An interplay between non-Hermiticity and disorder plays an important role in condensed matter physics. Here, we report the universal critical behaviors of the Anderson transitions driven by non-Hermitian disorders for three dimensional (3D) Anderson model and 3D U(1) model, which belong to 3D class ${\rm AI}^{\dagger}$ and 3D class A in the classification of non-Hermitian systems, respectively. Based on level statistics and finite-size scaling analysis, the critical exponent for length scale is estimated as $ν=0.99\pm 0.05$ for class ${\rm AI}^{\dagger}$, and $ν=1.09\pm 0.05 $ for class A, both of which are clearly distinct from the critical exponents for 3D orthogonal and 3D unitary classes, respectively. In addition, spectral rigidity, level spacing distribution, and level spacing ratio distribution are studied. These critical behaviors strongly support that the non-Hermiticity changes the universality classes of the Anderson transitions.
We study the Anderson transition for three-dimensional (3D) $N \times N \times N$ tightly bound cubic lattices where both real and imaginary parts of onsite energies are independent random variables distributed uniformly between $-W/2$ and $W/2$. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25\% smallest-modulus complex eigenvalues we find the average ratio $r$ of distances to the first and the second nearest neighbor as a function of $W$. F…
We report improved numerical estimates of the critical exponent of the Anderson transition in Anderson's model of localization in $d=4$ and $d=5$ dimensions. We also report a new Borel-Padé analysis of existing $ε$ expansion results that incorporates the asymptotic behaviour for $d\to \infty$ and gives better agreement with available numerical results.
We investigate localization of noninteracting particles with spins higher than 1/2 in a two-dimensional random potential in presence of spin-orbit coupling. We consider an integer spin ($s=1$) and a half-integer spin ($s=3/2$) belonging to orthogonal and symplectic symmetry classes, respectively. We show that particles with integer spin are localized and those with half-integer spin exhibit Anderson transition. The transition belongs to universality class of conventional symplectic model for spin-1/2 particles.
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 report the superuniversality notion of Anderson localization transitions in non-Hermitian two-dimensional (2D) systems with exceptional points in which the critical exponents do not depend on the symmetries. The Anderson localization transitions are numerically studied by using the finite-size scaling analysis of the participation ratios. At the exceptional points of eithe…
We consider the localization properties of a lattice of coupled masses and springs with random mass and spring constant values. We establish the full phase diagrams of the system for pure mass and pure spring disorder. The phase diagrams exhibit regions of stable as well as unstable wave modes. The latter are of interest for the instantaneous-normal-mode spectra of liquids and the nascent field of acoustic metamaterials. We show the existence of delocalization-localization transitions throughout the phase diagram and establish, by high-precision numerical studies, that the universality of thes…
Disordered non-interacting systems are classified into ten symmetry classes, with the unitary class being the most fundamental. The three and four dimensional unitary universality classes are attracting renewed interest because of their relation to three dimensional Weyl semi-metals and four dimensional topological insulators. Determining the critical exponent of the correlation/localistion length for the Anderson transition in these classes is important both theoretically and experimentally. Using the transfer matrix technique, we report numerical estimations of the critical exponent in a U(1…
We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris and Economou (TME). This is a $L\times L \times L$ tightly bound cubic lattice, where both real and imaginary parts of on-site energies are independent random variables uniformly distributed between $-W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W= W_c \simeq 6$ in three dimension. Here we numerically diagonalize TME $L \times L \times L$ cubic lattice matrices and calculate the nu…
We study the Anderson transition in the SU(2) model and the Ando model. We report a new precise estimate of the critical exponent for the symplectic universality class of the Anderson transition. We also report numerical estimation of the $β$ function.
We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition. We estimate the critical exponent and verify its universality and that of the critical conductance distribution for box, Gaussian and Lorentzian distributions of the random potential.
