties, i.e., topological invariants, thus robust against local perturbations and disorders [1], making them ideal candidates for various promising applications in electronics strict in dissipation, especially in quantum-computing systems [2], and have thus attracted significant attention. Moreover, in recent years, these topologically protected non-dissipative localized states have inspired studies of classical analogs of TIs in photonic [3,4,5,6,7], magnetic [8,9,10], and mechanical [11,12,13,14,15,16,17] systems.

In mechanical systems, a similar topological invariant can also be derived from the spectral evolution of eigenvectors of the dynamical matrix (or the compatibility matrix) from a unit cell analysis. The integer topological invariant can then inform of the numbers and types of topologically protected surface/edge/corner states confining static deformation [18,19,20,21,22] or vibration [11,13,14,15,16,17,23,24,25] within a bulk bandgap (which is a frequency range with no eigenfrequency solutions, $i.e.$, bulk wave propagation) when a material/structure with a non-zero topological invariant, i.e., a topologically non-trivial phase, forms a domain wall with another domain characterized by a zero-topological invariant, i.e., a topologically trivial phase, including vacuum. Such a bulk property reflected as surface/edge/corner states at a domain wall or edge is usually referred to as the bulk-edge correspondence.

A commonly used example to illustrate this bulk-edge correspondence is the one-dimensional (1D) Su-Schrieffer-Heeger (SSH) lattice, as shown in Figure 1(a). With alternating spring constants connecting identical masses, m, a bulk bandgap can be opened as shown in Figure 1(b). It has been well established that the larger the difference in these spring constants is, the wider the bandgap we can obtain. In the existing literature, the finite-frequency in-gap topological modes are commonly created by seaming two domains with a difference in topology, here, we refer to as the topologically protected domain-wall states (TPDWSs) [15,17,26], or by simply placing a lattice with a free or fixed edge, usually cited as the “topologically protected edge states" (TPESs) [27,28,29,30,31]. However, as discussed in Refs. [28,29], the existence of these edge states and their frequencies (in case they exist) depend heavily on the boundary conditions in finite lattices, making these edge states less robust. This is because such boundaries perturb the chiral symmetry of the system’s dynamical matrix. Strictly speaking, these edge states are no longer topologically protected due to the loss of chiral symmetry. In fact, this is also the case if a domain boundary is formed between two domains with different topologies when a large perturbation to the unit cell symmetry is introduced, which, however, has received much less attention since most discussions on TPDWSs introduce a small perturbation in spring stiffness to open the bandgap in order to preserve the chiral symmetry to some extent. Nonetheless, a large bandgap is usually desirable in many applications requiring vibration mitigation within a certain frequency range [32,33,34,35]. Hence, it is important to identify the conditions for the survivability of the in-gap states. Since these in-gap states are deviations from TPESs or TPDWSs, some topological features at these boundaries still exist even with large chiral symmetry perturbation.

In this work, we will delve into SSH systems with large symmetry perturbations at the domain boundary and the unit cell. Simple SSH chains with nearest-neighbor interactions only and the more complex SSH networks with beyond-nearest-neighbors (BNNs) will both be taken into account, the latter of which is gaining rising attention lately due to their roton-like dispersion relations similar to those in correlated quantum systems [36,37,38,39,40,41], as well as their unconventional topological states associated with the BNN coupling [17,42,43,44]. The survivability of the in-gap states will be analyzed, which will further instruct the design of large-bandgap structures in the continuum regime.

2. Results and Discussion

2.1. Su-Schrieffer-Heeger Systems with Nearest-Neighbor Interactions

To simplify the illustration, we only consider displacements of identical masses, m, connected by alternating nearest-neighbor springs with spring constants $k_1$ and $k_2$ along the chain, as shown in Figure 1(a). The governing equations of a lattice unit cell can be expressed as

$$\begin{array}{c}\hfill m{\ddot{x}}_1^p=k_1(x_2^p-x_1^p)-k_2(x_1^p-x_2^{p-1}),\end{array}$$

