Three-dimensional isometric tensor networks

Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor network, an operation needed to calculate quantum expectation values. Here we introduce a method for the time evolution of three-dimensional isometric tensor networks which respects the isometric structure and therefore renders contraction simple through a special canonical form. Our method involves a tetrahedral site splitting which allows one to move the orthogonality center of an embedded tree tensor network in a simple cubic lattice to any position. Using imaginary time evolution to find an isometric tensor network representation of the ground state of the three-dimensional transverse field Ising model across the entire phase diagram, we perform a systematic benchmark study of this method in comparison with exact Lanczos and quantum Monte Carlo results. We show that the obtained energy matches the exact ground-state result accurately deep in the ferromagnetic and polarized phases, while the regime close to the critical point requires larger bond dimensions. This behavior is in close analogy with the two-dimensional case, which we also discuss for comparison.

Figure 1

A generic 3D PEPS ansatz for the cubic lattice, where the tensors $T_{i\alpha \beta \gamma \delta \kappa \lambda }^{{\sigma }_i}$ are represented by spheres. The blue legs denote the physical degrees of freedom ${\sigma }_i$ and the gray legs denote the virtual degrees of freedom. The connections depict contractions between the virtual legs of neighboring tensors.

Figure 2

The isometry constraints are encoded by decorating the tensor legs with arrows (top row). Here panel (a) corresponds to Eq. (2) and panel (b) corresponds to Eq. (3), where in the bottom figures the tensor shown in the top panel is contracted with its conjugate to yield identities.

Figure 3

Multiple stages of the reduction of $\langle \text{isoTNS}|\text{isoTNS}\rangle$ to a contraction of just two tensors, illustrated for a 2D isoTNS on a $3\times 3$ square lattice. Panel (a) shows the amplitude before employing the isometry constraints, where the gray bonds have dimension $D$, the red bonds have dimension $\chi$, and the blue bonds have dimension $d$. Panel (b) shows the canonical MPS that is obtained after utilizing a large part of the isometry structure. Panel (c) shows the final pair of contracted tensors that remains after fully utilizing the isometry structure.

Figure 4

The final product of the isometry reductions when calculating the local two-body correlator $\langle \text{isoTNS}\left|O_1{O}_2\right|\text{isoTNS}\rangle$, leaving us with a MPS correlator that has to be fully computed.

Figure 5

The reduction of a particular 3D isoTNS configuration as it would occur upon contraction with its conjugate. In panel (a) we show the initial network. In panel (b) we show the canonical TTN that would remain after utilizing most of the isometry structure. In panel (c) we show how the canonical TTN further reduces to a single site, i.e., the orthogonality center.

Figure 6

The evolution of a slice in the 3D isoTNS, where for clarity we omit the transverse legs in most of the panels. In panel (a) we show the initial slice, which is the middle slice from Fig. 5. In panel (b) we apply a two-body gate at the orthogonality center, after which in panel (c) we shift the orthogonality center downwards using a SVD. In panel (d) we repeat this after which the orthogonality center is at the bottom of the first column. In panel (e) we perform the triangle splitting from Fig. 8 on the orthogonality center, which we repeat in panel (f). Now that we have reached the top of the column we perform a truncated SVD to finalize the column splitting, after which we have transferred the TTN strand to a temporary column that has only virtual legs, as shown in panel (g) where we restored the transverse legs to emphasize that the virtual column has no transverse legs. By absorbing the virtual column into the neighboring isometry column, yielding the configuration in panel (h), we have finally moved the TTN strand to the middle column. The increased vertical bond dimension $D\chi$ is truncated back to $\chi$ before evolving the new column. Repeating the column splitting and evolution once more we will have evolved all columns in the slice.

Figure 7

The reduced time evolution used in ${\mathrm{TEBD}}^3$. The bond depicted in panel (a) is to be evolved, to which end we first create a reduced bond by applying a QR decomposition at each tensor. This yields a pair of transient bonds with dimension $d\chi$ that are colored orange. Because these orange bonds are much smaller than the dark-red bonds, the gate application shown in panel (b) and the subsequent truncated SVD that yields panel (c) are much cheaper than when working in the gauge from panel (a). Having evolved the reduced bond we reabsorb the reduced tensors to get the original bond shown in panel (d), for which the orthogonality center has been shifted relative to panel (a).

Figure 8

The triangle splitting of the orthogonality center that is used multiple times in succession to shift a TTN strand to a neighboring column. By performing two truncated SVDs we split the orthogonality center shown in panel (a) into the triangle shown in panel (b), where the upper tensor is the new orthogonality center and where the two lower tensors are isometries. Note that the top and bottom-right tensors have only virtual legs. The amount of truncation during the SVDs is based on the usual color coding, and we have inserted a pair of disentanglers $U^{†}U=\mathbb{1}$, depicted as yellow cylinders, in order to minimize the entanglement across the red bond. After the splitting we absorb the top tensor upwards, as shown in panel (c). To now perform the splitting from panel (b) on the new orthogonality center we first temporarily combine the tilted legs as indicated by the arrows, so that the orthogonality center again looks as in panel (a).

