Anderson Localization Transition Research Articles

Quantumk-core conduction on the Bethe lattice

Classical and quantum conduction on a bond-diluted Bethe lattice is considered. The bond dilution is subject to the constraint that every occupied bond must have at least $k\ensuremath{-}1$ neighboring occupied bonds, i.e., $k$-core diluted. In the classical case, we find the onset of conduction for $k=2$ is continuous while for $k=3$, the onset of conduction is discontinuous with the geometric random first-order phase transition driving the conduction transition. In the quantum case, treating each occupied bond as a random scatterer, we find for $k=3$ that the random first-order phase transition in the geometry also drives the onset of quantum conduction giving rise to a new universality class of Anderson localization transitions.

Anderson Localization for a Supersymmetric Sigma Model

We study a lattice sigma model which is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices in 3D. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. The existence of a diffusive phase in 3 dimensions was proved in Disertori et al. (Commun. Math. Phys., doi:10.1007/s00220-010-1117-5, 2009) [2] for low temperatures. Here we prove localization at high temperatures for any dimension d ≥ 1. Our analysis uses Ward identities coming from internal supersymmetry.

The Anderson localization transition with long-ranged hoppings: analysis of the strongmultifractality regime in terms of weighted Lévy sums

For Anderson tight-binding models in dimensiond with random on-site energies and critical long-ranged hoppings decaying typically asVtyp(r) ∼ V/rd, we show that the strong multifractality regime corresponding to smallV can bestudied via the standard perturbation theory for eigenvectors in quantum mechanics. The inverse participationratios Yq(L), which are the order parameters of Anderson transitions, can be written in terms ofweighted Lévy sums of broadly distributed variables (as a consequence of the presence ofon-site random energies in the denominators of the perturbation theory). Wecompute at leading order the typical and disorder-averaged multifractal spectraτtyp(q) andτav(q) as afunction of q. For q < 1/2, we obtain the non-vanishing limiting spectrumτtyp(q) = τav(q) = d(2q − 1) as . For q > 1/2, this method yields the same disorder-averaged spectrumτav(q) of orderO(V) asobtained previously via the Levitov renormalization method by Mirlin and Evers (2000 Phys. Rev. B 627920). In addition, it allows us to compute explicitly the typical spectrum, also of orderO(V), but with a differentdependence on q, for all q > qc = 1/2. As a consequence, we find that the corresponding singularity spectraftyp(α) andfav(α) differ even in the positiveregion f > 0, and vanishat different values α + typ > α + av, in contrast to the standard picture. We also obtain that the saddle valueαtyp(q) of the Legendre transform reaches the termination pointα + typ whereftyp(α + typ) = 0 only in the limit .

Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

We generalize universal relations between the multifractal exponent \alpha_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 \Lambda_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 \Lambda_c^o is universally related to the boundary multifractal exponent \alpha_0^s for the typical wave function amplitude along a straight boundary (surface). (ii) We further propose a generalization of these universal relations to those symmetry classes whose density of states vanishes at the transition. (iii) We verify our generalized relations [Eqs. (6) and (7)] numerically for the following four types of two-dimensional Anderson transitions: (a) the metal-to-(ordinary insulator) transition in the spin-orbit (symplectic) symmetry class, (b) the metal-to-(Z_2 topological insulator) transition which is also in the spin-orbit (symplectic) class, (c) the integer quantum Hall plateau transition, and (d) the spin quantum Hall plateau transition.

Random cascade models of multifractality: real-space renormalization and travellingwaves

Random multifractals occur in particular at critical points of disordered systems.For Anderson localization transitions, Mirlin and Evers (2000 Phys. Rev. B 627920) have proposed the following scenario: (a) the inverse participation ratios (IPR)Yq(L) display the following fluctuations between the disordered samples of linear sizeL: with respect to the typical value that involve the typical multifractal spectrumτtyp(q), the rescaledvariable y = Yq(L)/Yqtyp(L) is distributed with a scale-invariant distribution presenting the power-law tail1/y1 + βq, so the disorder-averaged IPR have multifractal exponents τav(q) that differ from the typical onesτtyp(q) wheneverβq < 1; (b) the tailexponents βq and the multifractal exponents are related by the relationβqτtyp(q) = τav(qβq). Here we show that this scenario can be understood by considering thereal-space renormalization equations satisfied by the IPR. For the simplestmultifractals described in terms of random cascades, these renormalizationequations are formally similar to the recursion relations for disordered modelsdefined on Cayley trees and they admit travelling-wave solutions for the variable(lnYq) in the effectivetime teff = lnL: theexponent τtyp(q) represents the velocity, whereas the tail exponentβq represents the usual exponential decay of the travelling-wave tail. In addition, we obtainthat the relation in (b) above can be obtained as a self-consistency condition from theself-similarity of the multifractal spectrum at all scales. Our conclusion is thus that theMirlin–Evers scenario should apply to random critical points of other types, and even torandom multifractals occurring in other fields.

