Anderson Transition Research Articles

Multifractality at Anderson Transitions with Coulomb Interaction

We explore mesoscopic fluctuations and correlations of the local density of states (LDOS) near localization transition in a disordered interacting electronic system. It is shown that the LDOS multifractality survives in the presence of the Coulomb interaction. We calculate the spectrum of multifractal dimensions in 2+ϵ spatial dimensions and show that it differs from that in the absence of interaction. The multifractal character of fluctuations and correlations of the LDOS can be studied experimentally by scanning tunneling microscopy of two-dimensional and three-dimensional disordered structures.

Transition between localized and extended states in the hierarchical Anderson model

We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping between their phases and those of models with short range hopping in higher dimensions, and also because the renormalization group can be applied exactly without the approximations that generally are required in other models. In the hierarchical Anderson model we find a finite critical disorder strength Wc where the average inverse participation ratio goes to zero; at small disorder W < Wc the model lies in a delocalized phase. This result is based on numerical calculation of the inverse participation ratio in the infinite volume limit using an exact renormalization group approach facilitated by the model's hierarchical structure. Our results are consistent with the presence of an Anderson transition in short-range models with D > 2 dimensions, which was predicted using renormalization group arguments. Our finding should stimulate interest in the hierarchical Anderson model as a simplified and tractable model of the Anderson localization transition which occurs in finite-dimensional systems with short-range hopping.

Phase diagram of the anisotropic Anderson transition with the atomic kicked rotor: theory and experiment

We realize experimentally a cold-atom system, the quasiperiodic kicked rotor, equivalent to the three-dimensional Anderson model of disordered solids where the anisotropy between the x direction and the y–z plane can be controlled by adjusting an experimentally accessible parameter. This allows us to study experimentally the disorder versus anisotropy phase diagram of the Anderson metal–insulator transition. Numerical and experimental data compare very well with each other and a theoretical analysis based on the self-consistent theory of localization correctly describes the observed behavior, illustrating the flexibility of cold-atom experiments for the study of transport phenomena in complex quantum systems.

Finite-size scaling of entanglement entropy at the Anderson transition with interactions

We study the Anderson transition with interactions in one dimension from the perspective of quantum entanglement. Extensive numerical calculations of the entanglement entropy (EE) of the systems are carried out through the density matrix renormalization group algorithm. We demonstrate that the EE can be used for the finite-size scaling (FSS) to characterize the Anderson transition in both noninteracting and interacting systems. From the FSS analysis we can obtain a precise estimate of the critical parameters of the transition. The method can be applied to various one-dimensional models, either interacting or noninteracting, to quantitatively characterize the Anderson transitions.

Anticorrelations from power-law spectral disorder and conditions for an Anderson transition

We resolve an apparent contradiction between numeric and analytic results for one-dimensional disordered systems with power-law spectral correlations. The conflict arises when considering rigorous results that constrain the set of correlation functions yielding metallic states to those with non-zero values in the thermodynamic limit. By analyzing the scaling law for a model correlated disorder that produces a mobility edge, we show that no contradiction exists as the correlation function exhibits strong anticorrelations in the thermodynamic limit. Moreover, the associated scaling function reveals a size-dependent correlation with a smoothening of disorder amplitudes as the system size increases.

Electronic disorder of P- and B-doped Si at the metal–insulator transition investigated by scanning tunnelling microscopy and electronic transport

The (111)-2 × 1 surface of in situ cleaved heavily P- or B-doped Si is investigated by scanning tunnelling microscopy and spectroscopy at room temperature and at low temperature. P atoms have been identified on different sites of the Si(111)-2 × 1 surface by their characteristic voltage-dependent contrast for positive as well as negative buckling of the π-bonded chains. The distributions of dopants per surface area and of nearest-neighbour distances are found to be in agreement with a random arrangement of dopants in Si up to doping levels well above the metal–insulator transition. In addition, P atoms have been identified by their depth-dependent contrast down to the third layer beneath the surface with a volume density in agreement with the bulk doping density. The random electronic disorder supports the view of an Anderson transition driven by disorder close to the critical concentration or critical uniaxial stress.