Figure 9

The tetrahedron splitting used in the slice splitting that transfers the TTN strands to the next slice. In panel (a) we show the initial orthogonality center. In panel (b) we show the tetrahedron that results from three consecutive SVDs, where the pair of tripartite disentanglers is depicted as a yellow triangle, which reduces the tripartite entanglement $S{3}_{\alpha }\left(A\right|B\left|C\right)$. Note that only one of the four produced tensors has a physical leg. In panel (c) we have absorbed $A$ upwards and $B$ backwards. To repeat the tetrahedron splitting on the new orthogonality center we temporarily combine the tilted legs as indicated by the arrows.

Figure 10

The slice splitting by which we transfer the two TTN strands to the next slice, after evolving the vertical bonds of a slice. In panel (a) we show the initial isoTNS, which is obtained from Fig. 5 by evolving the middle slice as in Fig. 6. In panel (b) we show the first step of the slice splitting, where we employ the tetrahedron splitting from Fig. 9 on the orthogonality center, thereby moving it upwards by one site. After temporarily combining the tilted legs with the horizontal legs as indicated by the arrows in Fig. 9 we repeat the tetrahedron splitting to get panel (c). Finally we apply the triangle splitting from Fig. 8 to the uppermost tensor in panel (d). We then move the orthogonality center to the bottom of the column, resulting in panel (d). We repeat this combination of evolving and column splitting until we reach the final column, which is instead split via the procedure described in Figs. 6–6, yielding the split-off slice in panel (e). To finalize the transfer we absorb the transient slice into the neighboring slice, yielding panel (f). The increased bond dimensions are truncated before continuing the evolution.

Figure 11

A single iteration of ${\mathrm{TEBD}}^3$. Starting from the initial configuration in panel (a) we evolve all columns to get panel (b). Here the evolved gray bonds are colored green and the evolved red bonds are colored dark green. To evolve the next set of columns we rotate the network as indicated by the numbering of the corners, yielding panel (c), which becomes panel (d) after evolving the new columns. To evolve the final set of columns we rotate the network to panel (e), which turns into panel (f) upon evolving the columns. With all bonds evolved this concludes a single iteration of ${\mathrm{TEBD}}^3$.

Figure 12

Top: The energy density $E/N$ and its accuracy relative to the exact Lanczos value for the $3\times 3\times 3$ cubic TFIM with OBC at various field strengths $h$ across the critical point. Each curve belongs to a different set of bond dimensions $(D,\chi )$, and in the main plot we show the exact values with a red dotted line. Bottom: The $x$ magnetization $m_x$ and its error $\delta =m_x^e-m_x$ relative to the exact Lanczos value $m_x^e$ for various field strengths around the critical point, for the same TFIM system as in the top panel.

Figure 13

Top: The energy density $E/N$ and its accuracy relative to the exact QMC value for the $4\times 4\times 4$ cubic TFIM with OBC at various field strengths $h$ across the critical point. Each curve belongs to a different set of bond dimensions $(D,\chi )$, and in the main plot we show the exact values with a red dotted line. Bottom: The $x$ magnetization $m_x$ for various field strengths around the critical point, for the same TFIM system as in the top panel.

Figure 14

Top: The energy density $E/N$ of the (2,4) isoTNS and its accuracy relative to the exact QMC value for the $10\times 10\times 10$ cubic TFIM with OBC at various field strengths $h$ across the critical point. The exact values are shown as a red dotted line. Bottom: The $x$ magnetization $m_x$ for various field strengths around the critical point, for the same TFIM system as in the top panel.

Figure 15

Top: The energy density $E/N$ and its accuracy relative to the exact Lanczos value for the $5\times 5$ square TFIM with OBC at various field strengths $h$ across the critical point. Each curve belongs to a different set of bond dimensions $(D,\chi )$, and in the main plot we show the exact values with a red dotted line. Bottom: The $x$ magnetization $m_x$ and its error $\delta =m_x^e-m_x$ relative to the exact Lanczos value $m_x^e$ for various field strengths around the critical point, for the same TFIM system as in the top panel.

Figure 16

A comparison of the bipartite and tripartite disentanglers, showing the energy accuracy for the $4\times 4\times 4$ TFIM ground state at various $h$ and two $(D,\chi )$. We see that the improvement in using a tripartite over a bipartite disentangler is most significant in the critical region.

Figure 17

The $d\tau$ dependency of the relative error in energy density of the $3\times 3\times 3$ TFIM at $h=3.5$, for various isoTNS configurations $(D,\chi )$.