THE LOCALIZATION TRANSITION AT FINITE TEMPERATURES: ELECTRIC AND THERMAL TRANSPORT

The Anderson localization transition is considered at finite temperatures. This includes the electrical conductivity as well as the electronic thermal conductivity and the thermoelectric coefficients. An interesting critical behavior of the latter is found. A method for characterizing the conductivity critical exponent, an important signature of the transition, using the conductivity and thermopower measurements, is outlined.

SELF-CONSISTENT THEORY OF ANDERSON LOCALIZATION: GENERAL FORMALISM AND APPLICATIONS

The self-consistent theory of Anderson localization of quantum particles or classical waves in disordered media is reviewed. After presenting the basic concepts of the theory of Anderson localization in the case of electrons in disordered solids, the regimes of weak and strong localization are discussed. Then the scaling theory of the Anderson localization transition is reviewed. The renormalization group theory is introduced and results and consequences are presented. It is shown how scale-dependent terms in the renormalized perturbation theory of the inverse diffusion coefficient lead in a natural way to a self-consistent equation for the diffusion coefficient. The latter accounts quantitatively for the static and dynamic transport properties except for a region near the critical point. Several recent applications and extensions of the self-consistent theory, in particular for classical waves, are discussed.

SPECKLE STATISTICS IN THE PHOTON LOCALIZATION TRANSITION

We review the statistics of speckle in the Anderson localization transition for classical waves. Probability distributions of local and integrated transmission and of the evolution of the structure of the speckle pattern are related to their corresponding correlation functions. Steady state and pulse transport can be described in terms of modes whose speckle patterns are obtained by decomposing the frequency variation of the transmitted field. At the same time, transmission can be purposefully manipulated by adjusting the incident field and the eigenchannels of the transmission matrix can be found by analyzing sets of speckle patterns for different inputs. The many aspects of steady state propagation are reflected in diverse, but simply related, parameters so that a single localization parameter encapsulates the character of transport on both sides of the divide separating localized from diffusive waves.

Supersymmetric virial expansion for time-reversal invariant disordered systems

We develop a supersymmetric virial expansion for two-point correlation functions of almost diagonal Gaussian random matrix ensembles (ADRMT) of the orthogonal symmetry. These ensembles have multiple applications in physics and can be used to study the universal properties of time-reversal invariant disordered systems which are either insulators or close to the Anderson localization transition. We derive a two-level contribution to the correlation functions of the generic ADRMT and apply these results to the critical (multifractal) power law banded ADRMT. Analytical results are compared with numerical ones.

Criticality and entanglement in random quantum systems

We review studies of entanglement entropy in systems with quenched randomness, concentrating on universal behavior at strongly random quantum critical points. The disorder-averaged entanglement entropy provides insight into the quantum criticality of these systems and an understanding of their relationship to non-random (‘pure’) quantum criticality. The entanglement near many such critical points in one dimension shows a logarithmic divergence in subsystem size, similar to that in the pure case but with a different universal coefficient. Such universal coefficients are examples of universal critical amplitudes in a random system. Possible measurements are reviewed along with the one-particle entanglement scaling at certain Anderson localization transitions. We also comment briefly on higher dimensions and challenges for the future.

Statistical properties of two-particle transmission at an Anderson transition

The ensemble of L × L power-law random banded matrices, where the random hopping Hi,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 T2 for two distinguishable particles, two bosons or two fermions, in the non-interacting case. We find that the statistics of T2 is multifractal, i.e. the probability to have T2(L) ∼ 1/Lκ behaves as , where the multifractal spectrum Φ2(κ) for fermions is different from the common multifractal spectrum concerning distinguishable particles and bosons. However, in the three cases, the typical transmission Ttyp2(L) is governed by the same exponent κtyp2, which is much smaller than the naive expectation 2κtyp1, where κtyp1 is the typical exponent of the one-particle transmission T1(L).

Observation of Multifractality in Anderson Localization of Ultrasound

We report the experimental observation of strong multifractality in wave functions below 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 real data depends on the excitation scheme used in the experiment.

Termination of typical wave-function multifractal spectra at the Anderson metal-insulator transition: Field theory description using the functional renormalization group

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 differs markedly from that obtained from the disorder-averaged IPRs. This phenomenon is known as the termination of the multifractal spectrum. We provide a field theoretical derivation for the termination of the typical multifractal spectrum, by combining the non-linear sigma model framework, conventionally used to access the MIT in d = 2 + epsilon dimensions, with a functional renormalization group (FRG) technique. The FRG method deployed here was originally pioneered to study the properties of the two-dimensional (2D) random phase XY model [D. Carpentier and P. Le Doussal, Nucl. Phys. B 588, 565 (2000)]. The same method was used to demonstrate the termination of the multifractal spectrum in the very special problem of 2D Dirac fermions subject to a random Abelian vector potential. Our result shows that the typical multifractal wavefunction spectrum and its termination can be obtained at a generic Anderson localization transition in d > 2, within the standard field theoretical framework of the non-linear sigma model, when combined with the FRG.

