Entanglement and fluctuations of the fractonic phases in a one-dimensional dipole-conserving Bose-Hubbard chain

We study the entanglement and bipartite fluctuations of fractonic phases in a one-dimensional dipole-conserving Bose-Hubbard chain using the density matrix renormalization group method. We demonstrate that bipartite charge fluctuations fail to detect quantum phase transitions in fractonic systems. Instead, we introduce bipartite dipole fluctuations, ${\mathcal{F}}^{d}$, which exhibit scaling laws identical to the entanglement entropy $S$. For example, at integer filling $n=2$, both ${\mathcal{F}}^{d}$ and $S$ show logarithmic scaling in dipole Luttinger liquids and area-law scaling in dipole Mott insulators. Interestingly, in the dipole supersolid phase at noninteger filling $n=3/2$, we reveal that ${\mathcal{F}}^{d}$ and $S$ obey area laws despite the phase being compressible and hosting long-range dipole-dipole order. Moreover, we identify another phase at $n=3/2$, previously called the fractional Mott insulator phase, as a dipole-conserving charge density wave state. In this phase, ${\mathcal{F}}^{d}$ and $S$ follow area laws, but ${\mathcal{F}}^{d}$ exhibits an additional period-2 oscillation. These findings provide insights into the interplay between dipole conservation, quantum fluctuations, and entanglement in the fractonic phases.