Quantum entanglement and quantum phase transition for the Ising model on a two-dimension square lattice

•The zero-temperature quantum entanglement around the quantum critical point for the Ising model on a square lattice is investigated.•The quantum entanglement exhibits some interesting behaviors such as nonanalytic and scaling behaviors.•The scaling relationship between the entanglement exponent and the correlation length exponent is also found.

The quantum entanglement and quantum phase transition of the transverse-field Ising model on a two-dimensional square lattice were investigated by applying the quantum renormalization group method. The quantum critical point (QCP) and the correlation length exponent, νν, were obtained. By taking the concurrence as a measure of entanglement, the entanglement between spin blocks near the QCP is calculated as the size of the system becomes large. The entanglement reaches a maximum close to QCP, and can exist in a small range around QCP just at the limit of thermodynamics. The nonanalytic behavior of the derivative of the entanglement with the external field shows that the system undergoes a second order quantum phase transition from a ferromagnetic phase to a paramagnetic phase. The finite-size scaling behavior of the entanglement is described, and the relationship between the entanglement exponent, θθ, the correlation length exponent, νν, and the dimension of the system dd is also found, i.e., θ=1/(νd)θ=1/(νd).