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

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