Perturbative study of Supercritical Crossover in Noncommutative-corrected Spacetime

We analytically study the Widom line and supercritical crossover of noncommutative charged AdS black holes. Treating the noncommutative parameter $\alpha$ perturbatively, we compute thermodynamic quantities and the scaled variance $\Omega$ in both canonical and extended ensembles. The Widom line is identified as the extremum of $\Omega$. Using a Landau expansion near the critical point, we derive the two symmetric crossover branches $L^{\pm}$, which obey $\delta T\sim \left|\Delta Q\right|^{\beta+\gamma}$, $\delta S\sim \left|\Delta Q \right|^\beta$ in the canonical ensemble and $\delta P\sim \left|\Delta T\right|^{\beta+\gamma}$, $\delta \rho\sim \left|\Delta T\right|^{\beta}$ in the extended ensemble. These scaling relations conform to the mean-field universality class ($\beta=1/2$, $\gamma=1$), and the noncommutative parameter only shifts subleading amplitudes without altering the universality class. Numerical verification and complete supercritical phase diagrams are also presented using supercritical crossover lines. Our results show that noncommutative corrections preserve the mean-field universality of black hole supercriticality.