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Parent Lindbladians for matrix product density operators

Yuhan Liu1,2,*, Alberto Ruiz-de-Alarcón3,4, Georgios Styliaris1,2, Xiao-Qi Sun1,2, David Pérez-García3,5, and J. Ignacio Cirac1,2

  • *Contact author: yuhan.liu@mpq.mpg.de
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Phys. Rev. Research 8, 013210 – Published 25 February, 2026

DOI: https://doi.org/10.1103/1qyd-59z7

Abstract

Understanding quantum phases of matter is a fundamental goal in physics. For pure states, the representatives of phases are the ground states of locally interacting Hamiltonians, which are also renormalization fixed points (RFPs). These RFP states are exactly described by tensor networks. Extending this framework to mixed states, matrix product density operators (MPDOs) that are RFPs are believed to encapsulate mixed-state phases of matter in one dimension, where nontrivial topological phases have already been shown to exist. However, to better motivate the physical relevance of those states, and in particular the physical relevance of the recently found non trivial phases, it remains an open question whether such MPDO RFPs can be realized as steady states of local Lindbladians. In this work, we resolve this question by analytically constructing parent Lindbladians for MPDO RFPs. These Lindbladians are local, frustration free, and exhibit minimal steady-state degeneracy. Interestingly, we find that parent Lindbladians possess a rich structure that distinguishes them from their Hamiltonian counterparts. In particular, we uncover an intriguing connection between the noncommutativity of the Lindbladian terms and the fact that the corresponding MPDO RFP belongs to a nontrivial phase.

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Phys. Rev. Research 8, 013210– Published 25 February, 2026
  • Received 28 February 2025
  • Revised 10 October 2025
  • Accepted 15 January 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. Open access publication funded by Max Planck Society.

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