Symmetry- and energy-resolved entanglement dynamics in a disordered Bose-Hubbard model

The phenomenology of many-body localization (MBL) develops mainly from tackling the one-dimensional spin or fermion systems. The situation when interacting bosons get clustered in a random potential remains less explored. Using numerical quantum quenches with special emphasis on the integration of both symmetry and energy resolutions, we comprehensively study the dynamics of symmetry-resolved entanglement in a disordered Bose-Hubbard (dBH) model, concentrating on the two types of inhomogeneous initial states to target the lower- and higher-energy sections of its dynamical phase diagram. (i) Motivated by the recent experiment [A. Lukin et al., Science 364, 256 (2019)] which focused on the lower-energy dynamical behaviors of the dBH chain, we first show that at low energies, for a thermalizing state, although the second law of thermodynamics prohibits the decrease of the total entropy over time, for part of the channel-resolved entropies, a long-term entropic reduction may arise at weak disorder. (ii) A companion channel-resolved analysis at strong disorder further hints that the prior observed double-logarithmic growth of the number entropy might not directly indicate the breakdown of MBL in spin or fermion chains, providing a refreshing perspective on this major controversy in the community. (iii) From time-evolving the line-shape low-energy product state, we subsequently reveal an abrupt formation of a novel ``entropy imbalance pattern'' across the different symmetry channels. Intriguingly, this imbalance melts in the strong-disorder limit. We conjecture that the melting of the entropic pattern, together with the freezing of a concurrent particle-density wave, embodies a dual trait inherent to MBL. (iv) Conversely, the higher-energy section of the dynamical phase diagram is where the dBH model differs most significantly from the spin or fermion systems. This parametric space was not included in the previous literature. Specifically, we find a cluster MBL regime, unique to the Bose statistics, emerging from the higher-energy section. This cluster MBL regime, realizable even at weak disorder, does not appear to suffer from the finite-size drift and is distinguished by its absence of the hallmark of MBL---the unbounded growth of the entanglement entropy. Our theoretical predictions are, by and large, testable via the present experimental facilities.