  The entanglement in quantum XY spin chains of arbitrary length is
investigated via the geometric (measure of) entanglement. The emergence of
entanglement is explained intuitively from the perspective of perturbations.
The model is solved exactly and the energy spectrum is determined and analyzed
in particular for the lowest two levels. The entanglement is obtained over the
entire phase diagram, and its behavior can be used to delineate the boundaries
in the phase diagram. For example, the field-derivative of the entanglement
becomes singular along the critical line. The form of the divergence turns out
to be dictated by the universality class controlling the quantum phase
transition. The entanglement near criticality can be understood via a scaling
hypothesis, analogous to that for free energies. The entanglement density
vanishes along the so-called disorder line in the phase diagram, the ground
space is doubly degenerate and spanned by two product states. The entanglement
for the superposition of the lowest two states is also investigated. Even
though the exact value of the entanglement depends on the specific form of
superposition, in the thermodynamic limit the entanglement density is
independent of the superposition, showing that the entanglement density is
insensitive to whether the ground state is chosen to be the spontaneously $Z_2$
symmetry broken one or not. The finite-size scaling of entanglement at critical
points is also investigated from two different view points: (1) the maximum in
the field-derivative of the entanglement density vs. the system size, used to
deduce the correlation length exponent for the Ising class by the behavior of
entanglement; (2) the correction to the entanglement density vs. the system
size, with the coefficient being universal (but with different values for the
two lowest states).

Quantum Physics;Statistical Mechanics