Renormalization group contraction of tensor networks in three dimensions

We present a new strategy for contracting tensor networks in arbitrary geometries. This method is designed to follow as strictly as possible the renormalization group philosophy, by first contracting tensors in an exact way and, then, performing a controlled truncation of the resulting tensor. We benchmark this approximation procedure in two dimensions against an exact contraction. We then apply the same idea to a three-dimensional quantum system. The underlying rational for emphasizing the exact coarse graining renormalization group step prior to truncation is related to monogamy of entanglement.

Figure 1

Graphical representation of a single RG step on the square lattice. Each circle represents a tensor $E^{(m,n)}={\sum }_i{A}^{*(m,n)i}\otimes A^{(m,n)i}$, and the connecting lines are the bond indices between neighboring tensors. We form a plaquette joining four tensors $E^{(m,n)}$ in adjacent positions (left) to form tensor $P^{(m,n)}$. An RG operation is performed over $P^{(m,n)}$ to reduce its bond dimension to ${\chi }^2$ (right).

Figure 2

Decomposition of the plaquette $P$: Along each side we perform a SVD, $P_{{\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4}=U_{{\alpha }_1\mu }{\lambda }^{\mu }{V}_{\mu {\alpha }_2{\alpha }_3{\alpha }_4}$, to obtain the four matrices $W_{\alpha \mu }=U_{\alpha \mu }\sqrt{{\lambda }^{\mu }}$. $U$ and $V$ are unitary matrices and ${\lambda }^{\mu }$ are the singular values. Indices ${\alpha }_i$ span through each direction. After four successive decompositions, we construct the new tensor decomposition $\tilde{P}=W_1^{-1}{W}_2^{-1}{W}_3^{-1}{W}_4^{-1}P$.

Figure 3

Detailed realization of the approximation scheme to reduce the rank of a plaquette as sketched in Fig. 1. We perform a truncation of the matrices $W$ to obtain the set of isometries $I_S$ of rank $\chi$. This truncation is based on the SVD of $W_S$ and $W_S^{\prime }$ matrices sitting on neighboring plaquettes (see text). The resulting isometries are applied to $\tilde{P}$ to construct the renormalized plaquette tensor $\tilde{E}=I_1{I}_2{I}_3{I}_4\tilde{P}$.

Figure 4

Graphical representation of the identity $P{P}^{†}=U{D}^2{U}^{†}$, where $P=U\lambda V$. This operation is used to compute $W_S$ by skipping a complete computation of the singular value decomposition. By carefully selecting the order of the contracting operations, this operation can be performed efficiently to avoid an uncontrolled growth of open indices.

Figure 5

We test our method in a finite square lattice, with a PEPS representing the ground state of a quantum Ising model with transverse field $H=-\sum {\sigma }_i^z{\sigma }_j^z+h\sum {\sigma }_i^x$. The tensor inside the shaded regions are contracted exactly, and the renormalization procedure is performed between these contracted plaquettes. For larger lattices this step is iterated. The final renormalized lattice, of size $3\times 3$, is exactly contracted.

Figure 6

For an Ising model with transverse field in a finite 2D square lattice of size $6\times 6$, we plot the error of the magnetization between the exact contraction and our renormalization group contraction of the same PEPS, that is, $m_z(\chi ,\mathrm{exact})-m_z(\chi ,\mathrm{RG})$, for different values of $\chi$. (Inset) Same for a $12\times 12$ lattice with $\chi =2$.

Figure 7

(Left panel) Magnetization $m_z$ vs local field $h$ for the Ising model with transverse field in an infinite 2D square lattice. The critical point is found at $h_c\approx 3.2$ and the critical exponent of the magnetization $m_z\sim {|h-h_c|}^{\beta }$ is $\beta =0.38$. These calculations are run for $\chi =4$. (Right panel) Same as the left panel for the magnetization $m_x$.

Figure 8

For a 3D square lattice we form a plaquette by the contraction of the common indices of eight tensors, forming a cubic plaquette to be renormalized. After an efficient SVD, we isolate a set of matrices $W_S$ which are transformed (see text) onto a set $I_S$ of isometries of rank $\chi$. Each of these six matrices $I_S$ is applied to reduce the effective bond dimension of the plaquette. We depict here the substitution $W_S\to I_S$ for a single direction.

Figure 9

Magnetization $m_z$ vs local field $h$ for the Ising model with transverse field in an infinite 3D square lattice. The critical point is found at $h_c\approx 5.29$. The rank of the tensor network is $\chi =2$. (Inset) Computation of the critical exponent $\beta$.