Analytic Time Evolution, Random Phase Approximation, and Green Functions for Matrix Product States

This chapter summarizes the Hartree–Fock (HF) and Matrix product states (MPS) approaches to stationary states to establish notation and illustrate the parallel structure of the theories. It derives analytic equations of motion for MPS time evolution using the Dirac–Frenkel variational principle. The chapter shows that the resulting evolution is optimal for MPS of fixed auxiliary dimension. It discusses the relationship of this approach to time evolution to schemes currently in use. The chapter explains how excitation energies and dynamical properties can be obtained from a linear eigenvalue problem. The relationship of this MPS random phase approximation (RPA) to other dynamical approaches for matrix product states is discussed. Finally, the chapter explores the site‐based Green functions that emerge naturally within the theory of MPS and use the fluctuation‐dissipation theory to analyze the stationary‐state correlations introduced at the level of the MPS RPA.