(1)

$$\begin{array}{c}\hfill m{\ddot{x}}_2^p=k_2(x_1^{p+1}-x_2^p)-k_1(x_2^p-x_1^p),\end{array}$$

(2)

where displacements of the two masses in the p-th cell are denoted as $x_1^p$ and $x_2^p$, respectively. The displacements of masses in any unit cell p at time t can be expressed by the ones within a reference unit cell using a plane-wave solution in combination with Bloch-Floquet periodic boundary conditions:

$$\begin{array}{c}\hfill x^p\left(t\right)=\tilde{x}\left(q\right)e^{i(pqa-\omega t)},\end{array}$$

(3)

where $\omega$ is the vibration frequency, $x^p$ are the displacements of the p-th cell with $x^p=[x_1^p,x_2^p]$, q is the wave number, which is inversely proportional to the wavelength $\lambda$, i.e., $q=2\pi /\lambda$, a denotes the lattice constant, $\tilde{x}\left(q\right)$ are displacements within the reference unit cell. Substituting this expression in Eqns. 1 and gives:

$$\begin{array}{c}\hfill [D\left(q\right)-{\omega }^{2}m]\tilde{x}\left(q\right)=0,\end{array}$$

(4)

where $D\left(q\right)$ is the stiffness (or dynamical) matrix of the periodic system:

$$D\left(q\right)=\left[\begin{array}{cc}{k}_1+k_2& -k_1-k_2{e}^{-iqa}\\ -k_1-k_2{e}^{iqa}& k_1+k_2\end{array}\right],$$

(5)

The square root of the eigenvalues of $D\left(q\right)$ gives the phonon band diagram, as presented in Figure 1(b). With $k_1=k_2$, there is no bandgap between the two bands. When $k_1\ne k_2$, a bandgap is created between the two phonon bands due to the breaking of the unit cell space-inversion-symmetry (SIS). Although the bandgap does not alter if $k_1>k_2$ or vice versa, The two choices of gauge result in two different topological states since the transition from one to another unavoidably passes the non-gap state, $i.e.$, non-gapped transition, as shown in the middle panels of Figure 1(b–d). To characterize the topology of such a 1D system, the Zak phase [45] measuring the rotation of eigenvectors in the unit cell, or the winding number n are often evaluated, as shown in Figure 1(d). The latter of which is defined as

$$n={\int }_{-\pi /a}^{\pi /a}\frac{1}{4\pi i}\mathrm{tr}\left[{\sigma }_z\tilde{D}{\partial }_q\tilde{D}\right]dq,$$

(6)

where ${\sigma }_z$ is the unitary matrix, $\tilde{D}$ is the chiral matrix $D\left(q\right)$ after subtracting all diagonal elements:

$$\tilde{D}\left(q\right)=D\left(q\right)-(k_1+k_2)\mathbf{I},$$

(7)

where $\mathbf{I}$ is the identity matrix. The eigenvalues of $\tilde{D}\left(q\right)$ present to be symmetric about the x-axis, as shown in Figure 1(c).

As we can see, when $k_1>k_2$, $n=0$, indicating a trivial intra-cell-hopping phase. On the contrary, when $k_1<k_2$, $n=1$, suggesting a topologically nontrivial inter-cell-hopping phase. Such a difference in topology due to gauge choice suggests that if connecting two domains with these two different topologies, there exists a localized mode with a frequency located within the bandgap at the domain wall.

In this section, we will discuss the effect of the boundary conditions on the existence of in-gap states when a simple mass-spring lattice shown in Figure 1(a) is interfaced with a vacuum (Subsection In-Gap Edge States in Finite Lattices) and with a domain with an opposite gauge in the unit cell (Subsection In-Gap Domain-Wall States in Infinite Lattices) with various $\delta k$ characterizing the difference between $k_1$ and $k_2$, defined as $k_1=1-\delta k$ and $k_2=1+\delta k$. Here, we consider a wide range of $\delta k$, up to $\pm 0.9$, which is $90\%$ of the average spring stiffness (we set $\frac{k_1+k_2}{2}=1$). Such a big difference in the spring constants will create a large bulk bandgap, which, as mentioned in the Introduction, is desirable for many applications. However, the survivability of the in-gap states depends on the specific boundary condition.

