Impurity-induced topological decomposition

TL;DR

The work reveals that local on-site impurities can act as precise levers to progressively decompose and reconfigure global topological properties in non-Hermitian lattices with spectral winding, converting winding numbers into discrete quantized-response plateaus. By formulating a Green’s-function-based quantized response $\nu_{\alpha,m}$ and showing that each impurity reduces the winding by one, the authors establish a universal, impurity-driven pathway to manipulate edge states and topological features without bulk parameter changes. The framework is extended to Hermitian systems through a doubled Hamiltonian $H_{\rm H}$, where impurities induce sequentially emerging pairs of topological edge states, linking non-Hermitian winding decomposition to Hermitian boundary phenomena. The results demonstrate robust, programmable control of topological states across 1D platforms, enabling reconfigurable topological responses and edge-state engineering via local perturbations.

Abstract

Controlling topological phases is a central goal in quantum materials and related fields, enabling applications such as robust transport and programmable edge states. Here we uncover a mechanism in which local on-site impurities act as knobs to decompose global topological properties in discrete steps. In non-Hermitian lattices with spectral winding topology, we show that each impurity sequentially reduces the winding number by one, which is directly manifested as a stepwise decomposition of quantized plateaus in the steady-state response. Based on this principle, we further develop a scheme that sequentially induces topological edge states under impurity control, in a class of Hermitian topological systems constructed by doubling the non-Hermitian ones. Our findings reveal a general scheme to tune global topological properties with local perturbations, establishing a universal framework for impurity-controlled topological phases and offering a foundation for future exploration of reconfigurable topological phenomena across diverse physical platforms.

Figures (11)

