Topological Phase Transition in a One-Dimensional Elastic String System

: We show that topological interface mode can emerge in a one-dimensional elastic string system which consists of two periodic strings with di ﬀ erent 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.


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).

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.
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 where the matrix elements T11, T12, T21, and T22 can be found in Ref. [32].

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.
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 Due to the periodicity of density ρ, a Bloch theorem u = e −iqz u q (z) is introduced with the time convention e iωt , where u q z) = u q (z ± ) is the periodic Bloch wave, q is Bloch wave vector, and is lattice constant. The unit cell contains two homogeneous string segments with lengths 1 and 2 , and their densities are ρ 1 and ρ 2 . By imposing the boundary conditions u and ∂u/∂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 1 ) of the jth unit cell is expressed as u j (z) = a j e −ik 1 (z−j ) + b j e ik 1 (z−j ) , via periodic boundary condition u(z + ) = e -iq u(z), the matrix equation reads [32]: where the matrix elements T 11 , T 12 , T 21 , and T 22 can be found in Ref. [32]. Determined by the condition det T − e iq I = 0, the dispersion relation yields cos q = cos k 1 1 cos k 2 2 − 1 2 where k α = ω/v α , v α = τ/ρ α , and α = 1 or 2. Assuming that the parameters of tension and materials are τ = 477.46 Pa, ρ A = 2704 kg/m 3 , ρ B = 8100 kg/m 3 , A = 0.6759 and B = 0.3241 , the result of the band structure is shown in Figure 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 ρB from 8100 kg/m 3 to 9580 kg/m 3 as well as varying the segment lengths ℓA and ℓB in one-unit cell, the result shows that the gap reopens as shown in Figure 3. Meanwhile, following the condition (see Appendix A) 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 ( ) q π ±  . 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.

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/m 3 to 9580 kg/m 3 as well as varying the segment lengths A and B in one-unit cell, the result shows that the gap reopens as shown in Figure 3. Meanwhile, following the condition (see Appendix A) where m 1 and m 2 are two co-prime integers, the band crossing occurs within the (m 1 + m 2 )th band-gap at Brillouin center and boundary (±π/q ). 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.

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 ρB from 8100 kg/m 3 to 9580 kg/m 3 as well as varying the segment lengths ℓA and ℓB in one-unit cell, the result shows that the gap reopens as shown in Figure 3. Meanwhile, following the condition (see Appendix A) 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 ( ) q π ±  . 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.

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].
where A Zak n (q) = i unit cell ρ(z)u * n,q (z) ∂ q u n,q (z)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.

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. 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.

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.

Conclusion
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

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.

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. Acknowledgments: The authors would like to thank the reviewers for their nice comments and suggestions to improve the quality of the paper.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Band Crossing Condition
In this appendix, we prove that the band crossing condition given in Equation 4 (main text). Since there is no absorption in each segment string, the passband satisfies the condition that the absolute value of the r.h.s. of Equation 4 (main text) is smaller than or equal to 1. When we define γ ≡ k A A + k B B into Equation 4 (main text), it can be rewritten as where γ is the phase delay in a unit cell, κ ≡ -1 + (k B /k A + k A /k B )/2, and κ > 0. If γ = mπ, the absolute value of the r.h.s. of Equation (A1) is larger than 1 on the two conditions: (i) cos γ = -1 and sink A A sink B B ≥ 0 or (ii) cos γ = 1 and sink A A sink B B ≤ 0. Thus, the frequency must locate in the gap if the two bands do not cross. Nevertheless, if the two bands cross at γ = mπ, both of sink A A and sink B B equal to 0 so that Equation 5 in the main text can be derived. Moreover, when sin(k A A ) = 0, sin(k B B ) = 0, cos(k A A ) = (-1) m1 and cos(k B B ) = (-1) m2 , we have cos(q ) = (-1) m1+m2 . Thus, q = 0 when m 1 + m 2 is even, and q = ±π/ when m 1 + m 2 is odd. In the main text, the parameters of the band crossing system are listed in Figure 3, which gives k 1,band crossing 1,band crossing k 2,band crossing 2,band crossing = √ ρ 1,band crossing 1,band crossing √ ρ 2,band crossing 2,band crossing = 4 3 . (A2) As shown in Figure 3b, with m 1 + m 2 = 7 the band crossing occurs within the 7th band gap at Brillouin boundary ±π/q .

Appendix B. Zak Phase Calculation for Each Band
With inversion symmetry, we prove that if Zak connection evolution for an isolated band exists a singularity, then the Zak phase of this band must be π. Assuming that an open curve exists in the parameter space, all the components of u(R) on the curve are smooth. Firstly, we consider the relative phase between two neighboring states on the open curve. The relative phase factor corresponding to the two parameters R and R + dR is e −i∆Θ = u(R) u(R+dR) .
(A3) Thus, the phase difference ∆Θ is therefore given by When dR→0, to the first-order expansion Equation (A3) becomes, Thereby, the phase difference ∆Θ is where −Im u(R) ∇ R u(R) = i u(R) ∇ R u(R) is Berry connection. For a closed loop, Zak phase Θ is expressed as Since the Brillouin zone of a 1D periodic system can be regarded as a closed curve, Equation (A6) can be rewritten as Θ Zak n = −Im u n (q) ∂ q u n (q) · dq = −Im π/ −π/ u n (q) ∂ q u n (q) · dq (A8) where n is the band index, is the lattice constant and u(q) is Bloch wave function, respectively. If Zak connection evolution for the nth band is smooth in q space, the corresponding Zak phase must vanish. However, if Zak connection evolution experiences a singularity point at q 0 , Zak phase is then given by u n,q ∂ q u n,q + q 0 +δq q 0 −δq dq u n,q ∂ q u n,q As δq → 0, the states locate at q 0 + δq and q 0 -δq are two neighboring states. As a result, we can apply Equation (A4) to the third integration term in (A9), which reads u n,q ∂ q u n,q + ln u n,q 0 +δq u n,q 0 −δq According to the gauge transformation, the state around the singularity point satisfies u n,q 0 + = − u n,q 0 − (A11) Finally, substituting Equation (A11) into Equation (A10), Zak phase of the nth band is obtained by  u n,q ∂ q u n,q + ln − u n,q 0 +δq u n,q 0 +δq