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 average of the generalized inverse participation ratios. We show that the best symmetry in f(alpha) for typical averaging is achieved by system-size scaling, following a strategy that emphasizes using larger system sizes even if this necessitates fewer disorder realizations.
We report the first experimental observation of strong multifractality in wave functions at the Anderson localization transition in open three-dimensional elastic networks. Our results confirm the recently predicted symmetry of the multifractal exponents. We have discovered that the result of multifractal analysis of the real data depends on the excitation scheme used in the experiment.
Using the well-known "algebra of multifractality", we derive the functional equation for anomalous dimensions Δ_q, whose solution Δ_q=aq(q-1) corresponds to strict parabolicity of the multifractal spectrum. This result demonstrates clearly that a correspondence of σ-models with the initial disordered systems is not exact.
We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wavefunction intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\times10^6$ independently generated wavefunctions of system sizes up to $L^3=150^3$, yields accurate estimates for the critical exponent of the locali…
We propose a generalization of multifractal analysis that is applicable to the critical regime of the Anderson localization-delocalization transition. The approach reveals that the behavior of the probability distribution of wavefunction amplitudes is sufficient to characterize the transition. In combination with finite-size scaling, this formalism permits the critical parameters to be estimated without the need for conductance or other transport measurements. Applying this method to high-precision data for wavefunction statistics obtained by exact diagonalization of the three-dimensional Ande…
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…
The probability density function (PDF) for critical wavefunction amplitudes is studied in the three-dimensional Anderson model. We present a formal expression between the PDF and the multifractal spectrum f(alpha) in which the role of finite-size corrections is properly analyzed. We show the non-gaussian nature and the existence of a symmetry relation in the PDF. From the PDF, we extract information about f(alpha) at criticality such as the presence of negative fractal dimensions and we comment on the possible existence of termination points. A PDF-based multifractal analysis is hence shown to…
The disorder induced metal--insulator transition is investigated in a three-dimensional simple cubic lattice and compared for the presence and absence of time-reversal and spin-rotational symmetry, i.e. in the three conventional symmetry classes. Large scale numerical simulations have been performed on systems with linear sizes up to $L=100$ in order to obtain eigenstates at the band center, $E=0$. The multifractal dimensions, exponents $D_q$ and $α_q$, have been determined in the range of $-1\leq q\leq 2$. The finite-size scaling of the generalized multifractal exponents provide the critical …
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-…
The scaling theory of Anderson localization is based on a global conductance $g_L$ that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these …
We study the multifractal analysis (MFA) of electronic wavefunctions at the localisation-delocalisation transition in the 3D Anderson model for very large system sizes up to $240^3$. The singularity spectrum $f(α)$ is numerically obtained using the \textsl{ensemble average} of the scaling law for the generalized inverse participation ratios $P_q$, employing box-size and system-size scaling. The validity of a recently reported symmetry law [Phys. Rev. Lett. 97, 046803 (2006)] for the multifractal spectrum is carefully analysed at the metal-insulator transition (MIT). The results are compared to…
For Anderson Localization models with multifractal eigenvectors on disordered samples containing $N$ sites, we analyze in a unified framework the consequences for the statistical properties of the Green function. We focus in particular on the imaginary part of the Green function at coinciding points $G^I_{xx}(E-i η)$ and study the scaling with the size $N$ of the moments of arbitrary indices $q$ when the broadening follows the scaling $η=\frac{c}{N^δ}$. For the standard scaling regime $δ=1$, we find in the two limits $c \ll 1$ and $c \gg 1$ that the moments are governed by the anomalous expone…
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…
Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the multifractal exponents with indices $q<1/2$ to those with $q>1/2$. The second relation connects the wave function multifractality to that of Wigner delay times in a system with a lead attached.
We revisit the problem of wavefunction statistics at the Anderson metal-insulator transition (MIT) of non-interacting electrons in d > 2 spatial dimensions. At the transition, the complex spatial structure of the critical wavefunctions is reflected in the non-linear behavior of the multifractal spectrum of generalized inverse participation ratios (IPRs). Beyond the crossover from narrow to broad IPR statistics, which always occurs for sufficiently large moments of the wavefunction amplitude, the spectrum obtained from a typical wavefunction associated with a particular disorder realization dif…
We investigate the Anderson transition found in the spectrum of the Dirac operator of Quantum Chromodynamics (QCD) at high temperature, studying the properties of the critical quark eigenfunctions. Applying multifractal finite-size scaling we determine the critical point and the critical exponent of the transition, finding agreement with previous results, and with available results for the unitary Anderson model. We estimate several multifractal exponents, finding also in this case agreement with a recent determination for the unitary Anderson model. Our results confirm the presence of a true …
We show that quantum wavepackets exhibit a sharp macroscopic peak as they spread in the vicinity of the critical point of the Anderson transition. The peak gives a direct access to the mutifractal properties of the wavefunctions and specifically to the multifractal dimension $d\_2$. Our analysis is based on an experimentally realizable setup, the quantum kicked rotor with quasi-periodic temporal driving, an effectively 3-dimensional disordered system recently exploited to explore the physics of the Anderson transition with cold atoms.
