Anderson Metal-insulator Transition Research Articles

Mott-Hubbard and Anderson metal-insulator transitions in correlated lattice fermions with binary disorder

Strongly correlated fermions in a crystal or in an optical lattice in the presence of binary-alloy disorder are investigated. We employ the statistical dynamical mean-field theory, which incorporates both fluctuations due to disorder and local correlations due to interaction, to solve the Anderson-Hubbard model. Localization due to disorder is studied by means of the probability distribution function of the local density of states. We obtain a complete paramagnetic ground state phase diagram consisting of disordered correlated metal, Anderson-Mott insulator, and band insulator.

Poisson statistics for eigenvalues of continuum random Schrödinger operators

We show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the local- ization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point pro- cess with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues. We derive a Minami estimate for continuum An- derson Hamiltonians. We also give a simple and transparent proof of Minami's estimate for the (discrete) Anderson model. In this article we show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the localiza- tion region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues in that region. Local fluctuations of eigenvalues of random operators is believed to distinguish between localized and delocalized regimes, indicating an Anderson metal-insulator transition. Exponential decay of eigenfunctions implies that disjoint regions of space are uncorrelated and create almost independent eigenvalues, and thus absence of energy levels repulsion, which is mathematically translated in terms of a Poisson point process. On the other hand, extended states imply that distant regions have mutual influence, and thus create some repulsion between energy levels. Local fluctuations of eigenvalues have been studied within the context of random matrix theory, in particular Wigner matrices and GUE matrices, cf. (B, DiPS, ESY1, ESY2, J1, J2, SS) and references therein. It is challenging to understand random hermitian band matrices from the perspective of their eigenvalues fluctuations, by proving a transition between Poisson statistics and a semi-circle law for the density of states (a signature of energy levels repulsion), and relate this to the (discrete) Anderson model, cf. (B, DiPS). CMV matrices are another class of random matrices for which Poisson statistics and a transition to energy levels repulsion have been proved (KS, St1, St2). For random Schrodinger operators, Poisson statistics for eigenvalues was first proved by Molchanov (Mo2) for the same one-dimensional continuum random Schro- dinger operator for which Anderson localization was first rigorously established

Two-dimensional Anderson-Hubbard model in the DMFT + Σ approximation

The density of states, the dynamic (optical) conductivity, and the phase diagram of the paramagnetic two-dimensional Anderson-Hubbard model with strong correlations and disorder are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are accounted by the DMFT, while disorder is taken into account via the appropriate generalization of the self-consistent theory of localization. We consider the two-dimensional system with the rectangular “bare” density of states (DOS). The DMFT effective single-impurity problem is solved by numerical renormalization group (NRG). The “correlated metal,” Mott insulator, and correlated Anderson insulator phases are identified from the evolution of the density of states, optical conductivity, and localization length, demonstrating both Mott-Hubbard and Anderson metal-insulator transitions in two-dimensional systems of finite size, allowing us to construct the complete zero-temperature phase diagram of the paramagnetic Anderson-Hubbard model. The localization length in our approximation is practically independent of the strength of Hubbard correlations. But the divergence of the localization length in a finite-size two-dimensional system at small disorder signifies the existence of an effective Anderson transition.

Observation of the Anderson metal-insulator transition with atomic matter waves: Theory and experiment

Using a cold atomic gas exposed to laser pulses---a realization of the chaotic quasiperiodic kicked rotor with three incommensurate frequencies---we study experimentally and theoretically the Anderson metal-insulator transition in three dimensions. Sensitive measurements of the atomic wave function and the use of finite-size scaling techniques make it possible to unambiguously demonstrate the existence of a quantum phase transition and to measure its critical exponents. By taking proper account of systematic corrections to one-parameter scaling, we show the universality of the critical exponent $\ensuremath{\nu}=1.59\ifmmode\pm\else\textpm\fi{}0.01$, which is found to be equal to the one previously computed for the Anderson model.

Distribution of conductance for Anderson insulators: A theory with a single parameter

We obtain an analytic expression for the full distribution of conductance for a strongly disordered threedimensional conductor within a perturbative approach based on transfer-matrix formulation. Our results confirm the numerical evidence that the log-normal limit of the distribution is not reached even in the deeply insulating regime. We show that the variance of the logarithm of the conductance scales as a fractional power of the mean, while the skewness changes sign as one approaches the Anderson metal-insulator transition from the deeply insulating limit, all described as a function of a single parameter. The approach suggests a possible single parameter description of the Anderson transition that takes into account the full nontrivial distribution of conductance.

Critical Metal Phase at the Anderson Metal-Insulator Transition with Kondo Impurities

It is well known that magnetic impurities can change the symmetry class of disordered metallic systems by breaking spin and time-reversal symmetry. At low temperature, these symmetries can be restored by Kondo screening. It is also known that at the Anderson metal-insulator transition, wave functions develop multifractal fluctuations with power-law correlations. Here, we consider the interplay of these two effects. We show that multifractal correlations open local pseudogaps at the Fermi energy at some random positions in space. When dilute magnetic impurities are at these locations, Kondo screening is strongly suppressed. When the exchange coupling J is smaller than a certain value J;{*}, the metal-insulator transition point extends to a critical region in the disorder strength parameter and to a band of critical states.

