Matrix product states with backflow correlations

By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging $S=1/2$ two-dimensional (2D) $J_1\text{−}{J}_2$ model, demonstrating that it is competitive with the state-of-the-art methods in 2D.

Figure 1

Graphical representations of MPS and MPBS applied to (a) 1D and (b) 2D systems. The $F$ tensors encode correlations between different lattice sites. The pictures represent only one of the terms in which $F$ is involved (see Supplemental Material [36]).

Figure 2

MPBS of bond dimension $\chi =5$ tested with the modified 1D HS model: energy density convergence (left) and ${(c_{xx}+c_{yy})}_{\text{c}}$ connected correlator (right). The system size is $N=70$.

Figure 3

MPBS of bond dimension $\chi =7$ tested with the 2D Ising model ($N_x=11, N_y=11$): energy density convergence (left) and $c_{zz}$ correlator (right). The PEPS result is taken from Ref. [45].

Figure 4

MPBS of bond dimension $\chi =12$ tested with the $J_1\text{−}{J}_2$ model on a square lattice ($N_x=8, N_y=8$). PEPS and EPS results are taken from Ref. [28]. In the right plot, solid (open) markers represent DMRG results for $c_{\text{ver}}\left(r\right) \left[c_{\text{hor}}\left(r\right)\right]$, and red points are MPBS results.