[Paper Review] Impact of impurities on the topological boundaries and edge state localization in a staggered chain of atoms: SSH model and its topoelectrical circuit realization

Julio César Pérez-Pedraza, José Eduardo Barrios-Vargas+1 more|arXiv (Cornell University)|Feb 7, 2024

Advanced Physical and Chemical Molecular InteractionsChemistry3 citations

TL;DR

This paper investigates the impact of impurity superlattices on topological boundaries and edge state localization in the Su-Schrieffer-Heeger (SSH) model using both tight-binding theory and topolectrical circuit simulations. It demonstrates that impurities alter the topological phase boundary condition and significantly enhance edge state localization depending on impurity position within the supercell, with resonant impedance responses confirming topological protection in circuit realizations.

ABSTRACT

We study the Su-Schrieffer-Hegger model, perhaps the simplest realization of a topological insulator, in the presence of an embedded impurity superlattice. We consider the impact of the said impurity by changing the hopping amplitudes between them and their nearest neighbors in the topological boundaries and the edge state localization in the chain of atoms. Within a tight-binding approach and through a topolectrical circuit simulation, we consider three different impurity-hopping amplitudes. We found a relaxation of the condition between hopping parameters for the topologically trivial and non-trivial phase boundary and a more profound edge state localization given by the impurity position within the supercell.

Motivation & Objective

• To understand how impurities in a superlattice structure affect the topological phase transition in the SSH model.
• To analyze the localization of edge states in relation to impurity position within the supercell.
• To establish a topolectrical circuit realization of the SSH model with engineered impurities for experimental validation.
• To explore the modification of the topological phase boundary condition due to impurity-induced hopping amplitude changes.
• To demonstrate resonant impedance responses in topolectrical circuits as a signature of topological edge states.

Proposed method

• The SSH model is extended to an $N,i$-super-SSH model with impurity superlattices, where hopping amplitudes between impurities and neighbors are modified.
• A tight-binding Hamiltonian is constructed with position-dependent hopping parameters $v$, $w$, $v'$, and $w'$ to model impurity effects.
• Topolectrical circuits are realized using LC networks, where capacitances $C_1$, $C_2$, $C_3$, $C_4$ represent hopping amplitudes and inductances define node connectivity.
• The admittance matrix $J$ is derived from circuit Laplacian, with $J_{ab}( au) = i au [N_{ab}( au) + au ext{diag}(W_a)]$, linking to the tight-binding Hamiltonian via $J oldsymbol{V} = oldsymbol{I}$.
• Impedance $Z_{ab}$ is computed via the regularized Green's function $G = ext{sum} rac{|oldsymbol{ ilde{ ho}}_n|^2}{j_n}$, with $j_n$ as admittance eigenvalues.
• Edge state localization and topological phase transitions are probed by measuring voltage responses and impedance resonances at varying driving frequencies.

Figure 1: (a) Scheme of the $N,i$ -super-SSH model for $N=3$ , $i=3$ . The shaded area denotes the SSH-unit cell, the supercell number is given by the parameter $m$ , the supercell position by $p$ , the species parameter is $\alpha$ and the impurity position is denoted by $i$ . (b) and (c): Supercel

Experimental results

Research questions

• RQ1 How do impurities in a superlattice structure modify the topological phase boundary condition in the SSH model?
• RQ2 How does the position of an impurity within a supercell affect the localization of edge states?
• RQ3 To what extent can topolectrical circuits accurately simulate topological edge states and their response to impurities?
• RQ4 What is the relationship between the admittance eigenmodes and impedance resonances in the presence of impurities?
• RQ5 How do modified hopping amplitudes ($v', w'$) alter the topological invariant and mid-gap mode formation?

Key findings

• Impurities relax the condition between hopping parameters required for the topological phase transition, broadening the parameter space for nontrivial topology.
• Edge state localization becomes more pronounced when the impurity is positioned at specific sites within the supercell, particularly near the chain ends.
• Topolectrical circuits exhibit a sharp impedance resonance at the topological phase transition frequency, confirming the presence of protected edge modes.
• The voltage response in the circuit shows anti-phase oscillations between capacitors, consistent with the predicted mid-gap mode wavefunction $\psi_0(n) \propto (1,0,-t,0,t^2,0,-t^3,\dots)$ with $t = C_1/C_2$.
• When $C_1 < C_2$, a resonant impedance peak emerges due to a zero-mode edge state; when $C_1 > C_2$, no such resonance is observed, confirming the topological phase transition.
• The impedance $Z_{ab}$ diverges when admittance eigenmodes are localized at the boundary with small $j_n$, confirming topological protection in the circuit realization.

Figure 2: Berry phase in the space of free parameters of the periodic $N,i$ -super-SSH model for Case I with $w=1.0$ , $v^{\prime}=w^{\prime}$ and for different values of $N$ .