On the critical level-curvature distribution

The parametric motion of energy levels for non-interacting electrons at the Anderson localization critical point is studied by computing the energy level-curvatures for a quasiperiodic ring with twisted boundary conditions. We find a critical distribution which has the universal random matrix theory form ${\bar P}(K)\sim |K|^{-3}$ for large level-curvatures $|K|$ corresponding to quantum diffusion, although overall it is close to approximate log-normal statistics corresponding to localization. The obtained hybrid distribution resembles the critical distribution of the disordered Anderson model and makes a connection to recent experimental data.

The parametric motion of energy levels for non-interacting electrons at the Anderson localization critical point is studied by computing the energy level-curvatures for a quasiperiodic ring with twisted boundary conditions. We find a critical distribution which has the universal random matrix theory formP (K) ∼ |K| −3 for large level-curvatures |K| corresponding to quantum diffusion, although overall it is close to approximate log-normal statistics corresponding to localization. The obtained hybrid distribution resembles the critical distribution of the disordered Anderson model and makes a connection to recent experimental data. The Landauer-Buttiker(LB) scattering approach [1] nowadays is almost exclusively used for computing the conductance g for many materials in nanoscience. This very appealing formalism although not extremely rigorous is particularly suitable for small systems, something which became obvious following the discovery of conductance steps in ballistic point contacts more than a decade ago [2]. In such a scattering approach one gets the nonequilibrium transport for any suffiently realistic system if it is connected to two ideal leads. The conductance is simply the transmission probability through the sample scatterer which can be evaluated at any energy monitored by gates, e.g. via Green's function techniques. Although g is strictly speaking a property of open systems it can be also probed for a closed system from the eigensolutions of the stationary Schroedinger's equation. This is done via Thouless's intuitive definition of conductance which involves a measure of the sensitivity of eigenvalues to twisted boundary conditions (BC) [3]. In the later approach, the conductance is related to the so-called levelcurvatures and is usually expressed by the ratio of the geometric mean of the absolute curvature at zero twist over the local mean level-spacing. The scattering approach for open systems and the eigenvalue shift approach for closed systems are two established definitions of conductance, which compare rather favorably with each other for electrons in disordered systems (see Ref. [4]).
The search for universal features in the statistics of stationary electronic levels and their motion as a function of some parameter is a large area of study with obvious consequences to quantum transport. For example, avoided level-crossings which characterize the spectrum of a chaotic system under a perturbation, such as the application of external fields, evolve with universal random matrix theory (RMT) laws [5]. The very small levelspacings for avoided level-crossings correspond to large absolute level-curvatures so that the tail of the levelcurvature distributionP (K) is intimately related to the small part of the level-spacing distribution. Therefore, in the absence of symmetry breaking mechanisms the linear part of the Wigner distribution for small levelspacings implies large level-curvatures having a decreasing asymptotic behaviorP (K) ∼ 1/|K| 2+β , with β = 1 [6,7,8]. This has been confirmed experimentally for level-curvature distributions of acoustic resonance spectra from quartz blocks [9].
We focuse on the critical distribution of levelcurvatures by obtaining the response of a quasiperiodic complex system's energy spectrum to different BC [10], within Thouless's approach to quantum transport. For this purpose a phase factor exp(i φ) is imposed to the hopping probability connecting the first and the last sites of a ring and the resulting parametric dependence of the energy levels to changes of φ is obtained. This allows to explore the nature of the corresponding eigenstates from extended to localized, since extended states simply feel any changes in BC having large curvatures while localized states are insensitive to BC having curvatures which approach zero. Nevertheless, the studied model is interesting only at the metal-insulator transition point since it satisfies duality for the extended (quasiballistic) and the localized (nonrandom) regimes [11]. One may wonder what the level-curvature distribution might be at the metal-insulator transition for such a quasiperiodic system which has Cantor set-like fractal electronic structure and a semi-Poisson hybrid distribution of level-spacings [10].
Deviations around the maximum peak between levelcurvatures obtained in experiment [9] and RMT were attributed to the presence of hidden symmetries [12,13]. In our study the critical distribution of level-curvatures is obtained in a convenient one-dimensional setting which allows to discuss both the three-dimensional disordered system and questions of universality in general. Our finding is a scale-invariantP (K) which resembles the distribution obtained for the normalized level-curvatures in 3D critical disordered systems [14,15], having both a diffusive tail and overall localized behavior [4]. For a finite system, the transition from extended to localized states involves broadening of the distributionP (K) and lowering its maximum peak as the curvatures move to lower values by increasing disorder. Our study, on one one hand, is in agreement with the semi-Poisson levelspacing distribution for the same system [10]. On the other hand, the coexistence of the localized almost lognormal broad form with the diffusive tail might be the reason for the lowering of the maximum peak observed in the experiment [9].
We have studied the distribution function of levelcurvatures, defined as for the energy levels ε α of a quasiperiodic ring in the presence of phase exp(iφ), obtained from the critical tightbinding model Hamiltonian [10] The sum is taken over all sites n, c n (c + n ) is the annihilation (creation) operator on site n and the potential V n = 2cos(2πσn) is chosen at criticality with σ the golden mean irrational. The eigenvalues ε α have corresponding eigenvectors |α = x=1,L ψ α (x)|x with amplitudes ψ α (x) = x|α for a finite chain of size L = F i with σ = F i−1 /F i , the ratio of two successive Fibonacci numbers F i−1 , F i . The imposed general boundary condition to the wave function ψ(x + L) = e iφ ψ(x) is equivalent to piercing the ring by the Aharonov-Bohm magnetic flux Φ via φ = 2πΦ/Φ 0 , with Φ 0 = c /e the flux quantum.
