Convergence radius of perturbative Lindblad-driven nonequilibrium steady states

We address the problem of analyzing the radius of convergence of perturbative expansion of nonequilibrium steady states of Lindblad-driven spin chains. A simple formal approach is developed for systematically computing the perturbative expansion of small driven systems. We consider the paradigmatic model of an open $XXZ$ spin-$\frac{1}{2}$ chain with boundary-supported ultralocal Lindblad dissipators and treat two different perturbative cases: (i) expansion in the system-bath coupling parameter and (ii) expansion in the driving (bias) parameter. In the first case (i) we find that the radius of convergence quickly shrinks with increasing the system size, while in the second case (ii) we find that the convergence radius is always larger than 1, and in particular it approaches 1 from above as we change the anisotropy from an easy-plane ($XY$) to an easy-axis (Ising) regime.

Figure 1

Radius of convergence $\lambda$ dependence on system size $N$. We take $\mu =1/2$ for driving parameter and different values for anisotropy: $\mathrm{\Delta }=0$ (black diamonds), $\mathrm{\Delta }=1/2$ (red $\times$), $\mathrm{\Delta }=1$ (blue circles), $\mathrm{\Delta }=2$ (green squares), $\mathrm{\Delta }=4$ (cyan triangles), and $\mathrm{\Delta }=10$ (magenta asterisks). The lines correspond to the best linear fit for $ln\lambda$ vs $N$ in each case.

Figure 2

Radius of convergence $\lambda$ dependence on anisotropy $\mathrm{\Delta }$. We take $\varepsilon =1$ for bath-system coupling strength and different values for system size: $N=4$ (red circles) and $N=5$ (blue asterisks). The dashed black line is included as a ($-1$) slope guideline.