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Integrable systems offer rare examples of solvable many-body problems in thequantum world. Due to the fine-tuned structure, their realization in nature andexperiment is never completely accurate, therefore effects of integrability areobserved only transiently. One way to surpass that is to couple nearlyintegrable systems to baths and driving: these will stabilize integrableeffects up to arbitrary time, as encoded in the time dependent, and eventually,the stationary state of form of a generalized Gibbs ensemble. However, thedescription of such driven dissipative nearly integrable models is challengingand no exact analytical methods have been proposed so far. Here we develop aniterative scheme in which integrability breaking perturbations (baths)determine the most necessary conserved quantities to be added into a truncatedgeneralized Gibbs ensemble description. Our scheme significantly reduces thecomplexity of the problem, paving the way for thermodynamic results.

Iterative construction of conserved quantities in dissipative nearly integrable systems