Isometric Tensor Network States in Two Dimensions

Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS—which we demonstrate for the 2D transverse field Ising model.

Figure 1

Schematic representation of the canonical form in 1D and 2D. (a) Left and right isometries are represented by arrows whose orientation indicates whether $A^{†}A=B{B}^{†}=\mathbb{1}$. We view the isometry as an RG-like procedure from the large Hilbert space (incoming arrows) to the smaller one (outgoing arrows). In the case of higher-rank tensors, the contraction $A^{†}A=\mathbb{1}$ is always over all the incoming arrows. (b) A 1D MPS can be brought into a mixed canonical form with orthogonality center $\mathrm{\Lambda }$. Note that each dangling physical index implicitly has an incoming arrow. (c) Expectation values of local operators can be directly obtained from $\mathrm{\Lambda }$. (d) 2D canonical form with “orthogonality hypersurfaces” $\mathrm{\Lambda }$ (column and row highlighted in red). The orthogonality center $\lambda$ is marked by a red dot. In blue we indicate an example of a subregion with only outgoing arrows, whose boundary map is consequently an isometry.

Figure 2

The Moses move. (a) The orthogonality hypersurface ${\mathrm{\Lambda }}^{\ell }$ is split into the product of a left isometry $A^{\ell }$ and a zero-column state $\mathrm{\Lambda }$ with no physical indices. The unzipping is performed by successively applying the splitting procedure shown in panel (b). The legs of the center site $\lambda$ are grouped into a tripartite state $|ABC⟩$ which is “split” into three tensors in two steps: first find $|ABC⟩\approx a_0{U}^{†}|A{B}_L{B}_{R}C⟩$ for an initial guess of the isometry $a_0$ and unitary $U$ which minimizes the entanglement across the vertical bond highlighted in red; second set $a=a_0{U}^{†}$ and split $|A{B}_L{B}_{R}C⟩$ in two via SVD. The resulting $a$ comprise the tensors in $A^{\ell }$, and the choice of $U$ will produce a $\mathrm{\Lambda }$ with minimal vertical entanglement.

Figure 3

The MPS to isoTNS algorithm: (a) The MPS ${\mathrm{\Lambda }}^{1:L_x}$ for an $L_x\times L_y$ strip is fed into the MM by treating the legs of the first column as the left ancilla and the remaining columns as the right ancilla to obtain ${\mathrm{\Lambda }}^{1:L_x}=A^1{\mathrm{\Lambda }}^{2:L_x}$. The renormalized wave function ${\mathrm{\Lambda }}^{2:L_x}$ is then reshaped by viewing the legs of the second column as physical, and its vertical arrows are reversed downwards using the standard 1D MPS canonicalization algorithm. Applying the MM again, we can repeat to obtain a canonical TNS. (b) Entanglement entropy $S_{\ell }$ for the sequence of orthogonality hypersurfaces (highlighted in red) after $\ell$ iterations. $y$ runs from bottom right, to top right, to top left. (c) $S_{\ell }$ for a cut at $y\sim L_y/2$, compared against the bulk area law determined from DMRG.

Figure 4

The ${\mathrm{TEBD}}^2$ algorithm: (a) Trotterization of $e^{-\tau H_{r/c}}$ into a product of two-site terms acting on a single row or column of the isoTNS. (b) To complete one time step, the 1D update is applied to all rows or columns by first applying the TEBD sweep and then sequentially shifting the orthogonality center ${\lambda }^{c,r}$ using the MM. Note that the update of one row or column reverses the arrows twice, e.g., the TEBD sweep moves ${\lambda }^{c,r}$ from the top to the bottom and then MM moves it up again. (c) Error densities of the energy of the transverse field Ising model with $g=3.5$ for different system sizes and maximal bond dimensions $\chi$ as function of the Trotter step size $d\tau$.