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

Probability distribution for the ratio ($r$) of consecutive level spacings of the eigenvalues of a Poisson (generating regular spectra) spectrum and that of a GOE random matrix ensemble are given recently. Going beyond these, for the ensemble generated by the Hamiltonian $H_λ= (H_0+λV)/\sqrt{1+λ^2}$ interpolating Poisson ($λ=0$) and GOE ($λ\rightarrow \infty$) we have analyzed the transition curves for $\langle r\rangle$ and $\langle \tilde{r}\rangle$ as $λ$ changes from $0$ to $\infty$; $\tilde{r} = min(r,1/r)$. Here, $V$ is a GOE ensemble of real symmetric $d \times d$ matrices and $H_0$ is a diagonal matrix with a Gaussian distribution (with mean equal to zero) for the diagonal matrix elements; spectral variance generated by $H_0$ is assumed to be same as the one generated by $V$. Varying $d$ from 300 to 1000, it is shown that the transition parameter is $Λ\sim λ^2\,d$, i.e. the $\langle r\rangle$ vs $λ$ (similarly for $\langle \tilde{r}\rangle$ vs $λ$) curves for different $d$'s merge to a single curve when this is considered as a function of $Λ$. Numerically, it is also found that this transition curve generates a mapping to a $3 \times 3$ Poisson to GOE random matrix ensemble. Example for Poisson to GOE transition from a one dimensional interacting spin-1/2 chain is presented.