Amplitude Of Eigenfunctions Research Articles

Random Cantor sets and mini-bands in local spectrum of quantum systems

In this paper we give a physically transparent picture of singular-continuous spectrum in disordered systems which possess a non-ergodic extended phase. We present a simple model of identically and independently distributed level spacing in the spectrum of local density of states and show how a fat tail appears in this distribution at the broad distribution of eigenfunction amplitudes. For the model with a power-law local spacing distribution we derive the correlation function K(ω) of the local density of states and show that depending on the relation between the eigenfunction fractal dimension D2 and the spectral fractal dimension Ds encoded in the power-law spacing distribution, a singular continuous spectrum of a random Cantor set or that of an isolated mini-band may appear. In the limit of an infinite number of degrees of freedom the function K(ω) in the non-ergodic extended phase is singular at ω=0 with the branch-cut singularity for the case of a random Cantor set and with the δ-function singularity for the case of an isolated mini-band. For an absolutely continuous spectrum K(ω) tends to a finite limit as ω→0. For an arbitrary local spacing distribution function we formulated a criterion of fractality of local spectrum and tested it on simple examples.

Eigenfunction and eigenmode-spacing statistics in chaotic photonic crystal graphs.

The statistical properties of wave chaotic systems of varying dimensionalities and realizations have been studied extensively. These systems are commonly characterized by the statistics of the eigenmode spacings and the statistics of the eigenfunctions. Here, we propose photonic crystal (PC) defect waveguide graphs as a physical setting for chaotic graph studies. Photonic crystal waveguides possess a dispersion relation for the propagating modes, which is engineerable. Graphs constructed by joining these waveguides possess junctions and bends with distinct scattering properties. We present numerically determined statistical properties of an ensemble of such PC graphs including both eigenfunction amplitude and eigenmode-spacing studies. Our proposed system is compatible with silicon nanophotonic technology and opens chaotic graph studies to a new community of researchers.

The Heat Kernel on the Diagonal for a Compact Metric Graph

We analyze the heat kernel associated with the Laplacian on a compact metric graph, with standard Kirchhoff–Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin–Potthoff–Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an “edge” heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for graphs which are symmetric about each vertex.

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

Direct Observations of Surface‐Wave Eigenfunctions at the Homestake 3D Array

AbstractDespite the theory for both Rayleigh and Love waves being well accepted and the theoretical predictions accurately matching observations, the direct observation of their quantifiable decay with depth has never been measured in the Earth’s crust. In this work, we present observations of the quantifiable decay with depth of surface‐wave eigenfunctions. This is done by making direct observations of both Rayleigh‐wave and Love‐wave eigenfunction amplitudes over a range of depths using data collected at the 3D Homestake array for a suite of nearby mine blasts. Observations of amplitudes over a range of frequencies from 0.4 to 1.2 Hz are consistent with theoretical eigenfunction predictions. They show a clear exponential decay of amplitudes with increasing depth and a reversal in sign of the radial‐component Rayleigh‐wave eigenfunction at large depths, as predicted for fundamental‐mode Rayleigh waves. Minor discrepancies between the observed eigenfunctions and those predicted using estimates of the local velocity structure suggest that the observed eigenfunctions could be used to improve the velocity model. Our results confirm that both Rayleigh and Love waves have the depth dependence that they have long been assumed to have. This is an important direct validation of a classic theoretical result in geophysics and provides new observational evidence that classical seismological surface‐wave theory can be used to accurately infer properties of Earth structure and earthquake sources.

Exact eigenfunction amplitude distributions of integrable quantum billiards

The exact probability distributions of the amplitudes of eigenfunctions, Ψ(x, y), of several integrable planar billiards are analytically calculated and shown to possess singularities at Ψ = 0; the nature of this singularity is shape-dependent. In particular, we prove that the distribution function for a rectangular quantum billiard is proportional to the complete elliptic integral, K(1 − Ψ2), and demonstrate its universality, modulo a weak dependence on quantum numbers. On the other hand, we study the low-lying states of nonseparable, integrable triangular billiards and find the distributions thereof to be described by the Meijer G-function or certain hypergeometric functions. Our analysis captures a marked departure from the Gaussian distributions for chaotic billiards in its survey of the fluctuations of the eigenfunctions about Ψ = 0.

