# One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Intrinsic Topological Phononic Structures

**,**$C=\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 1& -1\\ 0& 0& -1& 1\end{array}\right)$. The symbol i refers to $\sqrt{-1}$. The wave functions $\mathsf{\Psi}=\left(\begin{array}{c}{\psi}_{1}\\ {\psi}_{2}\\ {\psi}_{3}\\ {\psi}_{4}\end{array}\right)$ and $\overline{\mathsf{\Psi}}=\left(\begin{array}{c}{\overline{\psi}}_{1}\\ {\overline{\psi}}_{2}\\ {\overline{\psi}}_{3}\\ {\overline{\psi}}_{4}\end{array}\right)$ are 4 vector solutions of $\left(\left[A\frac{\partial}{\partial t}+\beta B\frac{\partial}{\partial x}\right]-i\frac{\alpha}{\sqrt{2}}C\right)\mathsf{\Psi}=0$ and $\left(\left[A\frac{\partial}{\partial t}+\beta B\frac{\partial}{\partial x}\right]+i\frac{\alpha}{\sqrt{2}}C\right)\overline{\mathsf{\Psi}}=0$, respectively. $\mathsf{\Psi}$ and $\overline{\mathsf{\Psi}}$ are non-self-dual solutions. These equations do not satisfy time reversal symmetry ($t\to -t)$, T-symmetry, nor parity symmetry ($x\to -x)$ separately. In the language of Quantum Field Theory, $\mathsf{\Psi}$ and $\overline{\mathsf{\Psi}}$ represent “particles” and “anti-particles”. Let us now seek solutions of $\text{}\left(\left[A\frac{\partial}{\partial t}+\beta B\frac{\partial}{\partial x}\right]-i\frac{\alpha}{\sqrt{2}}C\right)\mathsf{\Psi}=0$ in the plane wave form: ${\psi}_{j}={a}_{j}{e}^{ikx}{e}^{i\omega t}$ with j = 1,2,3,4. This gives the eigen value problem:

## 3. Extrinsic Phononic Structure

_{I}) is subjected to a spatio-temporal modulation, i.e., $\alpha ={\alpha}_{0}+{\alpha}_{1}2\mathrm{sin}\left(Kx+\mathsf{\Omega}t\right)$ where ${\alpha}_{0}$ and ${\alpha}_{1}$ are constants (see Figure 4). Here, $K=\frac{2\pi}{L}$ where $L$ is the period of the modulation. $\mathsf{\Omega}$ is the frequency modulation and its sign determines the direction of propagation of the modulation.

^{−9}kg, with Born-Von Karman boundary conditions. The masses are equally spaced by h = 0.1 mm. The parameters ${K}_{0}$ = 0.018363 kg·m

^{2}·s

^{−2}and ${K}_{I}$ = 2295 kg·s

^{−2}. The spatial modulation has a period L = 100 h and an angular frequency $\mathsf{\Omega}$ = 1.934 × 10

^{5}rad/s. We have also chosen the magnitude of the modulation: ${\alpha}_{1}=\frac{1}{10}{\alpha}_{0}$. The dynamics of the modulated system is amenable to the method of molecular dynamics (MD). The integration time step is dt = 1.624 × 10

^{-9}s. The dynamical trajectories generated by the MD simulation are analyzed within the framework of the Spectral Energy Density (SED) method [43] for generating the band structure. To ensure adequate sampling of the system’s phase-space the SED calculations are averaged over 4 individual MD simulations, each simulation lasting 2

^{20}time steps and starting from randomly generated initial conditions. We report in Figure 5, the calculated band structure of the modulated system.

_{gap}) but only backward propagating waves (−k

_{gap}), may lead again to Boson-like behavior with no restriction on the amplitude of the backward propagating waves.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic illustration of the phononic structure composed of two coupled 1-D harmonic crystals. The atoms in the lower and upper 1-D harmonic crystals have mass m and M, respectively. The force constant of the springs of each 1-D harmonic crystal is taken to be the same, K

_{0}. The force constant of the coupling springs is K

_{I}. The periodicity of the crystal is a.

