Topologically switchable transport in a bundled cable of wires

Abstract

Advances in the next generation of mesoscopic electronics require an understanding of topological phases in inhomogeneous media and the principles that govern them. Motivated by the nature of motifs available in printable conducting inks, we introduce and study quantum transport in a minimal model that describes a bundle of one-dimensional metallic wires that are randomly interconnected by semiconducting chains. Each of these interconnects is represented by a Su-Schrieffer-Heeger chain, which can reside in either a trivial or a topological phase. Using a tight-binding approach, we show that such a system can transit from an insulating phase to a robust metallic phase as the interconnects undergo a transition from a trivial to a topological phase. In the latter, despite the random interconnectedness, the metal evades Anderson localization and exhibits a ballistic conductance that scales linearly with the number of wires. We show that this behavior originates from hopping renormalization in the wire network. The zero-energy modes of the topological interconnects act as effective random dimers, giving rise to an energy-dependent localization length that diverges as $\sim 1/E^2$. Our work establishes that random networks provide a yet-unexplored platform to host intriguing phases of topological quantum matter.

Figures (4)

• Figure 1: Model and Phase Diagram: (a) Schematic of $N$ one-dimensional metallic wires (solid lines) randomly connected by SSH interconnects (dashed lines) between two rectangular contacts. (b) At $E=0$, the two-terminal conductance per wire ($G/N$) is unity for $v/w<1$ (topological) and vanishes for $v/w>1$ (trivial). We refer to this as topological switching (TS). For $v < w$, the $G/N=1$ regime is the ballistic metal (bM). When $v>w$, the interconnects effectively break the wires, resulting in fractured metal (fM). (c) Schematic phase diagram in the $(E/w, v/w)$ plane. When $E\neq0,$ the system evolves into an Anderson insulator (AI) induced by either SSH zero modes ($\mathrm{AI_{ZM}}$) or due to SSH bulk bands, denoted $\mathrm{AI_{B}}$ ($t=1$).
• Figure 2: Characterizing the metallic chain and SSH junction: (a) Junction geometry: two metallic chains connected through an SSH segment; the inset illustrates the chosen hopping amplitudes. The dashed line marks the entanglement cut. (b) Energy spectrum of the junction as a function of $v/w$, computed with periodic boundary conditions and parameters $L = 10$ and $L_{\mathrm{SSH}} = 20$. (c) Conductance and entanglement entropy as functions of $v/w$, using $L = 100$ and $L_{\mathrm{SSH}} = 80$. The entanglement entropy is evaluated at the cut shown in panel (a). The insets highlight the qualitative change in the Wannier tight-binding description: for $v/w < 1$, a dimer forms at the junction, whereas for $v/w > 1$ the junction site effectively disappears and the chains become fragmented. (d) Comparison between the dimer hopping $t'$ extracted from the Wannier tight-binding model and the analytical expression, using $L = 10$ and $L_{\mathrm{SSH}} = 10$. The inset shows the indexing convention of the junction and the effective renormalization of the hopping $t \to t \sqrt Z$. All calculations in this panel use $w/t = 8$ and $t = 1$.
• Figure 3: Numerical data for multiple wire randomly connected by SSH chains: (a) Phase diagram: Density plot for conductance per wire ($G/N$) as a function of $v/w$ and $E/w$ ($N=5$, $\rho=0.1$ and $L=50$). (b) Behaviour of $G/N$ with $L$ for different values of $E$. ($N=5$, $\rho=0.1$, $v/t=0.5$). Solid lines are corresponding best fit curves with the form $Ae^{-L/\xi}$. (c) $G/N$ as a function of $v/w$ for different numbers of wires $N$ for a fixed $\rho=0.1$ and $L=50$. (d) $G/N$ as a function of $v/w$ for different $\rho$ ($N=5$, $L=50$). For the plots (c) and (d),the shaded region denotes the $v/w$ range where the SSH chains are topological. For all of the plots, $w/t=1$, and the $G/N$ is averaged over $100$ configurations. The number of sites in the SSH chain is taken to be $L_{\mathrm{SSH}}=30$.
• Figure 4: Random-dimer and dense limit: (a) When two SSH interconnects sit on consecutive sites (top), the effective dimers fuse and renormalize the hoppings of three consecutive bonds (bottom). This can be generalized for a higher number of fused dimers. (b) The localization length $\xi$ extracted from \ref{['fig:fig3']}(b) and from the effective random–dimer hopping model showing the same qualitative divergence $\xi \sim 1/E^2$. (c) Conductance as a function of energy at $\rho = 1$ (system shown in inset), with $v = \bar{v} + \delta v$ and $w = \bar{w} + \delta w$, where $\delta v$ and $\delta w$, both are randomly sampled from the interval $[-\delta/2, \delta/2]$. The calculation is done for $N=5$, $L=20$, $L_{\mathrm{SSH}}=30$, $\bar{v}/t=0.5$ and $\bar{w}/t=1.0$. Conductance is averaged over $200$ configurations.