Topological Anderson insulator phases in one dimensional quasi-periodic mechanical SSH chains

In this paper, we explore the transition between different topological phases, in a Su-Schrieffer-Heeger (SSH) model composed of springs and masses in which the intracellular Aubry-Andr & eacute; disorder modulates the spring constants. We analytically compute the eigenvectors and eigenvalues of the dynamical matrix for both periodic and fixed boundary conditions, and compare them with the dispersion spectrum of the original tight-binding SSH model. We observe the presence of a topological Anderson insulating (TAI) phase within a specific range of quasi-periodic modulation strength and calculate the phase transition boundary analytically. We analyze the localization properties of normal modes by examining the inverse participation ratio (IPR) of eigenstates, of the dynamical matrix, and the corresponding fractal dimension associated with quasiperiodic modulation. We also examine the stability of the TAI phase across a range of modulation strengths and comments on the presence of mobility edge that separate localized modes from non-localized ones. We expand our analysis of analytically calculating the expression for the anomalous mobility edge in the context of Aubry-Andr & eacute; modulation of intracellular spring stiffness. This involves computing the Lyapunov exponent of all the eigenmodes related to intracellular dynamics. We demonstrate that specific analytical techniques are required to obtain an exact expression for the mobility edges when considering Aubry-Andr & eacute; modulation in intra-cellular spring stiffness.