Finite steady-state current defies non-Hermitian many-body localization

TL;DR

This paper investigates whether non-Hermitian many-body localization (NH MBL) occurs in a disordered XXZ chain by focusing on transport rather than spectral indicators. Using exact diagonalization for small systems and tensor-network methods for large systems, the authors measure the steady-state spin current $J_\infty$ and the gap $\Delta$ to the first excited state, finding that $J_\infty$ decays exponentially with disorder strength $h$ but remains finite up to disorder values well beyond the spectral crossover near $h_c\simeq 4.5$, suggesting the absence of a true localization transition in observables. They also examine the fraction of real spectra $f_{noSS}$ and the non-real-eigenvalue fraction $f_{Im}$, observing a drift of spectral indicators with system size, which contrasts with the robust transport signatures. The work highlights a possible noncommutativity of infinite-time and thermodynamic limits and argues that dynamical probes are essential to correctly characterize localization in non-Hermitian settings, calling for a reevaluation of NH MBL criteria.

Abstract

Non-Hermitian many-body localization (NH MBL) has emerged as a possible scenario for stable localization in open systems, as suggested by spectral indicators identifying a putative transition for finite system sizes. In this work, we shift the focus to dynamical probes, specifically the steady-state spin current, to investigate transport properties in a disordered, non-Hermitian XXZ spin chain. Through exact diagonalization for small systems and tensor-network methods for larger chains, we demonstrate that the steady-state current remains finite and decays exponentially with disorder strength, showing no evidence of a transition up to disorder values far beyond the previously claimed critical point. Our results reveal a stark discrepancy between spectral indicators, which suggest localization, and transport behavior, which indicates delocalization. This highlights the importance of dynamical observables in characterizing NH MBL and suggests that traditional spectral measures may not fully capture the physics of non-Hermitian systems. Additionally, we observe a non-commutativity of limits in system size and time, further complicating the interpretation of finite-size studies. These findings challenge the existence of NH MBL in the studied model and underscore the need for alternative approaches to understand localization in non-Hermitian settings.

Figures (4)

• Figure 1: (a) Steady-state current $J_\infty$ obtained via exact diagonalization ($N \leq 16$) and via time evolution for long times ($N=18$). Around $h \simeq 4.5$ (green vertical stripe), a non-Hermitian many-body localization transition was previously claimed (see also Fig. \ref{['fig:fractions']}). However, a weak current persists up to $h = 12$, signaling that the system is actually delocalized. (b) Gap to the first excited state $\Delta$, setting the (inverse) timescale at which the steady state is reached. The data is averaged over 15000 disorder realizations for $N\leq 16$, and over 3000 realizations for $N=18$.
• Figure 2: (a) The fraction $f_\mathrm{noSS}$ of disorder realizations with a completely real spectrum drifts towards larger values of the disorder strength $h$, as the system size is increased. Correspondingly, the spectrum becomes more and more delocalized. In the inset we show the scaling of $f_\mathrm{noSS}$ with system size at various values of $h$. (b) Considering instead the average fraction of eigenvalues with a nonzero imaginary part $f_\mathrm{Im}$, one might conclude that there is a NH MBL transition at the value $h_c \simeq 4.5$ (vertical green stripes). More details in the main text.
• Figure 3: (a): To avoid boundary effects, we study the current in the central $2\ell_0$ sites of a $N=560$ chain ($\ell_0 = 25,35,45$ for increasing disorder). The resulting $\langle \hat{J}_\text{mid}\rangle$ shows convergence in time to a plateau $J_0(h)$, before boundary effects eventually kick in. (b): The value $J_0$ is obtained by averaging the current within a large time window and we further evaluate the current per site $j_0 = J_0/\ell_0$ to be able to compare with exact diagonalization results. The value of $j_0$ obtained from dynamics also decays exponentially with $h$, but with a different slope than the one resulting from exact diagonalization.
• Figure 4: (a): We compare the mid current for various system sizes, showing how the current plateau expands as $N$ increases. This suggests that the deviations from it are due to boundary effects and that the plateau corresponds to the asymptotic value as $N\to\infty$. (b): The value of the plateau is converged in bond dimension, ensuring the accuracy of our numerical simulations. We notice however that the extent of the plateau is affected by the choice of bond dimension.