Anomalous minimization for critical velocity of superflow along a step potential

To reveal a microscopic mechanism for the anomalous minimization and dependence of the superfluid critical velocity on a moving obstacle potential in a atomic Bose-Einstein condensate [\href{https://link.aps.org/doi/10.1103/PhysRevA.91.053615}{Phys.~Rev.~A \textbf{91}, 053615 (2015)}], we introduce a considerably simplified model of superflow along a step potential. The energy spectrum and wave functions of the lowest-energy excitations in this system are well described by the semi-classical analysis based on the Bogoliubov theory. We found that the critical velocity is minimized and becomes zero when the potential height equals the hydrostatic chemical potential, which corresponds to the critical point of the local condensation phase transition inside the step potential. In a finite-size system, the critical velocity $v_\mathrm{c}$ obeys a power-law scaling with the system size $L_x$ as $v_\mathrm{c}\propto L_x^{-0.963}$. This criticality provides an explanation of the power-law scaling of the minimum critical velocity observed in the experiment.