Anderson Localization Transition Research Articles

Universality in Anderson localization on random graphs with varying connectivity

We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder W_{E}WE which is strictly less than the Anderson localization critical disorder W_CWC, and that of other works which suggest that there is no such region. We find that, although no separate W_{E}WE exists from W_CWC, the length scale at which fully developed ergodicity is found diverges like |W-W_C|^{-1}|W−WC|−1, while the critical length over which delocalization develops is \sim |W-W_C|^{-1/2}∼|W−WC|−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.

Anderson localization in doped semiconductors

We theoretically consider the problem of doping induced insulator to metal transition in bulk semiconductors by obtaining the transition density as a function of compensation, assuming that the transition is an Anderson localization transition controlled by the Ioffe-Regel-Mott (IRM) criterion. We calculate the mean free path, on the highly doped metallic side, arising from carrier scattering by the ionized dopants, which we model as quenched random charged impurities. The Coulomb disorder of the charged dopants is screened by the carriers themselves, leading to an integral equation for localization, defined by the density-dependent mean free path being equal to the inverse of the Fermi wave number, as dictated by the IRM criterion. Solving this integral equation approximately analytically and exactly numerically, we provide detailed results for the localization critical density for the doping induced metal-insulator transition.

Anderson localization transitions in disordered non-Hermitian systems with exceptional points

The critical exponents of continuous phase transitions of a Hermitian system depend on and only on its dimensionality and symmetries. This is the celebrated notion of the universality of continuous phase transitions. Here we numerically study the Anderson localization transitions in non-Hermitian two-dimensional (2D) systems with exceptional points by using the finite-size scaling analysis of the participation ratios. At the exceptional points of either second order or fourth order, two non-Hermitian systems with different symmetries have the same critical exponent $\ensuremath{\nu}\ensuremath{\simeq}2$ of correlation lengths, which differs from all known 2D disordered Hermitian and non-Hermitian systems. These feature is reminiscent of the superuniversality notion of Anderson localization transitions. In the symmetry-preserved and symmetry-broken phases, the non-Hermitian models with time-reversal symmetry and without spin-rotational symmetry, and without both time-reversal and spin-rotational symmetries, are in the same universality class of 2D Hermitian electron systems of Gaussian symplectic and unitary ensembles, where $\ensuremath{\nu}\ensuremath{\simeq}2.7$ and $\ensuremath{\nu}\ensuremath{\simeq}2.3$, respectively. The universality of the transition is further confirmed by showing that the critical exponent $\ensuremath{\nu}$ does not depend on the form of disorders and boundary conditions.

Mobility edges and critical regions in a periodically kicked incommensurate optical Raman lattice

Conventionally the mobility edge (ME) separating extended states from localized ones is a central concept in understanding Anderson localization transition. The critical state, being delocalized and non-ergodic, is a third type of fundamental state that is different from both the extended and localized states. Here we study the localization phenomena in a one dimensional periodically kicked quasiperiodic optical Raman lattice by using fractal dimensions. We show a rich phase diagram including the pure extended, critical and localized phases in the high frequency regime, the MEs separating the critical regions from the extended (localized) regions, and the coexisting phase of extended, critical and localized regions with increasing the kicked period. We also find the fragility of phase boundaries, which are more susceptible to the dynamical kick, and the phenomenon of the reentrant localization transition. Finally, we demonstrate how the studied model can be realized based on current cold atom experiments and how to detect the rich physics by the expansion dynamics. Our results provide insight into studying and detecting the novel critical phases, MEs, coexisting quantum phases, and some other physics phenomena in the periodically kicked systems.

Generalized multifractality at metal-insulator transitions and in metallic phases of two-dimensional disordered systems

We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Anderson-localization transitions are considered. By using the nonlinear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator). At the same time, the generalized parabolicity is strongly violated at critical points of metal-insulator transitions, signaling violation of local conformal invariance. Moreover, in classes D and DIII, even the Weyl symmetry breaks down at critical points of metal-insulator transitions. This last feature is related to a peculiarity of the sigma-model manifolds in these symmetry classes: they consist of two disjoint components. Domain walls associated with these additional degrees of freedom are crucial for ensuring Anderson localization and, at the same time, lead to the violation of the Weyl symmetry.

Crossover from renormalized to conventional diffusion near the three-dimensional Anderson localization transition for light

We report on anomalous light transport in the strong scattering regime. Using low-coherence interferometry, we measure the reflection matrix of titanium dioxide powders, revealing crucial features of strong optical scattering which can not be observed with transmission measurements: (i) a subdiffusive regime of transport at early times of flight that is a direct consequence of predominant recurrent scattering loops, and (ii) a crossover to a conventional, but extremely slow, diffusive regime at long times. These observations support previous predictions that near-field coupling between scatterers prohibits Anderson localization of light in three-dimensional disordered media.

