Variational optimization with infinite projected entangled-pair states

We present a scheme to perform an iterative variational optimization with infinite projected entangled-pair states, a tensor network ansatz for a two-dimensional wave function in the thermodynamic limit, to compute the ground state of a local Hamiltonian. The method is based on a systematic summation of Hamiltonian contributions using the corner-transfer-matrix method. Benchmark results for challenging problems are presented, including the two-dimensional Heisenberg model, the Shastry-Sutherland model, and the $t\text{−}J$ model, which show that the variational scheme yields considerably more accurate results than the previously best imaginary-time evolution algorithm, with a similar computational cost and with a faster convergence towards the ground state.

Figure 1

(a) iPEPS ansatz with a $3\times 2$ unit cell of tensors which is periodically repeated in the lattice. (b) The reduced tensor $a^{[x,y]}$ is obtained from contracting an iPEPS tensor $A^{[x,y]}$ with its conjugate $A^{†[x,y]}$ along the physical leg. (c) The norm $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$ is represented as an infinite square-lattice network of reduced tensors. The CTM approach yields the environment tensors surrounding a bulk tensor $a^{[x,y]}$ where the corner tensors $C_1,C_2,C_3,C_4$ take into account a quarter-infinite system and the edge tensors $T_1,T_2,T_3,T_4$ take into account an infinite half row or half column of the system. (d) A left move is done by inserting a new column of tensors, multiplying the tensors to the left, and performing a renormalization step (this is done for all coordinates $y)$. (e) Diagrams to compute the updated corner and edge tensors $C_1^{\prime },C_4^{\prime },T_4^{\prime }$ at coordinate $x$ (for all $y$ coordinates). Note that the coordinates are always taken modulo the unit cell size.

Figure 2

(a) The H-shaped tensor is obtained by contracting the entire network representing $\langle \mathrm{\Psi }\left|\widehat{H}\right|\mathrm{\Psi }\rangle$ except tensors $A$ and $A^{†}$. (b) Similarly, the N-shaped tensor contains all tensors representing the norm $\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle$ except tensors $A$ and $A^{†}$ and can be obtained by contracting all environment tensors in Fig. 1 together. Minimizing the energy with respect to tensor $A$ boils down to solving the generalized eigenvalue problem shown in (c) by reshaping the H- and N-shaped tensors into matrices and $A$ into a vector.

Figure 3

Representation of the expectation value of the Hamiltonian, where the blue tensors contain sums of local Hamiltonian terms, as illustrated in the bottom part of the figure. For example, the corner tensor ${\tilde{C}}_1$ contains all contributions of local Hamiltonian terms in the upper left corner of the infinite system, whereas the edge tensor ${\tilde{T}}_4$ contains all contributions from an infinite half row, as depicted in the bottom part of the figure. The vertical corner tensor ${\tilde{C}}_{v1}$ takes into account all Hamiltonian terms located between the corner $C_1$ and the edge tensor $T_4$ (see bottom image; a similar definition holds for the horizontal corner tensors ${\tilde{C}}_{h1})$. All the other dark blue tensors on the other corners/edges are defined in a similar way. Finally, there are four remaining Hamiltonian terms (light blue) between the center site and its nearest neighbors. The cross on top of a tensor indicates that the Hamiltonian term is connected to the corresponding physical legs which are not shown in this top view.

Figure 4

All relevant diagrams to perform a left move to update the H-environment tensors in the CTM method. Coordinates of the tensors relative to the unit cell have been omitted for simplicity. The projectors to perform the renormalization step (yellow triangles) are the same as the ones computed for the norm. The tensor ${\widetilde{TT}}_4$ contains Hamiltonian terms connecting the sites between two edges $T_4$. The tensor $T_4^o$ is an edge where the physical legs of the right outermost bulk tensors are left open (such that a Hamiltonian term can be connected to it). Similar diagrams are defined for a right move, top move, and bottom move. In this way one keeps track of all nearest-neighbor Hamiltonian terms in a systematic way.

Figure 5

(a) Relative error of the energy per site of the 2D $S=1/2$ Heisenberg model as a function of the bond dimension $D$ obtained with three different optimization methods for iPEPS. (b) The order parameter (staggered magnetization) as a function of $D$, compared to the extrapolated QMC result in the thermodynamic limit (horizontal line). (c) Full update result as a function of the Trotter imaginary time step $\tau$ $\left(D=4\right)$ compared to the variational result. (d) Evolution of the energy as a function of simulation runtime on a MacBook Pro laptop for $D=4$ and $\chi =50$. The results from the variational optimization are shown at each sweep (squares), whereas the energy obtained with the full update is computed after every 30 time steps with $\tau =0.02$.

Figure 6

Evolution of the energy as a function of imaginary time (full update) or iteration step (variational update) for $J^{\prime }/J=0.7$ (in the plaquette phase) for $D=4$. Both simulations have been initialized from an initial antiferromagnetic state which was obtained using the simple update (for $J^{\prime }/J=0.78)$. The initial state is shown in the top left inset, where the lengths of the arrows are proportional to the magnitude of the local spin, and the thickness of the lines on the $J^{\prime }$ bonds scales with the magnitude of the energy on that bond (i.e., the thicker a line is, the lower the energy is). The variational update (squares) correctly reproduces a plaquette state, shown by the bottom right inset, where the bond energies around a plaquette are lower than on the other bonds, and the values of the local spins vanish. The full update (circles) fails to reproduce the state and yields an antiferromagnetic state with higher energy and slightly suppressed antiferromagnetic order compared to the initial state (top right inset).

Figure 7

Results for the Shastry-Sutherland model in a finite magnetic field at magnetization $m=1/8$, obtained in a $4\times 4$ unit cell of dimers. (a) Spin structure of a bound state of two triplet excitations exhibiting a ${90}^{\circ }$ rotational symmetry. (b) The rotational symmetry of a bound state gets strongly broken when using an inaccurate optimization such as the simple update (here for $D=5)$. (c) Comparison of the variational energies as a function of $D$ obtained with the three optimization methods. (d) Comparison of the symmetry error, quantifying the degree of the broken rotational symmetry.

Figure 8

Energy per hole in the $t\text{−}J$ model for $J/t=0.4$ and doping $\delta =0.12$. For each value of $D$ the energies are substantially improved by the variational optimization.

Figure 9

(a) In order to update the horizontal bond between coordinates $[x,y]$ and $[x+1,y]$, we first split the involved tensors into smaller pieces using an SVD (the singular values are absorbed in the $p$ and $q$ tensors). (b) Solving the generalized eigenvalue problem for tensor $r$ yields the solution $\tilde{r}$. (c) Ansatz for the solution $r^{\prime }$ made of a linear combination of $\tilde{r}$ and the initial tensor $r$, depending on a parameter $\lambda$. (d) The updated tensors $A^{\prime [x,y]}$ and $A^{\prime [x+1,y]}$ are obtained from recombining $p^{\prime }$ with $X$ and $q^{\prime }$ with $Y$, respectively, where $p^{\prime }$ and $q^{\prime }$ are determined using the full update.