Non-Abelian symmetries in thermal tensor network states

Andreas Weichselbaum

Andreas Weichselbaum

Andreas Weichselbaum

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 open-source as of QSpace v4.0 on bitbucket (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.

Sukhwinder Singh

In this thesis we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically protect the symmetry and exploit it for computational gain in tensor network simulations. Our formalism is independent of the details of a specific tensor network decomposition since the symmetry constraints are imposed at the level of individual tensors. Moreover, the formalism can be applied to a wide spectrum of physical symmetries described by any discrete or continuous group that is compact and reducible. We describe in detail the implementation of the conservation of total particle number (U(1) symmetry) and of total angular momentum (SU(2) symmetry). Our formalism can also be readily generalized to incorporate more exotic symmetries such as conservation of total charge in anyonic systems.

Tao Xiang, Hui‐Hai Zhao, Z. Y. Xie +2 more

Andreas Weichselbaum

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.