Statistics of the two-point transmission at Anderson localization transitions

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 et al., Phys. Rev. B 59, 15836 (1999)]: (i) the probability to have ${T}_{L}\ensuremath{\sim}1/{L}^{\ensuremath{\kappa}}$ behaves as ${L}^{\ensuremath{\Phi}(\ensuremath{\kappa})}$, where the multifractal spectrum $\ensuremath{\Phi}(\ensuremath{\kappa})$ terminates at $\ensuremath{\kappa}=0$ as a consequence of the physical bound ${T}_{L}\ensuremath{\le}1$; (ii) the exponents $X(q)$ that govern the moments $\overline{{T}_{L}^{q}}\ensuremath{\sim}1/{L}^{X(q)}$ become frozen above some threshold: $X(q\ensuremath{\ge}{q}_{\text{sat}})=\ensuremath{-}\ensuremath{\Phi}(\ensuremath{\kappa}=0)$, i.e., all moments of order $q\ensuremath{\ge}{q}_{\text{sat}}$ are governed by the measure of the rare samples having a finite transmission $(\ensuremath{\kappa}=0)$. In the present paper, we test numerically these predictions for the ensemble of $L\ifmmode\times\else\texttimes\fi{}L$ power-law random-banded matrices, where the random hopping ${H}_{i,j}$ decays as a power law ${(b/|i\ensuremath{-}j|)}^{a}$. This model is known to present an Anderson transition at $a=1$ between localized $(ag1)$ and extended $(al1)$ states with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra ${\ensuremath{\Phi}}_{b}(\ensuremath{\kappa})$ for various $b$ are in agreement with the relation $\ensuremath{\Phi}(\ensuremath{\kappa}\ensuremath{\ge}0)=2[f(\ensuremath{\alpha}=d+\frac{\ensuremath{\kappa}}{2})\ensuremath{-}d]$ in terms of the singularity spectrum $f(\ensuremath{\alpha})$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation ${\ensuremath{\kappa}}_{\text{typ}}(b)=2[{\ensuremath{\alpha}}_{\text{typ}}(b)\ensuremath{-}d]$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.

Boundary criticality and multifractality at the two-dimensional spin quantum Hall transition

Multifractal scaling of critical wave functions at a disorder-driven (Anderson) localization transition is modified near boundaries of a sample. Here this effect is studied for the example of the spin quantum Hall plateau transition using the supersymmetry technique for disorder averaging. Upon mapping the spin quantum Hall transition to the classical percolation problem with reflecting boundaries, a number of multifractal exponents governing wave-function scaling near a boundary are obtained exactly. Moreover, additional exact boundary scaling exponents of the localization problem are extracted, and the problem is analyzed in other geometries.

Entanglement entropy and multifractality at localization transitions

The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.

Aspects of Disorder in QCD

We review the application of random matrix theory to QCD. After explaining why random matrix theory is successful to the description of the QCD vacuum, we discuss a possible extension of the model involving the Anderson localization transition. We study in an extended model how the transition occurs and the level statistics deviates from the universal limit. This analysis makes it clear what should be clarified to describe the QCD vacuum as a disordered system.

Two-eigenfunction correlation in a multifractal metal and insulator

We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are close but not exactly at the Anderson localization transition. We show that even far away from the critical point the eigenfunction correlation show the remnant of multifractality which is characteristic of the critical states. By a combination of the numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we were able to identify the Gaussian random matrix ensembles that describe the multifractal features in the metal and insulator phases. In particular those random matrix ensembles describe new phenomena of eigenfunction correlation we discovered from simulations on the Anderson model. These are the eigenfunction mutual avoiding at large energy separations and the logarithmic enhancement of eigenfunction correlations at small energy separations in the two-dimensional (2D) and the three-dimensional (3D) Anderson insulator. For both phenomena a simple and general physical picture is suggested.

Directed polymer in a random medium of dimension1+3: Multifractal properties at the localization-delocalization transition

We consider the model of the directed polymer in a random medium of dimension 1+3 , and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights w{L}(r[over]) of the endpoint of the polymer of length L via the moments [equation: see text]. We measure the generalized exponents tau(q) and tau[over](q) governing the decay of the typical values [equation: see text] and disorder-averaged values Y{q}(L)[over] approximately L{-tau[over](q)} , respectively. To understand the difference between these exponents, tau(q) not equal to tau[over](q) above some threshold q>q{c} approximately 2 , we compute the probability distributions of [equation: see text] over the samples: We find that these distributions becomes scale invariant with a power-law tail 1/y{1+x{q}} . These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent nu approximately 2 .

Boundary multifractality in critical one-dimensional systems with long-range hopping

Boundary multifractality of electronic wave functions is studied analytically and numerically for the power-law random banded matrix (PRBM) model, describing a critical one-dimensional system with long-range hopping. The peculiarity of the Anderson localization transition in this model is the existence of a line of fixed points describing the critical system in the bulk. We demonstrate that the boundary critical theory of the PRBM model is not uniquely determined by the bulk properties. Instead, the boundary criticality is controlled by an additional parameter characterizing the hopping amplitudes of particles reflected by the boundary.