To use Eq. (3) we only need the eigensolutions of the periodic BC problem, that is the solutions of the Hamiltonian of Eq. (2) with φ = 0. The problem simplifies further by the presence of the parity symmetric potential cos(x) = cos(−x) which allows to find symmetric and antisymmetric solutions separately, dealing with two tridiagonal matrices instead of the full matrix in the presence of BC [17]. This dramatic reduction in storage requirements is achieved if the one-electron basis is rearranged to run within −s, −s+ 1, ..., s− 1, s also by distinguishing between even and odd size L. For example, for odd L = 2s + 1 the problem is reduced to a tridiagonal matrix of size s+1 for the symmetric states which has diagonal matrix elements the potential values V 0 , V 1 , ..., V s−1 , V s + 1, unity elements lying next to the diagonal, except in the first (second) row second(first) column where is √ 2, and each eigenvector amplitude (χ(0), χ(1), ..., χ(s)) is related to the original via the symmetric rule ψ(0) = χ(0), ψ(−s) = ψ(s) = χ(s)/ (2). The second tridiagonal matrix for the antisymmetric states is of size s with matrix elements V 1 , ..., V s−1 , V s − 1 on the diagonal and unity next to diagonal with corresponding eigenvector amplitudes (χ(1), ..., χ(s)) related to the original via antisymmetry ψ(0) = 0, ψ(−s) = −ψ(s) = −χ(s)/ (2). By replacing ψ with χ Eq. (3) splits into a symmetric part labelled by α = 1, 2, ..., s + 1 and an antisymmetric part for α = s + 2, s + 3, ..., 2s + 1 In Eqs. (4), (5) the s-th last element of each eigenvector can be computed iteratively with no need to increase the storage requirements beyond that of a tridiagonal matrix. Moreover, the corresponding sums over β run over opposite kind of symmetric and antisymmetric states, respectively, being precisely zero for species of the same symmetry. A similar formula for even L is easily obtained.
A ballistic ring has K α = (8π 2 /L 2 ) cos(2πα/L) which give square-root singularities forP (K). These disappear for the critical ring studied where the curvature dis-tributionP (K) is much more complex so it turns out more convenient to consider the logarithmic distribution P (ln |K|) = |K|P (K), instead. The computed critical distribution presented in Fig. 1 for various sizes L is shown to be scale-invariant, overall approximated by a logarithmic normal form. In Fig. 2 we present a log-log plot of P (ln |K|) which demonstrates the diffusive universal tailP (K) ∼ |K| −3 for large-|K|. For small-|K| we approximately findP (ln |K|) ∼ |K| 2 . Unfortunately, rapid loss of accuracy for the too small curvatures does not permit to extract reliable exponents from fits ofP (K) to the form suggested in [14]. However, the overall behavior of the critical distribution shown in Fig. 3 is roughly similar to the critical distribution obtained for the 3D Anderson model [14,15].
The level-curvatures in disordered or chaotic systems exploit a sort of "dynamics" of the quantum stationary spectra as a function of flux which might be thought of "time". The conductance is then obtained from the level-curvatures in terms of eigenvalues and eigenvectors of the tight binding system with periodic BC. In our case the computational effort is minimized because the parity symmetry of the potential reduces the problem to two simple tridiagonal matrices for the determination of symmetric and antisymmetric states. The needed last element of each eigenvector can be also computed efficiently with no increase of storage so that the size of the matrices can easily exceed 10 5 . We remind the reader that for diffusive disordered systems the computed averaged LB conductance g was shown [4] to be related to the mean absolute curvature via g = π |K| /∆, where ∆ is the local mean-level spacing. For localized disordered systems the curvatures diminish and both distributions approach a log-normal form with ln g = π ln(|K|) . Our study of the unormalized level-curvatures for the quasiperiodic  [14,15]. The distributions for the quasiperiodic and the disordered system are rather similar although in the disordered case the curvatures are normalized divided by the local mean levelspacing ∆.
system enabled us to obtain the criticalP (K) for very large system sizes. The curvatures become very small for such large systems which set limits to the accuracy of the distribution. Our results are similar to that obtained in 3D critical disordered system. Moreover, such computations might suggest that the lowering of the maximum peak observed in experiment [9] could be thought of as due to an approach towards the critical region. Although the critical distribution shown in Fig. 3 still displays a diffusive tail of the log-normal form it is much broader having less height than the pure diffusive RMT one.
The obtained results for non-interacting fermions in a quasiperiodic ring complement and confirm previous studies at the metal-insulator transition of disordered systems. The distributionP (K) for the ensemble of critical states at the transition (there is no ensemble over disorder) is scale-invariant like the corresponding levelspacing distribution which is known to depend sensitively on BC [10]. After average over BC the critical distribution of the level-spacings can be described by the semi-Poisson curve which combines both extended and localized behavior. Similarly, the obtainedP (K) exhibits a hybrid character like what is obtained for critical 3D disordered systems. In closing, it is remarkable that such a simple one-dimensional non-random model can capture most features displayed at the Anderson metal-insulator transition of realistic disordered systems.