2.1.1. In-Gap Edge States in Finite Lattices

To start with, we take a finite lattice containing 20 unit cells presented in Figure 1(a) with free boundaries, as shown in Figure 2(a). The stiffness matrix of such a finite system is:

$$D\left(q\right)=\left[\begin{array}{ccccccc}{k}_1& -k_1& 0& & 0& 0& 0\\ -k_1& k_1+k_2& -k_2& \cdots & 0& 0& 0\\ 0& -k_2& k_1+k_2& & 0& 0& 0\\ & ⋮& & \ddots & & ⋮& \\ 0& 0& 0& & k_1+k_2& -k_2& 0\\ 0& 0& 0& \cdots & -k_2& k_1+k_2& -k_1\\ -0& 0& 0& & 0& -k_1& k_1\end{array}\right].$$

(8)

As we can see, $D\left(q\right)$ is not a chiral matrix due to non-identical diagonal elements at the two free boundaries. Hence, no in-gap states exist, as shown in Figure 2(a–d), regardless of $\delta k$.

The in-gap edge states can be easily acquired by fixing the two ending masses with a spring, $k_{fix}=k_2$, connected to a wall, as shown in Figure 2(e), making the $D\left(q\right)$ chiral:

$$D\left(q\right)=\left[\begin{array}{ccccccc}{k}_1+k_2& -k_1& 0& & 0& 0& 0\\ -k_1& k_1+k_2& -k_2& \cdots & 0& 0& 0\\ 0& -k_2& k_1+k_2& & 0& 0& 0\\ & ⋮& & \ddots & & ⋮& \\ 0& 0& 0& & k_1+k_2& -k_2& 0\\ 0& 0& 0& \cdots & -k_2& k_1+k_2& -k_1\\ -0& 0& 0& & 0& -k_1& k_1+k_2\end{array}\right].$$

(9)

Subtracting the diagonal elements as in Eqn. 7 also gives us a chiral matrix $\tilde{D}\left(q\right)$. In this case, as long as the SSH terminating cells present a nontrivial gauge, $i.e.$, $k_1<k_2$ or $\delta k>0$, and are interfaced with vacuum – a trivial domain, the in-gap edge states guarantee to exist regardless of the magnitude of $\delta k$, as evident in Figure 2(f–h). Moreover, due to the strict chiral symmetry Eqn. 9 presents, all the in-gap edge states are located exactly at $\omega /{\omega }_0=\sqrt{(k_1+k_2)/m}$ in the eigenfrequency plot in Figure 2(f), or ${\omega }^2/{\omega }_0^2=0$ in the $\tilde{D}\left(q\right)$ eigenvalue plot in Figure 2(g), which we refer to as the mid-gap state in our discussion. These in-gap edge states completely follow the definition of TPESs, also known as the Jackiw-Rebbi zero modes [46].

However, despite being topologically protected, these TPESs are prone to boundary conditions. A small perturbation at the edges will drift the TPESs away from the mid-gap state, or make them disappear, as shown in Figure 2(i,j). Thus, topology alone can no longer guarantee the existence of TPESs. A chiral symmetry of the entire system is crucial to achieve TPESs. On the other hand, even when the chain terminates with topologically trivial unit cells on both ends, one in-gap state on both ends may still emerge with a sufficiently large $k_{fix}$, as presented in Figure 2(k). This in-gap state, nonetheless, is merely a trivial edge state due to the termination of periodicity.

