Helical Quasiperiodic Chains with Engineered Dissipation: Liouvillian Rapidity Diagnostics of Transport and Localization

TL;DR

This work analyzes the relaxation dynamics of a quadratic spinless-fermion chain with a quasiperiodic Aubry-André potential and helical long-range hopping under engineered Markovian dissipation. Using exact third-quantization in the Majorana covariance formalism, the authors map the Liouvillian to a non-Hermitian damping matrix and study the smallest nonzero rapidity $\kappa=\min_{k\neq0}\Re\beta_k$ across various dissipation patterns and hopping strengths $t_N$. They find a clear dichotomy: uniform dissipation yields large, robust gaps, while sparse local dissipation leads to rapid suppression of $\kappa$ as localization sets in (with $\lambda$), and this suppression is mitigated by increasing $t_N$ which enhances mode overlap with dissipators. Finite-size scaling and level-statistics analyses reveal crossovers from Wigner-Dyson-like to Poisson statistics and show slow-mode weights concentrating on dissipative sites in the localized regime. Overall, Liouvillian rapidities emerge as a compact, experimentally relevant diagnostic of relaxation, transport, and localization in engineered open quantum lattices, with potential for reservoir-engineering probes and Liouvillian spectroscopy.

Abstract

We study relaxation spectra of a quadratic spinless--fermion helical chain with an Aubry--Andre--type quasiperiodic potential and a single N--th neighbor (helical) hopping. Dissipation and pumping are introduced via local linear Lindblad jump operators and treated exactly using the third--quantization / Majorana covariance formalism. Focusing on periodic boundary conditions (to avoid edge artefacts) we compute the Liouvillian rapidities and their smallest nonzero real part (the rapidity gap) for several spatial dissipation patterns: uniform (all), single--site (one--site) and two--site (two--site) placement, plus pairwise gain/loss on helical partner sites. We show that uniform dissipation yields large, weakly lambda--dependent gaps, while sparse local dissipation produces gaps that shrink rapidly as the quasiperiodic potential lambda induces localization. Increasing t_N enhances relaxation by improving mode overlap with dissipative channels. Finite--size scaling, rapidity level statistics (Poisson vs Wigner--Dyson), and spatial profiles of slow modes provide a consistent picture linking Liouvillian spectral structure to transport and localization. Our results highlight Liouvillian rapidities as compact, experimentally relevant diagnostics of relaxation and sensitivity in engineered open quantum lattices.

Figures (3)

• Figure 1: Linecuts of the Liouvillian rapidity $\kappa$ versus quasiperiodic potential strength $\lambda$ for several values of the long-range hopping amplitude $t_N$. Each curve corresponds to one $t_N$ (legend in each panel). All data were obtained under periodic boundary conditions. Shown are: (a) uniform dissipation on all sites, (b) single-site dissipation and (c) two-site dissipation. Uniform dissipation yields the largest rapidity gaps and only weak dependence on $\lambda$, reflecting strong coupling of all modes to the environment. In contrast, localized dissipation leads to much smaller gaps that decrease rapidly with increasing $\lambda$, consistent with localization suppressing the overlap between slow modes and dissipative sites. Increasing $t_N$ systematically enhances $\kappa$, indicating that long-range hopping improves transport to dissipators and accelerates relaxation.
• Figure 2: Spectral diagnostics at $\lambda=0.5$ (transport-dominated regime), $N=3$. Each PDF (panels left to right) is a compact 2×2 diagnostic combining: (top-left) finite-size scaling of the Liouvillian rapidity gap $\kappa$ vs chain length $L$ for several $t_N$ values; (top-right) ordered real parts of rapidities (highlighting modes near zero); (bottom-left) nearest-neighbour spacing distribution $P(s)$ for the rapidities' real parts with Poisson (dashed) and Wigner–Dyson GOE (dot-dash) references; and (bottom-right) site-resolved weight of the slowest-decaying mode. The three panels show, from left to right: (a) uniform dissipation on all sites; (b) a single localized dissipator ; and (c) two localized dissipators. At $\lambda=0.5$ the uniform case displays the largest gaps and spacing statistics closer to Wigner–Dyson (indicative of strong mode mixing), while sparse localized dissipators show smaller gaps and more extended-to-intermediate localization signatures.
• Figure 3: Spectral diagnostics at $\lambda=1.5$ (localized regime), $N=3$. Panel layout and diagnostic content are the same as in Fig. \ref{['fig:diagnostics_lambda05']}. From left to right the panels correspond to: (a) uniform dissipation on all sites; (b) a single localized dissipator ; and (c) two localized dissipators. Increasing $\lambda$ to 1.5 drives stronger localization: sparse dissipator placements exhibit substantially reduced rapidity gaps, spacing distributions closer to Poisson, and slow-mode profiles that concentrate weight on the dissipative site(s). The uniform ('all') placement retains relatively large $\kappa$ and more extended slow modes by comparison.