RESONANT AMPLIFICATION OF TURBULENCE BY THE BLAST WAVES

We discuss an idea whether spherical blast waves can amplify by a non-local resonant hydrodynamic mechanism inhomogeneities formed by turbulence or phase segregation in the interstellar medium. We consider the problem of a blast-wave-turbulence interaction in the Linear Interaction Approximation. Mathematically, this is an eigenvalue problem for finding the structure and amplitude of eigenfunctions describing the response of the shock-wave flow to forced oscillations by external perturbations in the ambient interstellar medium. Linear analysis shows that the blast wave can amplify density and vorticity perturbations for a wide range of length scales with amplification coefficients of up to 20, with amplification the greater, the larger the length. There also exist resonant harmonics for which the gain becomes formally infinite in the linear approximation. Their orbital wavenumbers are within the range of macro- ($l \sim 1$), meso- ($l \sim 20$) and microscopic ($l > 200$) scales. Since the resonance width is narrow: typically, $\Delta l <1$, resonance should select and amplify discrete isolated harmonics. We speculate as to a possible explanation of an observed regular filamentary structure of regular-shaped round supernova remnants such as SNR 1572, 1006 or 0509-67.5. Resonant mesoscales found ($l \approx 18 $) are surprisingly close to the observed scales ($l \approx 15$) of ripples in the shell's surface of SNR 0509-67.5.

Gyrokinetic theory of electrostatic lower-hybrid drift instabilities in a current sheet with guide field

A kinetic electrostatic eigenvalue equation for the lower-hybrid drift instability (LHDI) in a thin Harris current sheet with a guide field is derived based on the gyrokinetic electron and fully kinetic ion(GeFi) description. Three-dimensional nonlocal analyses are carried out to investigate the influence of a guide field on the stabilization of the LHDI by finite parallel wavenumber, k∥. Detailed stability properties are first analyzed locally, and then as a nonlocal eigenvalue problem. Our results indicate that at large equilibrium drift velocities, the LHDI is further destabilized by finite k∥ in the short-wavelength domain. This is demonstrated in a local stability analysis and confirmed by the peak in the eigenfunction amplitude. We find the most unstable modes localized at the current sheet edges, and our results agree well with simulations employing the GeFi code developed by Lin et al. [Plasma Phys. Controlled Fusion 47, 657 (2005); Plasma Phys. Controlled Fusion 53, 054013 (2011)].

SEARCH FOR GLOBAL f -MODES AND p -MODES IN THE 8 B NEUTRINO FLUX

The impact of global acoustic modes on the 8B neutrino flux time series is computed for the first time. It is shown that the time fluctuations of the 8B neutrino flux depend on the amplitude of acoustic eigenfunctions in the region where the 8B neutrino flux is produced: modes with low n (or order) that have eigenfunctions with a relatively large amplitude in the Sun's core, strongly affect the neutrino flux; conversely, modes with high n that have eigenfunctions with a minimal amplitude in the Sun's core have a very small impact on the neutrino flux. It was found that the global modes with a larger impact on the 8B neutrino flux have a frequency of oscillation in the interval 250 \mu Hz to 500 \mu Hz (or a period in the interval 30 minutes to 70 minutes), such as the f-modes (n=0) for the low degrees, radial modes of order n smaller or equal to 3, and the dipole mode of order n=1. Their corresponding neutrino eigenfunctions are very sensitive to the solar inner core and are unaffected by the variability of the external layers of the solar surface. If time variability of neutrinos is observed for these modes, it will lead to new ways of improving the sound speed profile inversion in the central region of the Sun.

Statistics of anomalously localized states at the center of band E = 0 in the one-dimensional Anderson localization model

We consider the distribution function P(|ψ|2) of the eigenfunction amplitude at the center-of-band (E = 0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with |ψ|2 much larger than the inverse typical localization length ℓ0. Using the recently found solution for the generating function Φan(u, ϕ) we obtain the ALS probability distribution P(|ψ|2) at |ψ|2ℓ0 ≫ 1. As an auxiliary preliminary step, we found the asymptotic form of the generating function Φan(u, ϕ) at u ≫ 1 which can be used to compute other statistical properties at the center-of-band anomaly. We show that at moderately large values of |ψ|2ℓ0, the probability of ALS at E = 0 is smaller than at energies away from the anomaly. However, at very large values of |ψ|2ℓ0, the tendency is inverted: it is exponentially easier to create a very strongly localized state at E = 0 than at energies away from the anomaly. We also found the leading term in the behavior of P(|ψ|2) at small |ψ|2 ≪ ℓ−10 and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner.

