Symmetry of asymmetric quantum Rabi models
Symmetry of asymmetric quantum Rabi models
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DOI：
10.1088/1751-8121/aa649b
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发表时间：
2017-01
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影响因子：
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通讯作者：
M. Wakayama
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文献类型：
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作者：
M. Wakayama
本文的目的是利用sl ~ 2的李代数表示来更好地理解非对称量子Rabi模型的本征态。我们定义了sl ~ 2（R）的泛包络代数U（sl ~ 2）的一个二阶元，它通过sl ~ 2（R）的一个无限维表示的作用，提供了一幅非对称量子Rabi模型的图像，等价于由合流Heun常微分方程绘制的图像.利用这一描述，我们证明了当破缺参数λ = 12时，非对称量子Rabi模型谱图中能级交叉的存在性，并猜想了一个公式，该公式同样保证了一般λ ∈ 12 Z时能级交叉的存在性. Li和Batchelor在2015年用数值方法证明了这一结果，并调查了Braak（2011）早期的经验观察。1985年，Kubernetes首次对对称量子Rabi模型的简并谱进行了分析。在我们的图像中，如果谱由sl 2的表示描述，我们发现对于n ∈ 12 Z有一定的互易性（或Z2-对称性）。我们进一步从sl ~ 2（R）具有最低权向量的无穷维表示的观点，简要地讨论了例外谱的非退化部分。
The aim of this paper is a better understanding for the eigenstates of the asymmetric quantum Rabi model by Lie algebra representations of sl2. We define a second order element of the universal enveloping algebra U(sl2) of sl2(R), which, through the action of a certain infinite dimensional representation of sl2(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 12, and conjecture a formula that ensures likewise the presence of level crossings for general ϵ∈12Z. 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∈12Z if the spectrum is described by representations of sl2. We further discuss briefly the non-degenerate part of the exceptional spectrum from the viewpoint of infinite dimensional representations of sl2(R) having lowest weight vectors.