Communications in Number Theory and Physics

Volume 16 (2022)

Number 3

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

Pages: 615 – 672



Cid Reyes-Bustos (Dept. of Mathematical & Computing Science, School of Computing, Tokyo Institute of Technology, Tokyo, Japan; and NTT Institute for Fundamental Mathematics, Tokyo, Japan)

Masato Wakayama (Dept. of Mathematics, Tokyo University of Science, Tokyo, Japan; Institute for Mathematics for Industry, Kyushu University, Fukuoka, Japan; and NTT Institute for Fundamental Mathematics, Tokyo, Japan)


The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias was uncovered in recent studies by the explicit construction of operators $J_\ell$ 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 conjectural relation between the symmetry and degeneracy for the AQRM given explicitly in terms of two polynomials appearing independently 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 hyperelliptic curve) between the AQRM Hamiltonian and its basic commuting operator $J_\ell$. 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 quantum Rabi model. Moreover, it allows us to describe a remarkable approximation of the first $\ell$ 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–Néron 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.


Weyl algebra, hidden symmetry, degeneracy, constraint polynomials, Heun ODE, representation of $\mathfrak{sl}_2$, hyperelliptic curves, elliptic surfaces

2010 Mathematics Subject Classification

Primary 81Q10. Secondary 11G05, 34L40, 81S05.

Dedicated to Roger Howe on the occasion of his 77th birthday

This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 20K03560, JST CREST JPMJCR14D6, and CREST JPMJCR2113, Japan.

Received 21 October 2021

Accepted 5 May 2022

Published 4 October 2022