General framework for non-abelian symmetries in tensor networks

Details General framework for non-abelian symmetries in tensor networks General framework for non-abelian symmetries in tensor networks

2012-02-25

arxiv.org/abs/1202.5664v1

Physics

Condensed Matter

Strongly Correlated Electrons

22 pages (main text) + 32 pages (more pedagogical detailed introduction to non-abelian framework)

Scientific paper

A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of local orthonormal 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 given 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. This includes the more traditional symmetry setting SU(2)^4, the larger symmetry SU(2)*U(1)*SU(3), and their much larger enveloping symplectic symmetry SU(2)*Sp(6). These three symmetry settings are compared in detail, including their respective dramatic gain in numerical efficiency. In the appendix, 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 symmetries considered can consist of abelian symmetries together with arbitrary non-abelian symmetries with compact, i.e. finite-dimensional, semi-simple Lie algebras.

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