Gradient optimization of finite projected entangled pair states

Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension $D$, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to $D=10$ without resorting to any symmetry. Benchmark tests of the method on the $J_1-J_2$ model give impressive accuracy compared with exact results.

Figure 1

Contraction method for a finite tensor network. (a) A double-layer tensor network composed of bra $\langle {\mathrm{\Psi }}_{\mathrm{PEPS}}|$ and ket $|{\mathrm{\Psi }}_{\mathrm{PEPS}}\rangle$ is needed to be contracted to obtain the physical quantities by summing over the physical index in the standard contraction scheme. (b) The boundary-MPO method is used to contract a double-layer tensor network with computational scaling $O\left(D^4{D}_{c2}^3+d{D}^6{D}_{c2}^2\right)$. (c) A single-layer tensor network $|{\mathrm{\Psi }}_{\mathrm{PEPS}}\rangle$ with some given spin configuration $|S\rangle$ is needed to be contracted to obtain the physical quantities in the MC scheme. (d) The boundary-MPS method is used to contract a single-layer tensor network with computational scaling $O\left(D^4{D}_{c1}^2\right)$.

Figure 2

GO for a Heisenberg model on a $10\times 10$ square lattice using PEPS with $D=8$. (a) In the first 50 GO steps, $\delta t$ is set to 0.005. In the next 50 steps, $\delta t$ decreases slowly from 0.005 to 0.001, and in the last 20 steps, $\delta t=0.001$. (b) The energy variation in the last 20 steps.

Figure 3

Benchmark tests on the total energies of the spin-$\frac{1}{2}$ Heisenberg antiferromagnetic model on square lattices as functions of PEPS bond dimension $D$. The energies are obtained by different optimization methods including the original update (OU), simple update (SU), full update (FU), and gradient optimization (GO) methods, on different lattice sizes and $J_2$ parameters: (a) a $4\times 4$ lattice with $J_2=0$, (b) a $10\times 10$ lattice with $J_2=0$, and (c) a $4\times 6$ lattice with $J_2=0.5$. The error bars are too small to show.

Figure 4

(a) The ground energies of PEPS $D=8$ and $D=10$, and (b) the staggered magnetization $M^2$ of the Heisenberg model, calculated by PEPS with $D=8$, on the $L=8$, 10, 12, 14, 16 square lattices. The staggered magnetization is calculated on the central $W\times W$ region, with $W=(L-2)$ and $(L-4)$, respectively, to reduce the boundary effect.

Figure 5

The convergence of the ground-state energies as functions of bond dimension cutoff $D_c$ at $D=8$, for a Heisenberg model on a $10\times 10$ lattice and a $J_1-J_2$ model with $J_2=0.5$ on a $4\times 6$ lattice. The MC sampling error is order of ${10}^{-6}$.