  We consider a spin-1/2 tube (a three-leg ladder with periodic boundary
conditions) with a Hamiltonian given by two projection operators - one on the
triangles, and the other on the square plaquettes on the side of the tube -
that can be written in terms of Heisenberg and four-spin ring exchange
interactions. We identify 3 phases: (i) for strongly antiferromagnetic exchange
on the triangles, an exact ground state with a gapped spectrum can be given as
an alternation of spin and chirality singlet bonds between nearest triangles;
(ii) for ferromagnetic exchange on the triangles, we recover the phase of the
spin-3/2 Heisenberg chain; (iii) between these two phases, a gapless
incommensurate phase exists. We construct an exact ground state with two
deconfined domain walls and a gapless excitation spectrum at the quantum phase
transition point between the incommensurate and dimerized phase.
