Tensor network decompositions in the presence of a global symmetry

By Sukhwinder Singh, R. Pfeifer, Guifré Vidal. 2009

214 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d073f15d8d5fef703ac4399dddb61d46ac4d5d43

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Citations in the corpus

Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) (2018, background): “he charge rule (and to exploit the block strucure for the speedup). In these notes, we focus exclusively on global, abelian symmetries which act locally in the computational basis. and refer to Refs. [13,14,16,17] for the non-abelian case, which requires a change of the computational basis and is mu”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “Furthermore, extra symmetries such as global U(1) or SU(2) symmetries can easily be in- corporated by defining tensors which transform according to some representation of the corresponding group (Perez-Garcia et al., 2010; Singh et al., 2010).”
The density matrix renormalization group for ab initio quantum chemistry (2014, methodology): “This opened the path to implement multiplicity-free non-Abelian symmetries also in TNSs [164–166].”