Efficient representation of long-range interactions in tensor network algorithms

We describe a practical and efficient approach to represent physically realistic long-range interactions in two-dimensional tensor network algorithms via projected entangled-pair operators (PEPOs). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. To obtain a smooth and radially isotropic interaction across all length scales, we map the physical lattice to an auxiliary lattice of expanded size. Our construction yields a long-range PEPO as a sum of ancillary PEPOs, each of small, constant bond dimension. This representation enables efficient numerical simulations with long-range interactions using projected entangled pair states.

Figure 1

(a) The construction of the nonzero parts of the CF-PEPO tensor ${\mathbf{W}}^{\left[k\right]}$ via the coupling of the finite state machine (FSM) tensor (red) with the Ising correlation function tensors (blue). Note that here the physical indices of ${\mathbf{W}}^{\left[k\right]}$ are explicitly shown, whereas they are suppressed in Eq. (3). (b) and (c) Two possible constructions of the long-range PEPO for a $3\times 3$ physical system with 1 fictitious Ising site (blue) in between adjacent physical sites (red) and a two-site buffer to help mitigate boundary effects in the encoding of the potential. Black bonds are $D_O^{\prime }=2$ and red bonds are $D_O=8$ (b) and 6 (c).

Figure 2

Convergence properties of Coulomb fitting. For all plots $r_{ij}=0$ is the central point on the lattice. (a) The upper envelope of $\left|V_{\mathrm{fit}}\left(r_{ij}\right)-1/r_{ij}\right|$ obtained with $N_t=12,r_{ij}=R_{ij}$, a least-squares weight function of $r_{ij}^{1.5}$, and Ising model lattices with different side lengths $L$. The fits were performed on a disk with radius equal to the maximum $r_{ij}$ displayed for a given curve. (b) and (c) The maximum fitting error $\left|{\tilde{V}}_{\mathrm{fit}}^{\left[N_f\right]}-1/r_{ij}\right|$ at selected values of $r_{ij}$ as functions of $N_t$ (b) and $N_f$ (c). In (b), the open circles correspond to $N_f=0$ and the closed circles to $N_f=10$. In (c), $N_t=12$. The fits in (b) and (c) were performed on disks of radius $r_{ij}=36$ with $L=199$ and a weight function of $r_{ij}^{1.5}$.

Figure 3

(a) Average accuracy of energy per site expectation values for $6\times 6$ FM and AFM trial PEPS with $D_S=1$. The solid triangular markers show FM states while the open circles show AFM states. ${\mathrm{\Psi }}_0$ is a true FM or AFM state, while the “$x$ flip” regions are ${\mathrm{\Psi }}_0$ perturbed by $x$ random spin flips. The average error is taken over five PEPS for each $x$ and each $N_f$. (b) The signed error $1/r_{ij}-{\tilde{V}}_{\mathrm{fit}}^{\left[0\right]}\left(r_{ij}\right)$, where $r_{ij}=0$ is the white square in the center, each adjacent square is $r_{ij}=1$, etc. For (a) and (b) the fitted potentials are obtained from Eq. (4) with $N_t=12$.

Figure 4

Top: The trajectories over the first 25 iterations of the energy optimization for the $4\times 4, D_S=2$ system using the PEPO and the explicit sum over all $O\left(A^2\right)$ terms in (5). The long tails of the trajectories are excluded for clarity. Bottom: Ground state energies per site $e_0$ for the Hamiltonian (5) with various system sizes and bond dimensions. The fifth column is the overlap of the normalized ground states obtained with the two different methods. In all cases $N_f=4$ and $N_t=12$. The “exact” rows are the results of converged DMRG calculations.

Figure 5

The four cases of rules needed to build the PEPO that encodes all the pairwise terms in ${\sum }_{i<j}{\widehat{A}}_i{\widehat{B}}_j$ for arbitrary operators $\widehat{A}$ and $\widehat{B}$. All virtual bonds are labeled with their index value, except those that are indexed 0 which are left unlabeled. The red path denotes the path of the signal from ${\widehat{A}}_i$ to ${\widehat{B}}_j$, which are signified by the two red tensors. Note that all the blue sites will be $\widehat{I}$ in these cases.

Figure 6

(a) The upper envelope of $\left|V_{\mathrm{fit}}\left(r_{ij}\right)-1/r_{ij}\right|$ for different least-squares weight functions $r_{ij}^{\alpha }$ with $N_t=12,L=199$, and $r_{ij}=R_{ij}$. (b) All the errors $\left|V_{\mathrm{fit}}\left(r_{ij}\right)-1/r_{ij}\right|$ at each $r_{ij}$ for the $N_t=12,\alpha =1.5, L=199,r_{ij}=R_{ij}$ fit. Note that most of the errors for a given $r_{ij}$ are significantly smaller than the upper envelope that was shown in Fig. 2. (c) The lattice discretized $V_{\mathrm{fit}}\left(r_{ij}\right)$ compared to the continuous Coulomb potential for the $N_t=12,\alpha =1.5,L=199,r_{ij}=R_{ij}$ fit. Note that at small values of $r_{ij}$ the values of $V_{\mathrm{fit}}$ visibly deviate from the exact solution, while as $r_{ij}$ grows the agreement gets significantly better.

Figure 7

Operations which occur during the evaluation of expectation values using the optimized contraction scheme. The top row shows contraction of the boundary MPS into the next row of the grid. The bottom row shows the corresponding object on which an SVD must be performed. (a) Operations on physical sites of the PEPS, and also on physical sites of the full 2D FSM CF-PEPO. (b) Operations on physical sites of the snake CF-PEPO, and also the operations on all fictitious or identity tensors which lie in the same row as the physical PEPO or PEPS tensors. (c) Operations on PEPO fictitious sites which do not lie in the row or column of any physical sites.