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

Tensor networks offer a variational formalism to efficiently represent wave functions of extended quantum many-body systems on a lattice. In a tensor network $\mathcal{N}$, the dimension $\ensuremath{\chi}$ 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 ${\ensuremath{\chi}}^{\mathrm{min}}$ required to represent a given many-body wave function $|\ensuremath{\Psi}\ensuremath{\rangle}$ leads to the most compact, computationally efficient tensor network description of $|\ensuremath{\Psi}\ensuremath{\rangle}$. In the presence of a global, on-site symmetry, one can use a tensor network ${\mathcal{N}}_{\mathrm{sym}}$ made of symmetric tensors. Symmetric tensors allow one 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 $\mathcal{N}$ with minimal bond dimension ${\ensuremath{\chi}}^{\mathrm{min}}$ and a tensor network ${\mathcal{N}}_{\mathrm{sym}}$ made of symmetric tensors, where the minimal bond dimension ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}$ might be larger than ${\ensuremath{\chi}}^{\mathrm{min}}$. 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 ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}={\ensuremath{\chi}}^{\mathrm{min}}$. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that ${\ensuremath{\chi}}_{\mathrm{sym}}^{\mathrm{min}}>{\ensuremath{\chi}}^{\mathrm{min}}$. 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.

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