  The low-energy effective Hamiltonian of three coupled spin chains with
periodic boundary conditions (spin tube) is expressed, in the limit of strong
interchain coupling, in terms of XXZ chains coupled by biquadratic exchange
interaction. A similar class of models was proposed to describe the coupling of
spins to orbital degrees of freedom in materials such as NaV2O5. We investigate
these models by means of bosonization and renormalization group techniques, and
find that the generic phase diagram comprises a gapless region and gapped
regions consisting of a spin liquid phase and various antiferromagnetic phases.
We discuss the properties of the spin liquid phase, in particular the nature of
the ground state and of the elementary excitations above it. We then study the
effect of a magnetic field, and conclude that a strong enough magnetic field
can suppress the dimerized phase leading to a two component Luttinger liquid.
The critical exponents at the transition gapful-gapless are calculated and
shown to be non-universal in the spin tube case as well as in the generic spin
orbital problem.