Multifractal analysis of the electronic states in the Fibonacci superlattice under weak electric fields

Influence of the weak electric field on the electronic structure of the Fibonacci superlattice is considered. The electric field produces a nonlinear dynamics of the energy spectrum of the aperiodic superlattice. Mechanism of the nonlinearity is explained in terms of energy levels anticrossings. The multifractal formalism is applied to investigate the effect of weak electric field on the statistical properties of electronic eigenfunctions. It is shown that the applied electric field does not remove the multifractal character of the electronic eigenfunctions, and that the singularity spectrum remains non-parabolic, however with a modified shape. Changes of the distances between energy levels of neighbouring eigenstates lead to the changes of the inverse participation ratio of the corresponding eigenfunctions in the weak electric field. It is demonstrated, that the local minima of the inverse participation ratio in the vicinity of the anticrossings correspond to discontinuity of the first derivative of the difference between marginal values of the singularity strength. Analysis of the generalized dimension as a function of the electric field shows that the electric field correlates spatial fluctuations of the neighbouring electronic eigenfunction amplitudes in the vicinity of anticrossings, and the nonlinear character of the scaling exponent confirms multifractality of the corresponding electronic eigenfunctions.

Commensurability effects in one-dimensional Anderson localization: Anomalies in eigenfunction statistics

The one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with random uncorrelated on-site energies, has statistical anomalies at any rational point f = 2 a λ E , where a is the lattice constant and λ E is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions ψ( r) at such commensurability points. The approach is based on an exact integral transfer-matrix equation for a generating function Φ r ( u, ϕ) ( u and ϕ have a meaning of the squared amplitude and phase of eigenfunctions, r is the position of the observation point). This generating function can be used to compute local statistics of eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at f = p q with q > 2. The descender of the generating function P r ( ϕ ) ≡ Φ r ( u = 0 , ϕ ) is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we derived a second-order partial differential equation for the r-independent (“zero-mode”) component Φ( u, ϕ) at the E = 0 ( f = 1 2 ) anomaly. This equation is nonseparable in variables u and ϕ. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for Φ( u, ϕ) explicitly in quadratures. Using this solution we computed moments I m = N〈∣ ψ∣ 2 m 〉 ( m ⩾ 1) for a chain of the length N → ∞ and found an essential difference between their m-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the “extrinsic” localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio (“intrinsic” localization length). This is not the case at the E = 0 anomaly where the extrinsic localization length is smaller than the intrinsic one. At E = 0 one also observes an anomalous enhancement of large moments compatible with existence of yet another, much smaller characteristic length scale.

Assignment and Extracting Dynamics from Experimentally and Theoretically Obtained Spectroscopic Hamiltonians in the Complex Spectral and Classically Chaotic Regions

An analysis of existing algebraic multiresonance spectroscopic Hamiltonians, derived by fitting to experimental data or from classical canonical or quantum Van Vleck perturbation theory, allows without any significant further classical or quantum calculation the assignment of quantum numbers and motions to states observed in spectra that were previously thought to be irregular or just unassignable. In such cases, inspection of the amplitude and phase of eigenfunctions previously calculated in the Hamiltonians derivation process but now transformed to a reduced dimension semiclassical action-angle representation reveals extremely simple albeit unfamiliar topologies that give quantum numbers by simply counting nodes and phase advances. The topology allows these simple wave functions to be sorted into dynamically different excitation ladders or classes of states which are associated with different regions of phase space. The rungs of these ladders or the states in the classes intersperse in energy causing the spectral complexity. No experimental procedure allows such sorting. The success of the work stems from (1) the qualitative insights of nonlinear dynamics, (2) the conversion of the quantum problem in full dimension to a semiclassical one in reduced dimension by use of a canonical transform that takes advantage of the polyad and other constants of the motion, and (3) the judicious choice of the reduced angle variables to reflect rational ratio resonance frequency conditions. This leads to localization of those semiclassical wave functions, which are affected by the particular resonance. In reverse, the localized appearance of the reduced dimension wave function reveals which resonances govern it and makes sorting visually simple. The success of the work also stems from (4) the revealing use of plots of phase advances as well as the usual densities of the eigenstates for sorting and assignment purposes. Even in classically chaotic regions, organizing trajectories, which correspond to averages over regional phase space structures that need not be computed, can easily be drawn as the structure about which eigenfunction localization takes place. The organizing trajectories when transformed back to the full dimensional configuration space reveal the internal molecular motions. The complexity of the usual quantum stationary and propagated wave functions and associated classical trajectories forbids most often such assignments and sorting. This procedure brings the ability to interpret complex vibrational spectra to a degree previously thought possible only for lower excitations. The new methodology replaces and extends the computationally more difficult parts of a procedure used by the authors that was applied successfully to several molecules during the past few years. The new methodology is applied to DCO, CHBrClF, and the bending of acetylene.

