Two-dimensional Anderson Model Research Articles

Localization challenges quantum chaos in the finite two-dimensional Anderson model

It is believed that the two-dimensional (2D) Anderson model exhibits localization for any nonzero disorder in the thermodynamic limit and it is also well known that the finite-size effects are considerable in the weak disorder limit. Here we numerically study the quantum-chaos to localization transition in the finite 2D Anderson model using standard indicators used in the modern literature such as the level spacing ratio, spectral form factor, variances of observable matrix elements, participation entropy and the eigenstate entanglement entropy. We show that many features of these indicators may indicate emergence of robust single-particle quantum chaos at weak disorder. However, we argue that a careful numerical analysis is consistent with the single-parameter scaling theory and predicts the breakdown of quantum chaos at any nonzero disorder value in the thermodynamic limit. Among the hallmarks of this breakdown are the universal behavior of the spectral form factor at weak disorder, and the universal scaling of various indicators as a function of the parameter $u = \left(W \ln V\right)^{-1}$ where $W$ is the disorder strength and $V$ is the number of lattice sites.

Facilitation Dynamics and Localization Phenomena in Rydberg Lattice Gases with Position Disorder

We explore the dynamics of Rydberg excitations in an optical tweezer array under antiblockade (or facilitation) conditions. Because of the finite temperature the atomic positions are randomly spread, an effect that leads to quenched correlated disorder in the interatomic interaction strengths. This drastically affects the facilitation dynamics as we demonstrate experimentally on the elementary example of two atoms. To shed light on the role of disorder in a many-body setting we show that here the dynamics is governed by an Anderson-Fock model, i.e., an Anderson model formulated on a lattice with sites corresponding to many-body Fock states. We first consider a one-dimensional atom chain in a limit that is described by a one-dimensional Anderson-Fock model with disorder on every other site, featuring both localized and delocalized states. We then illustrate the effect of disorder experimentally in a situation in which the system maps on a two-dimensional Anderson-Fock model on a trimmed square lattice. We observe a clear suppression of excitation propagation, which we ascribe to the localization of the many-body wave functions in Hilbert space.

The local semicircle law for random matrices with a fourfold symmetry

We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN−y,N−x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. It also occurs as the flip matrix model — an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.

Absence of localization on the 2d model with long-range correlated off-diagonal disorder

We study the one-electron eigenstates in a two-dimensional (2d) Anderson model with long-range correlated off-diagonal disorder, generated by a 2d discrete Fourier method. The dynamics of an initially localized wave packet is investigated by numerically solving the 2d time-dependent Schrodinger equation. In additional the participation number and its scaling behavior was obtained through direct diagonalization. Our numerical data suggest that the system exhibits a ballistic dynamics in the strongly correlated disorder regime. Moreover, the scaling analysis of mean participation number around the band center also indicates the presence of extended states for high degree of correlations.

Localization–delocalization transition in two-dimensional system with correlated disorder

We study the localization properties of the two-dimensional (2D) Anderson model with diagonal correlated disorder named visible vw and non-visible nw sites. Using the exact diagonalization of Anderson Hamiltonian, investigating the behavior of the density of states (DOS), we found that states in the center of the band E = 0 show critical behavior. The energy levels statistics (ELS) is also examined, we found a crossover of the nearest-neighbor level spacing distribution P ( s ) from Wigner Surmise distribution at small correlated disorder vw and nw (indicating a metallic behavior) to the Poisson distribution at large correlated disorder vw and nw characteristic for localized states. In addition, an analysis of width M dependence of the reduced localization length λ / M is obtained by the transfer-matrix method (TMM). For all investigated strengths of correlated disorder at E = 0 and E =+ 2 , the reduced localization length λ / M always decreases with increasing width. We found only delocalized states in suggested energy and disorder range.

Marginal intermediate statistics in the excited spectra of the E ⊗ (b1 + b2) Jahn–Teller system

