Non-abelian symmetries in tensor networks: A quantum symmetry space approach

A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of well-defined orthonormal local as well as effective basis sets. The two crucial ingredients, the Clebsch–Gordan algebra for multiplet spaces as well as the Wigner–Eckart theorem for operators, are accounted for in a natural, well-organized, and computationally straightforward way. The unifying tensor-representation for quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with standard renormalization group algorithms such as the numerical renormalization group (NRG), the density matrix renormalization group (DMRG), or also more general tensor networks such as the multi-scale entanglement renormalization ansatz (MERA). In this paper, the focus is on the application of the non-abelian framework within the NRG. A detailed analysis is presented for a fully screened spin- 3/2 three-channel Anderson impurity model in the presence of conservation of total spin, particle–hole symmetry, and SU(3) channel symmetry. The same system is analyzed using several alternative symmetry scenarios based on combinations of U(1)charge, SU(2)spin, SU(2)charge, SU(3)channel, as well as the enveloping symplectic Sp(6) symmetry. These are compared in detail, including their respective dramatic gain in numerical efficiency. In the Appendix A, Appendix B and Appendix C, finally, an extensive introduction to non-abelian symmetries is given for practical applications, together with simple self-contained numerical procedures to obtain Clebsch–Gordan coefficients and irreducible operators sets. The resulting QSpace tensors can deal with any set of abelian symmetries together with arbitrary non-abelian symmetries with compact, i.e. finite-dimensional, semi-simple Lie algebras.

► We introduce a transparent framework for non-abelian symmetries in tensor networks.
► The framework was successfully applied within the numerical renormalization group.
► Combinations of SU(N≤3) and Sp(6) are implemented and discussed in detail.
► Non-abelian symmetries allow to speed up calculations by orders of magnitude.
► The framework is equally applicable to other tensor networks such as DMRG or MERA.