Solution of the Lindblad equation for spin helix states

Using Lindblad dynamics we study quantum spin systems with dissipative boundary dynamics that generate a stationary nonequilibrium state with a nonvanishing spin current that is locally conserved except at the boundaries. We demonstrate that with suitably chosen boundary target states one can solve the many-body Lindblad equation exactly in any dimension. As solution we obtain pure states at any finite value of the dissipation strength and any system size. They are characterized by a helical stationary magnetization profile and a ballistic spin current which is independent of system size, even when the quantum spin system is not integrable. These results are derived in explicit form for the one-dimensional spin-1/2 Heisenberg chain and its higher-spin generalizations, which include the integrable spin-1 Zamolodchikov-Fateev model and the biquadratic Heisenberg chain.

Figure 1

von Neumann entropy $S$ (upper curve) and steady-state current $j^z$ (lower curve) versus dissipative amplitude $\mathrm{\Gamma }$ in the $XXZ$ chain. Parameters: $J=1,N=4,\eta =\phi =4\pi /3, {\epsilon }_R=-{\epsilon }_L=1/20$. The pure state with $S=0$ describing a spin helix state is seen for the predicted value $\mathrm{\Gamma }=20|sin\phi |\approx 17.32$.

Figure 2

von Neumann entropy $S$ (upper curve) and steady-state current $j^z$ (lower curve) versus anisotropy $\mathrm{\Delta }$. Parameters: $J=1,N=4,\phi =4\pi /3,\mathrm{\Gamma }=20sin\phi$, and ${\epsilon }_R=-{\epsilon }_L=1/20$. A pure SHS with $S=0$ is obtained for the predicted value $\mathrm{\Delta }=cos\phi =-0.5$.