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

TL;DR

This paper introduces a non-perturbative method to calculate moments of eigenvector overlaps between a deterministic Hamiltonian and a random perturbation, aiding understanding of quantum system dynamics.

Contribution

It develops a novel technique for computing mixed moments of eigenvector overlaps in large quantum systems with deterministic and random Hamiltonians.

Findings

Successfully recovers second and fourth order moments of overlaps

Provides insights into local quantum dynamics and system interactions

Validates predictions with numerical simulations

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.