### Abstract

The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum Rabi model by Lie algebra representations of sI_{2}. We define a second order element of the universal enveloping algebra U(sI_{2}) of sI_{2} (R), which, through the action of a certain infinite dimensional representation of sI_{2} (R), provides a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we prove the existence of level crossings in the spectral graph of the asymmetric quantum Rabi model when the symmetry-breaking parameter ϵ is equal to 1/2 , and conjecture a formula that ensures likewise the presence of level crossings for general ϵ ∈ 1/2 ℤ. This result on level crossings was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by Braak (2011). The first analysis of the degenerate spectrum was given for the symmetric quantum Rabi model by Kuś in 1985. In our picture, we find a certain reciprocity (or ℤ_{2}-symmetry) for ϵ ∈ 1/2 ℤ if the spectrum is described by representations of sI_{2}. We further discuss briefly the non-degenerate part of the exceptional spectrum from the viewpoint of infinite dimensional representations of sI_{2} (R) having lowest weight vectors.

Original language | English |
---|---|

Article number | 174001 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 50 |

Issue number | 17 |

DOIs | |

Publication status | Published - Mar 24 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

**Symmetry of asymmetric quantum Rabi models.** / Wakayama, Masato.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 50, no. 17, 174001. https://doi.org/10.1088/1751-8121/aa649b

}

TY - JOUR

T1 - Symmetry of asymmetric quantum Rabi models

AU - Wakayama, Masato

PY - 2017/3/24

Y1 - 2017/3/24

N2 - The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum Rabi model by Lie algebra representations of sI2. We define a second order element of the universal enveloping algebra U(sI2) of sI2 (R), which, through the action of a certain infinite dimensional representation of sI2 (R), provides a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we prove the existence of level crossings in the spectral graph of the asymmetric quantum Rabi model when the symmetry-breaking parameter ϵ is equal to 1/2 , and conjecture a formula that ensures likewise the presence of level crossings for general ϵ ∈ 1/2 ℤ. This result on level crossings was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by Braak (2011). The first analysis of the degenerate spectrum was given for the symmetric quantum Rabi model by Kuś in 1985. In our picture, we find a certain reciprocity (or ℤ2-symmetry) for ϵ ∈ 1/2 ℤ if the spectrum is described by representations of sI2. We further discuss briefly the non-degenerate part of the exceptional spectrum from the viewpoint of infinite dimensional representations of sI2 (R) having lowest weight vectors.

AB - The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum Rabi model by Lie algebra representations of sI2. We define a second order element of the universal enveloping algebra U(sI2) of sI2 (R), which, through the action of a certain infinite dimensional representation of sI2 (R), provides a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we prove the existence of level crossings in the spectral graph of the asymmetric quantum Rabi model when the symmetry-breaking parameter ϵ is equal to 1/2 , and conjecture a formula that ensures likewise the presence of level crossings for general ϵ ∈ 1/2 ℤ. This result on level crossings was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by Braak (2011). The first analysis of the degenerate spectrum was given for the symmetric quantum Rabi model by Kuś in 1985. In our picture, we find a certain reciprocity (or ℤ2-symmetry) for ϵ ∈ 1/2 ℤ if the spectrum is described by representations of sI2. We further discuss briefly the non-degenerate part of the exceptional spectrum from the viewpoint of infinite dimensional representations of sI2 (R) having lowest weight vectors.

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U2 - 10.1088/1751-8121/aa649b

DO - 10.1088/1751-8121/aa649b

M3 - Article

AN - SCOPUS:85016929801

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 17

M1 - 174001

ER -