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

Abstract

The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g. with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the Clebsch-Gordan coefficients of the symmetry group) and are organized as a one-dimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have 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 described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group G that is compact, completely reducible and multiplicity free, acting as a global internal symmetry. Then in (ii) we described the implementation of Abelian group symmetries in much more detail, considering a U(1) symmetry (e.g., conservation of global particle number) as a concrete example. In this paper we describe the implementation of non-Abelian group symmetries in great detail. For concreteness we consider an SU(2) symmetry (e.g., conservation of global quantum spin). Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)-invariant version of the multi-scale entanglement renormalization ansatz and apply it to study the low energy spectrum of a quantum spin chain with a global SU(2) symmetry.

Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases

“…Let us first discuss how to impose symmetries on tensor networks 53,[56][57][58][59][60][61][62] . We focus on the case where the state is a 1D representation of symmetry group SG:…”

“…Further, we require the single tensor to be inversion symmetric: W I I •T a = T a , where W I is given in Eq. (60). Then, by solving these linear equations, we obtain a D I = 74 dimensional (complex) Hilbert space.…”

Existence of featureless paramagnets on the square and the honeycomb lattices in 2+1 dimensions

“…In general, global symmetries impose strong constraints on the PEPS [12][13][14][15][16][17][18][19] . To serve our purpose, we choose the 'entangled pairs' to be multiplets of SO(3) symmetric spins and, thus, will refer to them as the 'slave-spin' PEPS.…”

Tree tensor networks and entanglement spectra

“…The representation we use is more convenient in the tree tensor network as it does not require to associate the direction of the flow (for a given node all charges flow into the node and they sum up to Q at each node) nor specify the starting and the ending node, which makes is easier to consider completely generic tree networks without any regular topology. See also 46,47 for a general treatment of symmetries in tensor networks algorithms.…”