Classification and symmetry properties of scaling dimensions at Anderson transitions

We develop a classification of composite operators without gradients at Anderson-transition critical points in disordered systems. These operators represent correlation functions of the local density of states (or of wave-function amplitudes). Our classification is motivated by the Iwasawa decomposition for the field of the pertinent supersymmetric \sigma-model: the scaling operators are represented by "plane waves" in terms of the corresponding radial coordinates. We also present an alternative construction of scaling operators by using the notion of highest-weight vector. We further argue that a certain Weyl-group invariance associated with the \sigma-model manifold leads to numerous exact symmetry relations between the scaling dimensions of the composite operators. These symmetry relations generalize those derived earlier for the multifractal spectrum of the leading operators.

An absence of the Anderson transition in high-resistance alloys with a high electron density

In three-dimensional electron systems with a high electron density it is not possible to reach the Anderson transition neither by increasing disorder nor by a combination of static disorder with a dynamic one (by high temperature), or by selecting a material with a large deformation potential and a large constant of electron-phonon interaction.

Poisson-random matrix transition in the QCD Dirac spectrum

At zero temperature the lowest part of the spectrum of the QCD Dirac operator is known to consist of delocalized modes that are described by random matrix statistics. In the present paper we show that the nature of these eigenmodes changes drastically when the system is driven through the finite temperature crossover. The lowest Dirac modes that are delocalized at low temperature become localized on the scale of the inverse temperature. At the same time the spectral statistics changes from random matrix to Poisson statistics. We demonstrate this with lattice QCD simulations using $2+1$ flavors of light dynamical quarks with physical masses. Drawing an analogy with Anderson transitions we also examine the mobility edge separating localized and delocalized modes in the spectrum. We show that it scales in the continuum limit and increases sharply with the temperature.

Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition

A field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the Z(2) topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an N=1-[symbol: see text] expansion in the number of replicas, where for small [symbol: see text] the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field.

Field theory of Anderson transition of the kicked rotor

We present a microscopic theory of Anderson transition in the quantum kicked rotor. The behavior of the system is shown to depend sensitively on the value of the effective Planck constant, . For the periodically kicked rotor, we obtain quantitative results for the time-dependent behavior of the rotor's energy that characterizes the system's localization/resonance properties. For the quasiperiodically kicked rotor, we find that for irrational values of , the quantum phase transition exhibited in this system falls into the universality class of Anderson (metal–insulator) transition in disordered electronic systems; for rational values, the rotor-Anderson insulator turns into a ‘supermetal’ (i.e. the static conductivity diverges) and the system exhibits metal–supermetal transition.

Testing the Order Parameter of the Anderson Transition

The typical value of the local density of states ρ typ , known as a candidate of the order parameter of the Anderson transition, of two-dimensional disordered electron systems with spin–orbit interactions is studied for assessing the validity of ρ typ as the order parameter. We show that ρ typ behaves critically as ρ typ ∝( E c - E ) β with β=0.466±0.095 near the transition point. It is also found that the exponent β satisfies the scaling relation with the Lipschitz–Holder exponent α 0 . Furthermore, we demonstrate that fluctuations in ρ typ at the Anderson transition obey the generalized Gumbel distribution, which is expected to be the universal distribution function of critical fluctuations of order parameters. These results suggest that ρ typ is a suitable quantity of the order parameter of the Anderson transition.

Statistical properties of power-law random banded unitary matrices in the delocalization-localization transition regime

Power-law random banded unitary matrices (PRBUM), whose matrix elements decay in a power-law fashion, were recently proposed to model the critical statistics of the Floquet eigenstates of periodically driven quantum systems. In this work, we numerically study in detail the statistical properties of PRBUM ensembles in the delocalization-localization transition regime. In particular, implications of the delocalization-localization transition for the fractal dimension of the eigenvectors, for the distribution function of the eigenvector components, and for the nearest neighbor spacing statistics of the eigenphases are examined. On the one hand, our results further indicate that a PRBUM ensemble can serve as a unitary analog of the power-law random Hermitian matrix model for Anderson transition. On the other hand, some statistical features unseen before are found from PRBUM. For example, the dependence of the fractal dimension of the eigenvectors of PRBUM upon one ensemble parameter displays features that are quite different from that for the power-law random Hermitian matrix model. Furthermore, in the time-reversal symmetric case the nearest neighbor spacing distribution of PRBUM eigenphases is found to obey a semi-Poisson distribution for a broad range, but display an anomalous level repulsion in the absence of time-reversal symmetry.

