We study effects of classical magnetic impurities on the Anderson metal-insulator transition numerically. We find that a small concentration of Heisenberg impurities enhances the critical disorder amplitude $W_{\rm c}$ with increasing exchange coupling strength $J$. The resulting scaling with $J$ is analyzed which supports an anomalous scaling prediction by Wegner due to the combined breaking of time-reversal and spin-rotational symmetry. Moreover, we find that the presence of magnetic impurities lowers the critical correlation length exponent $ν$ and enhances the multifractality parameter $α_0$. The new value of $ν$ improves the agreement with the value measured in experiments on the metal-insulator transition (MIT) in doped semiconductors like phosphor-doped silicon, where a finite density of magnetic moments is known to exist in the vicinity of the MIT. The results are obtained by a finite-size scaling analysis of the geometric mean of the local density of states which is calculated by means of the kernel polynomial method. We establish this combination of numerical techniques as a method to obtain critical properties of disordered systems quantitatively.
In the supplemental materials we justify our choice of the number of Chebychev moments used within the kernel polynomial method, show some preliminary results for the large coupling behavior, discuss possible correlation effects in the local density of states, estimate the spin relaxation length and introduce the goodness of fit probability that is used to assess the quality of the fits.
Although topological Anderson insulator has been predicted in 2009, the lasting investigations of this disorder established nontrivial state results in only two experimental observations in cold atoms [Science, {\bf 362 },929 (2018)] and in photonic crystals [Nature, {\bf 560}, 461 (2018)] recently. In this paper, we study the topological Anderson transition in electric circuits. By arranging capacitor and inductor network, we construct a disordered Haldane model. Specially, the disorder is introduced by the grounding inductors with random inductance. Based on non-commutative geometry method a…
Dyson insulators with random hoppings in a lattice approach localization faster compared to the usual Anderson insulators with site disorder. For even-$N$ lattice sites the Dyson insulators mimic topological insulators with a pseudo-gap at the band center and the energy-level statistics obtained via the $P(S)$ distribution is of an intermediate type close to the Anderson localized Poisson limit. For odd-$N$ level-repulsion and Wigner statistics appears as in the quasi-metallic regime of $2D$ Anderson insulators, plus a single $E=0$ mode protected by chiral symmetry. The distribution of the par…
We study the influence of scale-free correlated disorder on the metal-insulator transition in the Anderson model of localization. We use standard transfer matrix calculations and perform finite-size scaling of the largest inverse Lyapunov exponent to obtain the localization length for respective 3D tight-binding systems. The density of states is obtained from the full spectrum of eigenenergies of the Anderson Hamiltonian. We discuss the phase diagram of the metal-insulator transition and the influence of the correlated disorder on the critical exponents.
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 scre…
The possibility of driving an Anderson metal-insulator transition in the presence of scale-free disorder by changing the correlation exponent is numerically investigated. We calculate the localization length for quasi-one-dimensional systems at fixed energy and fixed disorder strength using a standard transfer matrix method. From a finite-size scaling analysis we extract the critical correlation exponent and the critical exponent characterizing the phase transition.
Transitions between the quantum Hall state and the Anderson insulator are studied in a two dimensional tight binding model with a uniform magnetic field and a random potential. By the string (anyon) gauge, the weak magnetic field regime is explored numerically. The regime is closely related to the continuum model. The change of the Hall conductance and the trajectoy of the delocalized states are investigated by the topological arguments and the Thouless number study.
