Exterior integrability: Yang-Baxter form of nonequilibrium steady state density operator

TL;DR

This paper introduces a novel quantum transfer matrix approach for analyzing the nonequilibrium steady states of boundary-driven spin chains, revealing new integrability structures and potential for Bethe Ansatz solutions.

Contribution

It constructs an explicit Yang-Baxter R-matrix for a non-Hermitian transfer matrix associated with the Lindblad equation, extending integrability concepts to dissipative quantum systems.

Findings

Constructed a new quantum transfer matrix for the steady state density operator.

Explicitly derived an infinite-dimensional Yang-Baxter R-matrix different from standard models.

Indicated the potential for Bethe Ansatz solutions and discovered possible quasi-local conserved quantities.

Abstract

A new type of quantum transfer matrix, arising as a Cholesky factor for the steady state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang-Baxter R-matrix which we construct explicitly and which is essentially different from the standard XXX R-matrix. We also discuss a possibility to construct Bethe Ansatz for the spectrum and eigenstates of the non-equilibrium steady state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinitely-dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.