Integrability of one-dimensional Lindbladians from operator-space fragmentation.

Fabian H L Essler,Lorenzo Piroli

doi:10.1103/physreve.102.062210

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 dimensions.

Highlights

In many physical situations, a quantum system weakly coupled to an environment can be described by a Lindblad master equation [1,2]
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 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

Summary

INTRODUCTION

A quantum system weakly coupled to an environment can be described by a Lindblad master equation [1,2]. This question is highly relevant in light of the important role that exactly solvable models have played in the recent efforts aimed at understand nonequilibrium dynamics in isolated quantum systems [11,12,13,14,15,16] It has been known for some time that certain Lindblad equations can be cast in the form of imaginary-time Schrödinger equations with non-Hermitian “Hamiltonians” that are quadratic in fermionic or bosonic field operators [17]. For the Lindblad equations studied in this paper, the projection of the Lindbladian onto each invariant subspace is mapped onto an integrable Hamiltonian, allowing us to obtain the full spectrum analytically. We consign the most technical aspects of our paper to a few Appendices

OPERATOR-SPACE FRAGMENTATION

THE ASEP LINDBLADIAN

The exact spectrum

The dissipative dynamics

HIGHER-SPIN GENERALIZATIONS

CONCLUSIONS