  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 $\boldsymbol \Gamma$.

Mesoscale and Nanoscale Physics;Materials Science;Computational Engineering, Finance, and Science;Computational Physics;Quantum Physics