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

TL;DR

The paper demonstrates an accessible SSH-based platform by realizing an electrical lattice with two-site unit cells and tunable couplings $v$ and $w$ to emulate topological edge modes. It derives the dispersion $\Omega^2(k)=v+w \pm \sqrt{v^2+w^2+2vw \cos k}$ and shows that edge states emerge in finite lattices when $v<w$, with frequency $\Omega=\sqrt{v+w}$ and robust localization due to chiral symmetry. The authors extend the model with symmetry-preserving long-range couplings, revealing richer topological phases with winding numbers $\nu=-2$ or $\nu=\pm1$ and additional edge modes, and they analyze domain-wall bound states with frequencies $\Omega=\sqrt{2v+w}$ and $\Omega=\sqrt{v+2w}$. The work provides a low-cost, modular platform for education and research in topological wave phenomena and suggests future directions into longer lattices, higher dimensions, and nonlinear extensions.

Abstract

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.

Figures (13)

• Figure 1: (a) Sketch of the SSH model, where the unit cell $(a_n,b_n)$ is in dotted box. Coupling constant is $v$ within the unit cell and $w$ between cells. (b) Sketch of the equivalent electric lattice.
• Figure 2: (a) and (b) Band structure. (a) $v = w = 1$, (b) $v \neq w$ ($v = 0.25$ and $w = 1$). The continuous blue line corresponds to the optical band in an infinite lattice, and the red line to the acoustic band (in an infinite lattice). Solid points correspond to the frequencies of the dispersion bands for a finite lattice of $N = 5$ unit cells with periodic boundary conditions. The same bands are obtained when $v = 1$ and $w = 0.25$, i.e., when the coupling strengths are swapped. Panels (c) and (d) show the circle in the complex plane described by $h(k)$ as $k$ varies between $-\pi$ and $\pi$. (c) $v = 1$ and $w = 0.25$, (d) $v = 0.25$ and $w = 1$.
• Figure 3: (a) Sketch of the ends of a finite lattice. Note that, in order to keep structure of the equations, we have to include extra inductors $L_2$ connected to the ground in both ends. (b) The lattice is driven sinusoidally at one end via a capacitor $C_d$ in order to induce the linear modes.
• Figure 4: Numerical Frequencies and localized edge modes calculated numerically diagonalizing the matrix $H$. (a) Frequencies (black circles), where a piece of the optical (blue line) and acoustic (red line) corresponding to the infinite lattice has been superimposed. In dashed black line we show the frequency corresponding to $\sqrt{v+w}$ (in kHz). (b) Same as (a), but with the frequencies switched ($v>w$). In this case, no edge modes appear in the lattice. (c) and (d) show the localized edge modes profile (blue circles) and the approximate solutions determined by (\ref{['m1']}) and (\ref{['m2']}) (red dotted line). Electric lattice where $L_1=1$ mH, $L_2=0.1$ mH, $C=1$ nF and $N=5$.
• Figure 5: Experimental voltage at the driven end and its neighboring sites as a function of the driving frequency when energy is supplied locally at the end by a sinusoidal voltage source with amplitude $V_d = 1 \,\text{V}$. Circuit parameters: $L_1 = 1 \,\text{mH}$, $L_2 = 0.1 \,\text{mH}$, and $C = 1 \,\text{nF}$. (a) Black points represent the voltage at the end node, red points the voltage at its first neighbor, and blue points the voltage at its second neighbor (see inset). (b) Voltage response at the second node. Dashed red lines indicate the theoretical resonance frequencies corresponding to edge modes in (a) and to acoustic and optical modes in (b).