Matrix product operator representations for the local conserved quantities of the Heisenberg chain

Abstract

We present the explicit expressions for the matrix product operator (MPO) representation for the local conserved quantities of the Heisenberg chain. The bond dimension of the MPO grows linearly with the locality of the charges. The MPO has more simple form than the local charges themselves, and their Catalan tree patterns naturally emerge from the matrix products. The MPO representation of local conserved quantities is generalized to the integrable $\mathrm{SU}(N)$ invariant spin chain.

18 pages, 2 figures; (v2 revised for SciPost) revised argument, added reference, added one appendix, corrected typos; (v3 revised for SciPost) revised argument, added subsection, corrected typos

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• Block encoding of matrix product operators
• Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains
• Absence of nontrivial local conserved quantities in the quantum compass model on the square lattice