The Cluster Mean Field theory calculation is extended for finite temperatures to study the combined effect of quantum and thermal fluctuations on various phases arising in the spin-1 Bose Hubbard model. This investigation finds that the polar nature of the superfluid phase persists, and the density of bosons with spin component σ = 0 increases with the temperature. The phase diagram is obtained and compared with that of the single-site mean-field theory. Compressibility shows a signature of divergence at the transition from the polar superfluid phase to the normal bose liquid phase. Small thermal fluctuations are sufficient to destroy the anti-ferromagnetic, maximally entangled odd density Mott insulator phase, whereas large thermal fluctuations are needed to break the singlet formed in the even density Mott insulator phase. However, in both the cases, the insulators melt to normal bose liquid at the same temperature.
Using the discrete truncated Wigner approximation (dTWA), we investigate the nonequilibrium dynamics of ultracold bosons confined in optical kagome lattices, focusing on both unfrustrated positive and frustrated negative hopping regimes. We consider a protocol in which the system is initialized in a Mott insulating state at unit filling, and the hopping amplitude is gradually increased from zero. For positive hopping, the melting of the Mott insulator is accompanied by the emergence of a sharp peak in the momentum distribution at the
Γ point of the lowest band, signaling the onset of superfluidity. In contrast, for negative hopping, the Mott insulator melts into a highly nontrivial state without long-range phase coherence, characterized instead by a broad momentum distribution within the flat band, consistent with recent experimental observations. These results demonstrate the applicability of dTWA to highly frustrated quantum systems and offer a new route for numerically exploring the dynamics of frustrated quantum magnets.