We investigate spectral correlations in quasi one-dimensional Anderson insulators with broken time-reversal symmetry. While energy levels are uncorrelated in the thermodynamic limit of infinite wire-length, some correlations remain in finite-size Anderson insulators. Asymptotic behaviors of level-level correlations in these systems are known in the large- and small-frequency limits, corresponding to the regime of classical diffusive dynamics and the deep quantum regime of strong Anderson localization. Employing non-perturbative methods and a mapping to the Coulomb-scattering problem, recently …
The localization length, $ξ$, in a 2--dimensional Anderson insulator depends on the electron spin scattering rate by magnetic impurities, $τ_s^{-1}$. For antiferromagnetic sign of the exchange, %constant, the time $τ_s$ is {\em itself a function of $ξ$}, due to the Kondo correlations. We demonstrate that the unitary regime of localization is impossible when the concentration of magnetic impurities, $n_{\tiny M}$, is smaller than a critical value, $n_c$. For $n_{\tiny M}>n_c$, the dependence of $ξ$ on the dimensionless conductance, $g$, is {\em reentrant}, crossing over to unitary, and back to …
In summary, we investigated the role of Coulomb interactions in the nature of eigenfunction multifractality of an Anderson metal-insulator transition, based on the Hartree-Fock approximation and the Ewald summation technique. As a result, we showed that two types of mobility edges appear near the Fermi energy and at a high energy, respectively, where the low-energy mobility edge results from Coulomb interactions while the high-energy one is nothing but the mobility edge of the Anderson localization transition without electron correlations. Indeed, not only multifractal scaling exponents but al…
We study the dynamical behavior of the Anderson insulator in the presence of a local noise. We show that the noise induces logarithmically slow energy and entanglement growth, until the system reaches an infinite-temperature state, where both quantities saturate to extensive values. The saturation value of the entanglement entropy approaches the average entanglement entropy over all possible product states. At infinite temperature, we find that a density excitation spreads logarithmically with time, without any signs of asymptotic diffusive behavior. In addition, we provide a theoretical pictu…
We show that the coherent forward scattering (CFS) interference peak amplitude sharply jumps from zero to a finite value upon crossing a metal-insulator transition. Extensive numerical simulations reveal that the CFS peak contrast obeys the one-parameter scaling hypothesis and gives access to the critical exponents of the transition. We also discover that the critical CFS peak directly controls the spectral compressibility at the transition where eigenfunctions are multifractal, and we demonstrate the universality of this property with respect to various types of disorder.
The multifractality of the critical eigenstate at the metal to insulator transition (MIT) in the three-dimensional Anderson model of localization is characterized by its associated singularity spectrum f(alpha). Recent works in 1D and 2D critical systems have suggested an exact symmetry relation in f(alpha). Here we show the validity of the symmetry at the Anderson MIT with high numerical accuracy and for very large system sizes. We discuss the necessary statistical analysis that supports this conclusion. We have obtained the f(alpha) from the box- and system-size scaling of the typical averag…
Recently, a metal-insulator transition (MIT) was found in the anisotropic Anderson model of localization by transfer-matrix methods (TMM). This MIT has been also investigated by multifractal analysis (MFA) and the same critical disorders $W_c$ have been obtained within the accuracy of the data. We now employ energy level statistics (ELS) to further characterize the MIT. We find a crossover of the nearest-neighbor level spacing distribution $P(s)$ from GOE statistics at small disorder indicating metallic behavior to the Poisson distribution at large disorder characteristic for localized states.…
We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(ln x)^2 potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have multifractal eigenstates. If the coefficient $a$ is small, spectral correlations in the bulk are universally governed by a translationally invariant, one-parameter generalization of the sine kernel. We provide analytic expressions for the level spacing distribution functions of this kernel, which are hybrids of the Wigner-Dyson and Poisson distributions. By tuning the single parameter, our results can b…
We consider periodically driven Anderson insulators. The short time behavior for weak, monochromatic, uniform electric fields is given by linear response theory and was famously derived by Mott. We go beyond this to consider both long times---which is the physics of Floquet late time states---and strong electric fields. This results in a `phase diagram' in the frequency-field strength plane, in which we identify four distinct regimes. These are: a linear response regime dominated by pre-existing Mott resonances, which exists provided Floquet saturation is not reached within a period; a non-lin…
This paper details the investigation of the influence of different disorders in two-dimensional topological insulator systems. Unlike the phase transitions to topological Anderson insulator induced by normal Anderson disorder, a different physical picture arises when bond disorder is considered. Using Born approximation theory, an explanation is given as to why bond disorder plays a different role in phase transition than does Anderson disorder. By comparing phase diagrams, conductance, conductance fluctuations, and the localization length for systems with different types of disorder, a consis…
Disorder plays an important role in two dimensions, and is responsible for striking phenomena such as metal insulator transition and the integral and fractional quantum Hall effects. In this paper, we investigate the role of disorder in the context of the recently discovered topological insulator, which possesses a pair of helical edge states with opposing spins moving in opposite directions and exhibits the phenomenon of quantum spin Hall effect. We predict an unexpected and nontrivial quantum phase termed "topological Anderson insulator," which is obtained by introducing impurities in a two-…
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, local 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.
