We formulate the non-Abelian Berry connection (tensor $\mathbb R$) and phase (matrix $\boldsymbol \Gamma$) for a multiband system and apply them to semiconductor holes under the coexistence of Rashba and Dresselhaus spin-orbit interactions. For this purpose, we focus on the heavy-mass holes confined in a SiGe two-dimensional quantum well, whose electronic structure and spin texture are explored by the extended $\boldsymbol{k}\cdot\boldsymbol{p}$ approach. The strong intersubband interaction in the valence band causes quasi-degenerate points except for point $\Gamma$ of the Brillouin zone center. These points work as the singularity and change the Abelian Berry phase by the quantization of $\pi$ under the adiabatic process. To explore the influence by the non-adiabatic process, we perform the contour integral of $\mathbb R$ faithfully along the equi-energy surface by combining the time-dependent Schr\"{o}dinger equation with the semi-classical equation-of-motion for cyclotron motion and then calculate the energy dependence of $\boldsymbol \Gamma$ computationally. In addition to the function as a Dirac-like singularity, the quasi-degenerate point functions in enhancing the intersubband transition via the non-adiabatic process. Consequently, the off-diagonal components generate both in $\mathbb R$ and $\boldsymbol \Gamma$, and the simple $\pi$-quantization found in the Abelian Berry phase is violated. More interestingly, these off-diagonal terms cause "resonant repulsion" at the quasi-degenerate energy and result in the discontinuity in the energy profile of $\boldsymbol \Gamma$.

翻译：我们构建了多能带系统的非阿贝尔贝里联络（张量$\mathbb R$）和相位（矩阵$\boldsymbol \Gamma$），并将其应用于Rashba与Dresselhaus自旋轨道耦合共存下的半导体空穴系统。为此，我们聚焦于约束在SiGe二维量子阱中的重空穴，通过扩展的$\boldsymbol{k}\cdot\boldsymbol{p}$方法探讨其电子结构和自旋织构。价带中强烈的子带间相互作用导致除布里渊区中心$\Gamma$点外存在准简并点。这些点作为奇点，在绝热过程中通过$\pi$量子化改变阿贝尔贝里相位。为探究非绝热过程的影响，我们结合含时薛定谔方程与回旋运动的半经典运动方程，沿等能面对$\mathbb R$进行严格围道积分，并计算$\boldsymbol \Gamma$随能量的变化规律。准简并点除具有类狄拉克奇点功能外，还通过非绝热过程增强子带间跃迁。因此，$\mathbb R$与$\boldsymbol \Gamma$中均产生非对角分量，阿贝尔贝里相位中发现的简单$\pi$量子化被打破。更有趣的是，这些非对角项在准简并能级处引发"共振排斥"，导致$\boldsymbol \Gamma$能量分布出现不连续性。