Exact spectrum of the XX spin chain with constrained non-diagonal boundary fields

We study the exact spectrum of the XX spin chain with constrained non-diagonal boundary fields, which can be analyzed by solving the associated Bethe Ansatz equations. In these equations, the number of Bethe roots has a definite parity, and all Bethe roots are located at the zeros of a unary function. We investigate the possible positions of the Bethe roots. Based on numerical observations, we analyze the Bethe root configurations for the ground state and the first excited state. Our results show that elementary excitations are characterized by the cooperative change of a pair of Bethe roots. Furthermore, we obtain an analytical expression for the ground state energy in the thermodynamic limit.