  Block entanglement entropy in the ground state of a quantum spin chain is
investigated. The spins have Kitaev-type nearest-neighbor interaction, of
strength J_x or J_y, through either x or y components of the spins on
alternating bonds, along with a transverse magnetic field h. An exact solution
is obtained through Jordan-Wigner fermionization, and it exhibits a
macroscopically degenerate ground state for h=0, and a non-degenerate ground
state for nonzero h and for all interaction strengths. For a chain of N spins,
we study the block entropy of a partition of L contiguous spins. The block
entanglement entropy needs the eigenvalues of the 2^L-dimensional reduced
density matrix. We employ an efficient method that reduces this problem to
evaluating eigenvalues of a L-dimensional matrix, which enables us to calculate
easily the block entanglement for large-N chains numerically. The entanglement
entropy grows as log L, at the degeneracy point h=0, and only for J_x=J_y. For
nonzero magnetic field, the entropy becomes independent of the size, thus
obeying the area law. For unequal J_x and J_y, the block entropy shows a
non-monotonic behavior for L<N/2.
