Gaussian Orthogonal Random Matrix Ensemble Research Articles

Gaussian orthogonal ensemble for quasiperiodic tilings without unfolding: r -value statistics

We study the level-spacing statistics for noninteracting Hamiltonians defined on the two-dimensional quasiperiodic Ammann-Beenker (AB) tiling. When applying the numerical procedure of ``unfolding,'' these spectral properties in each irreducible sector are known to be well described by the universal Gaussian orthogonal random matrix ensemble. However, the validity and numerical stability of the unfolding procedure has occasionally been questioned due to the fractal self-similarity in the density of states for such quasiperiodic systems. Here, using the so-called $r$-value statistics for random matrices, $P(r)$, for which no unfolding is needed, we show that the Gaussian orthogonal ensemble again emerges as the most convincing level statistics for each irreducible sector. The results are extended to random-AB tilings where random flips of vertex connections lead to the irreducibility.

Deviations from Poisson statistics in the spectra of free rectangular thin plates.

The counterintuitive fact that wave chaos appears in the bending spectrum of free rectangular thin plates is presented. After extensive numerical simulations, varying the ratio between the length of its sides, it is shown that (i) frequency levels belonging to different symmetry classes cross each other and (ii) for levels within the same symmetry sector, only avoided crossings appear. The consequence of anticrossings is studied by calculating the distribution of the ratio of consecutive level spacings for each symmetry class. The resulting ratio distributions disagree with the expected Poissonian result. They are then compared with some well-known transition distributions between Poisson and the Gaussian orthogonal random matrix ensemble. It is found that the distribution of the ratio of consecutive level spacings agrees with the prediction of the Rosenzweig-Porter model. Also, the normal-mode vibration amplitudes are found experimentally on aluminum plates, before and after an avoided crossing for symmetrical-symmetrical, symmetrical-antisymmetrical, and antisymmetrical-symmetrical classes. The measured modes show an excellent agreement with our numerical predictions. The expected Poissonian distribution is recovered for the simply supported rectangular plate.

Onset of chaos in finite interacting Boson systems with F-spin

For m bosons, carrying spin (f = 1 2) degree of freedom, in Ω single particle levels which are doubly degenerate, one-plus two-body Embedded Gaussian Orthogonal random matrix Ensemble which conserves spin (F) [BEGOE(1 + 2)-F] is defined and spectral properties are examined. Using mean field defined by random single particle energies, for fixed-(m,F) BEGOE(1 + 2)-F, density of states being close to Gaussian, we analyze spectral fluctuation properties, like Nearest Neighbour Spacing Distribution (NNSD) and Dyson-Mehta (Δ3) statistic, as a function of λ- the two-body interaction strength. Numerical calculations are used to demonstrate that for very small value of λ, NNSD and Δ3 are found close to Poisson form and they move steadily to GOE form as λ increases. Moreover, spin dependence of the transition point λ C is obtained. It is also verified using periodogram analysis that BEGOE(1 + 2)-F spectra shows 1/f–noise behaviour.

Number of principal components and localization length inE2andM1transition strengths inV46

Predictions of the embedded Gaussian orthogonal random matrix ensemble of one plus two-body interactions, in the Gaussian domain, for the number of principal components and localization length, measuring complexity and chaos in transition strengths originating from an eigenstate with energy $E$, are tested successfully for the first time for $E2$ and $M1$ transition strengths. Shell-model results for the $2p1f$-shell nucleus $^{46}\mathrm{V}$ are used as the example.

Universal level-spacing statistics in quasiperiodic tight-binding models

We studied statistical properties of the energy spectra of two-dimensional quasiperiodic tight-binding models. The multifractal nature of the eigenstates of these models is corroborated by the scaling of the participation numbers with the system size. Hence one might have expected ‘critical’ or ‘intermediate’ statistics for the level-spacing distributions as observed at the metal-insulator transition in the three-dimensional Anderson model of disorder. However, our numerical results are in perfect agreement with the universal level-spacing distributions of the Gaussian orthogonal random matrix ensemble, including the distribution of spacings between second, third, and fourth neighbour energy levels.

Average-fluctuations separation in energy levels in dense interacting Boson systems

Using embedded Gaussian orthogonal random matrix ensemble of two-body interactions (EGOE(2)), it is demonstrated that finite interacting boson systems in the dense limit exhibit average-fluctuations separation with the smoothed state density being a Gaussian (with corrections) and the fluctuations are of GOE type.

Energy levels of quasiperiodic Hamiltonians, spectral unfolding, and random matrix theory

We consider a tight-binding Hamiltonian defined on the quasiperiodic Ammann–Beenker tiling. Although the density of states (DOS) is rather spiky, the integrated DOS (IDOS) is quite smooth and can be used to perform spectral unfolding. The effect of unfolding on the integrated level-spacing distribution is investigated for various parts of the spectrum which show different behaviour of the DOS. For energy intervals with approximately constant DOS, we find good agreement with the distribution of the Gaussian orthogonal random matrix ensemble (GOE) even without unfolding. For energy ranges with fluctuating DOS, we observe deviations from the GOE result. After unfolding, we always recover the GOE distribution.

Level fluctuations in quantum systems with multifractal eigenstates

The relationship between universality of level fluctuation laws and multifractality of eigenstates is studied by analyzing energy spectra of tight-binding models of quantum billiards with multifractal eigenstates. We show that level fluctuations in our models are well described by the universal statistical laws such as Poisson statistics for integrable systems and statistics for the Gaussian orthogonal random matrix ensemble as long as the energy levels are located in bandlike spectra, which indicates that level statistics is irrelative to the multifractal properties of eigenstates. Our results are further confirmed by statistical properties of energy spectra in tight-binding models of two-dimensional quasicrystals.

Application of random matrix theory to quasiperiodic systems

We study statistical properties of energy spectra of a tight-binding model on the two-dimensional quasiperiodic Ammann–Beenker tiling. Taking into account the symmetries of finite approximants, we find that the underlying universal level-spacing distribution is given by the Gaussian orthogonal random matrix ensemble, and thus differs from the critical level-spacing distribution observed at the metal–insulator transition in the three-dimensional Anderson model of disorder. Our data allow us to see the difference to the Wigner surmise.

Level-Spacing Distributions of Planar Quasiperiodic Tight-Binding Models

We study the statistical properties of energy spectra of two-dimensional quasiperiodic tight-binding models. We demonstrate that the nearest-neighbor level spacing distributions of these non-random systems are well described by random matrix theory. Properly taking into account the symmetries of models defined on various finite approximants of quasiperiodic tilings, we find that the underlying universal level-spacing distribution is given by the Wigner-Dyson distribution of the Gaussian orthogonal random matrix ensemble. Our data allow us to see the differences between the Wigner surmise and the exact level-spacing distribution. In particular, our result differs from the critical level-spacing distribution computed at the metal-insulator transition in the three-dimensional Anderson model of disorder.