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Tensor network decompositions in the presence of a global symmetry · Around

2009

Tensor network decompositions in the presence of a global symmetry

Singh, Sukhwinder, Pfeifer, Robert N. C., Vidal, Guifre

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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.

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Similar

G. Vidal, Phys. Rev. Lett. 99
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C. Rovelli and L. Smolin, Physical Review D 52
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F. Verstraete and J. I. Cirac, arXiv:cond-mat/0407066v1. G. Sierra and M.A. Martin-Delgado, arXiv:cond-mat/9811170v3
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A set of states | Ψ t ⟩ |\Psi_{t}\rangle that transform covariantly
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In non-multiplicity free groups, such as SU(3), where N a b c N_{ab}^{c} might be larger than 1, the coupled basis | c , o c , μ ⟩ |c,o_{c},\mu\rangle and tensor S μ a b c S^{abc}_{\mu} must include an extra index μ = 1 , … , N a b c \mu=1,\ldots,N_{ab}^{c} . See, for example, R. N. C. Pfeifer et al., Phys. Rev. B 82
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When 𝒢 \mathcal{G} is an Abelian group, such as U(1), the tensor product 𝕍 a ⊗ 𝕍 b \mathbb{V}^{a}\otimes\mathbb{V}^{b} of two irreps only gives rise to one irrep 𝕍 c \mathbb{V}^{c} , so that no intermediate charges e 1 , e 2 , … , e t ′ e_{1},e_{2},\ldots,e_{t^{\prime}} need to be specified in Eq. 15
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S. Singh, R. Pfeifer, and G. Vidal, arXiv:1008.4774v1
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S. Singh and G. Vidal, in preparation
Original
Cited in the paper.

Then

S. R. White, Phys. Rev. Lett. 69
2009
Closest in time.
G. Evenbly and G. Vidal, Phys. Rev. Lett. 104
2010
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