The long-range spectral density correlations (spectral rigidities and related spectral compressibilities) of the E ⊗ (b1 + b2) Jahn–Teller model are found strongly nonuniversal with respect to the Hamiltonian parameters and inhomogeneous with respect to the choice of a partial energy segment. However, the partial spectral rigidities exhibit common features: an anomalous linear part for small and a saturation for large . The spectral compressibilities found for the partial spectral segments and averaged over a whole relevant part of the spectrum cumulate close to a well-defined limit pertaining to the semi-Poisson statistics. This is in accordance with similar tendencies revealed in the short-range averaged statistical characteristics of this model investigated in our previous paper (Majerníková and Shpyrko 2006 Phys. Rev. E 73 057202). These features document an inhomogeneous and nonuniversal weakening of level repulsions and nonuniversality of level fluctuations on both long and short energy scales. The nonuniversality and inhomogeneity of the statistical characteristics correspond to a similar behaviour of the chaoticity parameter (a fraction of the chaotic phase space of the trajectories) found for the corresponding semiclassical Hamiltonian. We ascribe the nonuniversal and inhomogeneous nonintegrability behaviour to the changing degree of the brokening of the rotation symmetry when changing parameters of our effectively two-dimensional model. It results in a random distribution of the respective localized wavefunctions at all scales up to the size of an available state space. The multifractal behaviour of the wavefunctions is implied from the analysis of their (averaged) fractal dimensions which range up to 1.5 ± 0.1 (for ). This might imply the concept of the chaos-assisted tunnelling between the regions of reduced degree of stochasticity through regions of high degree of stochasticity. It supports the analogy with the two-dimensional Anderson model with marginal-asymptotically far metal–insulator transition. The features found allow us to classify the present model as a member of the class with a multifractal eigenfunction statistics characteristic for the spectra with weakened level repulsion similar to the Anderson model close to the metal–insulator transition.

Critical behavior of the two-dimensional Anderson model with long-range correlateddisorder

We describe the critical behavior of the Anderson-like transition predicted to occur in the2D tight-binding model with isotropic scale-free long-range correlated disordercharacterized by a power-law spectral density. We explore the scale invarianceof the participation function relative fluctuation at the critical point to locatethe mobility edge as a function of the power-law spectral density exponentα2D. The statesnear the band center, which exhibit power-law localization for uncorrelated disorder, become delocalizedfor α2D>2. In addition, we consider the finite-size scaling hypothesis to estimate thecorrelation length critical exponent. We find that the critical exponentν dependson α2D, thus indicating that correlations in the disorder distribution are indeed relevant in thisregime, in agreement with the extended Harris criterion.

Universal Distribution Functions in Two-Dimensional Localized Systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.

Renormalization approach for the two-dimensional Anderson model at the band edge: Scaling of the localization volume

We study the localization volumes $V$ (participation ratio) of electronic wave functions in the $2d$-Anderson model with diagonal disorder. Using a renormalization procedure, we show that at the band edges, i.e., for energies $E\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}4$, $V$ is inversely proportional to the variance $⟨{ϵ}^{2}⟩$ of the site potentials. Using scaling arguments, we show that in the neighborhood of $E=\ifmmode\pm\else\textpm\fi{}4$, $V$ scales as $V={⟨{ϵ}^{2}⟩}^{\ensuremath{-}1}g((4\ensuremath{-}\ensuremath{\mid}E\ensuremath{\mid})∕⟨{ϵ}^{2}⟩)$ with the scaling function $g(x)$. Numerical simulations confirm this scaling ansatz.

Anderson transition in low-dimensional disordered systems driven by long-range nonrandom hopping.

The single-parameter scaling hypothesis predicts the absence of delocalized states for noninteracting quasiparticles in low-dimensional disordered systems. We show analytically, using a supersymmetric method combined with a renormalization group analysis, as well as numerically that extended states may occur in the one- and two-dimensional Anderson model with a nonrandom hopping falling off as some power of the distance between sites. The different size scaling of the bare level spacing and the renormalized magnitude of the disorder seen by the quasiparticles finally results in the delocalization of states at one of the band edges of the quasiparticle energy spectrum.

Divergences of the Localization Lengths in the Two-Dimensional, Off-Diagonal Anderson Model on Bipartite Lattices

We investigate the scaling properties of the two-dimensional (2D) Anderson model of localization with purely off-diagonal disorder (random hopping). Using the transfer-matrix method and finite-size scaling we compute the infinite-size localization lengths for bipartite square and hexagonal 2D lattices, non-bipartite triangular lattices and different distribution functions for the hopping elements. We show that for small energies the localization lengths in the bipartite case diverge with a power-law behavior. The corresponding exponents are in the range $0.2 - 0.6$ and seem to depend on the type and the strength of disorder.

Exponents of the localization lengths in the bipartite Anderson model with off-diagonal disorder

We investigate the scaling properties of the two-dimensional Anderson model of localization with purely off-diagonal disorder (random hopping). In particular, we show that for small energies the infinite-size localization lengths as computed from transfer-matrix methods together with finite-size scaling diverge with a power-law behavior. The corresponding exponents seem to depend on the strength and the type of disorder chosen.

Smoothed universal correlations in the two-dimensional Anderson model

We report on calculations of smoothed spectral correlations in the two-dimensional Anderson model for weak disorder. As pointed out in (M. Wilkinson, J. Phys. A: Math. Gen. 21, 1173 (1988)), an analysis of the smoothing dependence of the correlation functions provides a sensitive means of establishing consistency with random matrix theory. We use a semiclassical approach to describe these fluctuations and offer a detailed comparison between numerical and analytical calculations for an exhaustive set of two-point correlation functions. We consider parametric correlation functions with an external Aharonov-Bohm flux as a parameter and discuss two cases, namely broken time-reversal invariance and partial breaking of time-reversal invariance. Three types of correlation functions are considered: density-of-states, velocity and matrix element correlation functions. For the values of smoothing parameter close to the mean level spacing the semiclassical expressions and the numerical results agree quite well in the whole range of the magnetic flux.

