X-symbols for non-Abelian symmetries in tensor networks

The full exploitation of non-Abelian symmetries in tensor network states (TNSs) derived from a given lattice Hamiltonian is attractive in various aspects. From a theoretical perspective, it can offer deep insights into the entanglement structure and quantum information content of strongly correlated quantum many-body states. From a practical perspective, it allows one to push numerical efficiency by orders of magnitude. Physical expectation values based on TNSs require the full contraction of a given tensor network, with the elementary ingredient being a pairwise contraction. While well established for no or just Abelian symmetries, this can become quickly extremely involved and cumbersome for general non-Abelian symmetries. As shown in this paper, however, the elementary step of a pairwise contraction of tensors of arbitrary rank can be tackled in a transparent and efficient manner by introducing so-called X-symbols. These deal with the pairwise contraction of the generalized underlying Clebsch-Gordan tensors (CGTs). They can be computed deterministically once and for all, and hence they can also be tabulated. Akin to $6j$-symbols, X-symbols are generally much smaller than their constituting CGTs. In applications, they solely affect the tensors of reduced matrix elements and therefore, once tabulated, allow one to completely sidestep the explicit usage of CGTs, and thus to greatly increase numerical efficiency.