In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K=2 with $N$ generations, we consider the Miller-Derrida scattering geometry [J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root of the tree, and where $K^{N}$ outcoming wires are attached to the leaves of the tree. In terms of the $K^{N}$ transmission amplitudes $t_j$, the tota…
We demonstrate that by considering disordered single-particle Hamiltonians (or their random matrix versions) on ultrametric spaces one can generate an interesting class of models exhibiting Anderson metal-insulator transition. We use the weak disorder virial expansion to determine the critical value of the parameters and to calculate the values of the multifractal exponents for inverse participation ratios. Direct numerical simulations agree favourably with the analytical predictions.
We uncover field-theoretic underpinnings of symmetry relations for multifractal spectra at Anderson transitions and at critical points of other disordered systems. We show that such relations follow from the conformal invariance of the critical theory, which implies their general character. We also demonstrate that for the Anderson localization problem the entire probability distribution for the local density of states possesses a symmetry arising from the invariance of correlation functions of the underlying non-linear $σ$-model with respect to the Weyl group of the target space of the model.
Critical fluctuations of wave functions and energy levels at the Anderson transition are studied for the family of the critical power-law random banded matrix ensembles. It is shown that the distribution functions of the inverse participation ratios (IPR) $P_q$ are scale-invariant at the critical point, with a power-law asymptotic tail. The IPR distribution, the multifractal spectrum and the level statistics are calculated analytically in the limits of weak and strong couplings, as well as numerically in the full range of couplings.
The statistics of critical wave functions at the Anderson transition in three and four dimensions are studied numerically. The distribution of the inverse participation ratios (IPR) $P_q$ is shown to acquire a scale-invariant form in the limit of large system size. Multifractality spectra governing the scaling of the ensemble-averaged IPRs are determined. Conjectures concerning the IPR statistics and the multifractality at the Anderson transition in a high spatial dimensionality are formulated.
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…
The ensemble of $L \times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/| i-j |)^a$, is known to present an Anderson localization transition at $a=1$, where one-particle eigenfunctions are multifractal. Here we study numerically, at this critical point, the statistical properties of the transmission $T_2$ for two distinguishable particles, two bosons or two fermions. We find that the statistics of $T_2$ is multifractal, i.e. the probability to have $T_2(L) \sim 1/L^κ$ behaves as $L^{Φ_2(κ)}$, where the multifractal spectrum $Φ_2(κ)$ for fermi…
We study spatial structures of anomalously localized states (ALS) in tail regions at the critical point of the Anderson transition in the two-dimensional symplectic class. In order to examine tail structures of ALS, we apply the multifractal analysis only for the tail region of ALS and compare with the whole structure. It is found that the amplitude distribution in the tail region of ALS is multifractal and values of exponents characterizing multifractality are the same with those for typical multifractal wavefunctions in this universality class.
