  Tensor networks offer a variational formalism to efficiently represent
wave-functions of extended quantum many-body systems on a lattice. In a tensor
network N, the dimension \chi of the bond indices that connect its tensors
controls the number of variational parameters and associated computational
costs. In the absence of any symmetry, the minimal bond dimension \chi^{min}
required to represent a given many-body wave-function |\Psi> leads to the most
compact, computationally efficient tensor network description of |\Psi>. In the
presence of a global, on-site symmetry, one can use a tensor network N_{sym}
made of symmetric tensors. Symmetric tensors allow to exactly preserve the
symmetry and to target specific quantum numbers, while their sparse structure
leads to a compact description and lowers computational costs. In this paper we
explore the trade-off between using a tensor network N with minimal bond
dimension \chi^{min} and a tensor network N_{sym} made of symmetric tensors,
where the minimal bond dimension \chi^{min}_{sym} might be larger than
\chi^{min}. We present two technical results. First, we show that in a tree
tensor network, which is the most general tensor network without loops, the
minimal bond dimension can always be achieved with symmetric tensors, so that
\chi^{min}_{sym} = \chi^{min}. Second, we provide explicit examples of tensor
networks with loops where replacing tensors with symmetric ones necessarily
increases the bond dimension, so that \chi_{sym}^{min} > \chi^{min}. We further
argue, however, that in some situations there are important conceptual reasons
to prefer a tensor network representation with symmetric tensors (and possibly
larger bond dimension) over one with minimal bond dimension.

Strongly Correlated Electrons