Topological States Due to Third‐Neighbor Coupling in Diatomic Linear Elastic Chains

The vibrations of a diatomic linear chain exhibit two bands, the acoustic and the optical one. In the case of equal masses and nearest‐neighbor spring constants, no gap exists. The effects of nearest, next‐nearest, and third‐nearest neighbor coupling with different spring constants are analyzed. Nearest neighbor coupling leads to two topologically different bulk states with winding numbers ν = 0 and −1. Next‐nearest neighbor coupling does not change the topological properties and introduces only a trivial “Semenov” gap. Third‐nearest neighbor coupling leads to four distinct topological bulk phases with winding numbers ν=0,±1,and−2 .


Introduction
The concept of topological phases is nowadays considered for surprisingly many systems. The ideas were initially developed for solitons in polymers. [1][2][3] The concepts for the explanation of quantum Hall states [4] are closely related to the physics in quasi-1D conductors as pointed out in the study by Schrieffer and coworkers. [5,6] The lattice vibrations in a solid are typically introduced in textbooks with a linear chain model. To allow for acoustic and optical branches, a diatomic unit cell is used. [7][8][9] Herein, we like to point out some topological aspects of this rather simple classical system.
The diatomic linear chain with nearest-neighbor and thirdneighbor coupling is schematically shown in Figure 1. We will show that the diatomic chain with nearest-neighbor coupling has two different topological bulk states. It is generally known that going beyond nearest-neighbor coupling can change the topological properties. [10,11] The introduction of next-neighbor coupling (Figure 1a) only opens a trivial gap and is thus considered only briefly. The main aspect of this article is to analyze the effect of third-neighbor coupling (Figure 1b) on the dispersion and the bulk winding number; it will lead to four distinct topological bulk phases.
A number of general [12,13] and specific linear [14,15] mechanical systems have been already analyzed regarding their topological properties. However, we find the present system as the simplest mechanical one for such study and many explicit results can be given.
We set M 1 ¼ M 2 ¼ M ¼ 1 as different masses only introduce a trivial gap, as discussed briefly in the following paragraphs. The nearest-neighbor coupling spring constants are C 1 and C 2 . The next-neighbor coupling constants are C A and C B , the third-neighbor coupling constants are C 0 1 and C 0 2 , as shown in Figure 1. As unit cell, we chose an A-B dimer and the Brillouin zone runs from Àπ to π, i.e., the lattice constant is set to a ¼ 1. The dispersion relations are derived from setting up the total elastic energy and Newton's equations of motion for solutions of type ∝ exp½iðkx À ωtÞ. The related eigenvalue problems deliver eigenfrequencies and eigenmodes for the amplitudes u ¼ ðu A , u B Þ of the A and B sites. For the bulk modes, Bloch-like functions are taken, e.g., for neighboring unit cells, u nAE1 ¼ expðAEikÞu n .

Nearest Neighbor Coupling
It is well known that a gap forms at the X point for C 1 6 ¼ C 2 . The lower branch is termed "acoustic" branch, the upper one the "optical." At the X points (k ¼ AEπ), the lower and the upper branches have the frequencies (in units of the width of the gap ω g ðXÞ ¼ ω þ ðX Þ À ω À ðXÞ being At the Γ point

Next-Nearest Neighbor Coupling
At the X points, the lower and the upper branches have the frequencies DOI: 10.1002/pssb.202000176 The vibrations of a diatomic linear chain exhibit two bands, the acoustic and the optical one. In the case of equal masses and nearest-neighbor spring constants, no gap exists. The effects of nearest, next-nearest, and third-nearest neighbor coupling with different spring constants are analyzed. Nearest neighbor coupling leads to two topologically different bulk states with winding numbers ν ¼ 0 and À1. Next-nearest neighbor coupling does not change the topological properties and introduces only a trivial "Semenov" gap. Third-nearest neighbor coupling leads to four distinct topological bulk phases with winding numbers ν ¼ 0, AE1, and À2.
with a gap forming for C 1 6 ¼ C 2 and/or and (4) remain valid.

Third-Neighbor Coupling Dispersion Relation
Herein, we consider The dispersion is shown in Figure 2a for C 0 ¼ 1=2 and in addition to the zero gap at the X point (k ¼ AEπ), another degeneracy exists at k ¼ AE2π=3 which we denote as AEX 0 . Generally, for C 0 1 ¼ C 0 2 , the chain is monoatomic and has actually a periodicity of a/2 and the doubled Brillouin zone, as shown in Figure 2b, is actually folded. In Figure 2b, also the dispersions for C 0 ¼ 0, 1=8, 1=3, 1=2 and C 0 ¼ 1 are shown.
A variation of C 0 1 and C 0 2 will open these two gaps and generate topologically different states. Around the Γ point in the lower branch is a region of imaginary frequencies which indicates frustration.
A representation of the ðC 0 1 , C 0 2 Þ parameter space is shown in Figure 3b. Along the two diagonals, the band structure has no gap, as detailed in the following.
www.advancedsciencenews.com www.pss-b.com Therefore, in summary, a band structure with zero gap exists for C 0 2 ¼ C 0 1 and C 0 2 ¼ 1 À C 0 1 . These two cases are represented by the diagonal black lines in Figure 3b. For all other combinations of C 0 1 and C 0 2 , the band structure has a gap. For these cases, we will calculate the winding number as topological invariant in the next section.

