  The projection of time-dependent variational principle (TDVP) for matrix
product states enables us to perform long-time simulations of one-dimensional
quantum systems with the conservation of the total energy and the norm of wave
functions. We compare long-time dynamics after a quantum quench simulated by
TDVP with those by the exact diagonalization method in order to evaluate the
performance of TDVP. We show that in a nonintegrable model the projection of
TDVP clearly improves the long-time behaviors of global observables included in
the Hamiltonian, such as the kinetic and interaction energies. In contrast,
this projection can lead to larger error for other observables than that caused
by the truncation of states.

Strongly Correlated Electrons;Quantum Gases;Quantum Physics