2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/113397We study the zero temperature phase diagram of hard core bosons in two dimensions subjected to three types of background potentials: staggered, uniform, and random. In all three cases there is a quantum phase transition from a superfluid (at small potential) to a normal phase (at large potential), but with different universality classes. As expected, the staggered case belongs to the XY universality, while the uniform potential induces a mean field transition. The disorder driven transition is clearly different from both; in particular, we find z~1.4, ν~1, and β~0.6.4 pages (6 figures); published version-- 2 references added, minor clarificationsStatistical MechanicsSuperconductivitytext