Tensor network decompositions in the presence of a global symmetry
Tensor network decompositions in the presence of a global symmetry
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DOI：
10.1103/physreva.82.050301
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发表时间：
2009-07
影响因子：
2.9
通讯作者：
Sukhwinder Singh;R. N. C. Pfeifer;G. Vidal
中科院分区：
文献类型：
--
作者：
Sukhwinder Singh;R. N. C. Pfeifer;G. Vidal
张量网络分解为格子系统的某些多体状态提供了一种有效的描述，是大量数值模拟算法的基础。我们讨论了如何在张量网络分解和算法中结合紧致的、完全可约的群G所给出的整体对称性。这是通过考虑在群G的作用下不变的张量来实现的。每个对称张量被分解成两种类型的张量：包含所有自由度的简并张量和仅依赖于对称群的结构张量。在数值计算中，对称张量的使用确保了对称性的保持，允许选择特定的对称扇区，并显著降低了计算成本。另一方面，由此产生的张量网络可以解释为指数级的许多自旋网络的叠加。自旋网络广泛应用于循环量子引力中，它们代表量子几何的状态。我们的工作也强调了它们在张量网络算法方面的重要性，从而为这两个研究领域之间的相互促进奠定了基础。
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research.