$E\otimes e$ Jahn-Teller模型Berry曲率的渐近分析
Asymptotic analysis of the Berry curvature in the $E\otimes e$ Jahn-Teller model
摘要：
线性$E\otimes e$ Jahn-Teller模型的有效哈密顿量描述了分子或大块晶体杂质中两个电子态和两个振动模式之间的耦合。在Born-Oppenheimer近似下，Berry曲率在势能面的圆锥交点处具有δ函数奇异性，而精确的Berry曲率是光滑的尖峰函数.数值计算表明，该峰的特征宽度为$\hbar K^{1/2}/gM^{1/2}$，其中$M$是与相关核坐标相关的质量，$K$是有效核间弹性常数，$g$是电子-振动耦合.这一结果在这里证实了渐近分析的$M\rightarrow\infty$限制，一个有趣的结果是出现了分离的长度尺度。基于精确的电子-核因式分解，我们的分析不涉及绝热势能面或非绝热耦合。它还表明，该模型的Ham减少因子可以从精确的几何相位。
英文摘要：
The effective Hamiltonian for the linear $E\otimes e$ Jahn-Teller model describes the coupling between two electronic states and two vibrational modes in molecules or bulk crystal impurities. While in the Born-Oppenheimer approximation the Berry curvature has a delta function singularity at the conical intersection of the potential energy surfaces, the exact Berry curvature is a smooth peaked function. Numerical calculations revealed that the characteristic width of the peak is $\hbar K^{1/2}/gM^{1/2}$, where $M$ is the mass associated with the relevant nuclear coordinates, $K$ is the effective internuclear spring constant and $g$ is the electronic-vibrational coupling. This result is confirmed here by an asymptotic analysis of the $M\rightarrow\infty$ limit, an interesting outcome of which is the emergence of a separation of length scales. Being based on the exact electron-nuclear factorization, our analysis does not make any reference to adiabatic potential energy surfaces or nonadiabatic couplings. It is also shown that the Ham reduction factors for the model can be derived from the exact geometric phase.
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