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By quantizing the semiclassical motion of excitons, we show that the Berrycurvature can cause an energy splitting between exciton states with oppositeangular momentum. This splitting is determined by the Berry curvature fluxthrough the $\bm k$-space area spanned by the relative motion of theelectron-hole pair in the exciton wave function. Using the gappedtwo-dimensional Dirac equation as a model, we show that this splitting can beunderstood as an effective spin-orbit coupling effect. In addition, there isalso an energy shift caused by other "relativistic" terms. Our result revealsthe limitation of the venerable hydrogenic model of excitons, and highlightsthe importance of the Berry curvature in the effective mass approximation.

Berry phase modification to the energy spectrum of excitons