Second-order Peierls transition in the spin-orbital Kumar-Heisenberg model

We add a Heisenberg interaction term ? in the one-dimensional SU(2) - XY spin-orbital model introduced by Kumar. At ?=0 the spin and orbital degrees of freedom can be separated by a unitary transformation leading to an exact solution of the model. We show that a finite ?>0 leads to spontaneous dimerization of the system which in the thermodynamic limit becomes a smooth phase transition at ?->0, whereas it remains discontinuous within the first-order perturbation approach. We present the behavior of the entanglement entropy, energy gap, and dimerization order parameter in the limit of ?->0 confirming the critical behavior. Finally, we show the evidence of another phase transition in the Heisenberg limit, ?-> and give a qualitative analytical explanation of the observed dimerized states both in the limit of small and large ?.