# Convergence radius of perturbative Lindblad driven non-equilibrium steady states

## TL;DR

This paper investigates the convergence properties of perturbative expansions for non-equilibrium steady states in Lindblad-driven spin chains, revealing how convergence radius varies with system size and parameters.

## Contribution

Develops a systematic formal approach to compute perturbative expansions of non-equilibrium steady states in open quantum spin chains, analyzing convergence behavior in different regimes.

## Key findings

- Convergence radius shrinks with increasing system size in coupling expansion.
- Convergence radius exceeds 1 in driving parameter expansion.
- Radius approaches 1 as anisotropy shifts from XY to Ising regime.

## Abstract

We address the problem of analyzing the radius of convergence of perturbative expansion of non-equilibrium 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 1/2 chain with boundary supported ultralocal Lindblad dissipators and treat two different perturbative cases: (i) expansion in system-bath coupling parameter and (ii) expansion in 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 easy plane ($XY$) to easy axis (Ising) regime.