One-dimensional Anderson Model Research Articles

Exploring entanglement characteristics in disordered free fermion systems through random bi-partitioning

This study investigates the entanglement properties of disordered free fermion systems undergoing an Anderson phase transition from a delocalized to a localized phase. The entanglement entropy is employed to quantify the degree of entanglement, with the system randomly divided into two subsystems. To explore this phenomenon, one-dimensional tight-binding fermion models and Anderson models in one, two, and three dimensions are utilized. Comprehensive numerical calculations reveal that the entanglement entropy, determined using random bi-partitioning, follows a volume-law scaling in both the delocalized and localized phases, expressed as [Formula: see text], where D represents the dimension of the system. Furthermore, the role of short and long-range correlations in the entanglement entropy and the impact of the distribution of subsystem sites are analyzed.

Anderson localization in one-dimensional systems signified by localization tensor

A localization tensor in term of periodic position operator is recently proposed (de Aragão et al. (2019) [20]). We numerically study it in the one-dimensional Anderson model, Aubry-André-Harper model and slowly varying incommensurate potential ones, respectively. Due to its compatibility with periodic boundary conditions, we find it can properly reflect state localization properties, mobility edges and metal-insulator transitions in these famous models. In addition, it is better than the participation ratio to reflect localization properties of extended (delocalized) states. So it can be as a sensitivity index to characterize Anderson transitions.

Coexistence of extended and localized states in the one-dimensional non-Hermitian Anderson model

In one-dimensional Hermitian tight-binding models, mobility edges separating extended and localized states can appear in the presence of properly engineered quasi-periodical potentials and coupling constants. On the other hand, mobility edges don't exist in a one-dimensional Anderson lattice since localization occurs whenever a diagonal disorder through random numbers is introduced. Here, we consider a nonreciprocal non-Hermitian lattice and show that the coexistence of extended and localized states appears with or without diagonal disorder in the topologically nontrivial region. We discuss that the mobility edges appear basically due to the boundary condition sensitivity of the nonreciprocal non-Hermitian lattice.

Single parameter scaling in the non-Hermitian Anderson model

We numerically study the single parameter scaling (SPS) hypothesis in a non-interacting one-dimensional non-Hermitian Anderson model. We examine the role of non-Hermiticity in disorder potential on the SPS hypothesis at the band center. We report numerical calculations of the mean and variance of the distribution of the negative logarithmic conductance based on the linearized Landauer formalism in the perturbative regime at zero temperature. Our numerical finding indicates the violation of the SPS hypothesis for the non-Hermitian Anderson model. In particular, it turns out that the numerical SPS value of the Hermitian Anderson model is twice the magnitude of the SPS value of the non-Hermitian Anderson model for overall energies. Moreover, we obtain a relation between the localization length of the Hermitian and non-Hermitian Anderson models.

A fractional Anderson model

We examine the interplay between disorder and fractionality in a one-dimensional tight-binding Anderson model. In the absence of disorder, we observe that the two lowest energy eigenvalues detach themselves from the bottom of the band, as fractionality s is decreased, becoming completely degenerate at s=0, with a common energy equal to a half bandwidth, V. The remaining N−2 states become completely degenerate forming a flat band with energy equal to a bandwidth, 2V. Thus, a gap is formed between the ground state and the band. In the presence of disorder and for a fixed disorder width, a decrease in s reduces the width of the point spectrum while for a fixed s, an increase in disorder increases the width of the spectrum. For all disorder widths, the average participation ratio decreases with s showing a tendency towards localization. However, the average mean square displacement (MSD) shows a hump at low s values, signaling the presence of a population of extended states, in agreement with what is found in long-range hopping models.

Localization and delocalization properties in quasi-periodically-driven one-dimensional disordered systems.

