Topological Vibration Analysis of Elastic Lattices via Bloch Sphere Mapping

TL;DR

This work develops a compact, coefficient-based description of 1D elastic lattices that reveals topology through Bloch-sphere trajectories of intra-cell modal amplitudes. By representing the in-cell motion as a normalized two- or three-component state, the authors connect inversion-symmetry–driven Zak phases to geometric winding on the Bloch sphere, and extend the framework to time-modulated lattices where open-path (non-cyclic) geometric phases arise from Floquet sideband mixing. The diatomic case exhibits inversion-protected Zak-phase quantization ($0$ or $\pi$) with clear, gauge-robust phase locking, while the triatomic cell generalizes to SU(3)-type geometry and dequantization under inversion breaking. Time-dependent stiffness and spatiotemporal modulation expand the state-accessible region on the Bloch sphere to three dimensions and yield gate-like, classical rotations with open-path geometric phases, offering a unified lens for designing vibration and acoustic functionalities in engineered structures.

Abstract

Mechanical lattices support topological wave phenomena governed by geometric phases. We develop a compact Hilbert space description for one-dimensional elastic chains, expressing intra-cell motion as a normalized superposition of orthogonal eigenstates and tracking complex amplitudes as trajectories on a Bloch sphere. For diatomic lattices, this framework makes inversion symmetry protection explicit: the relative phase between in-phase and out-of-phase modes is piecewise locked, and the Zak phase is quantized with band-dependent jumps at symmetry points. Extending the analysis to triatomic lattices shows that restoring inversion retains quantization, whereas breaking it dequantizes the geometric phase while leaving the spectral origin invariant. Viewing norm-preserving transformations of the modal coefficient pair as Bloch sphere rotations, we demonstrate classical analogues of single-qubit logic gates. A pi-phase rotation about a transverse axis swaps the modal poles, and a longitudinal-axis phase flip maps balanced superpositions to their conjugates. These gate-like operations are realized by controlled evolution across wavenumber space and can be driven or reprogrammed through spatiotemporal stiffness modulation. Introducing space-time modulation hybridizes carrier and sideband harmonics, producing continuous phase winding and open-path geometric phases accumulated along the Floquet trajectory. Across static and modulated regimes, the framework unifies algebraic and geometric viewpoints, remains robust to gauge and basis choices, and operates directly on amplitude-phase data. The results clarify how symmetry, modulation, and topology jointly govern dispersion, modal mixing, and phase accumulation, providing tools to analyze and design vibration and acoustic functionalities in engineered structures.

Figures (11)

• Figure 1: Evolution of the modulus and phase of coefficient of the superposition of states at acoustic and optical branch at two different condition of stiffness of $\psi_1>\psi_2$ and $\psi_1<\psi_2$. Top Panel: Modulus of complex amplitudes $\hat{\alpha}$ and $\hat{\beta}$ of two mutually orthogonal states $\ket{E_1}$ and $\ket{E_2}$. $\hat{\alpha}$ is dominating at acoustic branch while $\beta$ is dominating at the optical branch at both stiffness conditions. Bottom Panel: Phase difference between the amplitude moduli $\Delta\phi=\arg(\hat{\alpha})-\arg(\hat{\beta})$, showing a locked $\pm\pi/2$ plateaus across the first Brillouin zone $kL \in [-\pi,\pi]$
• Figure 2: Bloch state demonstration of change of state between $\ket{E_1}$ and $\ket{E_2}$ at a two-mass system depicted in Hilbert Space at the acoustic branch with different stiffness matrices for the two cases (a) Stiffness ordering $\psi_1>\psi_2$, the trajectory of the Bloch states approaches the pure state $\ket{E_1}$(b) Stiffness ordering $\psi_1<\psi_2$, exploring larger region of the Bloch sphere covering both pure state $\ket{E_1}$ and $\ket{E_2}$.
• Figure 3: Analytical result of the real and imaginary component of the Berry vector corresponding to the stiffness of (a) $\psi_1>\psi_2$ and (b) $\psi_1<\psi_2$.
• Figure 4: Absolute phases of the complex coefficients at the acoustic branch at the set parameters (i) $\psi_1=\psi_2, \psi_3=2\psi_1$, (ii) $\psi_1=\psi_2, \psi_3=0.5\psi_1$, and (iii) $\psi_2=\psi_3, \psi_1=2\psi_2$.
• Figure 5: Elastic wave band structure for variable periodical sinusoidal modulation of stiffness at $V_m=350m/s$. The blue line defines the first fundamental branch (FFB), and the orange line represents the second fundamental branch (SFB).