• Figure 1: (a) The model of Eq. \ref{['nonH_H']} with nearest ($t_{\pm1}$ and next-nearest ($t_{\pm2}$) hopping and $N_{\mu}=2$. $\mu_1$ and $\mu_2$ represent the two impurities. $\nu_{+,2}$ defined in Eq. \ref{['eq:response']} measures the growth rate of the response at the two left-hand sites (the "output") as the impurity strength increases when an external field is added to the two right-hand sites (the "input"). (b) The model in (a) can be rearranged into a multi-chain lattice with only nearest-neighbor hopping along $x$ direction. Omitting the inter-chain coupling ($t_j$ with $|j|<{\rm max}[r,l]$), each chain corresponds to a loop-like spectrum under PBCs with winding number $|W(E_r)|=1$ (gray loop on the right), which can be trivialized into a line-spectrum by a single strong impurity that effectively induces OBCs (red line on the right). The overall spectral winding (summing over all chains) is thus reduced by $1$ for each impurity. (c) Energy spectra of the model under PBCs without (gray) and with two identical impurities ($\mu_1=\mu_2=e^{13.9}$, blue), and under OBCs (red). (d) Response quantity $\nu_{+,2}$ as a function of $\beta$, for a reference energy $E_r=-0.74+0.96i$ [black star in (b)]. The plateau at $\nu_{+,2}=2$ drops to $1$ when $\beta\approx 13.9$, where the spectrum in (b) goes through the reference energy $E_r$. Other parameters are $t_{1}=1$, $t_{-1}=0.7$, $t_{2}=2$,$t_j=0$ for other values of $j$, and $N=50$.
• Figure 2: Spectral winding and evolution with increasing impurity strengths. (a) Spectral winding for our model under PBCs in the absence of impurities, $\mu_1=\mu_2=0$. Gray dots are the spectrum on the complex energy plane, and orange dots and line show its trajectory with quasi-momentum $k$ varying from $0$ to $2\pi$. (b) Spectral evolution (lilac color) with $\mu_2=0$ and $\mu_1$ increasing from $0$ (blue) to $e^{18}$ (red). Further increasing $\mu_1$ does not significantly affect the spectrum in the absence of the second impurity. (c) The same as in (a) but with $\mu_1=e^{18}$, where the spectral winding number is reduced to $W(E_r)=1$ for the area enclosed by the complex spectrum. (d) The same as in (b), but with $\mu_1=10^{35}$ and $\mu_2$ increasing from $0$ to $e^{35}$. (e) The same as in (a), but with $\mu_1=\mu_2=10^{35}$. The spectrum no longer encloses any area and $W(E_r)=0$ for arbitrary $E_r$, as under OBCs. Other parameters are $t_{1}=1$, $t_{-1}=0.7$, $t_{2}=2$,$t_j=0$ for other values of $j$, and $N=50$. Impurity states with exponentially large eigenenergies are omitted in all panels.
• Figure 3: Quantized response quantity $\nu_{+,2}$ for impurities $\mu_2={\eta}\mu_1={\eta}e^{\beta}$ and reference energy $E_r=-1+0.15i$. (a) $\nu_{+,2}$ varying with $\beta$ for different ratio $\eta$ between the two impurities. The plateau at $\nu_{+,2}=2$ is decomposed to two at $\nu_{+,1}$ when $|\ln \eta|$ increases. (b) to (d) Numerical (blue solid lines) and analytical (red dash lines) results of the quantized response at $\ln \eta=-10, -20, -30$, respectively. The drops of the numerical plateaus correspond to the saturation of response amplification due to finite system size, thus they do not occur in analytical results obtained under thermodynamic limit. Other parameters are the same as in Fig. \ref{['fig2']}.
• Figure 4: Topological properties of the the doubled Hermitian Hamiltonian $H_{\rm H}$. (a) OBC spectrum of $H_{\rm H}$ with $E_r=-1+0.15i$ and $W(E_r)=2$, in comparison of the quantized response quantity $\nu^{\rm ana}_{+,2}$ for the non-Hermitian Hamiltonian $H$. A pair of zero-energy edge states appear when $\nu^{\rm ana}_{+,2}$ increases by $1$ at $\beta\approx 0$ and $40$. (b) Logarithm of the spectrum in (a), where impurity states and topological edge states feature exponentially increasing and decreasing eigenenergies, respectively. Note that these states are both two-fold degenerate in ${\rm log}[|E_{\rm}|]$. (c) Distribution of impurity and topological edge states marked by the same colors (squares or circles) in (b). Note that each mark corresponds to a pair of eigenstates; and $\bar{\rho}(x)$ represents the average density of them summed over the two pseudospin components in Eq. \ref{['eq:H_H']}. (d) to (f) the same as (a) to (c), but with $E_r=0.5+0.15i$ and $W(E_r)=1$. Impurity states appear similarly at the two critical values of $\beta$, while topological edge states and nonzero plateau of $\nu_{+,1}$ appear only at the latter ($\beta\approx 40$). Other parameters are $t_1=1$, $t_{-1}=0.7$, $t_2=2$, $t_{-2}=0$, $\eta=e^{-40}$, and $N=80$.
• Figure S 1: (a) Complex eigenenergies for our model with only nearest-neighbor hopping. The spectrum under PBCs (blue dots) possesses $W(E_r)=1$ for the chosen reference energy $E_r=-0.2+0.1i$ (black star), and passes through $E_r$ in the presence of an extra impurity $\mu_1=e^{\beta}$ with $\beta=10$ (red dots). (b) Quantized response quantity $\nu_{+,1}$ obtained numerically (blue solid line), and analytically (red dash line) from Eq. \ref{['eq:nu_i']}. The plateau at $\nu_{+,1}=1$ ends at $\beta\approx 10$, where the spectrum pass through $E_r$ in (a). In addition, we display eigenenergies of the doubled Hermitian Hamiltonian $H_{\rm H}$ (see Eq. (7) in the main text), where the emergence of zero-energy topological edge states concides with that of the quantized plateau. Other parameters are $t_1=1$, $t_{-1}=0.7$, $t_j=0$ for other values of $j$, and $N=80$.