  Density matrix renormalization group (DMRG) is one of the most powerful
numerical methods available for many-body systems. In this work, we develop a
perturbation theory of DMRG (PT-DMRG) to largely increase its accuracy in an
extremely simple and efficient way. By using the canonical matrix product state
(MPS) representation for the ground state of the considered system, a set of
orthogonal basis functions $\left\lbrace | \psi_i \rangle \right\rbrace$ is
introduced to describe the perturbations to the ground state obtained by the
conventional DMRG. The Schmidt numbers of the MPS that are beyond the bond
dimension cut-off are used to define such perturbation terms. The perturbed
Hamiltonian is then defined as $\tilde{H}_{ij}= \langle \psi_i | \hat{H} |
\psi_j \rangle$; its ground state permits to calculate physical observables
with a considerably improved accuracy as compared to the original DMRG results.
We benchmark the second-order perturbation theory with the help of
one-dimensional Ising chain in a transverse field and the Heisenberg chain,
where the precision of DMRG is shown to be improved $\rm O(10)$ times.
Furthermore, for moderate length $L$ the errors of DMRG and PT-DMRG both scale
linearly with $L^{-1}$. The linear relation between the dimension cut-off of
DMRG and that of PT-DMRG with the same precision shows a considerable
improvement of efficiency, especially for large dimension cut-off's. In
thermodynamic limit we show that the errors of PT-DMRG scale with
$\sqrt{L^{-1}}$. Our work suggests an effective way to define the tangent space
of the ground state MPS, which may shed lights on the properties beyond the
ground state. Such second-order PT-DMRG can be readily generalized to higher
orders, as well as applied to the models in higher dimensions.
