Integrability of $1D$ Lindbladians from operator-space fragmentation

TL;DR

This paper introduces exactly solvable one-dimensional Lindblad equations where the operator space fragments into invariant subspaces, each governed by an integrable Hamiltonian, exemplified by a quantum ASEP model.

Contribution

It presents a novel class of Lindblad equations with operator-space fragmentation, linking dissipative dynamics to integrable Hamiltonians in each invariant subspace.

Findings

Operator space splits into exponentially many invariant subspaces.

Each subspace's dynamics is described by an integrable XXZ Heisenberg chain.

Lindbladians with this property exist for arbitrary local physical dimensions.

Abstract

We introduce families of one-dimensional Lindblad equations describing open many-particle quantum systems that are exactly solvable in the following sense: $(i)$ the space of operators splits into exponentially many (in system size) subspaces that are left invariant under the dissipative evolution; $(ii)$ the time evolution of the density matrix on each invariant subspace is described by an integrable Hamiltonian. The prototypical example is the quantum version of the asymmetric simple exclusion process (ASEP) which we analyze in some detail. We show that in each invariant subspace the dynamics is described in terms of an integrable spin-1/2 XXZ Heisenberg chain with either open or twisted boundary conditions. We further demonstrate that Lindbladians featuring integrable operator-space fragmentation can be found in spin chains with arbitrary local physical dimension.