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Bulletin of the American Physical Society

Probing the many-body localization transition with matrix elements of local operators

Maksym Serbyn, Zlatko Papić, Dmitry A. Abanin

Published Mar 5, 2015• 0 citations
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Thermal and many-body localized phases are separated by a dynamical phase transition of a new kind. We analyze the distribution of off-diagonal matrix elements of local operators across this transition in two different models of disordered spin chains. We show that the behavior of matrix elements can be used to characterize the breakdown of thermalization and to extract the many-body Thouless energy. We find that upon increasing the disorder strength the system enters a critical region around the many-body localization transition. The properties of the system in this region are: (i) the Thouless energy becomes smaller than the level spacing, (ii) the matrix elements show critical dependence on the energy difference, and (iii) the matrix elements, viewed as amplitudes of a fictitious wave function, exhibit strong multifractality. This critical region decreases with the system size, which we interpret as evidence for a diverging correlation length at the many-body localization transition. Our findings show that the correlation length becomes larger than the accessible system sizes in a broad range of disorder strength values and shed light on the critical behavior near the many-body localization transition.
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.