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

Drawing on similarities in Hartree-Fock theory and the theory of matrix product states (MPS), we explore extensions to time evolution, response theory, and Green functions. We derive analytic equations of motion for MPS from the least action principle, which describe optimal evolution in the small time-step limit. We further show how linearized equations of motion yield a MPS random phase approximation, from which one obtains response functions and excitations. Finally we analyze the structure of site-based Green functions associated with MPS, as well as the structure of correlations introduced via the fluctuation-dissipation theorem.