Bose glass in Ca$_2$RuO$_4$ nanofilms
Weak localization of bosons can give rise to an exotic quantum phase known as a Bose glass, characterized by the absence of global phase coherence yet finite conductivity. This phase is crucial in understanding the interplay between disorder, interactions, and superconductivity, especially in two-dimensional and strongly correlated systems. Here we report the presence of the Bose glass phase in the weak localization region of ruthenium oxide Ca$_2$RuO$_4$. The electrical resistance exhibits a characteristic logarithmic temperature dependence in this phase, $\rho \sim \ln(1/T)$. Through $\beta$-function scaling analysis, we observed "vertical flow" indicating unconventional scaling behavior associated with localized bosonic states. Our results suggest the existence of bosons-Cooper pairs-persisting up to high temperatures around 220 K and that these bosons undergo weak localization. In the Bose glass phase, vortices are found to have a dual relationship with the localized Cooper pairs, enabling their motion and resulting in finite resistance despite the presence of bosonic order. We identified two quantum critical points: one between the Bose glass and superconducting phases and another between the Bose glass and Mott insulating phases, allowing us to extract the corresponding quantum sheet resistances. We revealed that the ground state of the Ca$_2$RuO$_4$ changes depending on the localization strength. Thinning the Ca$_2$RuO$_4$ corresponds to controlling the electronic correlation by relieving the distortion in RuO$_6$ octahedra. These findings offer significant insights into the interplay between electronic correlations and bosonic transport, with important implications for studying high-temperature superconductors based on perovskite structures.