2.1.2. In-Gap Domain-Wall States in Infinite Lattices

When seaming two topologically different domains, as shown in Figure 3(a), to obtain TPDWSs, a chiral symmetry of the dynamical matrix, $D\left(q\right)$, of the entire system must also be maintained. Without any modification, $D\left(q\right)$ of a supercell shown in Figure 3(a), in which the Bloch-Floquet boundary conditions are applied to the top and bottom masses, writes:

$$D\left(q\right)=\left[\begin{array}{cccccccccc}{k}_1+k_2& -k_1& 0& & & & & 0& 0& -k_2{e}^{-iqa}\\ -k_1& k_1+k_2& -k_2& & & \cdots & & 0& 0& 0\\ 0& -k_2& k_1+k_2& & & & & 0& 0& 0\\ & & & \ddots & & & & & & \\ & ⋮& & -k_1& & 2k_1& -k_1& & ⋮& \\ & & & & & & \ddots \\ 0& 0& 0& & & & & k_1+k_2& -k_2& 0\\ 0& 0& 0& & & \cdots & & -k_2& k_1+k_2& -k_1\\ -k_2{e}^{ika}& 0& 0& & & & & 0& -k_1& k_1+k_2\end{array}\right],$$

(10)

which is not a chiral matrix due to the domain-wall mass connected to $k_1$ on both sides. Although we can achieve a chiral matrix by pinning the domain-wall mass, it is not realistic in practice and highly challenging when $k_1>k_2$.

When the difference between $k_1$ and $k_2$, or $\left|\delta k\right|$, is small, $D\left(q\right)$ does not significantly deviate from a chiral matrix, thus the in-gap state is still adjacent to the middle of the bandgap, as shown in Figure 3(b,c). However, when $k_1$ and $k_2$ are largely different, $D\left(q\right)$ can no longer be approximated as a chiral matrix. Interestingly, when $\delta k<0$, $i.e.$, $k_1>k_2$, the in-gap state is intact. The breaking of the chiral symmetry of $D\left(q\right)$ results in an additional trivial edge mode arising above the upper limit of the bulk mode. On the contrary, when $\delta k>0$, the in-gap state drops quickly near the lower bulk mode as $\delta k$ increases. Although unlike in finite lattices, such an in-gap state always survives, it is not easy to detect due to its proximity to the bulk mode. Mode shapes when $\delta k=\pm 0.9$ are presented in Figure 3(d). The mid-gap state when $\delta k=-0.9$ decays rapidly due to a large bandgap. In contrast, when $\delta k=0.9$, the domain-wall state decays slowly into the bulk since its frequency is merely slightly above the bulk mode.

2.2. Su-Schrieffer-Heeger Systems with Beyond-Nearest-Neighbor Interactions

Introducing non-local or BNN interactions further complicates the discussion. In this section, we will discuss systems with identical third-nearest neighbors (TNNs) but the nearest neighbors are non-identical, $i.e.$, $k_1^{\prime }=k_2^{\prime }$ and $k_1\ne k_2$, and vice versa, as shown in Figure 4.

The additional TNNs connect the current cell, p, with the one beyond its immediate neighbors, $i.e.$, $p\pm 2$, modifying the governing equations to be:

$$\begin{array}{c}\hfill m{\ddot{x}}_1^p=k_1(x_2^p-x_1^n)-k_2(x_1^p-x_2^{p-1})+k_1^{\prime }(x_2^{p+1}-x_1^p)-k_2^{\prime }(x_1^p-x_2^{p-2}),\end{array}$$

(11)

$$\begin{array}{c}\hfill m{\ddot{x}}_2^p=k_2(x_1^{p+1}-x_2^p)-c_1(x_2^p-x_1^p)+k_2^{\prime }(x_1^{p+2}-x_2^p)-k_1^{\prime }(x_2^p-x_1^{p-1}).\end{array}$$

(12)

Plugging in Eqn. (3), we get the stiffness matrix as:

$$D\left(q\right)=\left[\begin{array}{cc}{k}_1+k_2+k_1^{\prime }+k_2^{\prime }& -k_1-k_2{e}^{-iqa}-k_1^{\prime }{e}^{iqa}-k_2^{\prime }{e}^{-2iqa}\\ -k_1-k_2{e}^{iqa}-k_1^{\prime }{e}^{-iqa}-k_2^{\prime }{e}^{2iqa}& k_1+k_2+k_1^{\prime }+k_2^{\prime }\end{array}\right].$$

(13)

