[Paper Review] Many-Body Localization Transition, Temporal Fluctuations of the Loschmidt Echo, and Scrambling

Jun Yang, Alioscia Hamma|arXiv (Cornell University)|Feb 1, 2017

Quantum many-body systems15 references3 citations

TL;DR

This paper investigates the many-body localization (MBL) transition in a disordered Heisenberg spin chain after a quantum quench, showing that temporal fluctuations of the Loschmidt Echo and its infinite-time average scale differently in the ETH and MBL phases. The key finding is that finite-size scaling of these fluctuations—particularly the inverse participation ratio IPR₂—acts as a dynamical order parameter for the MBL transition, with distinct scaling behaviors distinguishing the two phases despite similar finite-time dynamics.

ABSTRACT

We show that the transition between a ETH phase and a many-body localized phase is marked by the different finite size scaling behaviour of the decay of the Loschmidt Echo and its temporal fluctuations - after a quantum quench - in the infinite time limit, despite the fact that the finite time behaviour of such quantities is dramatically different approach the MBL phase, so that temporal fluctuations cannot be inferred from the infinite time average of the Loschmidt Echo. We also show the different scrambling powers of ETH and MBL Hamiltonians as a probe to the different approaches to equilibrium.

Motivation & Objective

• To identify dynamical signatures of the many-body localization (MBL) transition beyond equilibrium properties.
• To investigate how temporal fluctuations of the Loschmidt Echo differ between the ETH and MBL phases after a quantum quench.
• To assess whether the infinite-time average of the Loschmidt Echo or its fluctuations serve as reliable probes of the MBL transition.
• To compare the scrambling power of ETH and MBL Hamiltonians using trace distance between evolving orthogonal product states.
• To determine the finite-size scaling of the scrambling time T* in both phases and relate it to dynamical localization.

Proposed method

• Computes the Loschmidt Echo (LE) as the overlap |⟨ψ(0)|ψ(t)⟩|² between the initial state and its time-evolved state after a quantum quench.
• Analyzes the infinite-time average of the LE via the inverse participation ratio IPR₂ = ∑_n |C_n|⁴, which bounds temporal fluctuations of local observables.
• Uses the IPR₂ to probe localization: IPR₂ → 0 in ETH (ergodic) phase, IPR₂ → const in MBL phase, indicating persistent fluctuations.
• Defines scrambling time T* as the time when trace distance between two orthogonal, energy-similar product states stabilizes within ε = 5×10⁻⁷.
• Measures trace distance d(φ,ψ) = ½‖φ − ψ‖ on half-chain reduced density matrices, averaged over 50–1000 disorder realizations.
• Performs finite-size scaling of log T* with system size L to distinguish ETH (log T* bounded) and MBL (log T* ∝ L) behavior.

Experimental results

Research questions

• RQ1 How do the temporal fluctuations of the Loschmidt Echo differ between the ETH and MBL phases after a quantum quench?
• RQ2 Can the infinite-time average of the Loschmidt Echo or its fluctuations serve as a reliable order parameter for the MBL transition?
• RQ3 What is the finite-size scaling behavior of the scrambling time T* in the ETH and MBL phases?
• RQ4 How does the scrambling power—measured by the time to reach maximum distinguishability loss—differ between ETH and MBL Hamiltonians?
• RQ5 At what disorder strength h does the MBL transition occur, as indicated by the scaling of IPR₂ and T*?

Key findings

• The MBL transition is marked by a change in the finite-size scaling of the inverse participation ratio IPR₂: IPR₂ → 0 in ETH, IPR₂ → const in MBL, indicating persistent temporal fluctuations in the MBL phase.
• The critical disorder strength for the MBL transition is estimated at h_c ≈ 3 ± 0.3, consistent with prior studies.
• In the ETH phase, the logarithmic scrambling time log T* is bounded and independent of system size L, indicating fast scrambling.
• In the MBL phase, log T* scales linearly with system size L, indicating exponentially slow scrambling due to slow dynamics.
• Temporal fluctuations in the Loschmidt Echo cannot be inferred from its infinite-time average in the MBL phase, as equilibration is not reached in polynomial times.
• The IPR₂ serves as a dynamical order parameter for the MBL transition, with distinct scaling in the two phases despite similar finite-time LE behavior.