Non-Hermitian impurity problem

Abstract

The problem of a single Hermitian impurity has long served as a cornerstone in condensed matter physics, offering fundamental insights into the mechanisms of Anderson localization. Yet, despite the increased interest in the spectral and localization properties of non-Hermitian lattices with defects, the non-Hermitian extension of the single impurity problem remains largely unexplored. In this work, we investigate the role of a single complex impurity in one-, two-, and three-dimensional infinite tight-binding lattices. Our study reveals a series of counterintuitive phenomena, including regions where localization vanishes and re-emerges as the impurity strength varies. Next, we study the corresponding finite-sized lattices, which are highly relevant to experimental realizations in readily accessible photonic platforms, revealing a variety of exotic features, such as scale-free localized states, exceptional points, and peculiar cross-shaped localized eigenstates, whose profiles deviate from the conventional exponential localization. This work paves the way for future studies on transport phenomena in non-Hermitian disordered lattices.

Figures (8)

• Figure 1: Bound-state existence maps and trajectories for a complex impurity $\gamma = V_1 + iV_2$ as a function of $V_1$ and $V_2$. (a) Map of bound-state existence corresponding to 1D infinite-sized lattice, where regions colored gray indicate bound-state existence, while the respective green-colored regions indicate absence. (b) Similarly with (a), for 2D infinite-sized square lattice. (c) Similarly with (a), for 3D infinite-sized cubic lattice. (d) Trajectory of the bound-state eigenvalue $\epsilon_b$ in the complex plane, for the 1D lattice with impurity $\gamma = V + iV$ as a function of $V \in [0,4]$ (shown in the color bar). (e) Similarly with (d), with impurity $\gamma = 1.5 + iV_2$ as a function of $V_2 \in [0,3]$. (f) Similarly with (d), with impurity $\gamma = V_1 + 1.6i$ as a function of $V_1 \in [0,4]$. In each of (d)–(f), the positive part of the unperturbed eigenvalue band is shown dotted. Corresponding points in parameter space of (a)–(c) and eigenvalue trajectories of (d)–(f) are marked by bold capital letters.
• Figure 2: Spectral properties of 1D finite-sized lattices with a single imaginary impurity. (a)–(d) Eigenvalue spectra of 1D tight-binding finite-sized lattices ($N$ sites) with a single imaginary defect of strength $V$ located at site $m$. (a)/(c) Mirror-symmetric case ($N = 51$, $m = 26$) for $V = 1$ and $V = 2.1$, respectively; (b)/(d) non-mirror-symmetric case ($N = 50$, $m = 25$) for the same values of $V$. Notice that all eigenvalues in (b) and (d), and $(N+1)/2$ of them in (a) and (c), were affected by the presence of the single defect by acquiring an imaginary part. (e) Difference $\delta b_{1,2} = \Im(\epsilon_1) - \Im(\epsilon_2)$ as a function of $V$. (f) Mean imaginary part $\langle b \rangle$ of non-real eigenvalues, excluding $b_1$ for $V > V_{\text{th}}$, as a function of $V$. (g) Participation ratio $P_1$ of the eigenstate with the largest imaginary part, as a function of $V$. (h) Average participation ratios of eigenstates with purely real ($P_R$, green) and complex ($P_C$) eigenvalues (excluding $\ket{u_1}$ for $V > V_{\text{th}}$), as a function of $V$. In (e)–(h), blue and red curves correspond to the mirror- and non-mirror-symmetric lattice cases, respectively., while the green vertical line marks the threshold value $V_{\text{th}}$ for formation of a localized state.
• Figure 3: Amplitudes $|u_{j,n}|$ of representative right eigenmodes $\ket{u_j}$ for 1D finite-sized lattices with a single imaginary impurity. (a)/(c) correspond to the mirror-symmetric case ($N=51$, $m=26$), while (b)/(d) pertain to the non-mirror symmetric configuration ($N=50$, $m=25$), where $N$ is the number of lattice sites and $m$ the site of the impurity. For $V=1.8<V_{\text{th}}$ (a)-(b), extended (gray) and scale-free localized (SFL) modes (blue) are observed; for $V=2.5>V_{\text{th}}$ (c)-(d), an exponentially localized (red) state emerges. In (c) the SFL modes persist, whereas in (d) they become scattering states (green). In particular, the extended real-eigenvalue eigenmodes (gray) $\ket{u_R}$ of (a) and (c) correspond to ${u_{R,n}}\propto \sin(\frac{7\pi n}{13})$ and ${u_{R,n}}\propto \sin(\frac{\pi n}{13})$, respectively.
• Figure 4: Classification of eigenmodes and scale-free localization for 1D finite-sized lattices. (a)/(b) Eigenvalue spectra of 1D tight-binding finite-sized lattices with a single imaginary defect of strength $V=2.5$ for (a) mirror-symmetric lattice ($N = 51$) and the defect placed on site $m=26$ and (b) the non-mirror-symmetric lattice ($N = 50$) and the defect places on site $m=25$. In both subfigures the eigenvalues are colored according to the classification of their eigenstates; localized (red), SFL (blue), extended (gray) and scattering (green). (c) Ratios $r$ for eigenstates with complex eigenvalues are shown for the mirror-symmetric lattice ($r_{C}^{(51)}$) and the non mirror-symmetric lattice ($r_{C}^{(50)}$); for comparison, the corresponding ratio for real-eigenvalue states of the mirror-symmetric lattice ($r_{R}^{(51)}$) is included (gray line). The color for $r_{C}^{(51)}$ and $r_{C}^{(50)}$ indicates the percentage of complex-eigenvalue eigenstates with $r>1.2$.
• Figure 5: Spectral properties and representative eigenmodes of 2D finite-sized lattices with a single imaginary impurity. (a)–(b) Eigenvalue spectra for 2D finite-sized tight-binding lattices of size $51\times51$ with a single imaginary defect at $\ket{\mathbf{m}}=(26,26)$, for (a) $V=1$ and (b) $V=2$. The eigenvalue corresponding to a localized state is marked in red. (c) Difference $\delta b_{1,2}$ as a function of $V$. (d) Participation ratio $P_1$ of the eigenstate with the largest imaginary part versus $V$ (inset: $P_1$ vs. $N$ at $V=1$). (e)-(h) Amplitudes $|u_{j,(n_x,n_y)}|$ of representative eigenmodes for defect strength (e)-(f) $V=1$, (g)-(h) $V=2$. Panels (e)/(g) correspond to localized states, while the (f)/(h) are other complex-eigenvalue states. In all panels, the amplitudes are normalized in order to have maximum value equal to unity.