Unifying Anderson transitions and topological amplification in non-Hermitian chains

TL;DR

This work develops a unified framework linking non-Hermitian Anderson localization to topological amplification in disordered 1D chains by tying the spectral winding number to the Lyapunov exponent via the Thouless formula. The authors show that the sign of $\Re{L(E)}$ selects the bulk winding via $w(E)=\!-1$ for skin states and $w(E)=0$ for Anderson-localized states, defining mobility edges and three phases: skin, mixed, and Anderson-localized. They derive exact results for unidirectional hopping, including how disorder phase distributions shape the mobility-edge geometry (circular for uniform phase, ellipse-like for fixed phase) and the forceful separation of topological and point-gap transitions. The framework generalizes prior SVD and Dyson formulations, provides practical tools for disorder-resilient transport and quantum sensing, and is applicable to photonic, optomechanical, and superconducting systems. Overall, it offers a comprehensive link between non-Hermitian topology and localization in disordered chains with clear experimental relevance.

Abstract

Non-Hermitian systems with non-reciprocal hopping may display the non-Hermitian skin effect, where states under open boundary conditions localize exponentially at one edge of the system. This localization has been linked to spectral winding and topological gain, forming a bulk-boundary correspondence akin to the one relating edge modes to bulk topological invariants in topological insulators and superconductors. In this work, we establish a bulk-boundary correspondence for disordered Hatano-Nelson models. We relate the localization of states to spectral winding using the Lyapunov exponent and the Thouless formula. We identify two kinds of phase transitions and relate them to transport properties. Our framework is relevant to a broad class of 1D non-Hermitian models, opening new directions for disorder-resilient transport and quantum-enhanced sensing in photonic, optomechanical, and superconducting platforms.

Figures (7)

• Figure 1: (a) Surface plot of the real part of the Lyapunov exponent $\Re{L(E)}$ given by the Thouless formula in \ref{['eq:Thouless']} as a function of $E\in\mathds{C}$ for the clean Hatano-Nelson model with $J_R=1$ and $J_L=0.5$. The pink plane corresponds to $\Re{L(E)}=0$ and intersects the surface at the red curve (PBC eigenenergies). The black line segment corresponds to the OBC eigenenergies. (b) Contour plot of $\Re{L(E)}$ in the complex plane for the disordered Hatano-Nelson model. The black circles correspond to OBC eigenenergies and the red point to PBC eigenenergies. The PBC eigenvalues that have $\Re{L(E)}=0$ delineate a part of the complex plane that has $\Re{L(E)}>0$ (skin states) with one that has $\Re{L(E)}<0$ (Anderson-localized states). The black star indicates the average OBC eigenenergy. The Hamiltonians are generated by sampling uniformly from $J_{R,n}\in[0.7,1.3]$, $J_{L,n}\in[0,2,0.8]$ and $V_n=-ir_n$ with $r_n\in[0,e]$ and $e=2.718\dotso$ for a chain of 100 sites.
• Figure 2: Schematic of the non-Hermitian topology behind directional amplification discussed in \ref{['sec:Phases']}. Depending on the value of the spectral winding number around the origin $\nu$ there may be (a) amplification or (b) decay of particle number in the steady state.
• Figure 3: The PBC spectrum (red dots) can either lie on the mobility edge curve (black) or be an isolated point, depending on disorder strength. The mean on-site potential value $\mathds{E}(V_n)$ (depicted by a black star) is the point at which the real part of the Lyapunov exponent is greatest for a single-band model in the large-chain limit.
• Figure 4: Normalized absolute value of the Green's function in the steady state ($E=0$) for a unidirectional chain of 100 sites subject to disorder. The dots represent the numerical Green's function averaged over 1000 disorder realizations for on-site potentials $V_n=e^{i\phi_n}r_n$ with uniform radial probability distribution $g(r_n)=1/R_0$ over $r_n\in[0,R_0]$. The data is in excellent agreement with our analytical result \ref{['eq:SS_greenf']}, as demonstrated by the solid line segments.
• Figure 5: Real part of the Lyapunov exponent in terms of the energy $|E|$ for a uniform radial distribution of $V_n$ over $[0,R_0]$. (a) The solid line segments correspond to the curve given by \ref{['eq:ReL_uni']} obtained with a uniform phase distribution of $V_n$ over the domain of OBC eigenvalues. The dashed line is the curve of \ref{['eq:ReL_uni_outside']}. When $|E|>R_0$, it is the continuation of $\Re{L(E)}$ outside of the domain of OBC eigenvalues. (b) The solid line segments correspond to the curve given by \ref{['eq:ReL_dirac']} over the range of OBC eigenvalues obtained when the phase of the on-site potentials $V_n$ is constant, $\phi_n=\theta$. The dashed lines are the continuation of $\Re{L(E)}$ outside of the domain of OBC eigenvalues.