  The antiferromagnetic Heisenberg spin systems on the three-leg ladder are
investigated. Periodic boundary condition is imposed in the rung direction. The
system has an excitation gap for all antiferromagnetic inter-chain coupling
($J_{\perp}>0$). The estimated gap for the strong coupling limit
($J_{\perp}/J_1 \to \infty$) is 0.28$J_1$. Although the interaction is
homogeneous and only nearest-neighbor, the ground states of the system are
dimerized and break the translational symmetry in the thermodynamic limit.
Introducing the next-nearest neighbor coupling ($J_2$), we can see that the
system is solved exactly. The ground state wave function is completely
dimer-ordered. Using density matrix renomalization group algorithm, we show
numerically that the original model ($J_2=0$) has the same nature with the
exactly solvable model. The ground state properties of the ladder with a higher
odd number of legs are also discussed.
