Engineering edge states in topoelectric circuits

Abstract

We study the topological phase transition in a two-leg Su-Schrieffer-Heeger (SSH) ladder by redefining the unit-cell structure. For both identical hopping dimerization pattern (uniform) and alternate hopping dimerization pattern (staggered), we demonstrate that different unit-cell choices generate different topological phases and phase transitions. In the uniformly dimerized ladder, variation of the inter-leg coupling induces transitions between distinct topological phases mediated by a gapless region. In contrast, the staggered dimerization configuration exhibits a richer phase structure, supporting both topological-topological and trivial-topological transitions occurring through a single gap-closing point, depending on the unit-cell definition. The phases are characterized through bulk-boundary correspondence, edge-state analysis, and bulk invariant. These analyses reveal interesting phenomena, including regimes supporting four localized boundary modes in the weak inter-leg coupling limit and two boundary modes in the strong inter-leg coupling limit. We further perform an experimental implementation using topoelectric circuit simulation by mapping the lattice Hamiltonian to an equivalent LC circuit. Circuit impedance and voltage responses obtained from LTspice simulations confirm the predicted edge-state signatures. These results establish topolectrical platforms as experimentally accessible realizations of tunable lattice topological phases.

Figures (9)

• Figure 1: Schematic of two types of ladder configurations. The left (right) panel indicates the ladder in uniform (staggered) configuration. The red, blue, and magenta boxes highlight three possible unit cell configurations in both cases, which give rise to different edge structures as shown in the bottom panels.
• Figure 2: Energy spectra of a two-leg ladder as a function of the inter-leg hopping strength $t_p$. Panel (a) shows the energy spectrum when both chains are in the staggered dimerization configuration. Panels (b) and (c) show the energy spectrum for ladders with uniform dimerization on both chains, corresponding to $(t_1,t_2)=(0.2,1.0)$ and $(1.0,0.2)$, respectively. The system size is $L=200$ sites. Panels (g)–(h) show the corresponding winding number $\nu/\pi$ as a function of $t_p$ for the parameter regimes shown in panels (a)–(c). The isolated states in (a) are the edge states in the high energy gaps.
• Figure 3: The energy spectra $E$ as a function of the eigenstate index ($i$) for two representative cuts, $t_p=0.5$ and $t_p=2.0$ corresponding to grey dashed lines in Fig. \ref{['fig:energy_spectrum']}(a)–(c).
• Figure 4: On-site probability density $|\psi_i|^2$ of four states located at the center of the spectrum for a system of size $L=40$ sites at $t_p = 0.5$.
• Figure 5: On-site probability densities $|\psi_i|^2$ of two states located at the center of the spectrum for a system of size $L=40$ sites at $t_p = 2.0$.