Cluster mean-field signature of entanglement entropy in bosonic superfluid-insulator transitions

Entanglement entropy (EE), a fundamental concept in quantum information for characterizing entanglement, has been extensively employed to explore quantum phase transitions (QPTs). Although the conventional singlesite mean-field (MF) approach successfully predicts the emergence of QPTs, it fails to include any entanglement. Here, in the framework of a cluster MF treatment, we extract the signature of EE in bosonic superfluid-insulator (SI) transitions. We consider a trimerized Kagome lattice of interacting bosons, in which each trimer is treated as a cluster, and implement the clusterMF treatment by decoupling all intertrimer hopping. In addition to superfluid and integer insulator phases, we find that fractional insulator phases appear when the tunneling is dominated by the intratrimer part. To quantify the residual bipartite entanglement in a cluster, we calculate the second-order Renyi entropy, which can be experimentally measured by quantum interference of many-body twins. The second-order Renyi entropy itself is continuous everywhere, however, the continuousness of its first-order derivative breaks down at the phase boundary. This means that the bosonic SI transitions can still be efficiently captured by the residual entanglement in our clusterMF treatment. Besides to the bosonic SI transitions, our clusterMF treatment may also be used to capture the signature of EE for other QPTs in quantum superlattice models.