The chiral matrix $\tilde{D}\left(q\right)$ can then be obtained from,

$$\begin{array}{c}\hfill \tilde{D}\left(q\right)=D\left(q\right)-(k_1+k_2+k_1^{\prime }+k_2^{\prime })\mathbf{I},\end{array}$$

(14)

The additional exponential terms in Eqns. 11 and , or in the off-diagonal terms in $D\left(q\right)$ (Eqn. 13) due to TNNs yield the bending of the dispersion curves in the irreducible Brillouin zone (IBZ), as shown in Figure 4(b,c), also known as the roton-like acoustic dispersion, similar to those discovered at cryogenic temperatures in correlated quantum systems [36,37,38,39,40,41].

Although phonon dispersions due to different nearest neighbors and TNNs are almost identical, their winding paths differ significantly. When $k_1>k_2$ and $k_1^{\prime }=k_2^{\prime }$, $n=0$ with two extra local loops formed along the path, while when $k_1=k_2$ and $k_1^{\prime }>k_2^{\prime }$, $n=-1$ since the circuit winds about the origin in the opposite direction (clockwise) to that of the main path. Such a difference in winding number indicates that the former lattice is topologically trivial, while the latter one is nontrivial. On the contrary, when $k_1<k_2$ and $k_1^{\prime }=k_2^{\prime }$, $n=1$ with three local loops formed along the path, while when $k_1=k_2$ and $k_1^{\prime }<k_2^{\prime }$, the circuit winds about the origin twice counterclockwise, thus $n=2$, indicating both being non-trivial. This suggests that when interfacing the lattice of $n\ne 0$ with a vacuum or a topologically trivial domain, n edge/domain-wall states should exist. However, previously, we have demonstrated that this is not necessarily correct. We mathematically demonstrated that the number of domain-wall states when seaming two topologically different domains is determined by the number of band crossings when $k_1=k_2$ and $k_1^{\prime }=k_2^{\prime }$ within the IBZ, or Dirac points, which is the phonon realization of the Jackiw-Rebbi theory [46]. We found that the actual number of TPDWSs can be characterized by the Berry connection calculated from the integrant of Eqn. 6, or local winding numbers, instead of the total winding numbers over the IBZ [17]. Nonetheless, the total winding numbers across the whole IBZ as shown in Figure 4(d) may still inform the number of edge states when a lattice is terminated and interfaced with a vacuum.

In this section, we will discuss the effect of a combination of the spring difference (both $\delta k$ and $\delta k^{\prime }$, the latter of which is defined as $k_1^{\prime }=1+\delta k^{\prime }$ and $k_2^{\prime }=1-\delta k^{\prime }$) and boundary conditions on the existence of in-gap states when a mass-spring lattice with TNNs shown in Figure 4(a) is interfaced with a vacuum (Subsection In-Gap Edge States in Finite Lattices with Third-Nearest Neighbors) and with a topologically different domain (Subsection In-Gap Domain-Wall States in Infinite Lattices with Third-Nearest Neighbors). We consider the scenarios across both a wide range of $\delta k$ and $\delta k^{\prime }$, up to 0.9, which opens a large bandgap. The conditions of maintaining the in-gap states will be explored.

2.2.1. In-Gap Edge States in Finite Lattices with Third-Nearest Neighbors

Taking a finite lattice containing 20 unit cells with free boundaries, as shown in Figure 5(a), yields the stiffness matrix as:

$$D\left(q\right)=\left[\begin{array}{ccccccc}{k}_1+k_1^{\prime }& -k_1& 0& & 0& 0& 0\\ -k_1& k_1+k_2+k_2^{\prime }& -k_2& \cdots & 0& 0& 0\\ 0& -k_2& k_1+k_2+k_1^{\prime }& & 0& 0& 0\\ & ⋮& & \ddots & & ⋮& \\ 0& 0& 0& & k_1+k_2+k_1^{\prime }& -k_2& 0\\ 0& 0& 0& \cdots & -k_2& k_1+k_2+k_2^{\prime }& -k_1\\ -0& 0& 0& & 0& -k_1& k_1+k_1^{\prime }\end{array}\right].$$

