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The van der Waals-Maxwell phase transition theory is among the oldest and themost simplest theories describing thermodynamic phase transitions in physics.Studying the complex Sachdev-Ye-Kitaev (cSYK) model numerically and deriving anew black hole solution in Jackiw-Telteiboim (JT)gravity analytically, weinvestigate how these models behave like the van der Waals-Maxwell liquid-gasmodel. The cSYK model exhibits first order phase transition terminating at thecritical point, while the charged black hole in JT gravity shows a clear vander Waals-Maxwell like phase structure. In the absence of charge, thecorresponding neutral black hole still has the Hawking-Page phase transition.The black hole solution also provides the first example in JT gravity yieldingthe van der Waals-Maxwell like phase structure.

Thermodynamic phase structure of complex Sachdev-Ye-Kitaev model and charged black hole in deformed JT gravity