Theory and Applications of Time Dependent Density Matrix Renormalization Group

Density matrix renormalization group (DMRG) is well known as a powerful method to study low-dimensional highly-correlated systems and has been widely used for electronic structure calculation in quantum chemistry. In recent years,the theory of time dependent DMRG (TD-DMRG) has developed rapidly and TD-DMRG has gradually emerged as one of the most important methods of quantum dynamics simulation for complex systems. This review briefly sketches the basic theory of DMRG in the language of matrix product states (MPS) and matrix product operators (MPO) , and discusses several most common TD-DMRG time evolution algorithms at length,including the propagation and compression (P&C) algorithm, algorithms based on time dependent variational principle (TDVP) , and the time step targeting (TST) algorithm. This review also introduces the purification algorithm and minimally entangled typical thermal states (METTS) algorithm that enable TD-DMRG to study the finite temperature effect and summarizes the recent applications of TD-DMRG in quantum dynamics of complex systems.