  We introduce real-time density matrix embedding theory (DMET), a dynamical
quantum embedding theory for computing non-equilibrium electron dynamics in
strongly correlated systems. As in the previously developed static DMET,
real-time DMET partitions the system into an impurity corresponding to the
region of interest coupled to the surrounding environment, which is efficiently
represented by a quantum bath of the same size as the impurity. In this work,
we focus on a single-impurity time-dependent theory as a first step towards a
full multi-impurity theory. The equations of motion of the coupled impurity and
bath embedding problem in real-time DMET are then derived using the
time-dependent variational principle. The accuracy of real-time DMET is
compared to that of time-dependent complete active space self-consistent field
(TD-CASSCF) theory and time-dependent Hartree-Fock (TDHF) theory for a variety
of quantum quenches in the single impurity Anderson model (SIAM), in which the
Hamiltonian is suddenly changed (quenched) to induce a non-equilibrium state.
Real-time DMET shows a marked improvement over the meanfield TDHF, converging
to the exact answer even in the non-trivial Kondo regime of the SIAM. However,
as in static DMET, the increased dynamic flexibility of TD-CASSCF in comparison
to real-time DMET using a single impurity leads to faster convergence with
respect to active space size. Our results demonstrate that real-time DMET is an
efficient method well suited for the simulation of non-equilibrium electron
dynamics in which strong electron correlation plays an important role.
