The local characterization of global tensor network eigenstates

TL;DR

This paper establishes a local, necessary and sufficient condition for Matrix Product States to be exact eigenstates of local operators, unifying various quantum states and symmetries, with applications to numerical methods and higher dimensions.

Contribution

It introduces a local fixed-size equation that characterizes all exact MPS eigenstates across diverse quantum systems and symmetries.

Findings

Characterizes the full space of exact eigenstates using local equations.

Recovers quantum group symmetries of the XXZ model.

Extends the framework to 2D PEPS and numerical algorithms.

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 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 allows to characterize the full space of solutions in all of the aforementioned problems, and to identify them both analytically and numerically. We elaborate on the concrete application of this characterization to all of the aforementioned settings, and in particular exemplify the power of our local characterization by using it to recover 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).