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Local Characterization of Global Tensor Network Eigenstates

José Garre Rubio1,2, András Molnár2, Norbert Schuch2,3, and Frank Verstraete4,5

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Phys. Rev. Lett. 137, 200403 – Published 13 November, 2026

DOI: https://doi.org/10.1103/5m7t-q2f6

Abstract

We study the conditions under which matrix product states (MPSs) or matrix product operators are exact eigenvectors of an extensive local operator such as a Hamiltonian. By suitably choosing the local operator, this covers a wide range of settings: exact eigenstates of Hamiltonians, including scar states; exact MPS trajectories for driven quantum systems; steady states of local Lindbladians; generalized symmetries of either Hamiltonians or density matrices; and many more. Our key result is that a local, fixed-size equation—namely, how a single term in the operator acts on a block of tensors—provides a necessary and sufficient condition for exact solutions. This local characterization allows us to obtain the full space of MPS solutions to all of the aforementioned problems; in particular, we exemplify the power of our results by recovering the quantum group symmetries of the XXZ model. We also discuss applications to numerical algorithms with MPSs and the generalization of our results to 2D, i.e., projected entangled pair states.

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Phys. Rev. Lett. 137, 200403– Published 13 November, 2026
  • Received 6 May 2026
  • Revised 20 August 2026
  • Accepted 29 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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