Some aspects of quantum chaos in many body interacting systems. Quantum spin chain and random matrices.
Yasar Atas, LPTMS
My thesis is devoted to the study of some aspects of many body quantum interacting systems. In particular we focus on quantum spin chains. I addressed especially questions related to the structure of eigenfunctions, the level densities and the spectral properties of spin chain Hamiltonians.
It is known that the level densities of most integrable models tend to the Gaussian in the thermodynamic limit. However, it appears that in certain limits of coupling of the spin chain to the magnetic field and for finite number of spins on the chain, one observes peaks in the level density. I show that the knowledge of the first two moments of the Hamiltonian in the degenerate subspace associated with each peak gives a good approximation to the level density both in the case of integrable and non integrable models.
Next, I study the statistical properties of the eigenvalues of spin chain Hamiltonians. One of the main achievements in the study of the spectral statistics of quantum complex systems concerns the universal behaviour of the fluctuation of measure such as the distribution of spacing between two consecutive eigenvalues. By following the Wigner surmise for the computation of the level spacing distribution, I obtained approximation for the distribution of the ratio of consecutive level spacings in the three canonical ensembles of random matrices. The prediction are compared with numerical results obtained by exact diagonalization of spin Hamiltonians and with zeros of the Riemann zeta function showing excellent agreement.
Finally, I investigate eigenfunction statistics of some canonical spin-chain Hamiltonians. Eigenfunctions together with the energy spectrum are the fundamental objects of quantum systems: their structure is quite complicated and not well understood. Due to the exponential growth of the size of the Hilbert space, the study of eigenfunctions is a very difficult task from both analytical and numerical points of view. I demonstrate that the groundstate eigenfunctions of all canonical models of spin chain are multifractal, by computing numerically the Rényi entropy and extrapolating it to obtain the multifractal dimensions.
Key words: Quantum spin chains, quantum Ising model, spectral statistics, level density, quantum chaos, random matrices, Wigner surmise, spacing distribution, multifractality, Rényi entropy.