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Many-body localization in periodically driven systems
Pedro Ponte, Z. Papić, François Huveneers, Dmitry A. Abanin
TL;DR
The paper tests whether Floquet eigenstates exhibit ETH behavior across localized and delocalized regimes. It examines eigenstate-observable deviations and constructs quasi-local integrals of motion, finding finite deviations and quasi-locality in the MBL phase but vanishing deviations and non-locality in the delocalized phase.
Problem
The study asks whether eigenstate expectation values distinguish MBL from delocalized Floquet phases in the thermodynamic limit.
Method
The paper measures deviations of local-observable expectations in individual Floquet eigenstates and constructs quasi-local integrals of motion using infinite-time-averaged operators.
Results
At T1 = 0.4, ⟨∆O⟩ remains finite with increasing system size, while at T1 = 3.0, ∆O approaches zero; corresponding operators are quasi-local in MBL and non-local when delocalized.
Takeaways & Limitations
Finite ETH deviations and exponentially decaying operator support provide consistent eigenstate and operator signatures of the MBL phase.
Takeaways & Limitations
The thermodynamic-phase conclusions rely on extrapolating quantities measured across finite chain sizes.
Abstract
from arXiv · showhide
We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. MBL phase exhibits logarithmic in time growth of entanglement entropy for initial product states, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion (LIOM), which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggests a direct transition between the two phases. Our results show that many-body localization is not destroyed by sufficiently weak periodic driving.
Supplemental Online Material for “Many-body localization in periodically driven
The supplemental material reports direct ETH tests and an explicit construction of local integrals of motion for the driven MBL problem.
- The material accompanies the paper “Many-body localization in periodically driven” by Pedro Ponte, Z. Papić, François Huveneers, and Dmitry A. Abanin.
- The supplemental material tests the eigenstate thermalization hypothesis in both driven MBL and delocalized phases.
- It explicitly constructs local integrals of motion in the MBL phase for the Floquet problem.
TESTING THE ETH
The ETH test compares local-operator expectation values with infinite-temperature canonical predictions. Their finite-size behavior differs between the MBL and delocalized phases, supporting distinct eigenstate properties.
- TESTING THE ETH: In the delocalized phase, ETH predicts local-operator expectation values converge to the infinite-temperature canonical-ensemble value.
- TESTING THE ETH: The test examines deviations of local-operator expectation values in individual Floquet eigenstates from O∞.
- TESTING THE ETH: The deviation is expected to vanish with increasing chain size in the delocalized phase but remain finite in the MBL phase.
- TESTING THE ETH: The operators act on two neighboring middle-chain sites and conserve Sz.
- TESTING THE ETH: At T1 = 0.4, the deviation changes weakly with system size, whereas at T1 = 3.0 it approaches zero.
- TESTING THE ETH: Figure 5 compares deviations for different Sz-preserving local operators in the MBL and delocalized phases.
LOCAL INTEGRALS OF MOTION
The paper constructs quasi-local integrals of motion in the MBL phase by time-averaging local operators and testing their spatial localization. These operators retain local magnetization memory, whereas the corresponding operator becomes non-local in the delocalized phase.
- Construction: Time-averaging a local σz operator produces an integral of motion whose spatial structure can be tested through long-time magnetization and partial norms.The construction relates magnetization spreading to the time-averaged operator and examines whether it can be approximated with finite support.
- Delocalized phase: In the delocalized phase at T1 = 3, magnetization spreads nearly uniformly across sites and the time-averaged operator becomes non-local.Its stronger dependence on chain size contrasts with the localized spatial profile in the MBL phase.
- MBL phase: In the MBL phase at T1 = 0.4, long-time magnetization remains order unity on the initial site but decays over several orders of magnitude with distance.This spatial decay is consistent with the operator being quasi-local.
- MBL phase: The normalized difference between total and partial operator norms decays exponentially with distance in the MBL phase, demonstrating quasi-locality.The partial norm uses a region containing sites 1 through j and tests how much operator weight lies outside that region.
- Emergent LIOMs: Quasi-local integrals of motion can be constructed at other sites, forming an extensive set of LIOMs in the MBL phase.Their localized support and exponentially weak effect on remote degrees of freedom characterize the MBL construction.