Termination of typical wave-function multifractal spectra at the Anderson metal-insulator transition: Field theory description using the functional renormalization group

We revisit the problem of wavefunction statistics at the Anderson metal-insulator transition (MIT) of non-interacting electrons in d > 2 spatial dimensions. At the transition, the complex spatial structure of the critical wavefunctions is reflected in the non-linear behavior of the multifractal spectrum of generalized inverse participation ratios (IPRs). Beyond the crossover from narrow to broad IPR statistics, which always occurs for sufficiently large moments of the wavefunction amplitude, the spectrum obtained from a typical wavefunction associated with a particular disorder realization differs markedly from that obtained from the disorder-averaged IPRs. This phenomenon is known as the termination of the multifractal spectrum. We provide a field theoretical derivation for the termination of the typical multifractal spectrum, by combining the non-linear sigma model framework, conventionally used to access the MIT in d = 2 + epsilon dimensions, with a functional renormalization group (FRG) technique. The FRG method deployed here was originally pioneered to study the properties of the two-dimensional (2D) random phase XY model [D. Carpentier and P. Le Doussal, Nucl. Phys. B 588, 565 (2000)]. The same method was used to demonstrate the termination of the multifractal spectrum in the very special problem of 2D Dirac fermions subject to a random Abelian vector potential. Our result shows that the typical multifractal wavefunction spectrum and its termination can be obtained at a generic Anderson localization transition in d > 2, within the standard field theoretical framework of the non-linear sigma model, when combined with the FRG.

Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves

We realize experimentally an atom-optics quantum-chaotic system, the quasiperiodic kicked rotor, which is equivalent to a 3D disordered system that allows us to demonstrate the Anderson metal-insulator transition. Sensitive measurements of the atomic wave function and the use of finite-size scaling techniques make it possible to extract both the critical parameters and the critical exponent of the transition, the latter being in good agreement with the value obtained in numerical simulations of the 3D Anderson model.

Mott-Hubbard transition and Anderson localization: A generalized dynamical mean-field theory approach

The DOS, the dynamic (optical) conductivity, and the phase diagram of a strongly correlated and strongly disordered paramagnetic Anderson-Hubbard model are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are taken into account by the DMFT, and disorder is taken into account via an appropriate generalization of the self-consistent theory of localization. The DMFT effective single-impurity problem is solved by a numerical renormalization group (NRG); we consider the three-dimensional system with a semielliptic DOS. The correlated metal, Mott insulator, and correlated Anderson insulator phases are identified via the evolution of the DOS and dynamic conductivity, demonstrating both the Mott-Hubbard and Anderson metal-insulator transition and allowing the construction of the complete zero-temperature phase diagram of the Anderson-Hubbard model. Rather unusual is the possibility of a disorder-induced Mott insulator-to-metal transition.

Corner multifractality for reflex angles and conformal invariance at 2D Anderson metal–insulator transition with spin–orbit scattering

We investigate boundary multifractality of critical wave functions at the Anderson metal–insulator transition in two-dimensional disordered non-interacting electron systems with spin–orbit scattering. We show numerically that multifractal exponents at a corner with an opening angle θ = 3 π / 2 are directly related to those near a straight boundary in the way dictated by conformal symmetry. This result extends our previous numerical results on corner multifractality obtained for θ < π to θ > π , and gives further supporting evidence for conformal invariance at criticality. We also propose a refinement of the validity of the symmetry relation of A.D. Mirlin et al. [Phys. Rev. Lett. 97 (2006) 046803] for corners.

POSSIBLE QUANTITATIVE CRITERIA FOR THE MOTT AND ANDERSON TRANSITIONS IN DOPED UNCOMPENSATED SYSTEMS

Metal-insulator transitions (MITs) in doped uncompensated systems are investigated in the Mott–Hubbard and Anderson impurity models by considering the intercarrier correlation and screening effect of carriers in the same hydrogenic impurity center, the formation of the superlattices with different coordination numbers (z=6, 8 and 12) and by studying the effect of randomness in impurity distribution. We have obtained simple and quite general criteria for the Mott and Anderson transitions and used these criteria to describe quantitatively the correlation and disorder-induced MITs in doped semiconductors and high-T c cuprates. We examine the validity of the obtained criteria for the Mott and Anderson MITs in these doped systems. It is found that the newly derived criteria for the Mott MIT are well satisfied in doped semiconductors, but they cannot be used to describe the observed MITs in the hole-doped high-T c cuprates, whereas the newly derived criteria for the Anderson MIT are applicable equally to describe the MITs observed both in doped semiconductors (at weak and intermediate disorders) and in doped cuprates (at intermediate and strong disorders). The new criteria for the Anderson MIT are extended to the polaronic systems in p-type cuprates. Our results are in quantitative agreement with the existing well-established experimental data and shed more light on the different types of MITs that occur in doped uncompensated semiconductors and cuprates.