Although quantum phase transitions involved with Anderson localization had been investigated for more than a half century, the role of spin polarization in these metal-insulator transitions has not been clearly addressed as a function of both the range of interactions and energy scales. Based on the Anderson-Hartree-Fock study, we reveal that the spin polarization has nothing to do with Anderson metal-insulator transitions in three dimensions as far as effective interactions between electrons are long-ranged Coulomb type. On the other hand, we find that metal-insulator transitions appear with …
We report observations of the Coulomb drag effect between two effectively 2-d insulating a-Si_{1-x}Nb_{x} films. We find that there only exist a limited range of experimental parameters over which we can measure a sizable linear-response transresistivity (ρ_{d}). The temperature dependence of ρ_{d} is consistent with the layers being Efros-Shklovskii Anderson insulators provided that a 3-d density of states and a localization length smaller than that obtained from the DC layer-conductivity are assumed.
The phase diagram of correlated, disordered electrons is calculated within dynamical mean--field theory using the geometrically averaged (''typical'') local density of states. Correlated metal, Mott insulator and Anderson insulator phases, as well as coexistence and crossover regimes are identified. The Mott and Anderson insulators are found to be continuously connected.
We investigate the presence of percolating states in disordered two-dimensional topological insulators. In particular, we uncover a close connection between these states and the so-called topological Anderson insulator (TAI), which is a topologically non-trivial phase induced by the presence of disorder. The decay of this phase could previously be connected to a delocalization of bulk states with increasing disorder strength. In this work we identify this delocalization to be the result of a percolation transition of states that circumnavigate the hills of the bulk disorder potential.
We discuss conditions for the existence of the diffusion pole and its consequences in disordered noninteracting electron systems. Using only nonperturbative and exact arguments we find against expectations that the diffusion pole can exist only in the diffusive (metallic) regime. We demonstrate that the diffusion pole in the Anderson localization phase would lead to nonexistence of the self-energy and hence to a physically inconsistent picture. The way how to consistently treat and understand the Anderson localization transition with vanishing of the diffusion pole is presented.
We consider chiral electrons moving along the 1D helical edge of a 2D topological insulator and interacting with a disordered chain of Kondo impurities. Assuming the electron-spin couplings of random anisotropies, we map this system to the problem of the pinning of the charge density wave by the disordered potential. This mapping proves that arbitrary weak anisotropic disorder in coupling of chiral electrons with spin impurities leads to the Anderson localization of the edge states.
We study the effect of disorder on the semimetal -- Mott insulator transition in the half-filled repulsive Hubbard model on a honeycomb lattice, a system that features vanishing density of states at the Fermi level. Using the determinant quantum Monte Carlo method, we characterize various phases in terms of the bulk-limit antiferromagnetic (AF) order parameter, compressibility, and temperature-dependent DC conductivity. In the clean limit, our data are consistent with previous results showing a single quantum critical point separating the semi-metallic and AF Mott insulating phases. With the p…
Different disorders lead to various localization and topological phenomena in condensed matter and artificial systems. Here we study the topological and localization properties in one-dimensional Su-Schrieffer-Heeger model with spatially correlated random binary disorders. It is found that random binary disorders can induce the topological Anderson insulating phase from the trivial insulator in various parameter regions. The topological Anderson insulators are characterized by the disorder-averaged winding number and localized bulk states revealed by the inverse participation ratio in both rea…
We present an analytic theory of quantum criticality in the quasi one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters $(g,χ)$ representing localization and topological properties, respectively. Surfaces of half-integer valued $χ$ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow describing the class A quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entir…
It is shown that if impurities in superfluid 3He have form of infinitely long non-magnetic strands, which are straight, parallel to each other and reflect quasi-particles specularly, the temperature of transition of liquid 3He from the normal into the polar phase coincides with that for the bulk liquid without impurities. Magnetic scattering lowers transition temperature for the polar phase in analogy with the conventional superconductors. These results are discussed in connection with the recent experimental findings.
