Chaos in the Bose-Hubbard model and Random Two-Body Hamiltonians

We investigate the chaotic phase of the Bose-Hubbard model [L. Pausch et al, Phys. Rev. Lett. 126, 150601 (2021)] in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose-Hubbard chaotic eigenvector features, quantified by the expectation value and eigenstate-to-eigenstate fluctuations of fractal dimensions. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime.

On ℓp-Gaussian–Grothendieck Problem

For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell _p$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $p=\infty $, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty .$ For the former, we compute the limit of the $\ell _p$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $n^{-1}$.

Onset of universality in the dynamical mixing of a pure state

We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described by random matrix theory (RMT) and undergoes a crossover from the Gaussian orthogonal ensemble to the Gaussian unitary ensemble (GUE) for short and large times, respectively. Using a semi-analytical treatment relying on a power series of the density matrix as a function of time, we find that the crossover occurs in a characteristic time that scales as the inverse of the dimension. The RMT results are contrasted with a paradigmatic model of many-body localization in the chaotic regime, where the GUE statistics is reached at large times, while for short times the statistics strongly depends on the peculiarity of the considered subspace.

High-dimensional regimes of non-stationary Gaussian correlated Wishart matrices

We study the high-dimensional asymptotic regimes of correlated Wishart matrices [Formula: see text], where [Formula: see text] is a [Formula: see text] Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both [Formula: see text] and [Formula: see text] grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt–Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in [Formula: see text]. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.

Random matrices and controllability of dynamical systems

We introduce the concept of $\epsilon $-uncontrollability for random linear systems, i.e. linear systems in which the usual matrices have been replaced by random matrices. We also estimate the $\varepsilon $-uncontrollability in the case where the matrices come from the Gaussian orthogonal ensemble. Our proof utilizes tools from systems theory, probability theory and convex geometry.

Spin Polarization Oscillations and Coherence Time in the Random Interaction Approach

We study the time evolution of the survival probability and the spin polarization of a dissipative nondegenerate two-level system in the presence of a magnetic field in the Faraday configuration. We apply the Extended Gaussian Orthogonal Ensemble approach to model the stochastic system-environment interaction and calculate the survival and spin polarization to first and second order of the interaction picture. We present also the time evolution of the thermal average of these quantities as functions of the temperature, the magnetic field, and the energy-levels density, for ρ(ϵ)∝ϵs, in the subohmic, ohmic, and superohmic dissipation forms. We show that the behavior of the spin polarization calculated here agrees rather well with the time evolution of spin polarization observed and calculated, recently, for the electron-nucleus dynamics of Ga centers in dilute (Ga,N)As semiconductors.

The crossing rates, exceedance probabilities, and related statistical properties of the energy frequency response functions of a random built-up system

A method is derived for computing a number of key statistical properties of the vibrational energy of each component of a built-up system that has random properties. The energy is considered to be a random function of frequency, and the derived statistical properties include: the mean rate at which the energy crosses a specified level, the probability that the energy will exceed a specified level within a given frequency band, the mean trough-to-peak height, the rate of occurrence of peaks, and the mean quefrency (a measure of the rate of fluctuation of the energy). The analysis is based on combining statistical energy analysis with a non-parametric model of uncertainty based on the Gaussian orthogonal ensemble and avoids the use of Monte Carlo simulations or large computational models. By way of example, the method is applied to a number of coupled plate systems.

Nuclear level density with proton resonance using Gaussian orthogonal ensemble theory

The Gaussian orthogonal ensemble (GOE) version of the random matrix theory (RMT) has been used to study the level density following up the proton interaction with 44Ca, 48Ti and 56Fe. A promising analysis method has been implemented based on the available data of the resonance spacing, where widths are associated with Porter Thomas distribution. The calculated level density for the compound nuclei 45Sc,49Vand 57Co shows a parity and spin dependence, where for Sc a discrepancy in level density distinguished from this analysis probably due to the spin misassignment .The present results show an acceptable agreement with the combinatorial method of level density.