We report a numerical analysis of corrections to finite size scaling at the Anderson transition due to irrelevant scaling variables and non-linearities of the scaling variables. By taking proper account of these corrections, the universality of the critical exponent for the orthogonal universality class for three different distributions of the random potential is convincingly demonstrated.
The Anderson metal-insulator transition is a continuous phase transition driven by disorder. It remains a challenging problem to theoretically determine universal critical properties at the transition. The Anderson transition in a model with a discrete sublattice or particle-hole symmetry belongs to one of seven universality classes which are different from the three well-known standard ones. Here we review our recent theoretical work on these new universality classes in (quasi) one and two dimensions.
We report a finite size scaling study of the Anderson transition. Different scaling functions and different values for the critical exponent have been found, consistent with the existence of the orthogonal and unitary universality classes which occur in the field theory description of the transition. The critical conductance distribution at the Anderson transition has also been investigated and different distributions for the orthogonal and unitary classes obtained.
We investigate the Anderson localization in non-Hermitian Aubry-André-Harper (AAH) models with imaginary potentials added to lattice sites to represent the physical gain and loss during the interacting processes between the system and environment. By checking the mean inverse participation ratio (MIPR) of the system, we find that different configurations of physical gain and loss have very different impacts on the localization phase transition in the system. In the case with balanced physical gain and loss added in an alternate way to the lattice sites, the critical region (in the case with p-…
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…
We experimentally test the universality of the Anderson three dimensional metal-insulator transition. Nine sets of parameters controlling the microscopic details of this second order phase transition have been tested. The corresponding critical exponents are independent (within $2σ$) of these microscopic details, and the average value $1.63\pm0.05$ is in very good agreement with the numerically predicted value, $ν= 1.58$.
We study non-Hermitian Aubry-André-Harper models with p-wave pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. By analysing the $\mathcal{PT}$ symmetry breaking, winding numbers of energy spectra, localization and fractal dimensions of states, and fate of Majorana fermions, a complete phase diagram on Anderson localization and topological phase transitions is obtained. In particular, the non-Hermitian topological nature of Anderson localization phase transitions from extended to critical and then to localized phases is identified, using both analytic…
Quasiperiodic system is an intermediate state between periodic and disordered systems with unique delocalization-localization transition driven by the quasiperiodic potential (QP). One of the intriguing questions is whether the universality class of the Anderson transition (AT) driven by QP is similar to that of the AT driven by the random potential in the same symmetry class. Here, we study the critical behavior of the ATs driven by QP in the three-dimensional (3D) Anderson model, Peierls phase model, and Ando model, which belong to the Wigner-Dyson symmetry classes. The localization length a…
The spectrum of exponents of the transfer matrix provides the localization lengths of Anderson's model for a particle in a lattice with disordered potential. I show that a duality identity for determinants and Jensen's identity for subharmonic functions, give a formula for the spectrum in terms of eigenvalues of the Hamiltonian with non-Hermitian boundary conditions. The formula is exact; it involves an average over a Bloch phase, rather than disorder. A preliminary investigation of non-Hermitian spectra of Anderson's model in D=1,2 and on the smallest exponent is presented.