(15)

Again, $D\left(q\right)$ is not a chiral matrix due to non-identical diagonal elements close to the two free boundaries. Therefore, we expect there not to be any in-gap states. However, this is not the case when $k_1\ne k_2$ and $k_1^{\prime }=k_2^{\prime }=1$, as shown in Figure 5(b–d). On the other hand, when $k_1=k_2=1$ and $k_1^{\prime }\ne k_2^{\prime }$, there are no in-gap states, Figure 6(a–d), consistent with the observation when masses are only connected by nearest neighbors, as in Figure 2(a–d). In the former scenario, although these in-gap states are indeed edge states, the two in-gap mode numbers are 22 and 23 when $\delta k>0$, and 21 and 22 when $\delta k<0$. In the case of topological in-gap states, the mode numbers should be 20 and 21 for a finite chain containing 20 unit cells. In addition, there is a significant leakage of the edge modes into the bulk compared to those shown in Figure 5(h) when the edges are fixed so that $D\left(q\right)$ and $\tilde{D}\left(q\right)$ (as defined in Eqn. 14) are chiral:

$$\tilde{D}\left(q\right)=\left[\begin{array}{ccccccc}0& -k_1& 0& & 0& 0& 0\\ -k_1& 0& 0& \cdots & 0& 0& 0\\ 0& -k_2& 0& & 0& 0& 0\\ & ⋮& & \ddots & & ⋮& \\ 0& 0& 0& & 0& -k_2& 0\\ 0& 0& 0& \cdots & -k_2& 0& -k_1\\ -0& 0& 0& & 0& -k_1& 0\end{array}\right],$$

(16)

as shown in Figure 5(e). Hence, the two in-gap edge states in the lattice with free boundaries shown in Figure 5(a–d) are trivial edge modes, whose existence is independent of its topology.

On the other hand, when fixing the masses near the two edges with appropriate springs to the ground/wall to make $\tilde{D}\left(q\right)$ chiral as in Eqn. 16, the number of TPESs appear exactly as predicted by their corresponding winding numbers. When $k_1>k_2$ and $k_1^{\prime }=k_2^{\prime }$, there are no in-gap states since $n=0$. While when $k_1<k_2$ and $k_1^{\prime }=k_2^{\prime }$, one in-gap state exists at each edge, matching $n=1$, as shown in Figure 5(f,g). Similarly, when $k_1^{\prime }>k_2^{\prime }$ and $k_1=k_2$, $n=-1$, corresponding to the two mid-gap states when $\delta k^{\prime }<0$, Figure 5(g). The opposite arrangement of TNNs, $i.e.$, $\delta k^{\prime }>0$ yields two edge states at each end, as shown in Figure 5(h), matching its topological invariant, $n=2$.

2.2.2. In-Gap Domain-Wall States in Infinite Lattices with Third-Nearest Neighbors

By seaming two domains of lattices with different winding numbers, we can, again, form a domain wall, as shown in Figure 7(a). Nonetheless, with the incorporation of TNNs, the domain wall is not a single mass as shown in Figure 3(a). Here, the interfacial mass connected by $k_1$ on both sides serves as the domain-wall mass for the nearest neighbors. This domain-wall mass is also connected by $k_1^{\prime }$ on both sides. One additional mass on each side of this domain-wall mass is connected by $k_2^{\prime }$ on both sides. Hence, these three masses, altogether, serve as the domain-wall mass for the TNNs. $\tilde{D}\left(q\right)$ then writes:

