Integrability versus chaos in the steady state of many-body open quantum systems

The Lindblad description of an open quantum system gives rise to two types of integrability, since the nonequilibrium steady state can be integrable independently of the Liouvillian. Taking boundary-driven and dephasing spin chains as a representative example, we discriminate Liouvillian and steady-state chaos by combining level spacing statistics and an extension of the eigenstate thermalization hypothesis to open quantum systems. Moreover, we analyze the structure of the steady states by expanding it in the basis of Pauli strings and comparing the weight of strings of different lengths. We show that the natural expectation that integrable steady states are "simple" (i.e., built from few-body local operators) does not hold: the steady states of both chaotic and integrable models have relevant contributions coming from Pauli strings of all possible lengths, including long-range and many-body interactions. Nevertheless, we show that one can effectively use the operator-size distribution to distinguish chaotic and integrable steady states.