Topological Phase Transition in a One-Dimensional Elastic String System

Abstract

We show that topological interface mode can emerge in a one-dimensional elastic string system which consists of two periodic strings with different band topologies. To verify their topological features, Zak-phase of each band is calculated and reveals the condition of topological phase transition accordingly. Apart from that, the transmittance spectrum illustrates that topological interface mode arises when two topologically distinct structures are connected. The vibration profile further exhibits the non-trivial interface mode in the domain wall between two periodic string composites.

1. Introduction

The recent discovery of topological features in condensed matter systems [1,2,3,4,5,6,7,8], for example, chiral edge states [9,10,11] or effective magnetic fields [12] have attracted massive interests [13,14]. The topological phases in electronic systems can be characterized by the band topologies and the sum over their corresponding topological invariants below the Fermi level [12,15,16]. Among all the topological materials, a one-dimensional (1D) polyethylene chain governed by Su-Schrieffer-Heeger (SSH) model with staggered hopping parameters [17,18] is a well-studied example to exhibit topological nature. To study the band topology in a 1D periodic system with inversion symmetry, the Zak phase is introduced to facilitate the topological classification [2,19]. Furthermore, when two SSH chains with distinct staggered hopping are connected, interface states arise at the domain wall as long as the corresponding Zak phases for both subdomains are different. Recent researches have demonstrated that the Zak phase also relates to the reflection phase and surface impedance [20,21].

In a similar manner, the topological aspects can apply to classical systems. Several topological properties have been predicted and observed in photonic crystals, coupled with resonators and linear circuits [22,23,24,25,26,27,28]. While the research to topological phenomena has also been investigated in various classical wave models [29,30,31], topological properties of pre-tensioned strings remain unnoticed. Since the vibrating frequency and amplitude of a pre-tensioned string system can be easily measured, the system shows great potential to directly observe topological properties. The purpose of the study is to explore the topological phase transition in a 1D elastic string (ES) system through the analysis of the related topological interface mode. The main aims in this report are to identify the topological quantities in the band structure and to elucidate the relation between the interface mode and the band topologies of the two 1D periodic ES subsystems (See Figure 1).

2. A One-Dimensional Periodic Elastic String

2.1. Dispersion Relation

We begin by considering the wave equation of a 1D periodic ES in which the internal stress of the string is omitted. Assuming that the entire structure is stretched by an external force to provide the tension, along the z-axis, we can write down the Lagrangian density for the string wave.

$$L=\frac{\rho }{2}{\left({\partial }_{t}u\right)}^2-\frac{\tau }{2}{\left({\partial }_{z}u\right)}^2,$$

(1)

where u, τ and ρ are the transverse displacement, the string tension, and the linear density of the string, respectively. According to the Euler-Lagrange equation, the wave equation is therefore given by

$${\partial }_z\left(\tau {\partial }_{z}u\right)=\rho {\partial }_t^{2}u.$$

(2)

Due to the periodicity of density ρ, a Bloch theorem $u=e^{-iqz}{u}_q(z)$ is introduced with the time convention $e^{i\omega t}$, where $u_q(z)=u_q(z\pm \ell )$ is the periodic Bloch wave, q is Bloch wave vector, and ℓ is lattice constant. The unit cell contains two homogeneous string segments with lengths ℓ₁ and ℓ₂, and their densities are ρ₁ and ρ₂. By imposing the boundary conditions u and $\partial u/\partial z$ at the segment-segment interfaces, one can derive the transfer matrix $T$ for one-unit cell. Since the transverse displacement in the first string segment (with length ℓ₁) of the jth unit cell is expressed as $u_j(z)=a_j{e}^{-i{k}_1\left(z-j\ell \right)}+b_j{e}^{i{k}_1\left(z-j\ell \right)}$, via periodic boundary condition u(z + ℓ) = e^-iqℓu(z), the matrix equation reads [32]:

$$\left(\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right)\left(\begin{array}{c}{a}_j\\ b_j\end{array}\right)=e^{iq\ell }\left(\begin{array}{c}{a}_j\\ b_j\end{array}\right),$$

(3)

where the matrix elements T₁₁, T₁₂, T₂₁, and T₂₂ can be found in Ref. [32]. Determined by the condition $\det \left(T-e^{iq\ell }I\right)=0$, the dispersion relation yields

$$\cos q\ell =\cos k_1{\ell }_1\cos k_2{\ell }_2-\frac{1}{2}\left(\frac{k_2}{k_1}+\frac{k_1}{k_2}\right)\sin k_1{\ell }_1\sin k_2{\ell }_2,$$

(4)

where k_α = ω/v_α, $v_{\alpha }=\sqrt{\tau /{\rho }_{\alpha }}$, and α = 1 or 2. Assuming that the parameters of tension and materials are τ = 477.46 Pa, ρ_A = 2704 kg/m³, ρ_B = 8100 kg/m³, ℓ_A = 0.6759 ℓ and ℓ_B = 0.3241 ℓ, the result of the band structure is shown in Figure 2.

