Extended and critical states are two common phenomena in quasiperiodic systems. In this paper, we first reveal the difference between the extended phase and the critical phase in the one-dimensional extended-critical system with off-diagonal quasiperiodic modulations from the perspectives of quantum transport and Wigner distribution. We reveal that in the extended phase, the quantum states exhibit ballistic or superdiffusion transport phenomena. While in the critical phase, the quantum states present subdiffusion transport phenomenon. Further, the transport conductance indicates that the extended-critical transition resembles a metallic-insulating transition. Moreover, the Wigner distributions show that the Wigner distribution of the extended wave function is localized in the momentum direction of the phase space, while that of the critical wave function is subextended in the momentum direction of the phase space. Based on the results of entanglement entropy, the extended-critical transition is a thermodynamic phase transition because it is accompanied by decreasing entropy. We engineer a quantum heat cycle engine with the extended-critical system as the working medium, and find that there are rich working modes in the engine, such as quantum accelerator, quantum heater, and quantum heat engine. Importantly, the extended phase is more conducive to the realization of quantum heat engines, while the critical phase is more conducive to the realization of quantum heaters. Our work is an important step toward exploring the rich thermodynamic applications of quasiperiodic systems.

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• Cold atoms & matter waves
• Entanglement entropy
• Localization
• Quantum thermodynamics
• Quantum transport
• Quasicrystals