Realization of staircase topological Anderson phase transitions

TL;DR

This work demonstrates a disorder driven cascade of topological Anderson transitions in a 1D chiral system inspired by semiconducting carbon nanotubes, where long range intercell disorder induces a staircase increase of the real-space winding number omega_mu until saturation at extremal values. The authors combine theory and experiment by mapping the model to a topolectrical circuit, employing real-space winding numbers, mobility gaps, and localization metrics to reveal robust edge modes that persist at strong disorder. The results show that edge states emerge and remain protected by a mobility gap through multiple transitions, contrasting with conventional TAIs, and offer a route toward disorder-tunable topological devices, or disordertronics, with potential realization in SWNTs. These findings extend the TAI paradigm to nonreentrant higher winding number phases and establish a framework for disorder controlled topological transport in one-dimensional systems.

Abstract

One-dimensional topological Anderson insulators provide a paradigm for disorder-induced topological phases in which the underlying system turns from a trivial to a topological phase. It is widely recognized that the latter vanishes at large disorder amplitude. Here, and contrary to the general belief, we provide evidence for a successive disorder-driven topological transitions in a single-wall nanotube, culminating in a topological Anderson phase that remains unexpectedly robust at strong disorder. This phenomenon is confirmed by analysis of the corresponding topological invariant, which increases stepwise as disorder increases, giving evidence for the emergence of edge states. We experimentally implement these topological Anderson staircase phase transitions in a one-dimensional topolectrical circuit, where the persistence of edge states is revealed by node-voltage measurements. The robustness of the edge states is corroborated by numerical calculations of their localization properties. Our work opens the road to topological disordertronics, where topological phases can be tuned by disorder.

Figures (11)

• Figure 1: Robustness of topological Anderson insulating phases in SWNT. (a) Topological phase diagram as a function of the disorder amplitude $W$ and the asymmetry parameter $\beta$ (Eq. \ref{['J2J3']}) for a $(8, 6)$-SWNT with $L=600$ unit cells. The other parameter are $p = q = -1,$ and $\mu = 0$. (b) Real-space winding numbers $\omega_\mu$ and bulk gap $(\Delta E)$ versus $W$ for the same nanotube with size $L=10^3$ unit cells and $\beta=2.8$. (c) Corresponding disorder-averaged spectra of finite 1D SWNT chain with open boundary conditions as a function of $W$ for the same parameters sets. The different colored lines indicate edge states, while black lines correspond to bulk states. (e) Probability-density maps of the midgap modes $\psi_N$, $\psi_{N-1}$, $\psi_{N-2}$, and $\psi_{N-3}$ as a function of disorder strength $W$. In our numerical calculations, an average over $50$ disorder realizations was performed.
• Figure 2: Localization properties of midgap eigenstates. (a) Average winding number and localization length at $E=0$ as a function of disorder strength $W$ for a $(8,6)$-SWNT with $p=q=-1$, $\mu=0$. (b) Average $\overline{\text{IPR}}$ and mean Shannon entropy $\overline{\text{S}}/\ln{L}$ of the midgap states as a function of $W$. Other parameters are the same as those in Fig. \ref{['PhaseDiagDiso']}.
• Figure 3: Experimental topolectrical circuit realization of a robust TAI cascade in the reduced 1D SWNT model. (a) Artistic view of the two coupled 1D chains of resonant nodes representing the A and B sublattices of the effective 1D SWNT model. (b) Reduced 1D model of the $(8,6)$-SWNT. Red lines denote the forward long-range coupling $J_2$, black lines the backward long-range coupling $J_3$, and blue lines the direct intracell coupling $J_1$ between two sites within the same unit cell. (c) Top view of the equivalent 1D topolectrical circuit, together with enlarged views highlighting the resonator sites and grounding elements. A function generator provides the excitation signal at selected nodes, and a digital oscilloscope is used to detect the resulting voltage response, as schematically indicated on the right side. (d) Photograph of the experimental setup with a size of 2 × 95 nodes. The zoomed-in views show the detailed structure of two unit cells.
• Figure 4: Observation of successive emergence of disorder-induced edge modes in the topolectrical SWNT analogue. (a-d) Measured normalized voltage distributions at the edge-state resonance frequency for disorder strengths $W= 7.8t,\ 10.9t,\ 14.6t,$ and $22t$, respectively. (e-h) Corresponding LTspice-simulated voltage profiles for the same disorder strengths and drive conditions. The dashed red box highlights the edge area selected for display. The blue and purple arrows mark the excitation regions of the signal excitation. (i-l) Histogram representation of the experimental edge-state voltage profiles. The noramlized amplitude of $|V_j/V_{\text{max}}|$ is shown as a bar plot versus site index for disorder amplitudes corresponding to (a-d) cases.
• Figure 5: (a) Unrolled two-dimensional hexagonal lattice of the $(4,2)$ SWNT. Wrapping the chiral vector $\mathbf{C}= 4 \mathbf{a_1} + 2 \mathbf{a_2}$ onto itself results in the nanotube with axis along a translation vector $\mathbf{T}$. The elementary vector $\mathbf{C}_d\equiv\mathbf{C}/d$ corresponds to a rotation of $\frac{2\pi}{d}$ around the tubule axis. The dashed rectangle delimits the unit cell used to define the reduced 1D model, while the grey one between $\mathbf{C}$ and $\mathbf{T}$ is the larger translational unit cell. In this case $(p,q)=(1,0)$ and $d=gcd(m,n)=2$. (b) Schematic illustration of the corresponding two 1D chains. The black solid lines denote the intracell hopping $J_1$, while the blue lines and red dashed lines indicate the $\mu$-dependent long-range hopping parameters $J_{2,\mu}$ and $J_{3,\mu}$, respectively.