Size consistency of tensor network methods for quantum many-body systems

Recently developed tensor network methods demonstrate great potential for addressing the quantum many-body problem, by constructing variational spaces with polynomially, instead of exponentially, scaled parameters. Constructing such an efficient tensor network, and thus the variational space, is a subtle problem and the main obstacle of the method. We demonstrate the necessity of size consistency in tensor network methods for their success in addressing the quantum many-body problem. We further demonstrate that size consistency is independent of the entanglement criterion, thus providing a general and tight constraint to construct the tensor network method. We propose a general and easy rule to construct a size-consistent tensor network.

Figure 1

$A$ and $B$ are two independent 1D Heisenberg spin chains, each containing $L$ sites. (a) A schematic MPS representation of the $A\oplus B$ supersystem. (b) Schematic MPS representations of the subsystems $A$ and $B$. Each filled circle represents a tensor $A_{i_k}^i$ and the line connecting them represents their virtual bonds $l$ and $r$.

Figure 2

$A$ and $B$ are two independent 2D Heisenberg spin lattice, each containing $L\times 2L$ sites. (a) A schematic MPS representation of the $A\oplus B$ supersystem. (b) Schematic MPS representations of the subsystems $A$ and $B$. (c) A schematic PEPS representation of the $A\oplus B$ supersystem. (d) Schematic PEPS repre-sentations of the subsystems $A$ and $B$.

Figure 3

The relative SCEs of the 2D Heisenberg model as functions of the Schmidt cutoff for different lattice sizes. The relative SCEs of MPS are represented by in open labels, whereas the relative SCEs of PEPS are represented by solid labels.

Figure 4

The two types of string bond states (SBSs) proposed in Ref. 27: (a) horizontal and vertical lines and (b) a long line covers all the sites.