Implementing global Abelian symmetries in projected entangled-pair state algorithms

Due to the unfavorable scaling of tensor-network methods with the refinement parameter $M$, new approaches are necessary to improve the efficiency of numerical simulations based on such states, in particular for gapless, strongly entangled systems. In one-dimensional density matrix renormalization group methods, the use of Abelian symmetries has led to large computational gain. In higher-dimensional tensor networks, this is associated with significant technical efforts and additional approximations. We explain a formalism to implement such symmetries in two-dimensional tensor-network states and present benchmark results that confirm the validity of these approximations in the context of projected entangled-pair state algorithms.

Figure 1

Pictorial representation of a projected entangled-pair state (PEPS). Left panel: For the square lattice, a tensor of rank 5 will be associated with each lattice site. The index pointing down connects to the physical system, while the other indices connect to neighboring tensors in the state. Right panel: The panel shows the PEPS decomposition of a coefficient $c(\varphi )$ for a state $|\Psi \rangle =\sum c({\varphi }_1\dots {\varphi }_9)|{\varphi }_1\dots {\varphi }_9\rangle$ on a $3\times 3$ square lattice with open boundary conditions.

Figure 2

Pictorial representation of the contraction of two rank-4 tensors with $U(1)$ symmetry. The steps are explained in detail in Sec. 3b.

Figure 3

End of a matrix-product state invariant under some symmetry group $\mathcal{G}$. $|{\varphi }_i\rangle$ denote physical states in the local Hilbert space ${\mathcal{H}}_{\text{loc}}$. By $\mathcal{C}$, we denote the set of charges associated with sectors in ${\mathcal{H}}_{\text{loc}}$, and by ${\mathcal{C}}^n$ the set of charges associated with sectors in $\underset{i=1}{\overset{n}{\otimes }}{\mathcal{H}}_{\text{loc}}$. The first auxiliary bond to the left simply carries the physical charges of the first site. For the second bond, all possible fusion outcomes of charges on the first auxiliary bond with the physical charges have to be considered. This can be continued up to the middle of the chain, such that each auxiliary bond carries the possible combinations of charges to the left. Joining such a state with its reflection will yield a finite symmetric MPS.

Figure 4

Corner of a symmetric PEPS state with an environment as in the directional corner transfer matrix method (Ref. 21). Here, the large blue circles denote tensors $T_i$ of the ansatz state and their conjugate $T_i^{*}$, the red squares denote tensors of the corner transfer matrix, and the small orange circles represent single-site operators acting on the physical index of the PEPS tensor. In the infinite case, in general there are three sets of charges involved: (i) the physical charges ${\mathcal{C}}_{\text{phys}}$ carried on the black, solid lines in the figure, (ii) the auxiliary charges ${\mathcal{C}}_{\text{aux}}$ on the blue, dashed bonds, and (iii) the charges carried on the bonds of the environment ${\mathcal{C}}_{\text{corner}}$. This reflects the three independent bond dimensions involved in a PEPS: the physical dimension $d$, the bond dimension $M$, and the environment dimension $\chi$. Usually, $M>d$ and $\chi ~M^2$. Therefore, ${\mathcal{C}}_{\text{phys}}\subset {\mathcal{C}}_{\text{aux}}\subset {\mathcal{C}}_{\text{corner}}$. In principle, all charges could depend on the location in the PEPS or the environment.

Figure 5

Top: Decrease in the relative error in the energy as $M$ is increased, for different choices of the symmetry group. Bottom: The same data versus the number of variational parameters in the state (note the logarithmic scale on both axes). Clearly, the fact that the relative errors are similar between symmetry groups for a given $M$ shows that reducing the number of parameters and the computation time by using larger (finite) symmetry groups does not lead to any loss in accuracy.

Figure 6

Relative error in the ground state energy of the two-dimensional (2D) Heisenberg model with a $U(1)$-symmetric PEPS, as a function of (i) the total bond dimension and (ii) the number of parameters. The choice of sectors and dimensions is shown in Table . We show results with ${\mathbb{Z}}_2$ symmetry for comparison.