Anderson localization and multifractal spectrum at the transition point in a two-dimensional non-Hermitian AII† system

The Anderson localization transition in a two-dimensional AII† system is studied by eigenvalue statistics and then confirmed by the multifractal analysis of the wave functions at the transition point. The system is modeled by a two-dimensional lattice structure with real-quaternion off-diagonal elements and complex on-site energies, whose real and imaginary parts are two independent random variables. Via finite-size scaling analysis of eigenvalue spacing ratios, we find the non-Hermiticity reduces the critical disorder and give an estimate of the critical exponent ν = 1.89, showing the system belongs to a new universal class other than the AII class and probably shares the same exponent with two-dimensional Hermitian DIII systems although they have different symmetries. The Anderson localization transition is further confirmed by checking the linearity in the parametric representation of the singularity strength and by checking the universality of the forms of the singularity spectra of different system sizes. The generalized dimensions are obtained as and .

Multifractality in nonunitary random dynamics

We explore the multifractality of the steady state wave function in non-unitary random quantum dynamics in one dimension. We focus on two classes of random systems: the hybrid Clifford circuit model and the non-unitary free fermion dynamics. In the hybrid Clifford model, we map the measurement driven transition to an Anderson localization transition in an effective graph space by using properties of the stabilizer state. We show that the volume law phase with nonzero measurement rate is non-ergodic in the graph space and exhibits weak multifractal behavior. We apply the same method to the hybrid Clifford quantum automaton circuit and obtain similar multifractality in the volume law phase. For the non-unitary random free fermion system with a critical steady state, we compute the moments of the probability distribution of the single particle wave function and demonstrate that it is also weakly multifractal and has strong variations in real space.

Rare thermal bubbles at the many-body localization transition from the Fock space point of view

In this work we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space from the wave function maximum and relate the cumulants of this distribution to the properties of the quasi-local integrals of motion in the MBL phase. We demonstrate non-self-averaging property of the many-body fractal dimension $D_q$ and directly relate it to the jump of $D_q$ as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of $D_q$ in the whole range of parameters. Introducing a simple toy-model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to non-ergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the non-interacting case.

Topological field theory approach to intermediate statistics

Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are characterized by the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szeg"{o}’s limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of U(N)U(N) Chern-Simons theory on S^3S3, and the SFF of this ensemble is proportional to the HOMFLY invariant of (2n,2)(2n,2)-torus links with one component in the fundamental and one in the antifundamental representation. This is one example of a large class of ensembles with intermediate statistics arising from topological field and string theories. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.

Spectral properties of three-dimensional Anderson model

The three-dimensional Anderson model represents a paradigmatic model to understand the Anderson localization transition. In this work we first review some key results obtained for this model in the past 50 years, and then study its properties from the perspective of modern numerical approaches. Our main focus is on the quantitative comparison between the level sensitivity statistics and the level statistics. While the former studies the sensitivity of Hamiltonian eigenlevels upon inserting a magnetic flux, the latter studies the properties of unperturbed eigenlevels. We define two versions of dimensionless conductance, the first corresponding to the width of the level curvature distribution relative to the mean level spacing, and the second corresponding to the ratio of the Heisenberg time and the Thouless time obtained from the spectral form factor. We show that both conductances look remarkably similar around the localization transition, in particular, they predict a nearly identical critical point consistent with other well-established measures of the transition. We then study some further properties of those quantities: for level curvatures, we discuss particular similarities and differences between the width of the level curvature distribution and the characteristic energy studied by Edwards and Thouless (1972) in their pioneering work, in which the hopping at one lattice edge is changed from periodic to antiperiodic boundary conditions. In the context of the spectral form factor, we show that at the critical point it enters a broad time-independent regime, in which its value is consistent with the level compressibility obtained from the level variance. Finally, we test the scaling solution of the average level spacing ratio in the crossover regime using the cost function minimization approach introduced recently in Šuntajs et al. (2020). The latter approach seeks for the optimal scaling solution in the vicinity of the crossing point, while at the same time it allows for the crossing point to drift due to finite-size corrections. We find that the extracted transition point and the scaling coefficient agree with those from the literature to high numerical accuracy.

Non-local corrections to the typical medium theory of Anderson localization

We use the recently developed finite cluster typical medium approach to study the Anderson localization transition in three dimensions. Applying our method to the box and binary alloy disorder distributions, we find a fast convergence with the cluster size. We demonstrate the importance of the typical medium environment and the non-local spatial correlations for the proper characterization of the localization transition. As the cluster size increases, our typical medium cluster method recovers the correct critical disorder strength for the transition. Our findings highlight the importance of the non-local cluster corrections for capturing the localization behavior of the mobility edge trajectories. Our results demonstrate that the typical medium cluster approach developed here provides a consistent and systematic description of the Anderson localization transition in the framework of the effective medium embedding schemes.

