Disorder driven topological phase transitions in 1D mechanical quasicrystals

TL;DR

This work investigates disorder driven topological phase transitions in a 1D mechanical SSH chain under Aubry–André quasi-periodic modulation. It develops a real-space diagnostic framework using a local topological marker, inverse participation ratio, fractal dimension and Lyapunov exponent to map topology and localization. Analytical expressions yield mobility edges in the non disordered case and reveal a topological Anderson insulator and reentrant transitions when quasi periodic modulation and discrete chirality disorder interact, with phase boundaries shifted by the quasi periodic background. The results show that topological edge modes and bulk invariants survive in classical lattices, and provide a pathway to engineer disorder assisted topological phases in mechanical metamaterials.

Abstract

We examine the transition from trivial to non-trivial phases in a Su-Schrieffer-Heeger model subjected to disorder in a quasi-periodic environment. We analytically determine the phase boundary, and characterize the localization of normal modes using their inverse participation ratio. We compute energy-dependent mobility edges and provide evidence for the emergence of a topological Anderson insulator within specific parameter ranges. Whereas the phase transition boundary is affected by the quasi-periodic modulation, the topologically insulating Anderson phase is stable with respect to the chiral disorder in a quasi-periodic setup. Additionally, our results also uncover a re-entrant topological phase transition from non-trivial to trivial phases for certain values of quasi-periodic modulation with fixed chiral disorder.

Figures (15)

• Figure 1: Schematic depiction of a mechanical SSH chain, with dashed lines indicating intra-cellular springs ($K_{a}$) and solid lines indicating inter-cellular springs ($K_{b}$). A single unit cell is from masses $A$ and $B$. The black solid circles at the ends indicate fixed ends.
• Figure 2: Energy versus mode number plot for a 1D spring-mass chain in the case of a clean system. (a) System indicating mid-frequency states. Eigenvalues are computed from the dynamical matrix provided in Eq. (\ref{['eq:dynamicalmat1']}). The probability distribution of the mid-gap states is displayed as an inset, indicating localized modes at the boundaries. The color indicates the chirality of the edge modes. (b) System without mid-frequency states. The inset reveals that the probability distribution of mid-gap states extends across the entirety of the bulk, devoid of edge modes, with the loss of chirality, signifying a topologically trivial phase.
• Figure 3: Evolution of LTM plot with intra-cellular spring stiffness for a single unit cell away from the boundary without disorder or modulation. The graph shows plots for various system sizes. It is noted that there is a distortion in the sudden change of winding number during phase transition due to finite size effect. In a system of infinite size with translational invariance, LTM converges to winding number. The plot demonstrates that as system size grows, there is a shift in LTM development from gradual to sudden, in line with theoretical predictions.
• Figure 4: $\ln(IPR)$ as a function of the eigenvalue spectrum $\omega^{2}-\omega_{o}^{2}$ for $N=1597$ unit cells, $\delta_{d}=0$, $\phi=0$. For $K_{a}<K_{b}$, the transition from extended phase to localized phase in real space and localized phase to extended phase in momentum space is much more sharp, while for $K_{a}>K_{b}$, there are certain regions where the transition is highligted (i.e. either converges to $0$ or $1$), and regions where the IPR of the eigenstate does not converge to $1$ or $0$. This provides an indication of the presence of a critical phase, which we have quantitatively verified in Fig. \ref{['fig:combined_vector_plot3']}, Fig. \ref{['fig:combined_vector_plot4']}. (a): $K_{a}<K_{b}$ in real space, (b): $K_{a}>K_{b}$ in real space, (c): $K_{a}<K_{b}$ in momentum space, (d): $K_{a}>K_{b}$ in momentum space.
• Figure 5: Plot of fractal dimension as a function of $\delta^{AA}$ for $\delta_{d}=0$, $\phi=0$ for various system sizes. (a): $K_{b}>K_{a}$ in real space, (b): $K_{a}>K_{b}$ in real space, (c): $K_{b}>K_{a}$ in momentum space, (d): $K_{a}>K_{b}$ in momentum space. In cases (a) and (b) the system begins from an extended phase and transitions quickly into a localized phase as $\delta^{AA}$ increases. However, in case (b), after $\delta^{AA}>2$, the value remains constant with increasing $\delta^{AA}$ dependent of $N$. In cases (c) and (d), the system begins from being in a localized phase and make transitions slowly to an extended phase, but in case (d), for $\delta^{AA}>2$, the value of $d_{f}(N)$ remains constant with increasing $\delta^{AA}$ independent of system size $N$. The nature of the curve, especially in case (d), implies that there is a presence of critical phase in the system with these parameter values.