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

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 $H_{\lambda }=(H_0+\lambda V)/\sqrt{1+{\lambda }^2}$ interpolating Poisson ($\lambda =0$) and GOE ($\lambda \to \infty$) we have analyzed the transition curves for $⟨r⟩$ and $⟨\tilde{r}⟩$ as $\lambda$ 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 $\Lambda \sim {\lambda }^{2}d$, 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 $\Lambda$. 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.

9 Pages, 5 figures

References in corpus (6)

• Many-Body Physics with Ultracold Gases
• Localization of interacting fermions at high temperature
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Quantum Simulation of Antiferromagnetic Spin Chains in an Optical Lattice
• Many-Body Localization in a Quasiperiodic System
• Statistical properties of the spectrum the extended Bose-Hubbard model

Cited by in corpus (5)

• Distribution of the Ratio of Consecutive Level Spacings for Different Symmetries and Degrees of Chaos
• Deviations from Poisson statistics in the spectra of free rectangular thin plates
• Spectral Crossovers and Universality in Quantum Spin-chains Coupled to Random Fields
• Characterization of many-body mobility edges with random matrices
• Ordered Level Spacing Distribution in Embedded Random Matrix Ensembles