Topologically Protected Spatially Localized Modes: An Easy Experimental Realization of the Su--Schrieffer--Heeger Model

In this paper, we review the basic concepts of topologically protected edge modes using the Su Schrieffer Heeger (SSH) model, originally introduced to describe electrical conductivity in doped polyacetylene polymer chains. We then propose an electrical circuit that emulates this model, provide its mathematical description, and present its experimental realization. The experimental setup is described in detail, with explanations designed to be broadly accessible without much prior familiarity with lattice theory, thus offering an introduction to this active area of research. Both theoretical predictions and experimental results confirm the presence of these modes, showing very good overall agreement. Using this concrete experimental system as a motivating example, we highlight the key aspects of topological protection.

💡 Research Summary

This paper provides a pedagogical yet rigorous treatment of topologically protected edge modes using the Su‑Schrieffer‑Heeger (SSH) model, and demonstrates an inexpensive electrical‑circuit implementation that faithfully reproduces the model’s physics. The authors begin by reviewing the SSH model’s origin in polyacetylene, where alternating single and double bonds give rise to a one‑dimensional lattice with two inequivalent hopping amplitudes, conventionally denoted v (intra‑cell) and w (inter‑cell). By mapping each lattice site to a voltage node in a transmission line composed of identical capacitors C and two sets of inductors L₁ and L₂, the authors obtain the equations of motion (Eqs. 1‑2) that are mathematically identical to the tight‑binding Schrödinger equation for the SSH chain. The coupling constants are expressed as v = 1/(L₁C) and w = 1/(L₂C).

Theoretical analysis proceeds by assuming plane‑wave solutions aₙ(t)=a e^{i(k n−Ω t)} and bₙ(t)=b e^{i(k n−Ω t)}. Substituting into the equations yields a 2×2 Bloch Hamiltonian H(k) whose eigenvalues give the dispersion relation Ω²(k)=v+w ± √(v²+w²+2vw cos k). The two bands (acoustic and optical) are separated by a gap that closes when v = w, marking a topological phase transition. The complex function h(k)=v+we^{-ik} traces a circle in the complex plane; if the circle encloses the origin (v < w) the winding number is 1, indicating a non‑trivial topological phase. If the origin is outside (v > w) the winding number is 0, corresponding to a trivial phase.

For a finite chain with open boundaries, the authors add extra inductors L₂ to ground at both ends to preserve the form of the equations. The resulting Hamiltonian matrix (Eq. 9) is diagonalized numerically. In the non‑trivial regime (v < w) two zero‑energy eigenstates appear, localized at the left and right edges. Analytic expressions for these edge states (Eqs. 10‑11) show exponential decay into the bulk with a decay factor (v/w)ⁿ. The presence of a chiral symmetry operator Γ (Eq. 12) guarantees that the spectrum is symmetric about zero and that the edge modes are protected against perturbations that preserve the symmetry.

Experimentally, the authors construct a chain of five unit cells (ten voltage nodes) using L₁ = 1 mH or 0.470 mH, L₂ = 0.1 mH, and C = 1 nF. A small drive capacitor C_d = 40 pF injects a sinusoidal voltage at one end via a function generator (Agilent 33220A). The response is recorded with a 16‑channel NI PXI‑1033 data acquisition system while sweeping the drive frequency. In the v < w configuration, a pronounced resonance near f ≈ 511 kHz is observed at the driven end, with negligible response elsewhere, matching the predicted edge‑mode frequency Ω = √(v + w) ≈ 554 kHz after accounting for parasitic inductance and capacitance. When the coupling strengths are swapped (v > w), the edge resonance disappears and only bulk acoustic and optical modes are seen. The measured spectra, spatial voltage profiles, and numerical diagonalization results are in excellent quantitative agreement, confirming the existence and robustness of the topological edge states.

The paper also discusses extensions involving long‑range couplings that generate phases beyond the standard SSH model, illustrating the flexibility of the circuit platform. By varying inductance values, one can explore richer topological invariants and observe new localized modes. The authors argue that such tabletop implementations provide an accessible entry point for students and researchers to explore topological physics without sophisticated nanofabrication, while also offering a testbed for prototype devices that could exploit topological protection in electronic or photonic applications.

In summary, the work successfully bridges abstract topological band theory and concrete laboratory practice, delivering a clear demonstration that a simple LC ladder network can host SSH‑type edge states, exhibit a well‑defined winding number, and display the hallmark robustness associated with topological protection. This contributes both pedagogically and technically to the growing field of topological metamaterials.