Localization and delocalization of quantum diffusion in a time-continuous one-dimensional Anderson model perturbed by the quasiperiodic harmonic oscillations of M colors is investigated systematically, which has been partly reported by a preliminary Letter [H. S. Yamada and K. S. Ikeda, Phys. Rev. E 103, L040202 (2021)2470-004510.1103/PhysRevE.103.L040202]. We investigate in detail the localization-delocalization characteristics of the model with respect to three parameters: the disorder strength W, the perturbation strength ε, and the number of colors, M, which plays the similar role of spatial dimension. In particular, attention is focused on the presence of localization-delocalization transition (LDT) and its critical properties. For M≥3 the LDT exists and a normal diffusion is recovered above a critical strength ε, and the characteristics of diffusion dynamics mimic the diffusion process predicted for the stochastically perturbed Anderson model even though M is not large. These results are compared with the results of discrete-time quantum maps, i.e., the Anderson map and the standard map. Further, the features of delocalized dynamics are discussed in comparison with a limit model which has no static disordered part.

Anomalous band center localization in the one-dimensional Anderson model with a disordered distribution of infinite variance.

We perform a detailed numerical study of the influence of distributions without a finite second moment on the Lyapunov exponent through the one-dimensional tight-binding Anderson model with diagonal disorder. Using the transfer matrix parametrization method and considering a specific distribution function, we calculate the Lyapunov exponent numerically and demonstrate its relation with the fractional lower order moments of the disorder probability density function. For the lower order of moments of disorder distribution with an infinite variance, we obtain the anomalous behavior near the band center.

Probing band-center anomaly with the Kernel polynomial method

We investigate the anomalous behavior of localization length of a non-interacting one-dimensional Anderson model at zero temperature. We report numerical calculations of the Thouless expression of localization length, based on the Kernel polynomial method (KPM), which has an computational complexity, where N is the system size. The KPM results show excellent agreement with perturbative results in a large system size limit, confirming the validity of the Thouless formula. In the perturbative regime, we show that the KPM approximation of the Thouless expression produces the correct localization length at the band center in the thermodynamic limit. The Thouless expression relates localization length in terms of density of states in a one-dimensional disordered system. By calculating the KPM estimates of the density of states, we find a cusp-like behavior around the band center in the perturbative regime. This cusp-like singularity can not be obtained by approximate analytical calculations within the second-order approximations, reflects the band-center anomaly.

Dynamical Localization for the One-Dimensional Continuum Anderson Model in a Decaying Random Potential

We consider a one-dimensional continuum Anderson model where the potential decays in average like $|x|^{-\alpha}$, $\alpha>0$. We show dynamical localization for $0<\alpha<\frac12$ and provide control on the decay of the eigenfunctions.

Kondo effect under the influence of spin–orbit coupling in a quantum wire

The analysis of the impact of spin–orbit coupling (SOC) on the Kondo state has generated considerable controversy, mainly regarding the dependence of the Kondo temperature TK on SOC strength. Here, we study the one-dimensional (1D) single impurity Anderson model (SIAM) subjected to Rashba (α) and Dresselhaus (β) SOC. It is shown that, due to time-reversal symmetry, the hybridization function between impurity and quantum wire is diagonal and spin independent (as it is the case for the zero-SOC SIAM), thus the finite-SOC SIAM has a Kondo ground state similar to that for the zero-SOC SIAM. This similarity allows the use of the Haldane expression for TK, with parameters renormalized by SOC, which are calculated through a physically motivated change of basis. Analytic results for the parameters of the SOC-renormalized Haldane expression are obtained, facilitating the analysis of the SOC effect over TK. It is found that SOC acting in the quantum wire exponentially decreases TK while SOC at the impurity exponentially increases it. These analytical results are fully supported by calculations using the numerical renormalization group (NRG), applied to the wide-band regime, and the projector operator approach, applied to the infinite-U regime. Literature results, using quantum Monte Carlo, for a system with Fermi energy near the bottom of the band, are qualitatively reproduced, using NRG. In addition, it is shown that the 1D SOC SIAM for arbitrary α and β displays a persistent spin helix SU(2) symmetry similar to the one for a 2D Fermi sea with the restriction α = β.

