Non-Abelian Berry phase for semiconductor heavy holes under the coexistence of Rashba and Dresselhaus spin-orbit interactions

We formulate the non-Abelian Berry connection (tensor R) and phase (matrix Γ) for a multiband system and apply them to semiconductor holes in the presence of Rashba and Dresselhaus spin-orbit interactions (SOIs). For this purpose, we focus on heavy-mass holes confined in a SiGe two-dimensional quantum well, whose electronic structure and spin texture are explored by the extended k·p approach. To explore the influence of the nonadiabatic process, we perform the contour integral of R faithfully along the equienergy surface by combining the time-dependent Schrödinger equation with the semiclassical equation of motion for a cyclotron and then calculate the energy dependence of Γ computationally. The intersubband interactions in the valence band strongly modifies the SOIs. Accordingly, holes conserve the spin quasidegeneracy at several specific points, where the interstate hybridization generates off-diagonal components both of R and Γ, and the simple π quantization found in the Abelian Berry phase is violated. Moreover, these off-diagonal terms cause "resonant repulsion"at the quasidegenerate energy. Consequently, HH± exhibits a discontinuity in the energy dependence of Γ.