Parent Lindbladians for Matrix Product Density Operators

TL;DR

This paper constructs local Lindbladians that have matrix product density operators as their steady states, providing a physical realization of mixed-state phases of matter in one dimension.

Contribution

It analytically demonstrates that MPDO RFPs can be realized as steady states of local Lindbladians, revealing a new connection between non-commutativity and non-trivial phases.

Findings

Constructed parent Lindbladians for MPDO RFPs

Lindbladians are local, frustration-free, with minimal degeneracy

Discovered a link between non-commutativity and non-trivial phases

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) which are RFPs are believed to encapsulate mixed-state phases of matter in one dimension, where non-trivial 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 non-commutativity of the Lindbladian terms and the fact that the corresponding MPDO RFP belongs to a non-trivial phase.