Localization behavior induced by asymmetric disorder for the one-dimensional Anderson model.

The behavior of the Lyapunov exponent under a small asymmetric disorder distribution is investigated for the one-dimensional Anderson model in the vicinity of the band center and for finite in-band energies. The special effect that could be found in systems with an asymmetric disorder distribution is shown to be small through a perturbation calculation. We obtain a quadratic formula for the Lyapunov exponent and show the enhancement of localization close to the band center induced by asymmetric disorder distribution. We find zero correction for an asymmetric disorder distribution with finite in-band energies. This quantitative behavior of the Lyapunov exponent explains why various asymmetric factors could be neglected in weakly disordered real systems. It also shows in what situation the asymmetric property of the disorder distribution should be considered during study of the localization behavior with a higher accuracy.

Aperiodic space-inhomogeneous quantum walks: Localization properties, energy spectra, and enhancement of entanglement.

We study the localization properties, energy spectra, and coin-position entanglement of the aperiodic discrete-time quantum walks. The aperiodicity is described by spatially dependent quantum coins distributed on the lattice, whose distribution is neither periodic (Bloch-like) nor random (Anderson-like). Within transport properties we identified delocalized and localized quantum walks mediated by a proper adjusting of aperiodic parameter. Both scenarios are studied by exploring typical quantities (inverse participation ratio, survival probability, and wave packet width), as well as the energy spectra of an associated effective Hamiltonian. By using the energy spectra analysis, we show that the early stage the inhomogeneity leads to a vanishing gap between two main bands, which justifies the predominantly delocalized character observed for ν<0.5. With increase of ν arise gaps and flat bands on the energy spectra, which corroborates the suppression of transport detected for ν>0.5. For ν high enough, we observe an energy spectra, which resembles that described by the one-dimensional Anderson model. Within coin-position entanglement, we show many settings in which an enhancement in the ability to entangle is observed. This behavior brings new information about the role played by aperiodicity on the coin-position entanglement for static inhomogeneous systems, reported before as almost always reducing the entanglement when comparing with the homogeneous case. We extend the analysis in order to show that systems with static inhomogeneity are able to exhibit asymptotic limit of entanglement.

Level repulsion and dynamics in the finite one-dimensional Anderson model

This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest-neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal on-site disorder strength W. But this is not the case when the model is finite and the localization length is larger than the system size L, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter ξ∝(W^{2}L)^{-1}. As ξ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster than when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for W=0, when it is in fact deeper.

Phenomenological approach to transport through three-terminal disordered wires.

We study the voltage drop along three-terminal disordered wires in all transport regimes, from the ballistic to the localized regime. This is performed by measuring the voltage drop on one side of a one-dimensional disordered wire in a three-terminal setup as a function of disorder. Two models of disorder in the wire are considered: (i) the one-dimensional Anderson model with diagonal disorder and (ii) finite-width bulk-disordered waveguides. Based on the known β dependence of the voltage drop distribution of the three-terminal chaotic case, β being the Dyson symmetry index (β=1, 2, and 4 for orthogonal, unitary, and symplectic symmetries, respectively), the analysis is extended to a continuous parameter β>0 and uses the corresponding expression as a phenomenological one to reach the disordered phase. We show that our proposal encompasses all the transport regimes with β depending linearly on the disorder strength.

Spectral statistics for one-dimensional Anderson model with unbounded but decaying potential

In this work, we study the spectral statistics for Anderson model on [Formula: see text] with decaying randomness whose single-site distribution has unbounded support. Here, we consider the operator [Formula: see text] given by [Formula: see text], [Formula: see text] and [Formula: see text] are real i.i.d random variables following symmetric distribution [Formula: see text] with fat tail, i.e. [Formula: see text] for [Formula: see text], for some constant [Formula: see text]. In case of [Formula: see text], we are able to show that the eigenvalue process in [Formula: see text] is the clock process.