**Figure 2.**Schematic illustration of the harmonic chain grounded to a substrate via side springs (system in Figure 1 when $M\to \infty $).

**Figure 3.**Schematic representation of the manifold supporting $\mathsf{\Psi}$ and $\overline{\mathsf{\Psi}}$. The manifold exhibits a local quarter-turn twist around k = 0 of the square cross section of the manifold reflects the orthogonality of $\mathsf{\Psi}$ and $\overline{\mathsf{\Psi}}$. The clored arrows are parallel transported on the manifold along the direction of wave number. Their change in orientation is indicative of the phase change.

**Figure 4.**Schematic illustration of the time evolution (top to bottom) of the harmonic chain grounded to a substrate with spatial modulation of the side spring stiffness illustrated as a “pink glow” of varying width when $\alpha >{\alpha}_{0}$ and a “blue glow” when $\alpha <{\alpha}_{0}$.

**Figure 5.**Band structure of the mechanical model system of Figure 2 calculated using the Spectral Energy Density (SED) method. The band structure is reported as a contour plot of the natural logarithm of the SED (color bar) versus normalized frequency and reduced wave number. The frequency is normalized to the lowest value of unperturbed band, namely 1.507 × 10

^{5}Hz. The horizontal axis is extended to the right beyond the first Brillouin zone [−π,π] to highlight the asymmetry and therefore the modulation-induced symmetry breaking of the band structure. The brighter branches correspond to the usual zeroth-order type wave (${e}^{i{\omega}_{0}\left(k+g\right){\tau}_{0}}$). The fainter branches parallel to the brighter ones are characteristic of first-order waves (${e}^{i\left({\omega}_{0}\left(k+g\right)\pm \mathsf{\Omega}\right){\tau}_{0}}$).

${e}^{+ikx}{e}^{+i{\omega}_{k}t}$ | ${e}^{-ikx}{e}^{+i{\omega}_{k}t}$ | ${e}^{+ikx}{e}^{-i{\omega}_{k}t}$ | ${e}^{-ikx}{e}^{-i{\omega}_{k}t}$ | |

${\xi}_{k}$ | $\left(\begin{array}{c}\sqrt{\omega +\beta k}\\ \sqrt{\omega -\beta k}\end{array}\right)$ | $\left(\begin{array}{c}\sqrt{\omega -\beta k}\\ \sqrt{\omega +\beta k}\end{array}\right)$ | $\left(\begin{array}{c}-\sqrt{\omega -\beta k}\\ \sqrt{\omega +\beta k}\end{array}\right)$ | $\left(\begin{array}{c}-\sqrt{\omega +\beta k}\\ \sqrt{\omega -\beta k}\end{array}\right)$ |

${\overline{\xi}}_{k}$ | $\left(\begin{array}{c}\sqrt{\omega -\beta k}\\ -\sqrt{\omega +\beta k}\end{array}\right)$ | $\left(\begin{array}{c}\sqrt{\omega +\beta k}\\ -\sqrt{\omega -\beta k}\end{array}\right)$ | $\left(\begin{array}{c}\sqrt{\omega +\beta k}\\ \sqrt{\omega -\beta k}\end{array}\right)$ | $\left(\begin{array}{c}\sqrt{\omega -\beta k}\\ \sqrt{\omega +\beta k}\end{array}\right)$ |

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**MDPI and ACS Style**

Deymier, P.; Runge, K.
One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology. *Crystals* **2016**, *6*, 44.
https://doi.org/10.3390/cryst6040044

**AMA Style**

Deymier P, Runge K.
One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology. *Crystals*. 2016; 6(4):44.
https://doi.org/10.3390/cryst6040044

**Chicago/Turabian Style**

Deymier, Pierre, and Keith Runge.
2016. "One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology" *Crystals* 6, no. 4: 44.
https://doi.org/10.3390/cryst6040044