Entanglement and quantum phase transition in 1D, 2D, and 3D spin models

We study the properties of multi-qubit entanglement in quantum phase transition using information theoretic measure for genuine multi-qubit entanglement and global entanglement for several spin models. We determine the behavior and critical points of multi-qubit entanglement for different parameter regions. As an example, we determine the critical points of 4-qubit models in 1D and 2D (a quantum dot). We find the genuine multi-qubit entanglement is maximum for ground state and fall at for Ising and Heisenberg XX models and for antiferromagnetic Heisenberg XXX model in 1D. The corresponding values in 2D are twice their 1D values. Anisotropic models are also investigated where we determine genuine multi-qubit entanglement as a function of anisotropy. We also find a genuine three qubit entanglement in specific mixed states, show its critical points and conjecture its form for more general mixed states. In the end we use finite-size scaling theory to investigate the behavior of these systems for large number of qubits in various dimensions. These results help us in a better understanding of the ground state properties including entanglement sharing in these models.