张量网络中的非阿贝尔对称性：量子对称空间方法
Non-abelian symmetries in tensor networks: a quantum symmetry space approach
摘要：
本文提出了一个非阿贝尔对称性的一般框架，它适用于在正交局部基和有效基下的矩阵乘积和张量网络状态。两个关键的成分，多重态空间的Clebsch-Gordan代数以及算子的Wigner-Eckart定理，都是以一种自然的、组织良好的、计算简单的方式来解释的。量子对称空间的统一张量表示，被称为QSpace，特别适合于处理标准的重整化群算法，如数值重整化群（NRG），密度矩阵重整化群（DMRG），或者更一般的张量网络，如多尺度纠缠重整化矩阵（MERA）。在本文中，重点是在NRG的非阿贝尔框架的应用。在总自旋守恒、粒子-空穴对称性和SU（3）沟道对称性的条件下，对自旋为3/2的三沟道完全屏蔽Anderson杂质模型进行了详细的分析.使用几种不同的对称方案分析了相同的系统。这包括更传统的对称设置SU（2）^4，更大的对称SU（2）*U（1）*SU（3），以及它们更大的包络辛对称SU（2）*Sp（6）。这些进行了详细的比较，包括各自的显着增益的数值效率。在附录中，最后，一个广泛的介绍非阿贝尔对称性的实际应用，连同简单的自包含的数值程序，以获得Clebsch-Gordan系数和不可约算子集。由此产生的QSpace张量可以处理任何阿贝尔对称集以及任意非阿贝尔对称与紧致，即有限维，半单李代数。
英文摘要：
A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of 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 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 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 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.
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