Critical Fidelity at the Metal-Insulator Transition

Using a Wigner Lorentzian random matrix ensemble, we study the fidelity, F(t), of systems at the Anderson metal-insulator transition, subject to small perturbations that preserve the criticality. We find that there are three decay regimes as perturbation strength increases: the first two are associated with a Gaussian and an exponential decay, respectively, and can be described using linear response theory. For stronger perturbations F(t) decays algebraically as F(t) approximately t(-D2(mu)), where D2(mu) is the correlation dimension of the local density of states.

Localization transition on complex networks via spectral statistics

The spectral statistics of complex networks are numerically studied. The features of the Anderson metal-insulator transition are found to be similar for a wide range of different networks. A metal-insulator transition as a function of the disorder can be observed for different classes of complex networks for which the average connectivity is small. The critical index of the transition corresponds to the mean field expectation. When the connectivity is higher, the amount of disorder needed to reach a certain degree of localization is proportional to the average connectivity, though a precise transition cannot be identified. The absence of a clear transition at high connectivity is probably due to the very compact structure of the highly connected networks, resulting in a small diameter even for a large number of sites.

Thermopower of Semiconducting Glasses in the V2O5-P2O5 and V2O5-CdS-P2O5 Systems

Vanadium phosphate and cadmium sulfide-doped vanadium phosphate glasses containing up to 50–80 mol % V2O5 are prepared. The thermopower of these glasses is measured in the temperature range 303–473 K. The activation energies are determined from the linear dependences of the thermopower S and the logarithm of the conductivity logσ on the reciprocal of the temperature 1/T. It is demonstrated that the activation energies obtained by two methods differ from each other. The difference between the activation energies is referred to as the hopping energy Δ. The possibility of an Anderson metal-insulator transition occurring is examined. It is revealed that similar transitions are not observed in the glasses under investigation. The data obtained are analyzed within the small-polaron hopping model describing the electron conduction. It is established that hopping conduction proceeds over localized states in the tail of the density of states. All the glasses are characterized by electron conduction.

Critical level-spacing distribution for general boundary conditions

It is believed that the semi-Poisson function $P(S)=4S\exp(-2S)$ describes the normalized distribution of the nearest level-spacings $S$ for critical energy levels at the Anderson metal-insulator transition from quantum chaos to integrability, after an average over four obvious boundary conditions (BC) is taken (Braun {\it et} {\it al} \cite{1}). In order to check whether the semi-Poisson is the correct universal distribution at criticality we numerically compute it by integrating over all possible boundary conditions. We find that although $P(S)$ describes very well the main part of the obtained critical distribution small differences exist particularly in the large $S$ tail. The simpler crossover between the integrable ballistic and localized limits is shown to be universally characterized by a Gaussian-like $P(S)$ distribution instead.

A characterization of the Anderson metal-insulator transport transition

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E) and set the weak metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E)>0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial, then β(E)≥1/(2d), where d is the space dimension. These results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge.

Spectral correlation in incommensurate multiwalled carbon nanotubes.

We investigate the energy spectra of clean incommensurate double-walled carbon nanotubes, and find that the overall spectral properties are described by the critical statistics similar to that known in the Anderson metal-insulator transition. In the energy spectra, there exist three different regimes characterized by Wigner-Dyson, Poisson, and semi-Poisson distributions. This feature implies that the electron transport in incommensurate multiwalled nanotubes can be either diffusive, ballistic, or intermediate between them, depending on the position of the Fermi energy.

Scaling of the Level Compressibility at the Anderson Metal–Insulator Transition

Scaling of the Level Compressibility at the Anderson Metal–Insulator Transition

Boundary conditions, the critical conductance distribution, and one-parameter scaling

We study the influence of boundary conditions transverse to the transport direction for disordered mesoscopic conductors both at the Anderson metal-insulator transition and in the metallic regime. We show that the boundary conditions strongly influence the conductance distribution exactly at the metal-insulator transition and we discuss implications for the standard picture of one-parameter scaling. We show in particular that the scaling function that describes the change of conductance with system size depends on the boundary conditions from the metallic regime up to the metal-insulator transition. An experiment is proposed that might test the correctness of the one-parameter scaling theory.

Short-range plasma model for intermediate spectral statistics

We propose a plasma model for spectral statistics displaying level repulsion without long-range spectral rigidity, i.e. statistics intermediate between random matrix and Poisson statistics similar to the ones found numerically at the critical point of the Anderson metal-insulator transition in disordered systems and in certain dynamical systems. The model emerges from Dysons one-dimensional gas corresponding to the eigenvalue distribution of the classical random matrix ensembles by restricting the logarithmic pair interaction to a finite number k of nearest neighbors. We calculate analytically the spacing distributions and the two-level statistics. In particular we show that the number variance has the asymptotic form Σ2(L) ∼χL for large L and the nearest-neighbor distribution decreases exponentially when s→∞, P(s) ∼ exp(- Λs) with Λ = 1/χ = kβ + 1, where β is the inverse temperature of the gas (β = 1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of k = β = 1, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. P(s) = 4s exp(- 2s). Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare them to the Semi-Poisson statistics.