Pseudospin Berry phase as a signature of nontrivial band topology in a two-dimensional system

Electron motion in crystals is governed by the coupling between crystal momentum and internal degrees of freedom such as spin implicit in the band structure. The description of this coupling in terms of a momentum-dependent effective field and the resultant Berry phase has greatly advanced our understanding of diverse phenomena including various Hall effects and has lead to the discovery of new states of matter exemplified by topological insulators. While experimental studies on topological systems have focused on the gapless states that emerge at the surfaces or edges, the underlying nontrivial topology in the bulk has not been manifested. Here we report the observation of Berry's phase in magneto-oscillations and quantum Hall effects of a coupled electron-hole system hosted in quantum wells with inverted bands. In contrast to massless Dirac fermions in graphene, for which the Berry phase $\Gamma$ is quantized at $\pi$, we observe that $\Gamma$ varies with the Fermi level $E_\mathrm{F}$, passing through $\pi$ as $E_\mathrm{F}$ traverses the energy gap that opens due to electron-hole hybridization. We show that the evolution of $\Gamma$ is a manifestation of the pseudospin texture that encodes the momentum-dependent electron-hole coupling and is therefore a bulk signature of the nontrivial band topology.