Distribution of level curvatures in the lowest Landau level

Abstract

An analysis of the statistics of level curvatures for a system exhibiting the integer quantum Hall effect is presented. The curvatures k n are calculated numerically and their distribution P(k) is evaluated for energy eigenvalues E n belonging to insulating as well as to critical states. In the insulating region it is found that P(0)ϭ0 and that P(lnk) depends on the system size, albeit weaker than linearly. There is no clear cut evidence that the distribution is log-normal. The distribution of curvatures corresponding to critical states is scale invariant and appears to be way off the mark set by random matrix theory for the unitary ensemble.

Universal Voltage Fluctuations in Disordered Superconductors

“…Keeping track of all the influences of magnetic field on the vortex motion is a rather herculean task. Nevertheless, we can exploit the theory developed for correlations [40][41][42][43] and curvature distribution [44][45][46][47] of the spectral response to external parameters which show universal behavior of the derivative of the energies of the system with respect to an external parameter. Specifically, for the ith eigenvalue i (x) (where x is the value of the external parameter) of the Hamiltonian one may define a "velocity" j i = ∂ x i (x)/δ (where δ is the mean level spacing, i.e.…”

“…This leads to an unusual correlation curve since the correlation should be maximum at δX = 0, and approaches zero from below an large δX, which means that there is a negative minimum of the correlation at some intermediate value of δX. For the curvature, a universal distribution is expected [44][45][46][47]. Defining k = K i /| K i |, one obtains the distribution…”