Classical model emerges in quantum entanglement: Quantum Monte Carlo study for an Ising-Heisenberg bilayer

By developing a cluster sampling of the stochastic series expansion quantum Monte Carlo method, we investigate a spin-12 model on a bilayer square lattice with intralayer ferromagnetic (FM) Ising coupling and interlayer antiferromagnetic Heisenberg interaction. The continuous quantum phase transition which occurs at gc=3.045(2) between the FM Ising phase and the dimerized phase is studied via large-scale simulations. From analysis of the critical exponents we show that this phase transition belongs to the (2+1)-dimensional Ising universality class. In addition, the quantum entanglement is strong between the two layers, especially in the dimerized phase. The effective Hamiltonian of a single layer seems like a transverse-field Ising model. However, we found that the quantum entanglement Hamiltonian is a pure classical Ising model without any quantum fluctuations. Furthermore, we give a more general explanation about how a classical entanglement Hamiltonian emerges.