It has been proposed that disorder may lead to a new type of topological insulator, called topological Anderson insulator (TAI). Here we examine the physical origin of this phenomenon. We calculate the topological invariants and density of states of disordered model in a super-cell of 2-dimensional HgTe/CdTe quantum well. The topologically non-trivial phase is triggered by a band touching as the disorder strength increases. The TAI is protected by a mobility gap, in contrast to the band gap in conventional quantum spin Hall systems. The mobility gap in the TAI consists of a cluster of non-triv…
We present an exact analysis of two conductor-insulator transitions in the random graph model. The average connectivity is related to the concentration of impurities. The adjacency matrix of a large random graph is used as a hopping Hamiltonian. Its spectrum has a delta peak at zero energy. Our analysis is based on an explicit expression for the height of this peak, and a detailed description of the localized eigenvectors and of their contribution to the peak. Starting from the low connectivity (high impurity density) regime, one encounters an insulator-conductor transition for average connect…
We obtain an analytic expression for the full distribution of conductance for a strongly disordered three dimensional conductor within a perturbative approach based on the transfer-matrix formulation. Our results confirm 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 …
The realization of the quantum anomalous Hall (QAH) effect without magnetic doping attracts intensive interest since magnetically doped topological insulators usually possess inhomogeneity of ferromagnetic order. Here, we propose a different strategy to realize intriguing QAH states arising from the interplay of light and non-magnetic disorder in two-dimensional topologically trivial systems. By combining the Born approximation and Floquet theory, we show that a time-reversal invariant disorder-induced topological insulator, known as the topological Anderson insulator (TAI), would evolve into …
Density of states, dynamic (optical) conductivity and phase diagram of strongly correlated and strongly disordered paramagnetic Anderson-Hubbard model are analyzed within the generalized dynamical mean field theory (DMFT+Σapproximation). Strong correlations are accounted by DMFT, while disorder is taken into account via the appropriate generalization of self-consistent theory of localization. The DMFT effective single impurity problem is solved by numerical renormalization group (NRG) and we consider the three-dimensional system with semi-elliptic density of states. Correlated metal, Mott insu…
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $ν$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulat…
We study the characterization and realization of higher-order topological Anderson insulator (HOTAI) in non-Hermitian systems, where the non-Hermitian mechanism ensures extra symmetries as well as gain and loss disorder.We illuminate that the quadrupole moment $Q_{xy}$ can be used as the real space topological invariant of non-Hermitian higher-order topological insulator (HOTI). Based on the biorthogonal bases and non-Hermitian symmetries, we prove that $Q_{xy}$ can be quantized to $0$ or $0.5$. Considering the disorder effect, we find the disorder-induced phase transition from normal insulato…
Numerical studies of the Anderson transition are based on the finite-size scaling analysis of the smallest positive Lyapunov exponent. We prove numerically that the same scaling holds also for higher Lyapunov exponents. This scaling supports the hypothesis of the one-parameter scaling of the conductance distribution. From the collected numerical data for quasi one dimensional systems up to the system size 24 x 24 x infinity we found the critical disorder 16.50 < Wc < 16.53 and the critical exponent 1.50 < ν< 1.54. Finite-size effects and the role of irrelevant scaling parameters are discussed.
