[Paper Review] Apparent slow dynamics in the ergodic phase of a driven many-body localized system without extensive conserved quantities

Talía L. M. Lezama, Soumya Bera+1 more|arXiv (Cornell University)|Sep 8, 2018

Quantum chaos and dynamical systems4 citations

TL;DR

This study investigates slow dynamics in the ergodic phase of a periodically driven many-body localized system without global conservation laws using a fast Walsh-Hadamard transform for exact time evolution. Despite observing subballistic entanglement growth and stretched-exponential autocorrelation decay—signatures of slow dynamics—these effects diminish with increasing system size, suggesting they may be finite-size artifacts rather than intrinsic to the true thermodynamic limit.

ABSTRACT

We numerically study the dynamics on the ergodic side of the many-body localization transition in a periodically driven Floquet model with no global conservation laws. We describe and employ a numerical technique based on the fast Walsh-Hadamard transform that allows us to perform an exact time evolution for large systems and long times. As in models with conserved quantities (e.g., energy and/or particle number) we observe a slowing down of the dynamics as the transition into the many-body localized phase is approached. More specifically, our data is consistent with a subballistic spread of entanglement and a stretched-exponential decay of an autocorrelation function, with their associated exponents reflecting slow dynamics near the transition for a fixed system size. However, with access to larger system sizes, we observe a clear flow of the exponents towards faster dynamics and can not rule out that the slow dynamics is a finite-size effect. Furthermore, we observe examples of non-monotonic dependence of the exponents with time, with dynamics initially slowing down but accelerating again at even larger times, consistent with the slow dynamics being a crossover phenomena with a localized critical point.

Motivation & Objective

• To investigate whether slow dynamics, typically associated with many-body localized phases, can emerge in the ergodic phase of a driven many-body localized system without extensive conserved quantities.
• To determine whether the observed slow dynamics in such systems are genuine critical phenomena or finite-size artifacts.
• To explore the nature of entanglement growth and autocorrelation decay in the absence of global conservation laws in Floquet-driven systems.
• To assess whether the system exhibits a Griffiths-like phase or a localized critical point based on transient dynamics.

Proposed method

• Employed a fast Walsh-Hadamard transform to enable exact time evolution for large system sizes (up to L = 28) and long times (nτ > 10⁴).
• Simulated stroboscopic dynamics of spin-spin autocorrelation functions starting from an infinite-temperature initial state.
• Tracked entanglement entropy evolution after a quench from a Néel state to analyze the spread of quantum information.
• Used logarithmic derivatives of entanglement entropy to extract power-law exponents α and β for entanglement and autocorrelation decay.
• Analyzed the time and system size dependence of exponents to detect non-monotonic behavior and potential crossovers.
• Compared results across different disorder strengths Γ to probe the approach to the MBL transition point.

Experimental results

Research questions

• RQ1 Does the ergodic phase of a driven many-body localized system without global conservation laws exhibit slow dynamics similar to that seen in systems with conserved quantities?
• RQ2 Are the observed subballistic entanglement growth and stretched-exponential decay of autocorrelation functions indicative of a true critical regime or finite-size effects?
• RQ3 Can the observed slowing down of dynamics be attributed to a localized critical point, or is it a transient behavior that eventually accelerates?
• RQ4 How do the exponents of entanglement growth and autocorrelation decay evolve with increasing system size?
• RQ5 Is the slow dynamics in this model consistent with a Griffiths phase, or does it instead reflect a crossover phenomenon?

Key findings

• Subballistic entanglement growth with exponents α ≈ 0.5 to 1.0 was observed on the ergodic side of the transition, consistent with slow dynamics for finite system sizes.
• The autocorrelation function exhibited stretched-exponential decay with exponents β that decreased toward zero as the system approached the critical disorder strength Γc ≈ 0.3.
• For fixed system size, the dynamics initially slowed down but later accelerated at longer times, indicating non-monotonic behavior and possible transient effects.
• As system size increased, the exponents α and β flowed toward faster dynamics, suggesting that the observed slow dynamics may not persist in the thermodynamic limit.
• The observed behavior is consistent with a localized critical point, where small systems remain influenced by critical dynamics before flowing into fully ergodic behavior.
• The absence of extensive conserved quantities does not preclude the emergence of slow dynamics, but such behavior appears to be a finite-size crossover rather than a true phase.