Distribution of level curvatures for the Anderson model at the localization-delocalization transition

Abstract

We compute the distribution function of single-level curvatures, P (k), for a tight binding model with site disorder, on a cubic lattice. In metals P (k) is very close to the predictions of the random-matrix theory (RMT). In insulators P (k) has a logarithmically-normal form. At the Anderson localizationdelocalization transition P (k) fits very well the proposed novel distribution P (k) ∝ (1 + k µ ) 3/µ with µ ≈ 1.58, which approaches the RMT result for large k and is non-analytical at small k. We ascribe such a non-analiticity to the spatial multifractality of the critical wave functions.PACS numbers: 71.30.+h, 72.15.Rn, 05.60.+w An important consequence of the one-parameter scaling theory [1] of the Anderson transition is the existence of three universality classes for the energy level statistics of disordered systems. In the metallic regime they are described by the strongly correlated Wigner-Dyson (WD) distribution, while in the insulating regime they follow an uncorrelated Poissonic law. This difference has its fundamental origin in the underlying nature of the corresponding eigenstates, being extended and strongly overlapping in the first case but localized in the second.The properties of the third universal statistics, describing the spectral correlations at the critical point, have been only recently the subject of intense investigation, both analytical and numerical. Using the results of numerical simulations, Shklovskii et al.[2] suggested that the spacing distribution function P (s) has the WD form P (s) ∼ s for small s and the Poissonic tail P (s) ∼ e −s for large s. Further analytical investigations [3,4] showed that the two-level correlation function R(s) in the critical region has a novel power-law asymptotic decay R(s) = −c/s 2−γ with a nontrivial exponent γ = 1 − 1/νd. Here ν is the critical exponent of the correlation/localization length ξ, which depends on the dimensionality d. Thus the two-level correlation function in the critical region resembles qualitatively the WD function which applies to the metallic phase. On the other hand, the level number variance Σ 2 (N ) = (δN ) 2 in an energy strip of width N ∆, (∆ is the mean level spacing), still contains a dominant Poissonic term, [5][6][7] linear in N , which is typical for insulators.It can be shown [3] that the Poissonic term in Σ 2 (N ) is only possible if a normalization sum rule on R(s) is violated in the thermodynamic limit (TL)

Many-body-localization transition: sensitivity to twisted boundary conditions

“…This rescaling by ∆ n has been first introduced by Edwards and Thouless [4] in the context of the Anderson Localization where it is used systematically [11][12][13][14][15][16], and where it has played a very essential role in the elaboration of the scaling theory of localization [54]. So here we should stress an important difference between Anderson Localization models and Many-Body-Localized models :…”

“…24 is not the Cauchy distribution (that would correspond to Eq. 27 for β = 0), because the regular part P reg (k) describing the central part of the distribution has been found analytically [14] and numerically [13,[15][16][17] to be log-normal.…”

“…3 cannot be computed explicitly. However, it has been understood on the examples of Anderson localization models [4,11,13,[15][16][17] and of quantum chaotic systems and random matrices [6][7][8][9] that the probability distribution of the dimensionless curvature k n contains some singular power-law tail involving the level repulsion exponent β…”

“…Its statistical properties have been much studied in chaotic quantum systems and random matrices [6][7][8][9][10] and in Anderson Localization models [11][12][13][14][15][16][17]. The case of the ground state energy in interacting quantum models has been analyzed both without disorder [18,19] and with disorder [20][21][22], in particular to characterize the persistent currents induced by a magnetic flux in mesoscopic rings.…”

Distribution of level curvatures in the lowest Landau level

“…Some of these and other related problems were studied recently for a three-dimensional disordered system governed by a tight-binding Hamiltonian. [8][9][10] In the present work we address the corresponding physics for a two-dimensional disordered electronic system at strong magnetic field exhibiting the integer quantum Hall effect. Beside the fact that a disordered system in strong magnetic field belongs to the unitary symmetry class, let us mention one point that emerges in the present case pertaining to calculation of curvatures in the localized regime.…”