We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter $\mathrm{\ensuremath{\Lambda}}$. The behaviors apply equally well to few- and many-body systems, e.g., interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems, as long as their subsystems are quantum chaotic and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading-order behaviors depend on $\sqrt{\mathrm{\ensuremath{\Lambda}}}$. The marginal case of the $1/2$ moment, which is related to the distance to the closest maximally entangled state, is an exception having a $\sqrt{\mathrm{\ensuremath{\Lambda}}}ln\mathrm{\ensuremath{\Lambda}}$ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charv\'at-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e., the entanglement spectrum, show a transition from power laws and L\'evy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

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Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

We investigate the fluctuations of linear spectral statistics of a Wigner matrix $W_N$ deformed by a deterministic diagonal perturbation $D_N$, around a deterministic equivalent which can be expressed in terms of the free convolution between a semicircular distribution and the empirical spectral measure of $D_N$. We obtain Gaussian fluctuations for test functions in $\mathcal{C}_c^7(\R)$ ($\mathcal{C}_c^2(\R)$ for fluctuations around the mean). Furthermore, we provide as a tool a general method inspired from Shcherbina and Johansson to extend the convergence of the bias if there is a bound on the bias of the trace of the resolvent of a random matrix. Finally, we state and prove an asymptotic infinitesimal freeness result for independent GUE matrices together with a family of deterministic matrices, generalizing the main result from [Shl18].

Fluctuations of linear spectral statistics of deformed wigner matrices

La presente these est consacree a l’etude de l’effet d’une perturbation sur le spectre d’une matrice hermitienne perturbee par une matrice aleatoire de petite norme operateur et dont les entrees dans la base propre de la premiere matrice sont independantes, centrees et possedent un profil de variance. Ceci est realise au travers de developpements perturbatifs de divers types des lois spectrales des grandes matrices perturbees considerees. Dans un premier temps, nous demontrons differents developpements perturbatifs de la mesure spectrale empirique dans les cas du regime perturbatif et du regime semi-perturbatif et mettons en evidence des modeles heuristiques bien connus en physique, comme la transition entre les regimes semi-perturbatifs et perturbatifs. Dans un deuxieme temps, nous proposons une etude approfondie du regime semi-perturbatif et prouvons le fait nouveau que ce regime peut etre decompose en un nombre infini de sous-regimes. Enfin, nous demontrons, au travers d’un developpement perturbatif des mesures spectrales associees a un vecteur donne, un developpement perturbatif des coordonnees des vecteurs propres des matrices perturbees que nous considerons.

Eigenvalues and eigenvectors of large matrices under random perturbations

We study two spiked models of random matrices under general frameworks corresponding, respectively, to additive deformation of random symmetric matrices and multiplicative perturbation of random covariance matrices. In both cases, the limiting spectral measure in the direction of an eigenvector of the perturbation leads to old and new results on the coordinates of eigenvectors.

Spectral Measures of Spiked Random Matrices

We propose a general framework to study the stability of the subspace spanned by P consecutive eigenvectors of a generic symmetric matrix H0 when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (H0 is then the Hamiltonian) and financial risk control (in which case H0 is the assets' return covariance matrix). We argue that the problem can be formulated in terms of the singular values of an overlap matrix, which allows one to define an overlap distance. We specialize our results for the case of a Gaussian orthogonal H0, for which the full spectrum of singular values can be explicitly computed. We also consider the case when H0 is a covariance matrix and illustrate the usefulness of our results using financial data. The special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail. In particular, the dynamics of the angle made by the top eigenvector and its true direction defines an interesting class of random processes.

Eigenvector dynamics: General theory and some applications

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.

Empirical Spectral Distribution of a Matrix Under Perturbation

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile This is done through a perturbative expansion of spectral measures associated to the state defined by a given vector

Eigenvectors of a matrix under random perturbation

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with smal...

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are either related to the one-dimensional Gaussian free field or to free probability theory.

Alkéos Michaïl

Papers

Empirical Spectral Distribution of a Matrix Under Perturbation

Eigenvectors of a matrix under random perturbation

Eigenvectors of a matrix under random perturbation

Empirical spectral distribution of a matrix under perturbation