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 $H_\lambda = (H_0+\lambda V)/\sqrt{1+\lambda^2}$interpolatingPoisson($\lambda=0$)andGOE($\lambda \rightarrow \infty$)wehaveanalyzedthetransitioncurvesfor$\langle r\rangle$and$\langle \tilde{r}\rangle$as$\lambda$changesfrom$0$to$\infty$;$\tilde{r} = min(r,1/r)$\mathrm{.}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 $\mathrm{\Lambda }\sim {\lambda }^{2}d$, i.e. the $⟨r⟩$ vs $\lambda$ (similarly for $⟨\tilde{r}⟩$ vs $\lambda$) curves for different $d$’s merge to a single curve when this is considered as a function of $\mathrm{\Lambda }$. 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.