Two-dimensional isometric tensor networks on an infinite strip

The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an “orthogonality column” of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (${\mathrm{iTEBD}}^2$) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.

Figure 1

Splitting problem considered for iMM method benchmarking results in Table 1. This two-sided iMPS of $\chi =128$ and ${\chi }_{\ell }={\chi }_r=4$ represents the ground state of 2D TFI model on $L_x=4$ infinite strip, as described in Sec. 4a.

Figure 2

Projection error (red), ${\varepsilon }_p$, and truncation error (blue), ${\varepsilon }_t$, as a function of ${\chi }_v$, as calculated by Eq. (20), using the combination of iMM-Local and iMM-Polar. (Inset) Splitting problem considered. The two-sided iMPS is the ground state of the $L_x=4$ 2D TFI from iDMRG, similar to Fig. 1. But three physical indices are grouped to the left and one to the right.

Figure 3

Error between original isoTNS $|{\mathrm{\Psi }}^{\left(0\right)}\rangle$ and isoTNS $|{\mathrm{\Psi }}^{\left(n\right)}\rangle$ produced by repeated $n$ iMM sweeps. $|{\mathrm{\Psi }}^{\left(0\right)}\rangle$ is an isoTNS with $L_x=6, {\chi }_{\text{init}}=2, D_b=1$ produced by peeling of iDMRG GS of the 2D TFI model, as described in Sec. 5.

Figure 4

Peeling procedure to convert iMPS to isoTNS. First the iMPS unit cell is collapsed to form an iMPS with $L_x$ physical legs per site. Then iMM is applied $L_x-1$ times to strip off columns. Example shown is for $L_x=3$.

Figure 5

Entropy of $|{\mathrm{\Phi }}_{\ell }\rangle$ as a function of number of columns $\ell$ removed by peeling the ground state of the 2D TFI Hamiltonian with width $L_x=8$ and $g=3.50$. After an initial delay, iMM removes an amount of entanglement consistent with the area law.

Figure 6

bMPO energy evaluation. (a) boundary MPO representing fixed point of isoTNS transfer matrix. $L_x\times \infty$ strip is oriented so that infinite direction is vertical. Horizontal arrows are omitted for clarity. Fixed point only needs to be found against the direction of vertical arrows, as fixed point in the direction of arrows is the identity. (b) Evaluation of $\langle h_{1,2}$ using bMPO fixed point and trivial fixed point. The energy of a row is found by ${\sum }_c\langle h_{c,c+1}\rangle$.

Figure 7

Energy of the $L_x=8, \chi =4$ isoTNS produced by peeling evaluated with the iMM and bMPO methods. We compare the energies to the exact energy evaluated by exact iMPS contraction. The energy from the bMPO method is close to exact for large $D_{\mathrm{bMPO}}$ but is very costly as $D_{\mathrm{bMPO}}$ grows due to complexity of finding the fixed point. The accuracy of iMM energy increases with both ${\eta }^{\prime }$ and ${\chi }^{\prime }$ used in iMM. In the figure, we use $f\equiv {\eta }^{\prime }/{\chi }^{\prime }$.

Figure 8

${\mathrm{iTEBD}}^2$ Algorithm. (a) Subroutine acting on two columns, the left of which is the orthogonality column. First the two columns are fused. 1D iTEBD is applied to the physical legs (colored red) of the double column using gates $e^{-idt{h}_{c,c+1}}$, acting on a plaquette of four physical sites; note that the gate does not act on any virtual legs. iMM is then applied to split the doubled column and move the orthogonality center to the right. (b) The ${\mathrm{iTEBD}}^2$ algorithm involves $L_x-1$ iterations of the subroutines described in (a) to implement a total Trotterized time evolution step of $dt$. Subsequent ${\mathrm{iTEBD}}^2$ sweeps alternate sweep directions, left-to-right and vice-versa.

Figure 9

Ground-state energies achieved with ${\mathrm{iTEBD}}^2$ for paramagnetic 2D TFI with $g=3.50$ on an $L_x=4,8,$ and 20 strip. We compare the isoTNS energies against essentially exact energies from quantum Monte Carlo (QMC) extrapolated from strips of finite length. As a comparison, the dashed line is the result of an iDMRG calculation with bond dimension $\chi =512$. For $L_x=4$, the iDMRG result is below the bottom axis of the plot.

Figure 10

Area law properties of peeled isoTNS. (a) $S\left(|{\mathrm{\Phi }}_{\ell }\rangle \right)$ is the half-chain entropy of $|{\mathrm{\Phi }}_{\ell }\rangle$, including contributions from left virtual legs. $S\left(|{\mathrm{\Phi }}_{\ell }^{\prime }\rangle \right)$ is a physical entropy between physical legs above and below the bipartition, found by projecting out the left virtual legs via iMM. (b) Entropy of $|{\mathrm{\Phi }}_{\ell }\rangle$ as a function of number of columns $\ell$ removed by peeling the ground state of the 2D TFI Hamiltonian with width $L_x=8$ and $g=3.50$. After an initial delay, iMM removes an amount of entanglement consistent with the area law. Physical entropy $S\left(|{\mathrm{\Phi }}_{\ell }^{\prime }\rangle \right)$ and entropy from iDMRG GS of different width strips agree and obey area law.

Figure 11

Entropy of $|{\mathrm{\Phi }}_{\ell }\rangle$ as a function of number of columns $\ell$ removed by peeling the ground state of the 2D TFI Hamiltonian with width $L_x=8$ and $g=3.04438$. After an initial delay, explained by $S\left(|{\mathrm{\Phi }}_{\ell }\rangle \right)$ not being a physical entropy, iMM removes an amount of entanglement consistent with the area law. Physical entropy $S\left(|{\mathrm{\Phi }}_{\ell }^{\prime }\rangle \right)$ [as defined in Fig. 10] and entropy from iDMRG GS of different width strips agree and obey area law.

Figure 12

Ground-state energies achieved with ${\mathrm{iTEBD}}^2$ for critical $g=3.04438$ 2D TFI on $L_x=4,8,20$ strip. An intermediate $d\tau$ and $f$ yield the best energy, while $\chi =8$ outperforms $\chi =4$. We compare the isoTNS energies against essentially exact energies from quantum Monte Carlo (QMC) extrapolated from strips of finite length. As a comparison, the dashed line is the result of an iDMRG calculation with bond dimension $\chi =512$. For $L_x=4,8$, the iDMRG result is below the bottom axis of the plot.