Poisson to GOE transition in the distribution of the ratio of consecutive level spacings

• Quantum chaos refers to GOE (Gaussian Orthogonal Ensemble) spectral fluctuations.
• Regular motion corresponds to Poisson level statistics.
• The ratio of successive spacing is studied for Poisson to GOE transition.
• Transition parameter is identified and universal transition curves are constructed.
• Application to a one-dimensional interacting spin-1/2 chain is presented.

Using an interpolating matrix ensemble with a parameter λ   giving Poisson (λ=0λ=0) to GOE (λ→∞λ→∞) transition, average of the ratio (r  ) of consecutive level spacings has been analyzed numerically using matrices with dimension d≤1000d≤1000. Established by this analysis are (i) the transition parameter Λ∼λ2dΛ∼λ2d giving scaling, (ii) the universal form for the transition curves 〈r〉〈r〉, 〈r˜〉 vs. Λ  ; r˜=min⁡(r,1/r), and (iii) a mapping between d×dd×d and 3×33×3 matrix ensembles. Example for the transition curves from a one-dimensional (1D) interacting spin-1/2 chain is presented.