$$\tilde{D}\left(q\right)=\left[\begin{array}{ccccccccc}0& -k_1& 0& & & & -k_2^{\prime }{e}^{-iqa}& 0& -k_2{e}^{-iqa}\\ -k_1& 0& -k_2& & \cdots & & 0& -k_1^{\prime }{e}^{-iqa}& 0\\ 0& -k_2& 0& & & & 0& 0& -k_2^{\prime }{e}^{-iqa}\\ & & & \ddots & & & & & & \\ & & & k_2^{\prime }-k_1^{\prime }& -k_1& 0& & & & \\ & ⋮& & -k_1& k_1-k_2+k_1^{\prime }-k_2^{\prime }& -k_1& & ⋮& \\ & & & 0& -k_1& k_2^{\prime }-k_1^{\prime }& & & & \\ & & & & & \ddots & & & \\ -k_2^{\prime }{e}^{iqa}& 0& 0& & & & 0& -k_2& 0\\ 0& -k_1^{\prime }{e}^{iqa}& 0& & \cdots & & -k_2& 0& -k_1\\ -k_2{e}^{iqa}& 0& -k_2^{\prime }{e}^{iqa}& & & & 0& -k_1& 0\end{array}\right],$$

(17)

As we can see, the spring arrangements at the three domain-wall masses prevent $\tilde{D}\left(q\right)$ from being chiral. Unless connecting them with extra springs to a fixed wall to remove the three non-zero diagonal elements in Eqn. 17, which is not practical, especially when $k_2^{\prime }>k_1^{\prime }$ or $k_1+k_1^{\prime }>k_2+k_2^{\prime }$, we expect to observe strong deviation of the in-gap states from mid-gap.

In the case of $k_1^{\prime }=k_2^{\prime }=1$, we have previously demonstrated that the winding numbers shown in Figure 4(d) can no longer correctly predict the number of TPDWSs. Instead, local winding numbers derived from the Berry connection inform us of three TPDWSs if $\tilde{D}\left(q\right)$ is set to be chiral [17]. Without any modification around the domain wall, as $\delta k$ increases, although the three in-gap states still exist, as shown in Figure 7(b) and (c), when $\delta k<0$, only one in-gap state remains at the mid-gap, two of them get extremely close to the bulk modes, resulting in significant leakage of the domain-wall states into the bulk, as shown in Figure 7(d). When $\delta k>0$, all three in-gap states survive in the bandgap, but with a large deviation from the mid-gap. Thus, the domain-wall states still leak into the bulk, as shown in fig. Figure 7(e), compared to those with small $\delta k$.

As we vary $\delta k^{\prime }$ while keeping $k_1=k_2=1$, both the winding number and Berry connection calculation predict that there should be three in-gap states at the domain wall. However, from Figure 7(f,g), this is only the case when $\delta k^{\prime }\to 0$. As $|\delta k^{\prime }|$ increases, when $\delta k^{\prime }<0$, only one mode remains mid-gap, one other gets extremely close to the lower bulk band, resulting in slow spatial decay from the domain boundary, and the third one completely disappears, as shown in Figure 7(h). While when $\delta k^{\prime }>0$, the two surviving in-gap states are located at the mid-gap state, thus, with rapid spatial decay, Figure 7(i), and the third in-gap mode no longer exists within the bulk bandgap. It, instead, migrates to above the top bulk band.

3. Conclusion

In this work, we use Su-Schrieffer-Heeger models to thoroughly compare the effect of various boundary conditions on the survivability of in-gap states within bulk bandgaps and their relationship with topologically protected edge/domain-wall states. We find that the in-gap edge states never exist when lattice boundaries are free since the stiffness matrix largely deviates from being chiral. The occasional observed localized edge modes only occur by accident due to specific spring arrangements. They are not topologically protected and leak much into the bulk. On the other hand, fixing the free ends to the ground with proper spring constants to ensure the chirality of the stiffness matrix guarantees the topologically protected edge states when the lattice arrangement is topological. The edge states are also strongly localized on the edges. On the other hand, the in-gap domain-wall states formed by two topologically different domains are much more tolerant to the non-chirality of the stiffness matrix, although not all in-gap states will survive with a large difference in the third-nearest neighbors. Interestingly, when the domain-wall mass is connected by stiffer springs, the mid-gap states are always retained and preserve most of the topological features, such as rapid spatial decay, which is not the case when the domain-wall mass is connected by softer springs.

Although we use discrete mass-spring models in our study, the conclusions are expandable to the continuum regime and can be used to distinguish and explain the difference between topological and trivial edge/domain-wall states in continuous structures in practical applications.