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Bulletin of the American Physical Society

Behaviour of l-bits Near the Many-Body Localization Transition

Abishek K. Kulshreshtha, Arijeet Pal, Thorsten B. Wahl, Steven H. Simon

Published Mar 8, 2018• 2 citations
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Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators ${\ensuremath{\tau}}_{i}^{z}$ (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators ${\ensuremath{\tau}}_{i}^{z}$ and ${\ensuremath{\tau}}_{i}^{x}$ associated with l-bits $({\mathbf{\ensuremath{\tau}}}_{i})$ completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a nonperturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.

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Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N ≫1 localized bits subject to random fields. On increasing J , the system transitions from an MBL to a delocalized phase on the vanishing scale J c ( N )∼1/ N , up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either ‘on’ (strongly entangled) or ‘off’ (weakly entangled) in eigenstates. The clusters of ‘on’ bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.

arXiv:1707.00004•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Oct 30, 2017•57 citations
Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators ${\ensuremath{\tau}}_{i}^{z}$ (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators ${\ensuremath{\tau}}_{i}^{z}$ and ${\ensuremath{\tau}}_{i}^{x}$ associated with l-bits $({\mathbf{\ensuremath{\tau}}}_{i})$ completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a nonperturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.
Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N ≫1 localized bits subject to random fields. On increasing J , the system transitions from an MBL to a delocalized phase on the vanishing scale J c ( N )∼1/ N , up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either ‘on’ (strongly entangled) or ‘off’ (weakly entangled) in eigenstates. The clusters of ‘on’ bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.