# Symmetries in Tensor Networks

A variety of symmetries can be exactly preserved with tensor networks, most notably symmetries corresponding to the action of a group on the individual tensor indices. For cases such as infinite tensor networks, other symmetries like translation invariance can be enforced.

## Global “On Site” Symmetries

A case that has been studied in great detail are symmetries of tensors and tensor networks under the action of a group, where the representation of the group is a tensor product of the group action on each external, or “site” index of the tensor or tensors. (For an introduction to group theory, the book by Arovas is recommended.[1].)

The general theory of tensor networks which are symmetric under such operations is reviewed in Ref. 2. Many more details about the cases of the groups $U(1)$ and $SU(2)$ can be found in Ref. 3 and 4, respectively.

For tensor networks symmetric under Abelian groups such as $\mathbb{Z}_2$ or $U(1)$, the main consequence is that each tensor in the network can be taken to be block-sparse without loss of generality. Thus many of the elements of the tensor can be assumed to be zero, and then are not stored in memory or explicitly used in computations, leading to significant gains in performance.

For tensor networks which are symmetric under non-Abelian groups such as $SU(2)$, the consequences are similar but the internal structure of the tensors in the network can be constrained to a greater degree, yielding much larger (lossless) compression compared to the Abelian case. However, the non-Abelian case is much more technical to implement, so that some authors have published detailed discussions of the implementation of $SU(2)$ symmetries in code.[4][5][6]

## References

1. Lecture Notes on Group Theory in Physics, Daniel Arovas
2. Tensor network decompositions in the presence of a global symmetry, Sukhwinder Singh, Robert N. C. Pfeifer, Guifr'e Vidal, Phys. Rev. A 82, 050301 (2010), arxiv:0907.2994
3. Tensor network states and algorithms in the presence of a global U(1) symmetry, Sukhwinder Singh, Robert N. C. Pfeifer, Guifre Vidal, Phys. Rev. B 83, 115125 (2011), arxiv:1008.4774
4. Tensor network states and algorithms in the presence of a global SU(2) symmetry, Sukhwinder Singh, Guifre Vidal, Phys. Rev. B 86, 195114 (2012), arxiv:1208.3919
5. A programming guide for tensor networks with global SU(2) symmetry, Philipp Schmoll, Sukhbinder Singh, Matteo Rizzi, Roman Orus, Annals of Physics 419, 168232 (2020), arxiv:1809.08180
6. Diagrammatics for SU(2) invariant matrix product states, Andreas Fledderjohann, Andreas Kluemper, Karl-Heinz Muetter, Journal of Physics A: Mathematical and Theoretical 44, 475302