  We have analytically obtained all the density matrix elements up to six
lattice sites for the spin-1/2 Heisenberg XXZ chain at $\Delta=1/2$. We use the
multiple integral formula of the correlation function for the massless XXZ
chain derived by Jimbo and Miwa. As for the spin-spin correlation functions, we
have newly obtained the fourth- and fifth-neighbour transverse correlation
functions. We have calculated all the eigenvalues of the density matrix and
analyze the eigenvalue-distribution. Using these results the exact values of
the entanglement entropy for the reduced density matrix up six lattice sites
have been obtained. We observe that our exact results agree quite well with the
asymptotic formula predicted by the conformal field theory.
