Comments on boundary driven open XXZ chain: asymmetric driving and uniqueness of steady states

TL;DR

This paper extends explicit matrix-product solutions for the steady states of boundary-driven XXZ spin chains to asymmetric driving conditions and arbitrary spin-flip rates, also providing a proof of the uniqueness of these states.

Contribution

It generalizes previous symmetric solutions to asymmetric cases and offers a simple proof of steady state uniqueness in boundary-driven XXZ chains.

Findings

Perturbative steady state solutions for asymmetric boundary rates.

Exact steady state form for arbitrary spin-flip rate ratios.

Proof of uniqueness of the steady states.

Abstract

In this short note we provide two extensions on the recent explicit results on the matrix-product ansatz for the non-equilibrium steady state of a markovianly boundary-driven anisotropic Heisenberg XXZ spin 1/2 chain. We write a perturbative solution for the steady state density matrix in the system-batyh coupling for an arbitrary (asymmetric) set of four spin-flip rates at the two chain ends, generalizing the symmetric-driving ansatz of [Phys. Rev. Lett. 106, 217206 (2011)]. Furthermore, we generalize the exact (non-perturbative) form of the steady state for just two Lindblad channels (spin-up flipping on the left, and spin-down flipping on the right) to an arbitrary (asymmetric) ratio of the spin flipping rates [Phys. Rev. Lett. 107, 137201 (2011)]. In addition, we also indicate a simple proof of uniqueness of our steady states.