We study the phase diagram of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland (ULS) model. This is done with the help of a level-one SU (3) Wess-Zumino-Witten model with certain marginal perturbations. We find that the renormalization group flow has infrared stable and unstable trajectories divided by a critical line on which the ULS model is located. The infrared unstable trajectory produced by a marginally relevant perturbation generates an exponential mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at ULS point is identified with the Berezinskiȋ-Kosterlitz-Thouless type. Our results support recent numerical studies by Fáth and Sólyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU (ν)-symmetric and asymmetric models.

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.

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.

We calculate wide distance connected correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles by solving the loop equation in the 1/N-expansion. The multi-level correlator is shown to be universal in large N limit. We show the algorithm to obtain the connected correlator to an arbitrary order in the 1/N-expansion.1