Quantum Machine Learning (QML) algorithms to solve classifications problems have been made available thanks to recent advancements in quantum computation. While the number of qubits are still relatively small, they have been used for "quantum enhancement" of machine learning. An important question is related to the efficacy of such protocols. We evaluate this efficacy using common baseline data sets, in addition to recent coronavirus spread data as well as the quantum metal-insulator transition in three dimensions. For the computation, we used the 16 qubit IBM quantum computer. We find that th…
We study the magnetoresistance of two-dimensional bosonic Anderson insulators. We describe the change in spatial decay of localized excitations in response to a magnetic field, which is given by an interference sum over alternative tunnelling trajectories. The excitations become more localized with increasing field (in sharp contrast to generic fermionic excitations which get weakly delocalized): the localization length ξ(B) is found to change as ξ^{-1}(B)-ξ^{-1}(0)\sim B^{4/5}. The quantum interference problem maps onto the classical statistical mechanics of directed polymers in random media …
The topological Anderson and Mott insulators are two phases that have so far been separately and widely explored beyond topological band insulators. Here we combine the two seemingly different topological phases into a system of spin-1/2 interacting fermionic atoms in a disordered optical lattice. We find that the topological Anderson and Mott insulators in the noninteracting and clean limits can be adiabatically connected without gap closing in the phase diagram of our model. Lying between the two phases, we uncover a disordered correlated topological insulator, which is induced from a trivia…
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=3π/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< πto θ> π, and gives further supporting evidence for conformal invariance at criticality. We also propose a refinement o…
Recently, a new family of symmetry-protected higher-order topological insulators has been proposed and was shown to host lower-dimensional boundary states. However, with the existence of the strong disorder in the bulk, the crystal symmetry is broken, and the associated corner states are disappeared. It is well known that the emergence of robust edge states and quantized transport can be induced by adding sufficient disorders into a topologically trivial insulator, that is the so-called topological Anderson insulator. The question is whether disorders can also cause the higher-order topologica…
Using a three-frequency one-dimensional kicked rotor experimentally realized with a cold atomic gas, we study the transport properties at the critical point of the metal-insulator Anderson transition. We accurately measure the time-evolution of an initially localized wavepacket and show that it displays at the critical point a scaling invariance characteristic of this second-order phase transition. The shape of the momentum distribution at the critical point is found to be in excellent agreement with the analytical form deduced from self-consistent theory of localization.
A generalization of the single-parameter scaling theory of localization is proposed for the case when the random potential depends on temperature. The scaling equation describing the behavior of the resistance is derived. It is shown that the competition between the metallic-like temperature dependence of the Drude resistivity and localization leads to a maximum (minimum) at higher (lower) temperatures. An illustration of a metal-insulator transition in the model of charged traps, whose concentration depends on temperature, is presented.
Shift current is a photovoltaic current in bulk noncentrosymmetric insulator. Studies on shift current have so far focused on the extended Bloch waves, such as in semiconductors and perovskites. In contrast, it is unknown whether the localized wavefunctions support the dc photocurrent. Here, we show theoretically that the dc shift current appears in a noncentrosymmetric disordered one-dimensional insulator with random potential. When the light illuminates the entire sample, the photocurrent forms in presence of the electron-phonon coupling. We find this photocurrent remains robustly even when …
Time-periodic (Floquet) drive can give rise to novel symmetry breaking and topological phases of matter. Recently, we showed that a quintessential Floquet topological phase known as the anomalous Floquet-Anderson insulator is stable to noise on the timing of its Floquet drive. Here, we perturb the anomalous Floquet-Anderson insulator at a single incommensurate frequency, resulting in a quasiperiodic 2-tone drive. Our numerics indicate that a robust topological phase survives at weak noise with topological pumping that is more stable than the case of white noise. Within the topological phase, w…
We investigate quantum transport through strongly disordered barriers, made of a material with exceptionally high resistivity that behaves as an Anderson insulator or a ``bad metal'' in the bulk, by analyzing the distribution of Landauer transmission eigenvalues for a junction where such barrier is attached to two clean metallic leads. We find that scaling of the transmission eigenvalue distribution with the junction thickness (starting from the single interface limit) always predicts a non-zero probability to find high transmission channels even in relatively thick barriers. Using this distri…
We study disorder effects in a two-dimensional system with chiral symmetry and find that disorder can induce a quadrupole topological insulating phase (a higher-order topological phase with quadrupole moments) from a topologically trivial phase. Their topological properties manifest in a topological invariant defined based on effective boundary Hamiltonians, the quadrupole moment, and zero-energy corner modes. We find gapped and gapless topological phases and a Griffiths regime. In the gapless topological phase, all the states are localized, while in the Griffiths regime, the states at zero en…
We investigate the three-dimensional Anderson model of localization via a modified transfer-matrix method in the presence of scale-free diagonal disorder characterized by a disorder correlation function $g(r)$ decaying asymptotically as $r^{-α}$. We study the dependence of the localization-length exponent $ν$ on the correlation-strength exponent $α$. % For fixed disorder $W$, there is a critical $α_{\rm c}$, such that for $α< α_{\rm c}$, $ν=2/α$ and for $α> α_{\rm c}$, $ν$ remains that of the uncorrelated system in accordance with the extended Harris criterion. At the band center, $ν$ is indep…