Metallic Phase of the Quantum Hall Effect in Four-Dimensional Space

We study the phase diagram of the quantum Hall effect in four-dimensional (4D) space. Unlike in 2D, in 4D there exists a metallic as well as an insulating phase, depending on the disorder strength. The critical exponent ν≈1.2 of the diverging localization length at the quantum Hall insulator-to-metal transition differs from the semiclassical value ν=1 of 4D Anderson transitions in the presence of time-reversal symmetry. Our numerical analysis is based on a mapping of the 4D Hamiltonian onto a 1D dynamical system, providing a route towards the experimental realization of the 4D quantum Hall effect.

Anomalous quantum diffusion and the topological metal

Electron wave scattering off disorders provides a key to many fascinating transport phenomena recently observed in topological insulators. Here, we present a nonperturbative diagrammatic theory of this subject. Surprisingly, quantum superdiffusion is found on the surface of three-dimensional strong topological insulators regardless of disorder strength (but not vanishing), where the diffusion coefficient grows in time logarithmically. Such a transport anomaly serves as a main characteristic of the novel quantum metal, the so-called ``topological metal,'' and indicates that it is a hybridization of Ohmic and perfect metals. It washes out the Anderson transition occurring in two-dimensional normal metals with disordered spin-orbit coupling, and leads to a logarithmic divergence of the conductance in the sample size instead. Therefore, the present work provides an analytical proof of the transport anomaly discovered numerically [Nomura, Koshino, and Ryu, Phys. Rev. Lett. 99, 146806 (2007); Bardarson et al., Phys. Rev. Lett. 99, 106801 (2007)].

Transport properties of graphene containing structural defects

We propose an extensive report on the simulation of electronic transport in 2D graphene in presence of structural defects. Amongst the large variety of such defects in sp$^2$ carbon-based materials, we focus on the Stone-Wales defect and on two divacancy-type reconstructed defects. First, based on ab initio calculations, a tight-binding model is derived to describe the electronic structure of these defects. Then, semiclassical transport properties including the elastic mean free paths, mobilities and conductivities are computed using an order-N real-space Kubo-Greenwood method. A plateau of minimum conductivity ($\sigma^{min}_{sc}= 4e^2/\pi h$) is progressively observed as the density of defects increases. This saturation of the decay of conductivity to $\sigma^{min}_{sc}$ is associated with defect-dependent resonant energies. Finally, localization phenomena are captured beyond the semiclassical regime. An Anderson transition is predicted with localization lengths of the order of tens of nanometers for defect densities around 1%.

The destruction of the virtual Anderson transition in a narrow impurity band of doped quantum well structures

Earlier we reported an observation at low temperatures of activation conductivity with small activation energies in strongly doped uncompensated layers of p-GaAs/AlGaAs quantum wells. We attributed it to Anderson delocalization of electronic states in the vicinity of the maximum of the narrow impurity band due to the small width of the impurity band characterized by weak disorder. In this case the carriers were activated from the "band tail" while its presence was related to weak background compensation. Here we study an effect of the extrinsic compensation and of the impurity concentration on this "virtual" Anderson transition. It was shown that an increase of compensation initially does not affect the Anderson transition, however at strong compensations the transition is suppressed due to increase of disorder. However at larger impurity concentration the conductivity becomes to be metallic due to Mott transition.

Wonderful life at weak Coulomb interaction: increasing of superconducting/superfluid transition temperature by disorder

We have shown that in systems where the Coulomb interaction is strongly suppressed, the superconducting transition temperature may be enhanced by disorder close to the Anderson localization transition. This phenomenon is based on the enhancement by disorder of the matrix element of attraction in the Cooper channel. For enhancement to take place one needs (i) strong disorder which makes the single-particle wave functions strongly inhomogeneous in space and (ii) strong correlation of the patterns of inhomogeneity for different wavefunctions. One case where such correlation is well known is the system close to the Anderson transition. We review the notion of multifractality of wavefunctions in this region and show how the enhancement of Tc arises out of the multifractal correlations.

Extension of Anderson's Localization Theory to Interacting Systems

The paper adapts Anderson's original paper on localization to include interaction. For that purpose his perturbation expansion is extended to a many-body Rayleigh Schrödinger perturbation expansion. Two systems are compared which are similar except that one does and the other does not include Coulomb interactions. The latter is assumed to have all states localized. It is shown that interactions have two opposing effects – one enhancing localization, the other delocalization. Deep in the Anderson localization regime the first effect prevails, close to the Anderson transition the latter prevails