Plaquette renormalization scheme for tensor network states

We present a method for contracting a square-lattice tensor network in two dimensions based on auxiliary tensors accomplishing successive truncations (renormalization) of eight-index tensors for $2\times 2$ plaquettes into four-index tensors. Since all approximations are done on the wave function (which also can be interpreted in terms of different kinds of tensor networks), the scheme is variational, and thus, the tensors can be optimized by minimizing the energy. Test results for the quantum phase transition of the transverse-field Ising model confirm that even the smallest possible tensors (two values for each tensor index at each renormalization level) produce much better results than the simple product (mean-field) state.

Figure 1

Tensor networks on the 2D square lattice. (a) For each site $s$ there is a four-index tensor $T_{\mathit{ijkl}}^s({\sigma }_s)$, where ${\sigma }_s$ is the physical index (here a spin ${\sigma }_s=\pm 1$) and $i,j,k,l\in \{1,\dots ,m\}$. The wave function $\Psi ({\sigma }_1,\dots ,{\sigma }_N)$ is the $N$-tensor product (contraction), defined as a summation over all shared indices on the lattice bonds. (b) The norm $\langle \Psi |\Psi \rangle$ is obtained by also contracting over the physical indices.

Figure 2

(a) Truncation (renormalization) of an eight-index plaquette tensor into an effective four-index tensor. The indices $a,\dots ,h\in \{1,\dots ,m\}$ and $i,j,k,l\in \{1,\dots ,m_{\mathrm{cut}}\}$. The diagonal lines indicate physical indices, which, for an $S=1/2$ spin system, can take two values before the renormalization and $16$ values after (and are further multiplied by 16 after each successive renormalization). In (b) the physical indices are traced out first, leading to double tensors. To stay with the same level of truncation as in (a), the indices after the renormalization should then take values $i,j,k,l\in \{1,\dots ,m_{\mathrm{cut}}^2\}$.

Figure 3

Renormalization of an eight-index plaquette tensor using auxiliary three-index tensors. The solid circles denote either the original tensors $T=T^0$, which depend on the physical spins (which are not shown here), or tensors $T^n$ arising after $n$ renormalization steps have been carried out (and depend on the $2^{2n}$ spins within the block). The squares denote three-index tensors $S^n$ by which the external indices are decimated by contracting common $T^{n-1}$ and $S^n$ indices. The remaining four free indices take values $1,\dots ,m_{\mathrm{cut}}$.

Figure 4

The effective reduced tensor network for the wave function on an $8\times 8$ lattice. Here there are two sets of $S$ tensors, $S^1$ and $S^2$, denoted by the smaller and larger squares, respectively. The solid circles correspond to the original, spin-dependent tensor $T^0$. At the lowest level (the black circles), there is also a spin index of the tensors, which is not indicated here.

Figure 5

Steps in the contraction of the renormalized plaquette tensor of Fig. 3. Partial contractions are constructed from left to right. The summations carried out are indicated by $\Sigma$, and the scaling of each step with $m$ is shown beneath each diagram. Each summation and free index contributes a factor $m$.

Figure 6

(top) Spontaneous magnetization $m_z$ and (bottom) field-induced polarization $m_x$ vs the transverse field for different $L\times L$ lattices. The inset in the top panel shows the behavior close to the critical point in greater detail; the solid curves are power-law fits, as discussed in the text and shown on a different scale in Fig. 7.

Figure 7

Critical scaling of the magnetization (the order parameter). The behavior for the larger system sizes is consistent with mean-field behavior (exponent $\beta =1/2$), as shown with the straight lines.

Figure 8

Spin-spin correlation versus separation $r$ at fields $h$ close to $h_c$ such that the long-distance correlations are approximately the same for system sizes $L=16,32$, and $64$.

Figure 9

(top) Ground-state energy per spin obtained with the tensor network ansatz for an $L=32$ lattice compared with unbiased quantum Monte Carlo calculations for the same systems. (bottom) The relative energy error in the the tensor network calculation [i.e., the deviation from the quantum Monte Carlo (QMC) results] for system sizes $L=8,16$, and $32$.