Extended slow dynamical regime close to the many-body localization transition (1511.05141v2)

Abstract: Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to $L=28$ sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\propto t^{1/z}$ with a disorder-dependent exponent $z\ge1$, in contrast with the pure ballistic growth $z=1$ of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance ${\cal{I}}(t)\propto t^{-\zeta}$ with a continuously varying disorder-dependent exponent $\zeta$, vanishing at the transition. This provides a clear experimental signature for detecting this non-conventional regime.

• The paper shows that sub-ballistic entanglement growth follows a power-law (S(t) ∝ t^(1/z)) with a disorder-dependent exponent.
• It utilizes an exact Krylov space method to analyze time evolution in a random-field Heisenberg chain of up to 28 spins.
• The findings reveal anomalous spin relaxation and memory retention near the many-body localization transition.

An Essay on "Extended Slow Dynamical Regime Close to the Many-Body Localization Transition"

The paper "Extended slow dynamical regime close to the many-body localization transition," authored by David J. Luitz, Nicolas Laflorencie, and Fabien Alet, examines the dynamics near the many-body localization (MBL) transition. This research explores the influence of MBL phases on the dynamics within the delocalized ergodic regime, specifically in the context of the random-field Heisenberg chain. Using an exact Krylov space method, the paper analyzes the time-evolution of systems as large as 28 spins, presenting a thorough investigation into the disorder-driven transition characteristic of MBL.

The authors present detailed numerical results highlighting the anomalous dynamics observed in the delocalized phase approaching the MBL transition. An intriguing aspect of this paper is the identified sub-ballistic growth of entanglement entropy, $S(t) \propto t^{1/z}$, where the exponent $z$ is disorder-dependent and indicates a slower-than-ballistic spread of entanglement in the ergodic phase. Typical of clean systems, the entanglement growth is ballistic with $z=1$, contrasting sharply with the findings in disordered systems where $z \ge 1$. This result substantiates predictions of a sub-diffusive regime close to the MBL transition, caused by Griffiths regions characterized by rare, localized states.

Another significant result is the observed relaxation of an initial spin imbalance, ${\cal{I}}(t) \propto t^{-\zeta}$, where $\zeta$ is another disorder-dependent exponent. This finding is robust across the critical disorder strength $h_c \simeq 3.7$, providing a salient experimental signature for this non-standard regime. Notably, in the MBL phase, the imbalance does not fully decay, indicating a retention of memory of the initial state, a haLLMark characteristic of many-body localized systems.

The implications of these findings extend broadly across theoretical and experimental fronts of condensed matter physics. The work suggests that classical notions of thermalization do not apply ubiquitously in quantum systems as proximity to localized phases can distort expected dynamics even in delocalized regions. The sub-diffusive regime observed, characterized by power-laws with non-universal exponents, offers a fresh perspective to consider dynamically how and why thermalization may fail in quantum systems.

Experimental realizations in cold atom setups further dictate the relevance of this research, particularly given the experimental observation of MBL in optical lattices. The extended slow dynamical regime highlighted by the authors may be detectable through careful manipulation and measurement of entanglement and spin-density imbalance, much as demonstrated in the paper.

Looking forward, questions remain about the extent and universality of this anomalous sub-diffusive phase. The presence of a mobility edge within the system implies further investigation into how these anomalous dynamics depend on initial energy states and whether the energy range dictates the observed dynamical regimes. Additionally, exploring the theoretical basis and consequences of these findings in higher-dimensional systems stands to provide a fertile ground for future research, particularly using larger and more sophisticated computational methods.

Overall, this paper constitutes a significant contribution to the understanding of dynamics near the MBL transition. By delineating a clear framework grounded in extensive numerical simulations, it provides a basis from which further theoretical and experimental investigations may draw upon to unravel the complexities of quantum many-body systems exhibiting localization phenomena.