Localization transitions in an open quasiperiodic ladder

TL;DR

This work analyzes localization transitions in an open two-stranded Aubry-André-Harper ladder with quasiperiodic onsite potentials $ ext{(} ext{e}_{p,j}=V ext{cos}(2\pi b j) ext{)}$ under Lindblad dissipation. Using third quantization in the Majorana basis, the dynamics are mapped to a damping matrix $X$ with rapidities $eta_i$ that reproduce the Liouvillian spectrum, enabling computation of the inverse and normalized participation ratios $ ext{IPR}$ and $ ext{NPR}$ to diagnose localization. It shows that dissipation can induce a mixed phase between delocalized and localized states in a strictly 1D ladder without diagonal hopping, with the mixed-phase window and localization thresholds tunable by dissipation strengths $( ho^g, ho^l)$ and by the dissipation pattern (every-site vs alternating-site). These findings highlight non-Hermitian environmental effects as controllable factors for localization in quasiperiodic systems, with potential relevance for engineered open quantum platforms.

Abstract

We investigate localization transition in an open quasiperiodic ladder where the quasiperiodicity is described by the Aubry-André-Harper model. While previous studies have shown that higher-order hopping or constrained quasiperiodic potentials can induce a mixed-phase zone in one dimension, we demonstrate that the dissipation can induce mixed phase zone in a one dimensional nearest-neighbor system without imposing any explicit constraints on the quasiperiodic potential or hopping parameter. Our approach exploits an exact correspondence between the eigenspectrum of the Liouvillian superoperator and that of the non-Hermitian Hamiltonian, valid for quadratic fermionic systems under linear dissipation. Using third quantization approach within Majorana fermionic representation, we analyze two dissipation configurations: alternating gain and loss at every site, and at alternate sites under balanced and imbalanced conditions. By computing the inverse and normalized participation ratios, we show that dissipation can drive the system into three distinct phases: delocalizd, mixed, and localized. Notably, the mixed-phase zone is absent for balanced dissipation at every site but emerges upon introducing imbalance, while for alternate site dissipation it appears in both balanced and imbalanced cases. Furthermore, the critical points and the width of the mixed-phase window can be selectively tuned by varying the dissipation strength. These findings reveal that the dissipation plays a decisive role in reshaping localization transitions in quasiperiodic systems, offering new insight into the interplay between non-Hermitian effects and quasiperiodic order.

Figures (5)

• Figure 1: (Color online). Schematic representation of two-stranded Aubry-André-Harper ladders illustrating two dissipation configurations: (a) alternating gain and loss at every site, and (b) alternating gain and loss applied at every other site. $\gamma^g$ and $\gamma^l$ denote the strengths of gain and loss dissipation, respectively, while $t$ and $\lambda$ represent the intra-chain and inter-chain hopping amplitudes, respectively.
• Figure 2: (Color online). $\langle\mathrm{IPR}\rangle$ and $\langle\mathrm{NPR}\rangle$ as functions of the Aubry-André-Harper disorder strength, with dissipation applied at every site. Panels (a)–(c) correspond to: (a) $\gamma^g=1,\ \gamma^l=1$; (b) $\gamma^g=1,\ \gamma^l=0.5$; and (c) $\gamma^g=1,\ \gamma^l=0$. Here the system size is $2N=3000$.
• Figure 3: (Color online). With dissipation applied at every site: (a) $\langle$IPR$\rangle$ as a function of the Aubry-André-Harper disorder strength $V$ for $\gamma^l = 1, 0.75, 0.5, 0.25,$ and $0$, with fixed $\gamma^g = 1$ and system size $2N = 3000$. (b) Phase diagram in the parameter space spanned by $V$ and $\gamma^l$, where the color bar represents the $\langle$NPR$\rangle$. Here $\gamma^g = 1$ and the system size is $2N = 500$.
• Figure 4: (Color online). $\langle\mathrm{IPR}\rangle$ and $\langle\mathrm{NPR}\rangle$ as functions of the Aubry-André-Harper disorder strength, with dissipation applied at alternate site. Panels (a)–(c) correspond to: (a) $\gamma^g=1,\ \gamma^l=1$; (b) $\gamma^g=1,\ \gamma^l=0.5$; and (c) $\gamma^g=1,\ \gamma^l=0$. Here the system size is $2N=3000$.
• Figure 5: (Color online). With dissipation applied at alternate site: (a) $\langle$IPR$\rangle$ as a function of the Aubry-André-Harper disorder strength $V$ for $\gamma^l = 1, 0.75, 0.5, 0.25,$ and $0$, with fixed $\gamma^g = 1$ and system size $2N = 3000$. (b) Phase diagram in the parameter space spanned by $V$ and $\gamma^l$, where the color bar represents the $\langle$NPR$\rangle$. Here $\gamma^g = 1$ and the system size is $2N = 500$.