“…Thus, we investigated the effects of different quantum and topological phase transitions induced by the spin-couplings, as well as the dimensionality and geometry of the lattice on quantum correlation and entanglement in Refs. [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 ]. The presence of different types of coupling leads to the different quantum and topological phases that are studied using analytical techniques, such as the nonlinear sigma model, spin-wave theory, and SU(N) Schwinger bosons, which are adequate in treating each case at low energy limits, as well as numerical techniques, such as the quantum Monte Carlo method and the density matrix renormalization group (DMRG), and so on.…”