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The benefits of exploiting the presence of symmetries in tensor networkalgorithms have been extensively demonstrated in the context of matrix productstates (MPSs). These include the ability to select a specific symmetry sector(e.g. with a given particle number or spin), to ensure the exact preservationof total charge, and to significantly reduce computational costs. Compared tothe case of a generic tensor network, the practical implementation ofsymmetries in the MPS is simplified by the fact that tensors only have threeindices (they are trivalent, just as the Clebsch-Gordan coefficients of thesymmetry group) and are organized as a one-dimensional array of tensors,without closed loops. Instead, a more complex tensor network, one where tensorshave a larger number of indices and/or a more elaborate network structure,requires a more general treatment. In two recent papers, namely (i) [Phys. Rev.A 82, 050301 (2010)] and (ii) [Phys. Rev. B 83, 115125 (2011)], we describedhow to incorporate a global internal symmetry into a generic tensor networkalgorithm based on decomposing and manipulating tensors that are invariantunder the symmetry. In (i) we considered a generic symmetry group G that iscompact, completely reducible and multiplicity free, acting as a globalinternal symmetry. Then in (ii) we described the practical implementation ofAbelian group symmetries. In this paper we describe the implementation ofnon-Abelian group symmetries in great detail and for concreteness consider anSU(2) symmetry. Our formalism can be readily extended to more exotic symmetriesassociated with conservation of total fermionic or anyonic charge. As apractical demonstration, we describe the SU(2)-invariant version of themulti-scale entanglement renormalization ansatz and apply it to study the lowenergy spectrum of a quantum spin chain with a global SU(2) symmetry.

Tensor network states and algorithms in the presence of a global SU(2) symmetry