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg’s theorem. That is, a Schrödinger operator inℓ2(Z)\ell ^2(\mathbb {Z})whose potential is given by independent, identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions, and its unitary group exhibits exponential off-diagonal decay, uniformly in time. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogues of these models.

Fidelity susceptibility in one-dimensional disordered lattice models

We investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-Andr\'{e} model, from the fidelity susceptibility approach. First, we find that the fidelity susceptibility and the generalized adiabatic susceptibility are maximum at the quantum critical points of the disordered models, through which one can locate the quantum critical point in disordered lattice models. Second, finite-size scaling analysis of the fidelity susceptibility and of the generalized adiabatic susceptibility show that the correlation length critical exponent and the dynamical critical exponent at the quantum critical point of the one-dimensional Anderson model are respectively 2/3 and 2 and of the Aubry-Andr\'{e} model are respectively 1 and 2.375. Thus the quantum phase transitions in the Anderson model and in the Aubry-Andr\'{e} model are of different universality classes. Because the fidelity susceptibility and the generalized adiabatic susceptibility are directly connected to the dynamical structure factor which are experimentally accessible in the linear response regime, the fidelity susceptibility in quantum disordered systems may be observed experimentally in near future.

Unified Description on Behavior of Lyapunov Exponent for 1-D Anderson Model Near Band Center**The Numerical Integration was Performed on HPC Cluster of ITP-CAS and Supported by National Natural Science Foundation of China under Grant Nos. 1121403 and 11745006

We investigate the behavior for the Lyapunov exponent around the band center in one-dimensional Anderson model with weak disorder. Using a parametrization method we derive the corresponding differential equation and solve the associated invariant distribution. We obtain the coefficient for the leading correction term for small energy in band center anomaly. A high precision Padé approximation formula is applied to fully amend the anomalous behavior of Lyapunov exponent near band center.

Entanglement entropy distribution in the strongly disordered one-dimensional Anderson mode

The entanglement entropy (EE) distribution of strongly disordered one dimensional spin chains, which are equivalent to spinless fermions at half-filling on a bond (hopping) disordered one-dimensional Anderson model, has been shown to exhibit very distinct features such as peaks at integer multiplications of , essentially counting the number of singlets traversing the boundary. Here we show that for a canonical Anderson model with box distribution on-site disorder and repulsive nearest-neighbor interactions the EE distribution also exhibits interesting features, albeit different than the distribution seen for the bond disordered Anderson model. The canonical Anderson model shows a broad peak at low entanglement values and one narrower peak at . Density matrix renormalization group calculations reveal this structure and the influence of the disorder strength and the interaction strength on its shape. A modified real space renormalization group method was used to get a better understanding of this behavior. As might be expected the peak centered at low values of EE has a tendency to shift to lower values as disorder is enhanced. A second peak appears around the EE value of , this peak is broadened and no additional peaks at higher integer multiplications of are seen. We attribute the differences in the distribution between the canonical model and the broad hopping disorder to the influence of the on-site disorder which breaks the symmetry across the boundary.

Self-organized critical behavior and marginality in Ising spin glasses

We have studied numerically the states reached in a quench from various temperatures in the one-dimensional fully-connected Kotliar, Anderson and Stein Ising spin glass model. This is a model where there are long-range interactions between the spins which falls off as a power σ of their separation. We have made a detailed study in particular of the energies of the states reached in a quench from infinite temperature and their overlaps, including the spin glass susceptibility. In the regime where , where the model is similar to the Sherrington–Kirkpatrick model, we find that the spin glass susceptibility diverges logarithmically with increasing N, the number of spins in the system, whereas for it remains finite. We attribute the behavior for to self-organized critical behavior, where the system after the quench is close to the transition between states which have trivial overlaps and those with the non-trivial overlaps associated with replica symmetry breaking. We have also found by studying the distribution of local fields that the states reached in the quench have marginal stability but only when .