For Anderson tight-binding models in dimension $d$ with random on-site energies $ε_{\vec r}$ and critical long-ranged hoppings decaying typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality regime corresponding to small $V$ can be studied via the standard perturbation theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios $Y_q(L)$, which are the order parameters of Anderson transitions, can be written in terms of weighted Lévy sums of broadly distributed variables (as a consequence of the presence of on-site random energies in the denominators of…
Using numerical simulations, we investigate the distribution of Kondo temperatures at the Anderson transition. In agreement with previous work, we find that the distribution has a long tail at small Kondo temperatures. Recently, an approximation for the tail of the distribution was derived analytically. This approximation takes into account the multifractal distribution of the wavefunction amplitudes (in the parabolic approximation), and power law correlations between wave function intensities, at the Anderson transition. It was predicted that the distribution of Kondo temperatures has a power…
We investigate the emergence of complex dynamics in networks with heavy-tailed connectivity by developing a non-Hermitian random matrix theory. We uncover the existence of an extended critical regime of spatially multifractal fluctuations between the quiescent and active phases. This multifractal critical phase combines features of localization and delocalization and differs from the edge of chaos in classical networks by the appearance of universal hallmarks of Anderson criticality over an extended region in phase space. We show that the rich nonlinear response properties of the extended crit…
Random critical points are generically characterized by multifractal properties. In the field of Anderson localization, Mirlin, Fyodorov, Mildenberger and Evers [Phys. Rev. Lett 97, 046803 (2006)] have proposed that the singularity spectrum $f(α)$ of eigenfunctions satisfies the exact symmetry $f(2d-α)=f(α)+d-α$ at any Anderson transition. In the present paper, we analyse the physical origin of this symmetry in relation with the Gallavotti-Cohen fluctuation relations of large deviation functions that are well-known in the field of non-equilibrium dynamics: the multifractal spectrum of the diso…
In Refs. [1,2] we have shown how a combination of modern linear-scaling DFT, together with a subsequent use of large, effective tight-binding Hamiltonians, allows to compute multifractal wave functions yielding the critical properties of the Anderson metal-insulator transition (MIT) in doped semiconductors. This combination allowed us to construct large and atomistically realistic samples of sulfur-doped silicon (Si:S). The critical properties of such systems and the existence of the MIT are well known, but experimentally determined values of the critical exponent $ν$ close to the transition h…
Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: logarithmic multifractality and critical localization
We study the interaction driven localization transition, which a recent experiment in Ga_{1-x}Mn_xAs As has shown to come along with multifractal behavior of the local density of states (LDoS) and the intriguing persistence of critical correlations close to the Fermi level. We show that the bulk of these phenomena can be understood within a Hartree-Fock treatment of disordered, Coulomb-interacting spinless fermions. A scaling analysis of the LDoS correlation demonstrates multifractality with correlation dimension d_2=1.57, which is significantly larger than at a non-interacting Anderson transi…
Systems with the power-law quasiparticle dispersion $ε_{\bf k}\propto k^α$ exhibit non-Anderson disorder-driven transitions in dimensions $d>2α$, as exemplified by Weyl semimetals, 1D and 2D arrays of ultracold ions with long-range interactions, quantum kicked rotors and semiconductor models in high dimensions. We study the wavefunction structure in such systems and demonstrate that at these transitions they exhibit fractal behaviour with an infinite set of multifractal exponents. The multifractality persists even when the wavefunction localisation is forbidden by symmetry or topology and occu…
We generalize universal relations between the multifractal exponent α_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude Λ_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude Λ_c^o is universally related to the boundary multifractal exponent α_0^s for the typical wav…
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybr…
Generalized multifractality characterizes scaling of eigenstate observables at Anderson-localization critical points. We explore generalized multifractality in 2D systems, with the main focus on the spin quantum Hall (SQH) transition in superconductors of symmetry class C. Relations and differences with the conventional integer quantum Hall (IQH) transition are also studied. Using the field-theoretical formalism of non-linear sigma-model, we derive the pure-scaling operators representing generalizing multifractality and then "translate" them to the language of eigenstate observables. Performin…
We study the level-spacing distribution function $P(s)$ at the Anderson transition by paying attention to anomalously localized states (ALS) which contribute to statistical properties at the critical point. It is found that the distribution $P(s)$ for level pairs of ALS coincides with that for pairs of typical multifractal states. This implies that ALS do not affect the shape of the critical level-spacing distribution function. We also show that the insensitivity of $P(s)$ to ALS is a consequence of multifractality in tail structures of ALS.
Finite-size effects have been a major and justifiable source of concern for studies of many-body localization, and several works have been dedicated to the subject. In this paper, however, we discuss yet another crucial problem that has received much less attention, that of the lack of self-averaging and the consequent danger of reducing the number of random realizations as the system size increases. By taking this into account and considering ensembles with a large number of samples for all system sizes analyzed, we find that the generalized dimensions of the eigenstates of the disordered Hei…
The wavefunctions of a disordered two-dimensional electron gas at the quantum-critical Anderson transition are predicted to exhibit multifractal scaling in their real space amplitude. We experimentally investigate the appearance of these characteristics in the spatially resolved local density of states of a two-dimensional mixed surface alloy Bi_xPb_{1-x}/Ag(111), by combining high-resolution scanning tunneling microscopy with spin and angle-resolved inverse-photoemission experiments. Our detailed knowledge of the surface alloy electronic band structure, the exact lattice structure and the ato…
At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L \sim 1/L^κ$ behaves as $L^{Φ(κ)}$, where the multifractal spectrum $Φ(κ)$ terminates at $κ=0$ as a consequence of the physical bound $T_L \leq 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q} \sim 1/L^{X(q)}$ become frozen above some threshold: $X(q \geq q_{sat}) = - Φ(κ=0)$, i.e. all moments of order $q \geq q_{sa…
The dynamical heterogeneity in supercooled liquids measured by a molecular dynamics simulation has been quantified on the basis of the multifractal formalism. The singularity spectrum becomes broader as the glass transition is approached. This behavior is similar to that observed in the case of the Anderson-localization transition.
