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arXiv:1202.5664 · Annals of Physics

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

Andreas Weichselbaum

Published Jul 24, 2012• 274 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2012.07.009

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This is the documentation for the tensor library QSpace (v4.0) that provides a toolbox to exploit ‘quantum symmetry spaces’ in tensor network states in the quantum many-body context. QSpace permits arbitrary combinations of symmetries including the Abelian symmetries \mathbb{Z}_n ℤ n and U(1) U ( 1 ) , as well as all non-Abelian symmetries represented by the simple classical Lie algebras: A_n A n , B_n B n , C_n C n , and D_n D n , or respectively, the special unitary group SU( n n ), the odd orthogonal group SO( 2n+1 2 n + 1 ), the symplectic group Sp( 2n 2 n ), and the even orthogonal group SO( 2n 2 n ). The code (C++ embedded via the MEX interface into Matlab) is available (https://bitbucket.org/qspace4u) under the Apache 2.0 license. QSpace is designed as a bottom-up approach for non-Abelian symmetries. It starts from the defining representation and the respective Lie algebra. By explicitly computing and tabulating generalized Clebsch-Gordan coefficient tensors, QSpace is versatile in its operations across all symmetries. At the level of an application, much of the symmetry-related details are hidden within the QSpace C++ core libraries. Hence when developing tensor network algorithms with QSpace, these can be coded (nearly) as if there are no symmetries at all, despite being able to fully exploit general non-Abelian symmetries.
In this thesis I develop a formalism whereby a tensor network may be understood in terms of a unitary braided tensor category, and represented in a particularly efficient manner corresponding to the exploitation of this mathematical structure. This approach permits the exploitation of the global Abelian and non-Abelian internal group symmetries of a lattice model, both to facilitate the study of particular symmetry sectors and to provide a considerable reduction in computational cost. It also enables the study of models with non-trivial exchange statistics (fermions, Abelian and non-Abelian anyons) using tensor network algorithms which scale polynomially in the system size.