Guide to Exact Diagonalization Study of Quantum Thermalization

Exact diagonalization is a powerful numerical method to study isolated quantum many-body systems. This paper provides a review of numerical algorithms to diagonalize the Hamiltonian matrix. Symmetry and the conservation law help us perform the numerical study efficiently. We explain the method to block-diagonalize the Hamiltonian matrix by using particle number conservation, translational symmetry, particle-hole symmetry, and spatial reflection symmetry in the context of the spin-1/2 XXZ model or the hard-core boson model in a one-dimensional lattice. We also explain the method to study the unitary time evolution governed by the Schr^¨odinger equation and to calculate thermodynamic quantities such as the entanglement entropy. As an application, we demonstrate numerical results that support that the eigenstate thermalization hypothesis (ETH) holds in the XXZ model.