  What entanglement is present in naturally occurring physical systems at
thermal equilibrium? Most such systems are intractable and it is desirable to
study simple but realistic systems which can be solved. An example of such a
system is the 1D infinite-lattice anisotropic XY model. This model is exactly
solvable using the Jordan-Wigner transform, and it is possible to calculate the
two-site reduced density matrix for all pairs of sites. Using the two-site
density matrix, the entanglement of formation between any two sites is
calculated for all parameter values and temperatures. We also study the
entanglement in the transverse Ising model, a special case of the XY model,
which exhibits a quantum phase transition. It is found that the next-nearest
neighbour entanglement (though not the nearest-neighbour entanglement) is a
maximum at the critical point. Furthermore, we show that the critical point in
the transverse Ising model corresponds to a transition in the behaviour of the
entanglement between a single site and the remainder of the lattice.
