The local characterization of global tensor network eigenstates

Abstract:We study the conditions under which Matrix Product States (MPS) 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 MPS and the generalization of our results to 2D, i.e., projected entangled pair states (PEPS).

The new version expands the result significantly for 2D injective PEPS. New figure added. All proofs are contained in the appendices (23 pages)

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el)

arXiv:2603.28349 [quant-ph]

(or arXiv:2603.28349v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2603.28349 Focus to learn more arXiv-issued DOI via DataCite

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