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arXiv:1610.02389 · Physical review. B./Physical review. B

Thouless energy and multifractality across the many-body localization transition

Maksym Serbyn, Zlatko Papić, Dmitry A. Abanin

Published Sep 6, 2017• 158 citations
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Abstract

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.

DOI
10.1103/physrevb.96.104201

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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.
Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f(0≤α≤2)=α2f(0 \leq \alpha \leq 2)=\frac{\alpha}{2}f(0≤α≤2)=2α​, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered J_1 J 1 - J_2 J 2 model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.
We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, ${V}_{ij}\ensuremath{\propto}{\left|i\ensuremath{-}j\right|}^{\ensuremath{-}\ensuremath{\alpha}}$, where the exponent of the interaction range $\ensuremath{\alpha}$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\ensuremath{\alpha}$, the critical exponent experiences a sharp increase at about ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.2$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with an ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.
Abstract We study the details of the distribution of the entanglement spectrum (eigenvalues of the reduced density matrix) of a disordered spin chain exhibiting a many-body localization (MBL) transition. In the thermalizing region we identify the evolution under increasing system size of the eigenvalue distribution function, whose thermodynamic limit is close to (but possibly different from) the Marchenko–Pastur distribution. From the analysis we extract a correlation length L s ( h ) determining the minimum system size to enter the asymptotic region. We find that L s ( h ) diverges at the MBL transition. We discuss the nature of the subleading corrections to the entanglement spectrum distribution and to the entanglement entropy.
The transition between many-body localized states and the delocalized thermal states is an eigen-state phase transition at finite energy density outside the scope of conventional quantum statistical mechanics. In this work we investigate the properties of the transition by studying the behavior of the entanglement entropy of a subsystem of size LAL_ALA​ in a system of size L>LAL > L_AL>LA​ near the critical regime of the many-body localization transition. The many-body eigenstates are obtained by exact diagonalization of a disordered quantum spin chain under twisted boundary conditions to reduce the finite-size effect. We present a scaling theory based on the assumption that the transition is continuous and use the subsystem size LA/ξL_A /ξLA​/ξ as the scaling variable, where ξξξ is the correlation length. We show that this scaling theory provides an effective description of the critical behavior and that the entanglement entropy follows the thermal volume law at the transition point. We extract the critical exponent governing the divergence of ξξξ upon approaching the transition point. We also study the participation entropy in the spin-basis of the domain wall excitations and show that the transition point and the critical exponent agree with those obtained from finite size scaling of the entanglement entropy. Our findings suggest that the many-body localization transition in this model is continuous and describable as a localization transition in the many-body configuration space.