Exact matrix product solution for the boundary-driven Lindblad $XXZ$-chain

TL;DR

This paper provides an explicit matrix product solution for the non-equilibrium steady state of the boundary-driven XXZ spin chain, revealing connections to quantum algebra and demonstrating non-zero stationary currents.

Contribution

It introduces an exact matrix product ansatz for the steady state of the XXZ chain with boundary Lindblad operators, linked to quantum algebra $U_q[SU(2)]$, and extends to isotropic cases with boundary fields.

Findings

Exact steady state constructed via matrix product ansatz.

Stationary currents arise due to boundary twist.

Solution relates to quantum algebra $U_q[SU(2)]$.

Abstract

We demonstrate that the exact non-equilibrium steady state of the one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the non-equilibrium density matrix where the matrices satisfy a {\it quadratic algebra}. This algebra turns out to be related to the quantum algebra $U_q[SU(2)]$. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that this boundary twist leads to non-vanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.