Anderson localization transition in a robust PT -symmetric phase of a generalized Aubry-André model

We study a generalized Aubry-Andre model that obeys $\mathcal{PT}$-symmetry. We observe a robust $\mathcal{PT}$-symmetric phase with respect to system size and disorder strength, where all eigenvalues are real despite the Hamiltonian being non-hermitian. This robust $\mathcal{PT}$-symmetric phase can support an Anderson localization transition, giving a rich phase diagram as a result of the interplay between disorder and $\mathcal{PT}$-symmetry. Our model provides a perfect platform to study disorder-driven localization phenomena in a $\mathcal{PT}$-symmetric system.

Boundary-dependent self-dualities, winding numbers, and asymmetrical localization in non-Hermitian aperiodic one-dimensional models

We study a non-Hermitian Aubry-Andr\'e-Harper model with both nonreciprocal hoppings and complex quasiperiodical potentials, which is a typical non-Hermitian quasicrystal. We introduce boundary-dependent self-dualities in this model and obtain analytical results to describe its Asymmetrical Anderson localization and topological phase transitions. We find that the Anderson localization is not necessarily in accordance with the topological phase transitions, which are characteristics of localization of states and topology of energy spectrum respectively. Furthermore, in the localized phase, single-particle states are asymmetrically localized due to non-Hermitian skin effect and have energy-independent localization lengths. We also discuss possible experimental detections of our results in electric circuits.

Quantum chaos challenges many-body localization.

Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.

Fragile extended phases in the log-normal Rosenzweig-Porter model

In this paper we suggest an extension of the Rosenzweig-Porter (RP) model, the LN-RP model, in which the off-diagonal matrix elements have a wide, log-normal distribution. We argue that this model is more suitable to describe a generic many body localization problem. In contrast to RP model, in LN-RP model a fragile weakly ergodic phase appears that is characterized by broken basis-rotation symmetry which the fully-ergodic phase, also present in this model, strictly respects in the thermodynamic limit. Therefore, in addition to the localization and ergodic transitions in LN-RP model there exists also the transition between the two ergodic phases (FWE transition). We suggest new criteria of stability of the non-ergodic phases which give the points of localization and ergodic transitions and prove that the Anderson localization transition in LN-RP model involves a jump in the fractal dimension of the eigenfunction support set. We also formulate the criterion of FWE transition and obtain the full phase diagram of the model. We show that truncation of the log-normal tail shrinks the region of weakly-ergodic phase and restores the multifractal and the fully-ergodic phases.

Suppression of transport anisotropy at the Anderson localization transition in three-dimensional anisotropic media

We study the transport of classical waves through three-dimensional (3D) anisotropic media close to the Anderson localization transition. Time-, frequency-, and position-resolved ultrasonic measurements are performed on anisotropic slab-shaped mesoglass samples to probe the dynamics and the anisotropy of the multiple scattering halo, and hence to investigate the influence of disorder on the nature of wave transport and its anisotropy. These experiments allow us to address conflicting theoretical predictions that have been made about whether or not the transport anisotropy is affected by the interference effects that lead to Anderson localization. We find that the transport anisotropy is significantly reduced as the mobility edge is approached---a behavior similar to the one predicted recently for matter waves in infinite anisotropic 3D media.

Mesoscopic wave physics in fish shoals

Ultrasound scattered by a dense shoal of fish undergoes mesoscopic interference, as is typical of low-temperature electrical transport in metals or light scattering in colloidal suspensions. Through large-scale measurements in open sea, we show a set of striking deviations from classical wave diffusion, making fish shoals good candidates to study mesoscopic wave phenomena. The very good agreement with theories enlightens the role of fish structure in such a strong scattering regime that features slow energy transport and brings acoustic waves close to the Anderson localization transition.

Localization as an Entanglement Phase Transition in Boundary-Driven Anderson Models.

The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.

Locally self-consistent embedding approach for disordered electronic systems

We present a new embedding scheme for the locally self-consistent method to study disordered electron systems. We test this method in a tight-binding basis and apply it to the single band Anderson model. The local interaction zone is used to efficiently compute the local Green's function of a supercell embeded into a local typical medium. We find a quick convergence as the size of the local interaction zone which reduces the computational costs as expected. This method captures the Anderson localization transition and accurately predicts the critical disorder strength. The present work opens the path towards the development of a typical medium embedding scheme for the $O(N)$ multiple scattering methods.