Graded hopping screens nonreciprocity and reorganizes Stark asymptotics in a non-Hermitian Stark chain

TL;DR

This paper investigates a non-Hermitian Stark chain with graded hopping, revealing how boundary effects and localization phenomena are reorganized, leading to new insights into nonreciprocity screening and Stark localization.

Contribution

It introduces a similarity transformation that converts asymmetric hopping into algebraic boundary accumulation, clarifies the Stark threshold, and explores the interplay of nonreciprocity and localization.

Findings

The similarity transformation maps the asymmetric chain to a symmetric one with algebraic boundary accumulation.

The Stark threshold is identified at |F_1|=2|F_2|, separating different localization regimes.

Enhanced entanglement growth occurs near the Stark threshold, indicating a crossover in localization behavior.

Abstract

We study a one-dimensional non-Hermitian Stark chain in which nonreciprocal hopping, a linear potential, and linearly graded hopping act simultaneously. The central question is how boundary pumping and field-induced confinement are reorganized when the hopping amplitude itself grows with position. We show that the graded term separates the two localization channels at the level of the large-position asymptotics. An exact diagonal similarity transformation removes the bond asymmetry and converts the usual exponential skin factor into an algebraic boundary accumulation with exponent $\eta=\gamma/F_2$. The transformed symmetric chain then reduces asymptotically to a constant-coefficient recurrence, giving the Stark threshold $|F_1|=2|F_2|$. The original right eigenstates acquire the unified envelope $\psi_j^R\sim j^{\eta}\phi_j$, with oscillatory, double-root, and exponentially localized branches across the threshold. This form also yields two finite-size scales, one measuring the logarithmic screening of nonreciprocity and the other balancing the algebraic skin factor against the exponential Stark tail. A joint localization map in the $(\gamma,F_1/F_2)$ plane verifies this structure. The edge polarization bends near the Stark threshold and weakens on the localized side, while the inverse participation ratio of the most localized eigenstates rises rapidly for $F_1/F_2>2$. Using a normalized Gaussian projector appropriate for non-unitary evolution, we further show that the same threshold enhances half-chain entanglement growth after a charge-density-wave quench. These results identify graded hopping as a controlled mechanism for screening nonreciprocity, resetting Stark asymptotics, and organizing the finite-size crossover between algebraic skin accumulation and Stark localization.