Tensor network decompositions in the presence of a global symmetry

Abstract

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 also in the context of tensor network algorithms, thus setting the stage for cross-fertilization between these two areas of research.Comment: 4 pages, 4 figures, RevTeX 4-

Tensor network states and algorithms in the presence of a global U(1) symmetry

“…Following Ref. 52 we considered tensor networks constructed from tensors which were invariant under the action of the internal symmetry, and showed how each tensor may be decomposed according to a canonical form into degeneracy tensors (which contain all the degrees of freedom that are not affected by the symmetry) and structural tensors (which are completely determined by the symmetry). We then introduced a set of primitive operations P which may be used to carry out tensor network algorithms using Ansätze such as MPS, PEPS, and MERA, and showed how each of these operations can be implemented in such a way that the canonical form is both preserved and exploited for computational gain.…”

“…We will also discuss several practical aspects of the exploitation of Abelian symmetries not covered in Ref. 52. For concreteness we will concentrate on the U(1) symmetry, but extending our results to any Abelian group is straightforward.…”

“…52 we described a formalism for incorporating global internal symmetries into a generic tensor network algorithm. Both Abelian and non-Abelian symmetries were considered.…”

“…As explained in Ref. 52, a generic symmetric tensor can be decomposed into a degeneracy part, which contains all degrees of freedom not determined by symmetry, and a structural part, which is completely determined by symmetry and can be further decomposed as a trivalent network of Clebsch-Gordan coefficients.…”

Gapped spin liquid with Z2 topological order for the kagome Heisenberg model

“…6 and the ground-state energy is e 0 = −0.4369. Here, to compute the variational energy, we utilize the state-of-the-art SU(2)-symmetry implementation [82][83][84]96].…”