Interaction-dependent enhancement of the localisation length for two interacting particles in a one-dimensional random potential

We present calculations of the localisation length, $\lambda_{2}$, for two interacting particles (TIP) in a one-dimensional random potential, presenting its dependence on disorder, interaction strength $U$ and system size. $\lambda_{2}(U)$ is computed by a decimation method from the decay of the Green function along the diagonal of finite samples. Infinite sample size estimates $\xi_{2}(U)$ are obtained by finite-size scaling. For U=0 we reproduce approximately the well-known dependence of the one-particle localisation length on disorder while for finite $U$, we find that $ \xi_{2}(U) \sim \xi_2(0)^{\beta(U)} $ with $\beta(U)$ varying between $\beta(0)=1$ and $\beta(1) \approx 1.5$. We test the validity of various other proposed fit functions and also study the problem of TIP in two different random potentials corresponding to interacting electron-hole pairs. As a check of our method and data, we also reproduce well-known results for the two-dimensional Anderson model without interaction.

Ward-Type Identities for the Two-Dimensional Anderson Model at Weak Disorder

Using the particular momentum conservation laws in dimension d = 2, we rewrite the Anderson model in terms of low-momentum long-range fields, at the price of introducing electron loops. The corresponding loops satisfy a Ward-type identity, hence are much smaller than expected. This fact should be useful for the study of the weak-coupling model in the middle of the spectrum of the free Hamiltonian.

Transport regimes and critical energies in the two-dimensional Anderson model

We calculate numerically the energy level distributions of the two-dimensional Anderson model in the nearly clean, ballistic, diffusive and strongly localized regimes. The Poisson statistics governs nearest level spacings in the clean (in the absence of degeneracies due to the geometry) and localized regimes, whereas in the ballistic and diffusive regimes the level statistics follows the Wigner-Dyson distribution. In the diffusive and ballistic regimes, we study the critical energy , defined as the maximum energy up to which level fluctuations follow the logarithmic behaviour characteristic of random-matrix theory (RMT). A reasonably accurate determination of the ballistic-diffusive transition is achieved through the behaviour of the critical energies. We show how the level statistic is an adequate tool for characterizing the different regimes of disordered systems, and how the results obtained are in qualitative agreement with theoretical estimates. We finally analyse the behaviour of as a function of the energy within the band.

Critical Behavior in the Two-Dimensional Anderson Model of Localization with Random Hopping

We study a variant of the two-dimensional (2D) Anderson model of localization in which the disorder is represented by randomly chosen hopping terms. The density of states reveals an abnormally strong peak in the band center and an analysis of multifractal properties indicates that localization is less strong at E = 0 than at E ≠ 0. A finite-size-scaling analysis of localization lengths as obtained from the transfer-matrix method, shows that the state at E = 0 exhibits critical behavior up to a strip width M = 180. However, states outside the band center are localized and the critical state vanishes already for very small amounts of onsite potential disorder. Thus, there is no violation of the scaling theory of localization.

The two-dimensional Anderson model of localization with random hopping

We examine the localization properties of the 2D Anderson Hamiltonian with off-diagonal disorder. Investigating the behavior of the participation numbers of eigenstates as well as studying their multifractal properties, we find states in the center of the band which show critical behavior up to the system size $N= 200 \times 200$ considered. This result is confirmed by an independent analysis of the localization lengths in quasi-1D strips with the help of the transfer-matrix method. Adding a very small additional onsite potential disorder, the critical states become localized.

Decimation algorithm for the statistics of energy levels

We propose a new numerical method for the analysis of the statistical properties of the electron spectra of random systems at a given energy. By applying a decimation transformation to the two-dimensional Anderson model with diagonal disorder, the level spacing distribution at the band center is calculated for system sizes up to 1024×1024 lattice sites. The results show a crossover from the Wigner to the Poisson statistics when increasing the system size.

Insulating states of correlated electrons

The insulating states of three model systems, the two-chain Hubbard model, the two-layer two-dimensional Hubbard model, and the two-dimensional symmetric periodic Anderson model are discussed. All of these systems at half-filling have a charge gap but they can exhibit various types of spin gaps ranging from a vanishing spin gap in systems with strong antiferromagnetic correlations to a spin gap which is equal to the charge gap in the band-insulator regime. The interchain, interplane, or Anderson model f-d hybridization plays a key role in determining the spin-to-charge gap ratio and the nature of the insulating state.