We describe a new multifractal finite size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L^3=120^3 and involving nearly 10^6 independent wavefunctions have yielded unprecedented precision for the critical disorder W_c=16.530 (16.524,16.536) and the critical exponent nu=1.590 (1.579,1.602). We find that the multifractal exponents Delta_q exhibit a previously predicted symmetry relation and we confirm the non-…
We report a numerical analysis of the Anderson transition in a quantum-chaotic system, the quasiperiodic kicked rotor with three incommensurate frequencies. It is shown that this dynamical system exhibits the same critical phenomena as the truly random 3D-Anderson model. By taking proper account of systematic corrections to one-parameter scaling, the universality of the critical exponent is demonstrated. Our result $ν= 1.59 \pm 0.01$ is in perfect agreement with the value found for the 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 th…
We calculate the effective spatial dimension $d_\text{IR}$ of electron modes at critical points of 3D Anderson models in various universality classes (O,U,S,AIII). The results are equal within errors, and suggest the super-universal value $d_\text{IR} \!=\! 2.665(3) \!\approx\! 8/3$. The existence of such a unique marker may help identify natural processes driven by Anderson localization, and provide new insight into the spatial geometry of Anderson transitions. The recently introduced $d_\text{IR}$ is a measure-based dimension of Minkowski/Hausdorff type, designed to characterize probability-…
We study a non-Hermitian Aubry-André-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…
Uncorrelated disorder potential in one-dimensional lattice definitely induces Anderson localization, while quasiperiodic potential can lead to both localized and extended phases, depending on the potential strength. We investigate the Anderson localization in one-dimensional lattice with nonHermitian complex disorder and quasiperiodic potential. We present a non-Hermitian SSH chain to demonstrate that a nonzero imaginary disorder on-site potential can induce the standard Anderson localization. We also show that the non-Hermitian Aubry-Andre (AA) model exhibits a transition in parameter space, …
We provide an analytic theory of Anderson localization on a lattice with a weak short-range correlated disordered potential. Contrary to the general belief we demonstrate that even next-neighbor statistical correlations in the potential can give rise to strong anomalies in the localization length and the density of states, and to the complete violation of single parameter scaling. Such anomalies originate in additional symmetries of the lattice model in the limit of weak disorder. The results of numerical simulations are in full agreement with our theory, with no adjustable parameters.
Non-Hermitian systems can exhibit unique topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems by studying a non-Hermitian disordered Su-Schrieffer-Heeger model with nonreciprocal hoppings. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing energy gaps. Moreover, we uncover a topological phase which emerges only under both moderate non-Hermiticity and disorders, and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge mode…
We report a careful finite size scaling study of the metal insulator transition in Anderson's model of localisation. We focus on the estimation of the critical exponent $ν$ that describes the divergence of the localisation length. We verify the universality of this critical exponent for three different distributions of the random potential: box, normal and Cauchy. Our results for the critical exponent are consistent with the measured values obtained in experiments on the dynamical localisation transition in the quantum kicked rotor realised in a cold atomic gas.
Disorder and non-Hermiticity dramatically impact the topological and localization properties of a quantum system, giving rise to intriguing quantum states of matter. The rich interplay of disorder, non-Hermiticity, and topology is epitomized by the recently proposed non-Hermitian topological Anderson insulator that hosts a plethora of exotic phenomena. Here we experimentally simulate the non-Hermitian topological Anderson insulator using disordered photonic quantum walks, and characterize its localization and topological properties. In particular, we focus on the competition between Anderson l…
Non-Hermiticity enriches the 10-fold Altland-Zirnbauer symmetry class into the 38-fold symmetry class, where critical behavior of the Anderson transitions (ATs) has been extensively studied recently. Here, we propose a correspondence of the universality classes of the ATs between Hermitian and non-Hermitian systems. We illustrate that the critical exponents of the length scale in non-Hermitian systems coincide with the critical exponents in the corresponding Hermitian systems with additional chiral symmetry. A remarkable consequence of the correspondence is superuniversality, i.e., the ATs in …
The three-dimensional Anderson model with a rectangular distribution of site disorder displays two distinct localization-delocalization transitions, against varying disorder intensity, for a relatively narrow range of Fermi energies. Such transitions are studied through the calculation of localization lengths of quasi-- one-dimensional systems by transfer-matrix methods, and their analysis by finite-size scaling techniques. For the transition at higher disorder we find the localization-length exponent $ν=1.60(5)$ and the limiting scaled localization-length amplitude $Λ_0=0.57(1)$, strongly sug…
Products of random matrix products of $\mathrm{SL}(2,\mathbb{R})$, corresponding to transfer matrices for the one-dimensional Schrödinger equation with a random potential $V$, are studied. I consider both the case where the potential has a finite second moment $\langle V^2\rangle<\infty$ and the case where its distribution presents a power law tail $p(V)\sim|V|^{-1-α}$ for $0<α<2$. I study the generalized Lyapunov exponent of the random matrix product (i.e. the cumulant generating function of the logarithm of the wave function). In the high energy/weak disorder limit, it is shown to be given b…
We study the Anderson-type transition previously found in the spectrum of the QCD quark Dirac operator in the high temperature, quark-gluon plasma phase. Using finite size scaling for the unfolded level spacing distribution, we show that in the thermodynamic limit there is a genuine mobility edge, where the spectral statistics changes from Poisson to Wigner-Dyson statistics in a non-analytic way. We determine the correlation length critical exponent, $ν$, and find that it is compatible with that of the unitary Anderson model.