Winding Numbers
From the eigenproblem, the eigenvectors of type u AE ðkÞ ¼ ðu A,AE ðkÞ, u B,AE ðkÞÞ ¼ ð1, exp½∓iδϕðkÞÞ with the relative phase δϕ of A and B sites are obtained for both bands (AE indicating the upper and lower bands); the phases of the lower and upper bands are related by a factor of À1. The important quantity is the change of phase within a band when traversing the Brillouin zone from Àπ to þπ. As the physical properties are periodic with the Brillouin zone, this total phase change must be an integer multiple of 2π. To properly account for the change in phase, the property A AE a 1D Berry connection, is calculated and integrated over the Brillouin zone, resulting in the associated Berry phase. The resulting winding number is (for a gapped system) an integer as topological invariant of a 1D BDI-type system [12] ν ¼ 1 2π

Nearest Neighbor Coupling
The winding number for the diatomic chain with nearest neighbor coupling is (for the nonzero gap case) ν ¼ 0 for C 1 > C 2 and ν ¼ À1 for C 1 < C 2 . This is similar to the Su-Schrieffer-Heeger tight-binding model [1] for linear molecules with alternating bonds and nearest neighbor hopping parameters and related works. [16,17] Thus, these two cases of intra-and interdimer spring constants are topologically different. This seems to be largely ignored in literature so far and presents a simple example of topologically different states. In the zero-gap case (C 1 ¼ C 2 ), each band has a formal winding number of À1/2. We note that different masses M 1 6 ¼ M 2 introduce a gap ("Semenov" insulator [18] ) but do not lead to a change of winding number; in particular, for the gapless case C 1 ¼ C 2 , different masses leave the winding number for the two bands at À1/2 and topologically the bulk band structure is trivial. This can also be seen (without further discussion here) from the properties of a finite chain (due to the general bulk-boundary correspondence). For a topologically nontrivial gap as for C 1 6 ¼ C 2 , end/edge states can develop for a finite chain with fixed ends. For M 1 6 ¼ M 2 and C 1 ¼ C 2 , a frequency gap develops but no end states exist for any finite chain.

Next-Nearest Neighbor Coupling
The winding numbers for the cases C 1 6 ¼ C 2 do not change upon the introduction of next-nearest neighbor coupling. Thus, the topological properties are not changed, as shown in Figure 3a.
In the case of C 1 ¼ C 2 , values C A 6 ¼ C B introduce a frequency gap at the X points but the winding numbers of the two branches remain À1/2 as in the zero-gap case. Thus, the cases C 1 ¼ C 2 and C A 6 ¼ C B are topologically trivial (again a "Semenov" insulator [18] ).

Third-Nearest Neighbor Coupling
For the cases with nonzero gap, the winding number can take four values, ν ¼ 0, AE1, À2 as shown in Figure 3b. For selected parameters, these four cases are shown with their dispersion and the phase visualized as color in Figure 4. The phase of the lower band and for the four cases is shown also as line graphic in Figure 5. At the Γ and X points, the modes have defined even (δϕ ¼ 2nπ, n ∈ ℤ) or odd (δϕ ¼ ð2n þ 1Þπ, n ∈ ℤ) parity. Also, around the second gap, the parity is defined for a certain k. Across the Brillouin zone, the parity changes 2ν times. The case ν ¼ 0 is topologically the simplest as parity does not change within a band. This means that the typical "expectation," that the lower (acoustic) band has even parity (A and B sites are in phase) and the upper (optical) band has odd parity (A and B sites are out-of-phase), is fulfilled.
We note that, however, the choice of unit cell is free and could also be a B-A dimer. Then the winding numbers will be different as the role of C 0 1 and C 0 2 are exchanged but the general scheme of Figure 3b remains the same.

Higher-Order Neighbor Coupling
As generalization of Equation (6), coupling of higher-order n-th neighbors will introduce terms of type cos nk. Thus, in general, further degeneracy points are possible for certain elastic parameters. These can split into gaps and generate topological states with higher winding numbers; for odd n ¼ 2m þ 1 (m ≥ 0), we expect ν ¼ Àm À 1, : : : , m.

Conclusions and Outlook
The rather elementary model of the diatomic linear chain can serve as simple example to understand the concept of topologically different bulk states. For the trivial chain with equal nearestneighbor coupling, the third-neighbor coupling has gapless states for two conditions, namely C 0 2 ¼ C 0 1 and C 0 2 ¼ 1 À C 0 1 . For the other cases, the dispersion has a gap (and is an insulator in the terminology of topological states) and takes one of four distinct topological states. However, while our model can be executed with arbitrary model parameters, realistic parameters are C 0 ( C and the cases ν ¼ 0 and ν ¼ À1 known from nearestneighbor coupling dominate, whereas the cases ν ¼ þ1 and ν ¼ À2 will hardly occur naturally.
Our model has direct application to the maybe unexpected mechanical properties of linear molecules with alternating "short" and "long," i.e., stiffer and softer, bonds such as, for example, in trans-polyacetylene. However, the topological states arise only when the ends of the chain are fixed, and the first spring is the stiffer one. We note that if only one end is fixed like that, accordingly one topological state can exist at this end. Such could be the case for molecules in a confined space or if they are sufficiently strongly attached or bonded to another entity. As is known [12,13] 2D and 3D mechanical systems also can exhibit topological states, and it remains to be seen whether, e.g., topological vibrational motions in 2D molecules become an interesting field.

Conflict of Interest
The author declares no conflict of interest.
The inset represents the ðC 0 1 , C 0 2 Þ parameter space, as shown in Figure 3b. The circles depict the parameter positions for the five dispersions and match their frame colors. www.advancedsciencenews.com www.pss-b.com