Efficient perturbation theory to improve the density matrix renormalization group

The density matrix renormalization group (DMRG) is one of the most powerful numerical methods available for many-body systems. It has been applied to solve many physical problems, including the calculation of ground states and dynamical properties. In this work, we develop a perturbation theory of the DMRG (PT-DMRG) to greatly increase its accuracy in an extremely simple and efficient way. Using the canonical matrix product state (MPS) representation for the ground state of the considered system, a set of orthogonal basis functions $\left\{|{\psi }_i\rangle \right\}$ 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 cutoff are used to define these perturbation terms. The perturbed Hamiltonian is then defined as ${\tilde{H}}_{ij}=\langle {\psi }_i|\widehat{H}|{\psi }_j\rangle$; its ground state permits us to calculate physical observables with a considerably improved accuracy compared to the original DMRG results. We benchmark the second-order perturbation theory with the help of a one-dimensional Ising chain in a transverse field and the Heisenberg chain, where the precision of the DMRG is shown to be improved $O\left(10\right)$ times. Furthermore, for moderate $L$ the errors of the DMRG and PT-DMRG both scale linearly with $L^{-1}$ (with $L$ being the length of the chain). The linear relation between the dimension cutoff of the DMRG and that of the PT-DMRG at the same precision shows a considerable improvement in efficiency, especially for large dimension cutoffs. In the thermodynamic limit we show that the errors of the 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 light on the properties beyond the ground state. This second-order PT-DMRG can be readily generalized to higher orders, as well as applied to models in higher dimensions.

Figure 1

Graphical representation of the matrix product state (MPS).

Figure 2

Matrix product operator representation of $\widehat{H}$. At each site a four-rank tensor ${\widehat{W}}_{a_{l-1}{a}_l}^{{\sigma }_l{\sigma }_l^{\prime }}$ is defined.

Figure 3

Graphical representation of the overlap $\langle {\psi }_i|{\psi }_j\rangle$ represented in Eq. (12).

Figure 4

Error $\varepsilon$ of the 1D Ising model on a $32,44,64$ chain with open boundary conditions as a function of $h$. The error of the PT-DMRG method with bond dimension $\chi =4$ is more than $O\left(10\right)$ times smaller compared with the error of the standard DMRG.

Figure 5

Error $\varepsilon$ of the 1D Ising model as a function of the length in the quantum phase transition $h=1.0$, for different values of $\chi$: 2, 4, and 8. We show that the error for the PT-DMRG is much smaller. The PT-DMRG gives a systematic improvement in accuracy.

Figure 6

Error $\varepsilon$ of the 1D Ising model versus $\chi$ in the quantum phase transition $h=1.0$ for $L=64$. We show how the error of the PT-DMRG decreases more rapidly than the error of the standard DMRG.

Figure 7

The plot is the fit between ${\chi }^{\left(\mathrm{DMRG}\right)}$ and ${\chi }^{\left(\mathrm{PT}\right)}$ of the 1D Ising model in the quantum phase transition $h=1.0$ for $L=64$. We show how the PT-DMRG requires a smaller bond dimension $\chi$ than the DMRG.

Figure 8

Error $\varepsilon$ of the 1D Heisenberg model versus $\chi$ for $h=0.0$ and $L=64$. We show how the error of the PT-DMRG decrease more rapidly than the error of the standard DMRG.

Figure 9

The plot is the fit between ${\chi }^{\left(\mathrm{DMRG}\right)}$ and ${\chi }^{\left(\mathrm{PT}\right)}$ of the 1D Heisenberg model in the quantum phase transition $h=0.0$ for $L=64$. We show how the PT-DMRG requires a smaller bond dimension $\chi$ than the DMRG.

Figure 10

Error $\varepsilon$ of the 1D Ising model as a function of the length in the quantum phase transition $h=1.0$, for $\chi =8$. We show that the error of the PT-DMRG in the large-$L$ limit does not give a systematic improvement in accuracy.

Figure 11

Graphical representation of $\langle \psi |\psi \rangle$ through the two-rank tensors $T^A$ and $T^B$.

Figure 12

Graphical representation of the matrices $T^A$ and $T^B$ that contain the contraction.

Figure 13

Tensor network represented the quantity $\langle \psi \left|\widehat{H}\right|\psi \rangle$.

Figure 14

Graphical representation of the environment left $L$ and right $R$, where $L$ contains the contracted left part, while $R$ contains the contracted right part of the network.

Figure 15

Graphical representation of the effective Hamiltonian $H_{\mathrm{eff}}$ defined in Eq. (A21).