This paper explores the entanglement properties in a simple quantum phase transition system, specifically the 1D infinite-lattice anisotropic XY model. The authors use the Jordan-Wigner transform to solve the model exactly and calculate the two-site reduced density matrix for all pairs of sites. They then determine the entanglement of formation between any two sites for all parameter values and temperatures. The study also examines the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest neighbor entanglement is maximized at the critical point, while the nearest-neighbor entanglement is not. The critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the rest of the lattice. The paper discusses the implications of these findings for understanding complex quantum systems and the role of entanglement in quantum phase transitions.This paper explores the entanglement properties in a simple quantum phase transition system, specifically the 1D infinite-lattice anisotropic XY model. The authors use the Jordan-Wigner transform to solve the model exactly and calculate the two-site reduced density matrix for all pairs of sites. They then determine the entanglement of formation between any two sites for all parameter values and temperatures. The study also examines the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest neighbor entanglement is maximized at the critical point, while the nearest-neighbor entanglement is not. The critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the rest of the lattice. The paper discusses the implications of these findings for understanding complex quantum systems and the role of entanglement in quantum phase transitions.