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 maximally for ground state and fall at for Ising and Heisenberg XX models and for anti-ferromagnetic Heisenberg XXX model in 1D. The corresponding values in 2D are twice their 1D value. 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.