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The localization of one-electron states in the large (but finite) disorder limit is investigated. The inverse participation number shows a non--monotonic behavior as a function of energy owing to anomalous behavior of few-site localization. The two-site approximation is solved analytically and shown to capture the essential features found in numerical simulations on one-, two- and three-dimensional systems. Further improvement has been obtained by solving a three-site model.
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We evaluate the localization length of the wave (or Schroedinger) equation in the presence of a disordered speckle potential. This is relevant for experiments on cold atoms in optical speckle potentials. We focus on the limit of large disorder, where the Born approximation breaks down and derive an expression valid in the quasi-metallic phase at large disorder. This phase becomes strongly localized and the effective mobility edge disappears.
Exponential localization of wavefunctions in lattices, whether in real or synthetic dimensions, is a fundamental wave interference phenomenon. Localization of Bloch-type functions in space-periodic lattice, triggered by spatial disorder, is known as Anderson localization and arrests diffusion of classical particles in disordered potentials. In time-periodic Floquet lattices, exponential localization in a periodically driven quantum system similarly arrests diffusion of its classically chaotic counterpart in the action-angle space. Here we demonstrate that nonlinear optical response allows for clear detection of the disorder-induced phase transition between delocalized and localized states. The optical signature of the transition is the emergence of symmetry-forbidden even-order harmonics: these harmonics are enabled by Anderson-type localization and arise for sufficiently strong disorder even when the overall charge distribution in the field-free system spatially symmetric. The ratio of even to odd harmonic intensities as a function of disorder maps out the phase transition even when the associated changes in the band structure are negligibly small.
The disordered many-body systems can undergo a transition from the extended ensemble to a localized ensemble, known as many-body localization (MBL), which has been intensively explored in recent years. Nevertheless, the relation between Anderson localization (AL) and MBL is still elusive. Here we show that the MBL can be regarded as an infinite-dimensional AL with the correlated disorder in a virtual lattice. We demonstrate this idea using the disordered XXZ model, in which the excitation of $d$ spins over the fully polarized phase can be regarded as a single-particle model in a $d$ dimensional virtual lattice. With the increasing of $d$, the system will quickly approach the MBL phase, in which the infinite-range correlated disorder ensures the saturation of the critical disorder strength in the thermodynamic limit. From the transition from AL to MBL, the entanglement entropy and the critical exponent from energy level statics are shown to depend weakly on the dimension, indicating that belonging to the same universal class. This work clarifies the fundamental concept of MBL and presents a new picture for understanding the MBL phase in terms of AL.
The role of disorder on wave propagation through the universe is studied. Assuming space fluctuations of the order of the Planck length and the size of the universe as the corresponding localization length for the background radiation, we obtain the exponent (close to unity) in the power law relationship between these quantities. This suggests that the role of Anderson localization is not negligible at cosmological scales.
We study Anderson localization of single particles in continuous, correlated, one-dimensional disordered potentials. We show that tailored correlations can completely change the energy-dependence of the localization length. By considering two suitable models of disorder, we explicitly show that disorder correlations can lead to a nonmonotonic behavior of the localization length versus energy. Numerical calculations performed within the transfer-matrix approach and analytical calculations performed within the phase formalism up to order three show excellent agreement and demonstrate the effect. We finally show how the nonmonotonic behavior of the localization length with energy can be observed using expanding ultracold-atom gases.
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