Tensor operators: Constructions and applications for long-range interaction systems

We consider the representation of operators in terms of tensor networks and their application to the ground-state approximation and time evolution of systems with long-range interactions. We provide an explicit construction to represent an arbitrary many-body Hamilton operator in terms of a one-dimensional tensor network (i.e., as a matrix product operator). For pairwise interactions, we show that such a representation is always efficient and requires a tensor dimension growing only linearly with the number of particles. For systems obeying certain symmetries or restrictions we find optimal representations with minimal tensor dimension. We discuss the analytic and numerical approximation of operators in terms of low-dimensional tensor operators. We demonstrate applications for time evolution and the ground-state approximation, in particular for long-range interaction with inhomogeneous couplings. The operator representations are also generalized to other geometries such as trees and two-dimensional lattices, where we show how to obtain and use efficient tensor network representations respecting a given geometry.

Figure 1

Matrix product operator representation. An operator $O$ acting on $N$ particles is decomposed into $N$ low-rank tensors $A^{[k]}$. Each tensor has two physical indices (input $i_k$, output $j_k$) and one or two virtual indices ${\alpha }_{k-1},{\alpha }_k$ which are summed over.

Figure 2

Sketch of embedding local term and nearest-neighbor interaction into a next-nearest-neighbor Hamiltonian. There exist three possible resulting operators compatible with the set of rules: $C_i$, $X_i{Y}_{i+1}$, and $X_i{Y}_{i+2}$.

Figure 3

The quality of the MPO approximation for the Hubbard model (23). We choose the following parameters: $N=100$ particles (i.e., bond dimension 52 for the exact MPO); $\beta =1$, $\Omega =0.1,$ and $\delta =0$. The estimated errors are ${10}^{-15}$ for the operator overlap, ${10}^{-10}$ for the ground-state energy, and ${10}^{-10}$ for the ground-state fidelity. The bond dimension of the ground state is equal to 80. Dashed lines correspond to (i) the approximation of the operator by a sum of exponentially decaying functions; solid lines correspond to (ii) the MPO obtained by numerical truncation. Relative errors for the Hamiltonian (blue), ground-state fidelity (green), and ground-state energy (red) as a function of the bond dimension of the approximating MPO are plotted.

Figure 4

Same situation as in Fig. 3, except that the positions of the single particles are shifted away from the regular lattice by adding a normally distributed number with variance 0.2 in units of the lattice distance.

Figure 5

Long-range Ising with random couplings and transverse magnetic field with $B=1$. There is no chance to truncate an MPO such that the ground-state properties are conserved. The expected errors are the same as for the first example. The system size $N$ is 30. The bond dimension of the ground state is 80.

Figure 6

Comparison of a Taylor expansion and a Trotter decomposition for the $\mathit{XXZ}$ model on a small system, $N=12$. The squared distance between the exact operator to the MPO are compared for different minimal time steps $\Delta t$ for both MPO-generating methods, Taylor and Trotter. In addition, the squared distance of an exactly evolved state to the evolved MPS is measured at $t=10$ in appropriate units. The demonstrated model is defined in Eq. (23), with the parameters $\Theta =0.35$ and $\Delta =0.1$. The order for the Taylor series and the number of time-doubling steps are adjusted for different $\Delta t$ (see [30] for a guideline). The MPS exhibits maximal bond dimension and starts with all spins up. The bond dimension of the MPO $D_{\mathrm{MPO}}$ is restricted to 30. The Trotter method was performed with the same parameters.

Figure 7

Two-point quantum correlation of Eq. (26). For a chain of 100 particles the averaged correlations of the sites 40 to 60 were calculated. This snapshot was taken at $t=20$. The parameters of the model were $\beta =10,\Omega =1,$ and $\delta =-12$. The unitary time evolution operator was generated by a seventh-order Taylor polynomial, subsequently five times doubled to obtain $\Delta t=0.025$; $D_{\mathrm{MPO}}=50$. We repeated the calculations for $D_{\mathrm{MPS}}=90,$ 110, and 130. The differences in the results for different $D_{\mathrm{MPS}}$ are negligible for our demonstration.

Figure 8

Example for the definition of a tree tensor network operator on the basis of a given definition for a tree tensor network state. Boxes indicate tensors and the red lines correspond to the new open indices, while joint indices are contracted.

Figure 9

The network of $\langle \psi |O|\varphi \rangle$, where $O$ is represented as (a) an MPO or as (b) a TNO. The red tensors belong to the operator, and the black tensors to the vectors. The blue contour envelops the area which is contracted in the first step. The contraction using a TNO is more efficient, since the resulting tensor is of a lower rank.

Figure 10

The definition of a PEPO. Without the red bars, the network represents a state; adding them, one obtains an operator. The contraction leads to the standard notation. In the insert the indices are marked as an illustration for Eq. (27).

Figure 11

A sketch on how the information of the relative positions of $X$ and $Y$ are carried to the tensor $C$ in an optimal way.

Figure 12

Sketch showing the combination of rules for a long-range interaction with $N=10$ for the instance that $Y$ is to the right of $X$. The gray circles show the rule number of the tables. The “trivial” rules are shaded and the “communication” lines are thicker (compare also with Fig. 11); $C\equiv c_{2312}\mathbb{1}$.

Figure 13

Detail of a possible configuration of a vertical interacting pair. The gray circles indicate the rule number used.