Numerical Study of Disordered Noninteracting Chains Coupled to a Local Lindblad Bath

TL;DR

This study uses noninteracting disordered chains coupled to a local Lindblad bath to examine avalanche-like destabilization of localization. By analyzing the Lindbladian gap $Δ = -\mathrm{Re}(\lambda_1)$ and the left-edge eigenstate overlap, the authors reveal strong finite-size effects that produce nonmonotonic relaxation behavior in both Anderson and Aubry–André–Harper models. The results indicate that finite-size scaling can overestimate the avalanche threshold in finite systems, with Anderson showing particularly large finite-size drifts compared to the quasi-periodic case, and they connect these findings to a simple toy model of local baths. The work highlights caution in extrapolating boundary-bath conclusions to infinite systems and discusses implications for the stability of many-body localization in interacting chains. Overall, the paper provides a controlled platform to relate avalanche physics to boundary dissipation and emphasizes the role of system size in interpreting localization-delocalization transitions in disordered quantum systems.

Abstract

Disorder can prevent many-body quantum systems from reaching thermal equilibrium, leading to a many-body localized phase. Recent works suggest that nonperturbative effects caused by rare regions of low disorder may destabilize the localized phase. However, numerical simulations of interacting systems are generically possible only for small system sizes, where finite-size effects might dominate. Here we perform a numerical investigation of noninteracting disordered spin chains coupled to a local Lindblad bath at the boundary. Our results reveal strong finite-size effects in the Lindbladian gap in both bath-coupled Anderson and Aubry-André-Harper models, leading to a non-monotonic behavior with the system size. We discuss the relaxation properties of a simple toy model coupled to local Lindblad baths, connecting its features to those of noninteracting localized chains. We comment on the implications of our findings for many-body systems.

Figures (7)

• Figure 1: (a) Sketch of the local Lindblad bath framework in Eqs. \ref{['eq:lindblad_2']}-\ref{['eq:lindblad-operators_fermi']}. The chain (black circles) is characterized by the Hamiltonian $H$ that can be either the Anderson or the Aubry-André-Harper one [Eqs. \ref{['eq:spin-hamiltonian']}, \ref{['eq:anderson-disorder']}, \ref{['eq:AAH-disorder']}]. The bath on the left is coupled via $L_1$ and $L_2$ to the leftmost site of the chain. (b) Effect of the bath on localized noninteracting spin chains. The Lindblad jump operators give rise to hoppings from the bath to the single-particle Hamiltonian eigenstates labeled by $\alpha$, with strength $J_\alpha$ exponentially decaying with the localization center $n_\alpha$, as exemplified by the shading arrows. The onsite bars at different heights represent the random single-particle eigenenergies $\epsilon_\alpha$.
• Figure 2: $\langle \log_{10}(\Delta 2^{L}) \rangle$ for the Anderson [(a)-(b)] and the Aubry-André-Harper [(c)-(d)] chains coupled to a local Lindblad bath acting on the leftmost site as a function of system size $L$ and disorder strength $W$ or $\lambda$. Dotted lines in the insets of (b) and (d) mark the critical disorder strength of the closed chains: $W_c=0$ for the Anderson chain and $\lambda_c=1$ for the Aubry-André-Harper chain. We set $J = \gamma = 1$ and average over at least 2000 (1000) disorder realizations for the Anderson (Aubry-André-Harper) model for each data point. Error bars are 95% bootstrap confidence intervals.
• Figure 3: Percentage of single-particle Hamiltonian eigenstates with nonvanishing (that is, larger than $10^{-14}$) overlap on the leftmost physical site $p$ of the Anderson and Aubry-André-Harper chains as a function of system size [(a) and (c)] and disorder strength [(b) and (d)]. (a)-(b) For the Anderson chain, $p$ rapidly decreases as $L$ and $W$ increase. (c)-(d) For the Aubry-André-Harper chain, $p \approx 1$ for $\lambda \lesssim \lambda_c = 1$, whereas it rapidly decreases for $\lambda \gtrsim \lambda_c$. We set $J=1$ and average over 100 disorder realizations for each data point. Error bars are nearly invisible.
• Figure 4: (a) A three-site ring (trimer) pierced by a magnetic flux and coupled to three different local baths creating and annihilating spinless fermions on each site. (b) One site is spatially decoupled from the other two that form a dimer; a local bath acts on one of the sites of the dimer. (c) Same as (b) but with the bath acting on the decoupled site. (d) A trimer with periodic boundary conditions in which the bath acts only on a single site. (e) A trimer with open boundary conditions coupled to a bath acting on one boundary site. (f) Same as (e) but with a local bath acting only on the central site.
• Figure 5: $\langle \log_{10}(\Delta 4^L) \rangle$ for the Anderson [(a)-(b)] and Aubry-André-Harper [(c)-(d)] chain coupled to a local bath acting on the leftmost site as a function of system size $L$ and disorder strength $W$ or $\lambda$. Dotted lines in the insets of (b) and (d) mark the critical disorder strength of the closed chains: $W_{c}=0$ for the Anderson chain and $\lambda_{c}=1$ for the Aubry-André-Harper chain. We set $J=\gamma=1$, and average over at least 2000 disorder realizations for each data point. Error bars are 95% bootstrap confidence intervals.