Statistical diagonalization of a random biased Hamiltonian: the case of the eigenvectors

Abstract

We present a non perturbative calculation technique providing the mixed moments of the overlaps between the eigenvectors of two large quantum Hamiltonians: ${\hat{H}}_0$ and ${\hat{H}}_0+\hat{W}$, where ${\hat{H}}_0$ is deterministic and $\hat{W}$ is random. We apply this method to recover the second order moments or Local Density Of States in the case of an arbitrary fixed ${\hat{H}}_0$ and a Gaussian $\hat{W}$. Then we calculate the fourth order moments of the overlaps in the same setting. Such quantities are crucial for understanding the local dynamics of a large composite quantum system. In this case, ${\hat{H}}_0$ is the sum of the Hamiltonians of the system subparts and $\hat{W}$ is an interaction term. We test our predictions with numerical simulations.

arXiv comment15 pages, 7 figures

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• Typical relaxation of perturbed quantum many-body systems
• Taking snapshots of a quantum thermalization process: emergent classicality in quantum jump trajectories
• Many Body Density of States of a system of non interacting spinless fermions