2.2. Band Crossing Condition

As mentioned, the Zak phase plays an important role in the classification of topological insulators, and the interface mode arises around the domain wall of two topologically distinct band structures [21,33]. When we continuously tune the density of the B-segment ρ_B from 8100 kg/m³ to 9580 kg/m³ as well as varying the segment lengths ℓ_A and ℓ_B in one-unit cell, the result shows that the gap reopens as shown in Figure 3. Meanwhile, following the condition (see Appendix A)

$$\frac{k_A{\ell }_A}{k_B{\ell }_B}=\frac{m_1}{m_2},$$

(5)

where m₁ and m₂ are two co-prime integers, the band crossing occurs within the (m₁ + m₂)th band-gap at Brillouin center and boundary $\left(\pm \pi /q\ell \right)$. In the same manner, our design shows that the band inversion occurs around the 7th bandgap. In the subsequent content, we will focus on the band structure around this band.

2.3. Zak Phase

To classify the topological bands of ES-I and ES-II systems, the Zak phase of each band in both systems are evaluated. The Zak phase for nth band is given by [19].

$${\mathsf{\Theta }}_n^{Zak}={\displaystyle {\int }_{-\pi /\ell }^{\pi /\ell }{A}_n^{Zak}(q)\cdot dq}$$

(6)

where $A_n^{Zak}(q)=i{\displaystyle {\int }_{\mathrm{unit}\mathrm{cell}}\rho (z)u_{n,q}^{*}\left(z\right){\partial }_q{u}_{n,q}\left(z\right)dz}$ is Zak connection. In general, Zak connection is not globally defined in the whole q-space owing to the existence of a singularity. If a discontinuous point of Zak connection exists in q-space, a gauge transformation must be introduced to make Zak connection evolution analytic. With regard to inversion symmetry, the value of the Zak phase arises from the singularity within Zak connection evolution. Figure 4 shows the evolution of Zak connections for the 7th and 8th bands of the ES-I and the ES-II systems. As shown in Figure 4b,c, Zak phase vanishes when its evolution is continuous. Yet, if the evolution curve contains a singularity, to smoothen the evolution of u_q(z), Zak phase gives a value of π (see Appendix B). As a result, the singularities appearing in Figure 4a,d imply that the corresponding band has a topological phase π, showing the emergence of topological phase transition in the ES system. Moreover, u_q(z) can be visualized as the “2D field” which is the distribution of vibration mode. This provides an alternative method for comparing the difference of topologies between ES-I and ES-II, as depicted in Figure 5. Figure 5b,c present the field continuity in momentum space. In Figure 5a,d, however, the vibration modes show a discontinuous nature consistent with the locations of the singularities in Figure 4. Therefore, it must assign a minus sign to the vibration mode while q evolves across the singularity point. As the inversion symmetry holds, this sign change from this gauge transformation will give an extra phase π.

3. The Interface Mode

To verify the emergence of the interface mode, a 1D composite ES system consisting of 20 unit cells of ES-I and ES-II is discussed. The corresponding band structures of the ES-I and the ES-II systems are demonstrated in Figure 6a,b. Although the wave propagation within a bandgap is forbidden, the interface state appears as a sharp peak in the transmission spectrum. Figure 6c illustrates that the common gap (the 7th gap) of the ES-I and ES-II supports this topological interface mode at the frequency 2.83, which matches the location of the peak in the transmission spectrum. This interface mode can also be examined by investigating the vibrating amplitude u(z) with the frequency of a source 2.82 on the left end of the composite string as shown in Figure 7. Yet, due to the finite size effect and the limitation of numerical accuracy, the peak in Figure 6c and the corresponding interface resonance in Figure 7 have acceptable inaccuracy less than 1%. The amplitude distribution of the vibrating string in Figure 7 is also consistent with the bulk-edge correspondence principle, which says that interface mode or edge mode exists at the interface between two regions with distinct band topologies.

4. Possible Experimental Implementation

Since the proposed 1D ES system is made by aluminum, Cu-Zn alloy, and Cu-Pb alloy, it is feasible to experimentally observe the topological interface mode. For the topological interface mode excitation, vibration source can be set by a function generator connected to a commercial String Vibrator named PASCO WA-9857A [34]. Additionally, as a small amplitude of vibration waves can avoid metal fatigue, the interface mode may be excited without structure damage. In our design, the operating frequency of 2.82 corresponds to an ES lattice constant of about 1 cm, which is feasible to fabricate by using current alloy-wire fabrication technology.

5. Conclusions

In conclusion, we have theoretically studied the topological properties of a 1D composite ES system and reveal the topological phase transition arises from band crossing. The Zak connection evolutions of the corresponding periodic subsystems with and without the occurrence of the topological phase transition have been investigated. Moreover, the patterns of the ES vibration modes of the subsystems in q space further confirm the topological characteristics of the subsystems. Therefore, the interface mode emerges as indicated by the sharp peak in the transmission spectrum when both bandgaps align with subsystems and have different topologies. Finally, a viable way for future experimental demonstration is discussed.