Anderson localization of waves in disordered media was originally predicted fifty years ago, in the context of transport of electrons in crystals. The phenomenon is much more general and has been observed in a variety of systems, including light waves. However, Anderson localization has not been observed directly for matter waves. Owing to the high degree of control over most of the system parameters (in particular the interaction strength), ultracold atoms offer opportunities for the study of disorder-induced localization. Here we use a non-interacting Bose-Einstein condensate to study Anderson localization. The experiment is performed with a one-dimensional quasi-periodic lattice-a system that features a crossover between extended and exponentially localized states, as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated through investigations of the transport properties and spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. This controllable system may be used to investigate the interplay of disorder and interaction (ref. 7 and references therein), and to explore exotic quantum phases.

Anderson Localization of a Non-Interacting Bose-Einstein Condensate

The delocalized non-ergodic phase existing in some random $N \times N$ matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site $j_0$. The main output of this approach is the inverse $\Gamma_{j_0}(N)$ of the characteristic time to leave the state $j_0$ that provides some broadening $\Gamma_{j_0}(N) $ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening $\Gamma_{j_0}(N) $ is smaller in scaling than the level spacing $\Delta_{j_0}(N) \propto \frac{1}{N}$, while the delocalized non-ergodic phase corresponds to the region where the broadening $\Gamma_{j_0}(N) $ decays with $N$ but is bigger in scaling than the level spacing $\Delta_{j_0}(N) $. Then the number $\frac{\Gamma_{j_0}(N)}{\Delta_{j_0}(N)} $ of resonances grows only sub-extensively in $N$. This approach allows to recover the multifractal spectrum of the Generalized-Rosenzweig-Potter (GRP) Matrix model [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. We then consider the Levy generalization of the GRP Matrix model, where the off-diagonal matrix elements are drawn with an heavy-tailed distribution of Levy index $1<\mu<2$ : the dynamics is then governed by a stretched exponential of exponent $\beta=\frac{2 (\mu-1)}{\mu}$ and the multifractal properties of eigenstates are explicitly computed.

Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach

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 $f(0 \leq \alpha \leq 2)=\frac{\alpha}{2}$, 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.

pdf/many-body-localization-transition-strong-multifractality-4d12icg1ss.pdf

Many-Body-Localization Transition : strong multifractality spectrum for matrix elements of local operators

The theory of large deviations is already the natural language for the statistical physics of equilibrium and non-equilibrium. In the field of disordered systems, the analysis via large deviations is even more useful to describe within a unified perspective the typical events and the rare events that occur on various scales. In the present pedagogical introduction, we revisit various emblematic classical and quantum disordered systems in order to highlight the common underlying mechanisms from the point of view of large deviations.

Revisiting classical and quantum disordered systems from the unifying perspective of large deviations

The space of one-dimensional disordered interacting quantum models displaying a many-body localization (MBL) transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the localized phase and the Wigner–Dyson statistics of the delocalized phase. In this paper, we consider the strong disorder limit of the MBL transition, where the level statistics at the MBL critical point is close to the Poisson statistics. We analyze a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent 0 < α < 1 anywhere between the area law α = 0 and the volume law α = 1 , as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value ν = 1 for the correlation length exponent. Independently of the strong disorder limit, we explain why, for the many-body localization transition concerning individual eigenstates, the correlation length exponent ν is not constrained by the usual Harris inequality ν ≥ 2 / d , so that there is no theoretical inconsistency with the best numerical measure ν = 0 . 8 ( 3 ) obtained by Luitz et al. (2015).

pdf/many-body-localization-transition-in-the-strong-disorder-3y1b5ru0f1.pdf

Many-Body-Localization Transition in the Strong Disorder Limit: Entanglement Entropy from the Statistics of Rare Extensive Resonances

We develop a supersymmetric virial expansion for two point correlation functions of almost diagonal Gaussian Random Matrix Ensembles (ADRMT) of the orthogonal symmetry. These ensembles have multiple applications in physics and can be used to study universal properties of time-reversal invariant disordered systems which are either insulators or close to the Anderson localization transition. We derive a two-level contribution to the correlation functions of the generic ADRMT and apply these results to the critical (multifractal) power law banded ADRMT. Analytical results are compared with numerical ones.

Supersymmetric Virial Expansion for Time-Reversal Invariant Disordered Systems

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Supersymmetric Virial Expansion for Time-Reversal Invariant Disordered Systems