We compare different partitioning schemes for the box-counting algorithm in the multifractal analysis by computing the singularity spectrum and the distribution of the box probabilities. As model system we use the Anderson model of localization in two and three dimensions. We show that a partitioning scheme which includes unrestricted values of the box size and an average over all box origins leads to smaller error bounds than the standard method using only integer ratios of the linear system size and the box size which was found by Rodriguez et al. (Eur. Phys. J. B 67, 77-82 (2009)) to yield …
A Dyson hierarchical model for Anderson localization, containing non-random hierarchical hoppings and random on-site energies, has been studied in the mathematical literature since its introduction by Bovier [J. Stat. Phys. 59, 745 (1990)], with the conclusion that this model is always in the localized phase. Here we show that if one introduces alternating signs in the hoppings along the hierarchy (instead of choosing all hoppings of the same sign), it is possible to reach an Anderson localization critical point presenting multifractal eigenfunctions and intermediate spectral statistics. The a…
The phase diagram of the metal-insulator transition in a three dimensional quantum percolation problem is investigated numerically based on the multifractal analysis of the eigenstates. The large scale numerical simulation has been performed on systems with linear sizes up to $L=140$. The multifractal dimensions, exponents $D_q$ and $α_q$, have been determined in the range of $0\leq q\leq 1$. Our results confirm that this problem belongs to the same universality class as the three dimensional Anderson model, the critical exponent of the localization length was found to be $ν=1.622\pm 0.035$. T…
We examine the localization properties of the 2D Anderson Hamiltonian with off-diagonal disorder. Investigating the behavior of the participation numbers of eigenstates as well as studying their multifractal properties, we find states in the center of the band which show critical behavior up to the system size $N= 200 \times 200$ considered. This result is confirmed by an independent analysis of the localization lengths in quasi-1D strips with the help of the transfer-matrix method. Adding a very small additional onsite potential disorder, the critical states become localized.
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.
We study various box-size scaling techniques to obtain the multifractal properties, in terms of the singularity spectrum f(alpha), of the critical eigenstates at the metal-insulator transition within the 3-D Anderson model of localisation. The typical and ensemble averaged scaling laws of the generalised inverse participation ratios are considered. In pursuit of a numerical optimisation of the box-scaling technique we discuss different box-partitioning schemes including cubic and non-cubic boxes, use of periodic boundary conditions to enlarge the system and single and multiple origins for the …
Quantum multifractality is a fundamental property of systems such as non-interacting disordered systems at an Anderson transition and many-body systems in Hilbert space. Here we discuss the origin of the presence or absence of a fundamental symmetry related to this property. The anomalous multifractal dimension $Δ_q$ is used to characterize the structure of quantum states in such systems. Although the multifractal symmetry relation \mbox{$Δ_q=Δ_{1-q}$} is universally fulfilled in many known systems, recently some important examples have emerged where it does not hold. We show that this is the …
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…
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…
Anomalously localized states (ALS) at the critical point of the Anderson transition are studied for the SU(2) model belonging to the two-dimensional symplectic class. Giving a quantitative definition of ALS to clarify statistical properties of them, the system-size dependence of a probability to find ALS at criticality is presented. It is found that the probability increases with the system size and ALS exist with a finite probability even in an infinite critical system, though the typical critical states are kept to be multifractal. This fact implies that ALS should be eliminated from an ense…
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…
We develop a classification of composite operators without gradients at Anderson-transition critical points in disordered systems. These operators represent correlation functions of the local density of states (or of wave-function amplitudes). Our classification is motivated by the Iwasawa decomposition for the field of the pertinent supersymmetric σ-model: the scaling operators are represented by "plane waves" in terms of the corresponding radial coordinates. We also present an alternative construction of scaling operators by using the notion of highest-weight vector. We further argue that a …
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.…
