Combination of tensor network states and Green's function Monte Carlo

We propose an approach to study the ground state of quantum many-body systems in which tensor network states, specifically projected entangled pair states (PEPSs), and Green's function Monte Carlo (GFMC) are combined. PEPSs, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short-range interactions but are hindered by the high computational complexity scaling with bond dimension $D$. GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed-node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of the negative sign problem is the introduction of systematic error by the guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPSs are very accurate variational wave functions, while at the same time, only contractions of the single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of the $J_1-J_2$ Heisenberg model on the square lattice are provided.

Figure 1

PEPSs and the optimization process. (a) A PEPS on a square lattice, where there is a local tensor on each vertex. (b) The local tensor. The wave function is obtained by contracting all the auxiliary indexes. (c) PEPO for a plaquette. (d) The application of a PEPO to PEPS in a plaquette. (e) The bond dimension of PEPS is increased after the application of a PEPO; (f) The bond dimension is truncated back to the original value in (a).

Figure 2

Energy versus step in the GFMC calculation for a $4\times 4,J_2=0$ system with OBCs. A PEPS with bond dimension $D=4$ is used as the guiding function. The energy at step 0 is the variation energy of the $D=4$ PEPS, which is $-9.034333$. There is no sign problem in this case, so GFMC should give numerically exact energy. The exact energy is $-9.189207$, as indicated by the blue dashed line, while the GFMC energy is $-9.189\left(1\right)$. The inset shows a zoom of the energy after equilibrium.

Figure 3

Same as Fig. 2, but for energy of a $6\times 6$ system with OBCs and $J_2=0.5$. The guiding wave function is a $D=4$ PEPS with variational energy $-16.763\left(4\right)$ (the energy at step 0). The GFMC energy is $-16.965\left(1\right)$, and exact energy (the blue dashed line) for this system is $-17.24733$.

Figure 4

Comparison of the GFMC energy and the energy of the corresponding PEPS guiding wave functions for a range of bond dimensions. The relative error of the energy is shown. The system size is $6\times 6$, and $J_2=0.5$.