Tensor network decompositions in the presence of a global symmetry

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group $\mathcal{G}$, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group $\mathcal{G}$. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research.

Figure 1

(i) Four-site state $\Psi$ expressed in terms of a tensor network made of three tensors connected according to a directed graph. (ii) Invariance of tensor $T$ in Eq. (7). (iii) Invariance of a tensor network of symmetric tensors, Eq. (3).

Figure 2

Decomposition of tensors with one to four indices. The sums in (iv) run over the intermediate indices $(e,{\epsilon }_e,q_e)$ and $(f,{\zeta }_f,r_f)$ in Eqs. (11) and (12).

Figure 3

A TN for a symmetric state $|\Psi \rangle \in {\mathbb{V}}^{\otimes N}$ of lattice $\mathcal{L}$ (Fig. 1) is expressed as a linear superposition of spin networks. The sum runs over the intermediate indices that carry charges $e$ and $f$ (shown explicitly) as well as all indices shared by two tensors.

Figure 4

Product of two symmetric tensors. Only the intermediate charges $d$, $e,$ and $f$ are explicitly shown. Additional sums apply to all indices shared by two tensors. The computation involves evaluating spin networks.