[Paper Review] Local integrals of motion and the logarithmic lightcone in many-body localized systems

Isaac H. Kim, Anushya Chandran+1 more|arXiv (Cornell University)|Dec 9, 2014

Elasticity and Wave Propagation15 citations

TL;DR

This paper proposes defining many-body localization (MBL) via the quasi-locality of a complete set of local integrals of motion (LIMs), which leads to a logarithmic lightcone for information propagation. The authors prove that this structure implies logarithmic entanglement growth after a global quench and enables efficient classical simulation of typical disorder realizations, establishing a universal, model-independent characterization of the MBL phase.

ABSTRACT

We propose to define full many-body localization in terms of the recently introduced integrals of motion[Chandran et al., arXiv:1407.8480], which characterize the time-averaged response of the system to a local perturbation. The quasi-locality of such integrals of motion implies an effective lightcone that grows logarithmically in time. This subsequently implies that (i) the average entanglement entropy can grow at most logarithmically in time for a global quench from a product state, and (ii) with high probability, the time evolution of a local operator for a time interval $|t|$ can be classically simulated with a resource that scales polynomially in $|t|$.

Motivation & Objective

• To establish a rigorous, model-independent definition of many-body localization based on the locality of integrals of motion.
• To demonstrate that the quasi-locality of these integrals of motion implies a logarithmic lightcone for information propagation.
• To prove that the average entanglement entropy grows at most logarithmically after a global quench.
• To show that the time evolution of most disorder realizations can be classically simulated with polynomial resources in time.
• To provide a framework that avoids the technical challenges of rare resonant regions by focusing on averaged decay properties.

Proposed method

• Introduce a disorder-averaged decay condition (Definition 1) for local integrals of motion, replacing probabilistic bounds on rare events.
• Derive a modified Lieb-Robinson bound tailored to the decay of integrals of motion, yielding a logarithmic lightcone with effective speed approaching zero.
• Use the logarithmic lightcone to bound the rate of entanglement entropy growth, showing it is at most O(log t) for long times.
• Apply the bound to prove that typical disorder realizations allow efficient classical simulation of time evolution with resources polynomial in time.
• Leverage the average decay of integrals of motion to sidestep issues from rare resonant regions that plague probabilistic approaches.
• Connect the framework to the fractional moment method, using averaged decay to infer dynamical localization without requiring full probabilistic control.

Experimental results

Research questions

• RQ1 Can the locality of the integrals of motion introduced by Chandran et al. (2014) be used to derive universal features of the MBL phase?
• RQ2 Does the quasi-locality of these integrals of motion imply a logarithmic lightcone for information propagation in MBL systems?
• RQ3 What is the maximum possible growth rate of average entanglement entropy after a global quench in the MBL phase?
• RQ4 Can the time evolution of most disorder realizations in MBL systems be efficiently simulated on a classical computer?
• RQ5 Is the average decay of integrals of motion sufficient to establish dynamical localization without requiring rare-event control?

Key findings

• The average entanglement entropy after a global quench grows at most logarithmically in time, bounded by O(ξ log t), where ξ is the localization length.
• A modified Lieb-Robinson bound with a logarithmic lightcone is derived, showing information spreads with an effective speed approaching zero in the infinite time limit.
• The time evolution of a local operator over time |t| can be classically simulated with resources scaling polynomially in |t| for typical disorder realizations.
• The logarithmic lightcone implies that transport is absent in the MBL phase, consistent with the absence of thermalization.
• The approach avoids the need to control rare resonant regions by focusing on averaged decay, making it more robust than probabilistic methods.
• The framework is general and applies to any dimension, suggesting that entanglement growth in higher dimensions may scale as |∂A| log t, where |∂A| is the area of the entanglement cut.