Surface Critical Phenomena of a Free Bose Gas with Enhanced Hopping at the Surface

We study the Bose--Einstein condensation in a tight-binding model with a hopping rate enhanced only on a surface. We show that this model exhibits two different critical phenomena depending on whether the hopping rate on the surface $t_s$ exceeds the critical value $5t/4$, where $t$ is the hopping rate in the bulk. For $t_s/t<5/4$, normal Bose--Einstein condensation occurs, while the Bose--Einstein condensation for $t_s/t\geq5/4$ is characterized by the spatial localization of the macroscopic number of particles at the surface. By exactly calculating the surface free energy, we show that for $t_s/t<5/4$, the singularity of the surface free energy stems from diverging the correlation length in the bulk, while for $t_s/t\geq5/4$, it is induced by the coupling effects between the bulk and surface.