Exact ground states with deconfined gapless excitations for the 3 leg spin-1/2 tube

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.