Orbit bifurcations and the scarring of wave functions

Orbit bifurcations and the scarring of wave functions

Statistical properties of eigenstates in three-dimensional mesoscopic systems with off-diagonal or diagonal disorder

The statistics of eigenfunction amplitudes are studied in mesoscopic disordered electron systems of finite size. The exact eigenspectrum and eigenstates are obtained by solving numerically Anderson Hamiltonian on a three-dimensional lattice for different strengths of disorder introduced either in the potential on-site energy (``diagonal'') or in the hopping integral (``off-diagonal''). The samples are characterized by the exact zero-temperature conductance computed using real-space Green function technique and related Landauer-type formula. The comparison of eigenstate statistics in two models of disorder shows sample-specific details which are not fully taken into account by the conductance, shape of the sample, and dimensionality. The wave function amplitude distributions for the states belonging to different transport regimes within the same model are contrasted with each other as well as with universal predictions of random matrix theory valid in the infinite conductance limit.

Advanced Lanczos diagonalization for models of quantum disordered systems

An application of an effective numerical algorithm for solving eigenvalue problems which arise in modeling electronic properties of quantum disordered systems is considered. We study the electron states at the localization-delocalization transition induced by a random potential in the framework of the Anderson lattice model. The computation of the interior of the spectrum and corresponding wavefunctions for very sparse, Hermitian matrices of sizes exceeding 106×106 is performed by the Lanczos-type method especially modified for investigating statistical properties of energy levels and eigenfunction amplitudes.

Spatial structure of anomalously localized states in disordered conductors

The spatial structure of wave functions of anomalously localized states (ALS) in disordered conductors is studied in the framework of the σ–model approach. These states are responsible for slowly decaying tails of various distribution functions. In the quasi-one-dimensional case, properties of ALS governing the asymptotic form of the distribution of eigenfunction amplitudes are investigated with the use of the transfer matrix method, which yields an exact solution to the problem. Comparison of the results with those obtained in the saddle-point approximation to the problem shows that the saddle-point configuration correctly describes the smoothed intensity of an ALS. On this basis, the properties of ALS in higher spatial dimensions are considered. We also study the ALS responsible for the asymptotic behavior of distribution functions of other quantities, such as relaxation time, local and global density of state. It is found that the structure of an ALS may be different, depending on the specific quantity, for which it constitutes an optimal fluctuation. Relations between various procedures of selection of ALS, and between asymptotics of corresponding distribution functions, are discussed.

Non Saturation of Energy in a Quantum Kicked Oscillator

We present a quantum mapping for the kicked harmonic oscillator which relates the probability amplitudes of the undriven oscillator's eigenfunctions over successive kicks. We show how for various kick strengths the wave func- tions lave a linear energy increase up to the limit imposed b y the finite matrix size of the evolution matrix. We use this linear energy increase to define a quantum diffusion-like coefficient. We also show how this increase in energy causes the wave functions to spread out and become diifuse with fittle or no discernible structure. This model may serve as a paradigm for the study of quantum chaos.

Mesoscopic fluctuations of eigenfunctions and level-velocity distribution in disordered metals.

We calculate the distribution of eigenfunction amplitudes and the variance of the ``inverse participation ratio'' (IPR) in disordered metallic samples. A relation is established between the distribution function of IPR and that of ``level velocities'' -- parametric derivatives of energy levels with respect to a random perturbation.

STATISTICAL PROPERTIES OF EIGENFUNCTIONS OF RANDOM QUASI 1D ONE-PARTICLE HAMILTONIANS

The article reviews recent analytical results concerning statistical properties of eigenfunctions of random Hamiltonians with broken time reversal symmetry describing a motion of a quantum particle in a thick wire of finite length L. It is demonstrated that the problem is equivalent to the study of properties of large Random Banded Matrices in the limit of large width of the band. Matrices of this class are relevant for a number of problems in Solid State physics and in the domain of Quantum Chaos. We find the analytical expressions for the distribution of the following quantities: i) the eigenfunction amplitude |ψ(r)|2 at given point of the sample; ii) spatial extent of the eigenfunction measured by the “inverse participation ratio” P=∫V dr|ψ(r)|4; iii) the quantity R=|ψ(r)ψ(r′)|2, points r and r′ belonging to the opposite ends of the sample. For a long sample the quantity –(ln R)/L characterizes the decay rate of a localized eigenfunction (Lyapunov exponent). Relation with available numerical results is discussed.