We study a 1 dimensional spin-orbital model using both analytical and numerical methods. Renormalization group calculations are performed in the vicinity of a special integrable point in the phase diagram with SU(4) symmetry. These indicate the existence of a gapless phase in an extended region of the phase diagram, missed in previous studies. This phase is SU(4) invariant at low energies apart from the presence of different velocities for spin and orbital degrees of freedom. The phase transition into a gapped dimerized phase is in a generalized Kosterlitz-Thouless universality class. The phase diagram of this model is sketched using the density matrix renormalization group technique.

A fibre optic probe system was used to measure the profiles of vertical particle velocities in the spout and the fountain of a half‐column and a full‐column spouted bed. In addition, a fibre optic image probe was employed to measure vertical particle velocity profiles in the annulus of the full‐column. In the spout, radial profiles of vertical particle velocities were of near Gaussian distribution. Particle velocities along the spout axis in the half‐column were 30% lower than in the full‐column under identical operating conditions. In the half column, particle velocities adjacent to the front plate were approximately 24% lower than a few millimeters away. The fountain core expanded suddenly near the bed surface and then gradually contracted with height. The model of Grace and Mathur (1978) gave good predictions of fountain heights for the full‐column. In the annulus region, there was a 28% difference between particle velocities adjacent to the column wall and those only 2 mm away. The integrated upward solids mass flow in the spout and the downward solids flow in the annulus matched well at different bed levels.

We study a spin 1/2 Heisenberg zigzag spin chain model near decoupled two chains. Taking into account a symmetry breaking perturbation, we discuss the existence of an energy gap in the ferromagnetic interchain coupling as well as the antiferromagnetic one. In the ferromagnetic model, a marginally relevant fixed line reduces the gap strongly, so that the correlation length becomes an astronomical length scale even in order 1 coupling. This result agrees with density matrix renormalization group results.