[Paper Review] Absence of true localization in many-body localized phases

Maximilian Kiefer-Emmanouilidis, R. G. Unanyan+2 more|arXiv (Cornell University)|Oct 1, 2020

Quantum many-body systems4 citations

TL;DR

This paper provides numerical evidence that many-body localized (MBL) phases in disordered quantum spin chains do not exhibit true localization, as the particle number entropy $ S_N \sim \ln\ln t $ grows slowly and continuously over time, even at strong disorder. This persistent growth indicates ongoing particle number fluctuations across subsystems, contradicting the standard MBL picture where $ S_N $ saturates, and implies the system remains ergodic in the thermodynamic limit despite strong disorder.

ABSTRACT

We have recently shown that the logarithmic growth of the entanglement entropy following a quantum quench in a many-body localized (MBL) phase is accompanied by a slow growth of the number entropy, $S_N\sim\ln\ln t$. Here we provide an in-depth numerical study of $S_N(t)$ for the disordered Heisenberg chain and show that this behavior is not transient and persists even for very strong disorder. Calculating the truncated Rényi number entropy $S_N^{(α)}(t)=(1-α)^{-1}\ln\sum_n p^α(n)$ for $α\ll 1$ and $p(n)>p_c$ -- which is sensitive to large number fluctuations occurring with low probability -- we demonstrate that the particle number distribution $p(n)$ in one half of the system has a continuously growing tail. This indicates a slow but steady increase of the number of particles crossing between the partitions in the interacting case, and is in sharp contrast to Anderson localization, for which we show that $S_N^{(α o 0)}(t)$ saturates for any cutoff $p_c>0$. We show, furthermore, that the growth of $S_N$ is $\mathit not$ the consequence of rare states or rare regions but rather represents typical behavior. These findings provide strong evidence that the interacting system is never fully localized even for very strong but finite disorder.

Motivation & Objective

• To investigate whether the logarithmic growth of entanglement entropy in MBL phases is accompanied by sustained growth in number entropy $ S_N $, indicating incomplete localization.
• To determine if the observed $ S_N \sim \ln\ln t $ behavior is transient or persists at strong disorder, ruling out effects from rare regions or finite-size artifacts.
• To assess whether the growth of $ S_N $ is a typical feature of the MBL phase or due to rare states near the ergodic-MBL transition.
• To evaluate the reliability of finite-size extrapolations for $ S_N $ saturation values and challenge previous interpretations of decreasing $ S_N $ as evidence of localization.

Proposed method

• Numerical exact diagonalization of the disordered Heisenberg chain for system sizes up to $ L = 24 $, with disorder strengths up to twice the estimated critical value.
• Calculation of the truncated Rényi number entropy $ S_N^{(\alpha)}(t) = (1 - \alpha)^{-1} \ln \sum_n p^n(n) $ for $ \alpha \ll 1 $, focusing on large $ p(n) $ to detect slowly growing tails in the particle number distribution.
• Analysis of the full probability distribution $ p(n) $ of particle number in one subsystem to identify the emergence of a continuously growing tail, indicating persistent particle transfer.
• Use of double logarithmic fits $ S_N(t) \approx (\nu/2)\ln\ln t + b $ to extrapolate saturation values and estimate $ t_{\text{sat}} \sim \exp(L) $, accounting for numerical precision limits.
• Comparison with the non-interacting, off-diagonal disorder model to confirm that the observed behavior is not an artifact of integrability or special symmetry.
• Systematic study of $ S_N(t) $ across multiple disorder strengths and system sizes to assess scaling behavior and non-monotonicity in finite-size data.

Experimental results

Research questions

• RQ1 Does the number entropy $ S_N $ in the MBL phase grow indefinitely over time, or does it eventually saturate?
• RQ2 Is the observed $ S_N \sim \ln\ln t $ growth a transient effect or a robust feature of the MBL phase at strong disorder?
• RQ3 Can the persistent growth of $ S_N $ be attributed to rare regions or rare states, or is it typical behavior in the MBL phase?
• RQ4 Do finite-size extrapolations of $ S_N $ saturation values reliably indicate localization, or is the apparent decrease in $ S_N $ with system size misleading due to non-monotonic scaling?
• RQ5 How does the behavior of $ S_N^{(\alpha)} $ for $ \alpha \to 0 $ differ between interacting MBL systems and non-interacting Anderson-localized systems?

Key findings

• The number entropy $ S_N(t) $ grows as $ \ln\ln t $ over extended timescales, even at disorder strengths twice the estimated critical value, indicating persistent particle number fluctuations.
• The particle number distribution $ p(n) $ in one subsystem develops a continuously growing tail, which is a signature of ongoing particle transfer across the subsystem boundary.
• The truncated Rényi number entropy $ S_N^{(\alpha)} $ for $ \alpha \ll 1 $ shows sustained growth, confirming that large number fluctuations with low probability are not transient but accumulate over time.
• In contrast to Anderson localization, where $ S_N^{(\alpha \to 0)} $ saturates for any $ p_c > 0 $, the interacting MBL system shows no such saturation, indicating a fundamental difference in localization behavior.
• Finite-size scaling of $ S_N $ saturation values is non-monotonic and lacks a clear scaling function, making reliable extrapolation to the thermodynamic limit impossible with current methods.
• The observed behavior is not due to rare regions or rare states near the transition, but rather represents typical dynamics in the MBL phase, challenging the existence of a true MBL phase in the thermodynamic limit.