Exact and numerical results for the dimerized spin-orbital model

We establish exact results for the one-dimensional spin-orbital model for special values of the four-spin interaction $V$ and dimerization parameter $\delta$. The first exact result is at $\delta=1/2$ and $V=-2$. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with $V=-4$ and gives evidence that the system has a spontaneously dimerized ground state. Numerical results indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\leq V<-2$.