Effect of symmetry-breaking perturbations in the one-dimensional $\mathrm{SU}\mathrm{}\left(4\right)\mathrm{}$ spin-orbital model

We study the effect of symmetry-breaking perturbations in the one-dimensional $\mathrm{SU}\mathrm{}\left(4\right)\mathrm{}$ spin-orbital model. We allow the exchange in spin ${(J}_1)$ and orbital ${(J}_2)$ channel to be different and thus reduce the symmetry to $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(2\right).$ A magnetic field $h$ along the $S^z$ direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett. $83,$ 624 (1999)] we extend their analysis of the isotropic $J_1{=J}_2,$ $h=0\mathrm{}$ case and obtain the low-energy effective theory near the $\mathrm{SU}\mathrm{}\left(4\right)\mathrm{}$ point in the generic case $J_1\ne J_2,$ $h\ne 0.$ In zero magnetic field, we retrieve the same qualitative low-energy physics as in the isotropic case. In particular, the isotropic massless behavior found on the line $J_1{=J}_2<K/4\mathrm{}$ extends in a large anisotropic region. We discover, however, that the anisotropy plays its trick in allowing nontrivial scaling behaviors of the physical quantities. For example, the mass gap $M$ has two different scaling behaviors depending on the anisotropy. In addition, we show that in some regions, the anisotropy is responsible for anomalous finite-size effects and may change qualitatively the shape of the computed critical line in a finite system. When a magnetic field is present the effect of the anisotropy is striking. In addition to the usual commensurate-incommensurate phase transition that occurs in the spin sector of the theory, we find that the field may induce a second transition of the KT type in the remaining degrees of freedom to which it does not couple directly. In this sector, we find that the effective theory is that of an SO(4) Gross-Neveu model with an $h-\mathrm{dependent}$ coupling that may change its sign as $h$ varies.