We investigate the scaling and spatial distribution of genuine multiparticle entanglement in three- and four-spin reduced states of the one-dimensional
XY model at the quantum phase transition. We observe a logarithmic divergence, show that genuine three- and four-particle entanglement obeys finite-size scaling of the
XY model and demonstrate that the genuine three-particle entanglement has a finite spatial range.
The ground-state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum-phase-transition approach. The entanglement measure is taken as the order parameter and this is a nonlocal order parameter, which is different from the conventional order parameter of the Mott-insulator--superfluid-phase transitions. For this nonlocal order parameter, the scaling behavior corresponding to a continuous phase transition is obtained and a power-law divergence near the critical region follows it. This scaling behavior of quantum entanglement is analyzed by the finite-size scaling and the critical exponents are obtained as $\ensuremath{\nu}=1.01$ and $\ensuremath{\gamma}=0.86$. A close connection between quantum fluctuations and the phase transition of entanglement is also obtained.
We have studied here the evolution of entanglement towards criticality in the
XY model when quantum phase transition (QPT) is induced by a quench. To this end, we have utilized the relation of the Berry phase acquired by a spin state when it traverses a closed path at criticality with the concurrence measuring the entanglement for a pair of nearest-neighbor spins. It has been argued that when the spin state is rotated towards the generation of the closed path, the segments of noncyclic paths with open ends can be mapped onto cyclic paths enclosing flux lines associated with
q-deformed monopoles. This essentially gives rise to the sequence of Berry phases attained by the quasispin states belonging to the quantum group
SUq(2) when rotated in a closed path. As the standard Berry phase at criticality is obtained when the time-dependent deformation parameter
q attains the value 1, this formulation helps us to analyze the evolution of entanglement towards criticality when QPT is induced by a quench. From this analysis we also formulate a scaling law for the entanglement entropy for a block of
L spins with the rest at off-criticality. As in QPT off-critical points correspond to thermal fluctuations, this formulation represents the evolution of entanglement towards criticality with temperature as $T\ensuremath{\rightarrow}0$.