The three-dimensional Edwards-Anderson spin-glass model present strong spatial heterogeneities well characterized by the so-called backbone, a magnetic structure that arises as a consequence of the properties of the ground state and the low-excitation levels of such a frustrated Ising system. Using extensive Monte Carlo simulations and finite size scaling, we study how these heterogeneities affect the phase transition of the model. Although, we do not detect any significant difference between the critical behavior displayed by the whole system and that observed inside and outside the backbone,…
We investigate the topology and localization of one-dimensional Hermitian and non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping modulations. In the Hermitian case, phase diagrams are obtained by numerically and analytically calculating various topological and localization characters. We show the presence of topological extended, intermediate, and localized phases due to the coexistence of independent topological and localization phase transitions driven by the quasiperiodic disorder. Unlike the gapless and localized TAI phase in one-dimensional random disordered systems, we …
The one parameter scaling theory is a powerful tool to investigate Anderson localization effects in disordered systems. In this paper we show this theory can be adapted to the context of quantum chaos provided that the classical phase space is homogeneous, not mixed. The localization problem in this case is defined in momentum, not in real space. We then employ the one parameter scaling theory to: a) propose a precise characterization of the type of classical dynamics related to the Wigner-Dyson and Poisson statistics which also predicts in what situations Anderson localization corrections inv…
In a disordered environment, the probability of transmission of a wave reduces with increasing disorder, the ultimate limit of which is the near-zero transmission due to Anderson localization. Under localizing conditions, transport is arrested because the wave is trapped in the bulk of the sample with decaying-exponential coupling to the boundaries. Any further increase in disorder does not modify the overall transport properties. Here, we report the experimental demonstration of a hitherto-unrealized anomalous transport of hybrid particles under localizing disorder in a non-Hermitian setting.…
We report an analysis of the Anderson transition in an SU(2) model with chiral symmetry. Clear single parameter scaling behaviour is observed. We estimate the critical exponent for the divergence of the localization length to be $ν=2.72\pm.02$ indicating that the transition belongs to the symplectic universality class.
A review of recent progress in numerical studies of the Anderson transition in three dimensional systems is presented. From high precision calculations the critical exponent $ν$ for the divergence of the localization length is estimated to be $ν=1.57\pm 0.02$ for the orthogonal universality class, which is clearly distinguished from $ν=1.43\pm 0.03$ for the unitary universality class. The boundary condition dependences of some quantities at the Anderson transition are also discussed.
Anderson localization is a general phenomenon that applies to a variety of disordered physical systems. Recently, a novel manifestation of Anderson localization for wave packets launched with a finite average velocity was proposed, the Quantum boomerang effect (QBE). This phenomenon predicts that the disorder-averaged center of mass of a particle initially moves ballistically, then makes a U-turn, and finally slowly returns to its initial position. The QBE has been predicted to take place in several Hermitian models with Anderson localization and has been experimentally observed in the paradig…
We study the Anderson transition on a generic model of random graphs with a tunable branching parameter $1<K\le 2$, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase which is ergodic at large scales but strongly non-ergodic at smaller scales. In the critical regime, multifractal wavefunctions are located on few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusua…
