Investigation of excited-state quantum phase transitions in the U(5)↔SO(6) transitional region by the recurrence formula and Jacobi matrix

In this paper, we have theoretically studied the Excited State Quantum Phase Transition $\left(ESQPT\right)$. For this purpose, based on the formulas of the inverse relation and the Jacobian matrix from the theory of orthogonal polynomials, we have presented a numerical solution for the model transition between the two phases $U\left(5\right)$ and $O\left(6\right)$. We have obtained the energy spectrum of the excited states for the given N and we have observed the transition region well in it. One of the prominent features of the energy spectrum is that for a given N, all the excited states are acquired. The absence of level crossing between levels is another characteristic of the spectrum, the reason for which can be proven using orthogonal polynomials. In addition, the expected value of the number of bosons and the ratio of the energy gaps have been studied, yielding good results in relation to them.