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We examine the relationship between the second law of thermodynamics and theenergy eigenstates of quantum many-body systems that undergo cyclic unitaryevolution. Using a numerically optimized control protocol, we analyze how thework extractability is affected by the integrability of the system. Ourfindings reveal that, in nonintegrable systems the number of work-extractableenergy eigenstates converges to zero, even when the local control operationsare optimized. In contrast, in integrable systems, there are exponentially manyeigenstates from which positive work can be extracted, regardless of thelocality of the control operations. We numerically demonstrate that such astrikingly different behavior can be attributed to the number of athermalenergy eigenstates. Our results provide insights into the foundations of thesecond law of thermodynamics in isolated quantum many-body systems, which areexpected to contribute to the development of quantum many-body heat engines.

Work extractability from energy eigenstates under optimized local operations