Operator-space fragmentation and integrability in Lindblad open quantum systems

We investigate the extension of Hilbert-space fragmentation to Lindblad open quantum systems, in which it manifests in the dynamics of observables in the Hilbert-Schmidt space. Using bond and commutant algebras of superoperators, we formalise this general framework and discuss the different cases revealed, similar to the classification of classical and quantum fragmentation in the closed system case.

We highlight a particularly interesting example where the operator space decomposes into exponentially many fragments which range in sizes up to the square-root of the full Hilbert-Schmidt dimension. At the boundary, there are a number of edge modes and in the bulk the operator dynamics is controlled by the fragments. Each of these fragments is equivalent to a non-Hermitian Lee-Yang theory with perturbed boundary fields, which may be exactly solved. This model has a phase transition governed by a non-unitary conformal field theory with a negative central charge. In the full Lindbladian, these transitions correspond to a series of exceptional points in the spectrum. We study the consequence of this for the underlying open quantum system, dynamics of operators and correlation functions, and for the stability of the edge modes.