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

N. D. Chavda; H. N. Deota; V. K. B. Kota

doi:10.1016/j.physleta.2014.08.021

paper

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

arXiv:1405.6321 · doi:10.1016/j.physleta.2014.08.021

Abstract

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 $H_{λ} = (H_{0} + λV) / 1 + λ^{2}$ interpolating Poisson ( $λ = 0$ ) and GOE ( $λ \to \infty$ ) we have analyzed the transition curves for $⟨ r ⟩$ and $⟨ r ~ ⟩$ as $λ$ changes from $0$ to $\infty$ ; $r ~ = min (r, 1/ r)$ . 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 $⟨ r ⟩$ vs $λ$ (similarly for $⟨ r ~ ⟩$ 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.

9 Pages, 5 figures

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

Abstract

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