Degeneracy and hidden symmetry for the asymmetric quantum Rabi model with integral bias

摘要

The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias was uncovered in recent stud-ies by the explicit construction of operators JP commuting with the Hamiltonian. The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, the crossings on the energy curves. In this paper we propose a con-jectural relation between the symmetry and degeneracy for the AQRM given explicitly in terms of two polynomials appearing in-dependently in the respective investigations. Concretely, one of the polynomials appears as the quotient of the constraint polynomials that assure the existence of degenerate spectrum while the other determines a quadratic relation (in general, it defines a hyperel-liptic curve) between the AQRM Hamiltonian and its basic com-muting operator J. The significance of the conjecture is that it provides a concrete and unexpected realization of the presumed relation between the hidden symmetry and the degeneracy of the AQRM with a half-integral bias, and moreover, that the resulting equation leads to structural insights of the whole spectrum. For instance, the energy curves are naturally shown to lie on a surface determined by the family of hyperelliptic curves by considering the coupling constant as a variable. This geometric picture contains the generalization of the parity decomposition of the symmetric quan-tum Rabi model. Moreover, it allows us to describe a remarkable approximation of the first B energy curves by the zero-section of the corresponding hyperelliptic curve. These investigations naturally lead to a geometric picture of the (hyper-) elliptic surfaces given by the Kodaira-Ne & PRIME;ron type model for a family of energy curves over the projective line, which may be expected to contribute to a complex analytic proof of the conjecture.