Global symmetries in tensor network states: symmetric tensors versus minimal bond dimension

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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 χof 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 χmin required to represent a given many-body wave-function |Ψ> leads to the most compact, computationally efficient tensor network description of |Ψ>. In the presence of a global, on-site symmetry, one can use a tensor network Nsym 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 χmin and a tensor network Nsym made of symmetric tensors, where the minimal bond dimension χminsym might be larger than χmin. 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 χminsym = χmin. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that χsymmin > χmin. 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.