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.