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

We add a Heisenberg interaction term $\propto \lambda$ in the one-dimensional SU(2)$\otimes$XY spin-orbital model introduced by B. Kumar. At $\lambda =0$ 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 $\lambda >0$ leads to spontaneous dimerization of the system which in the thermodynamic limit becomes a smooth phase transition at $\lambda \to 0$, 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 $\lambda \to 0$ confirming the critical behavior. Finally, we show the evidence of another phase transition in the Heisenberg limit, $\lambda \to \mathrm{\infty }$, and give a qualitative analytical explanation of the observed dimerized states both in the limit of small and large $\lambda$.