Evolution of the Berry phase and topological properties in models for merging Dirac cones

Abstract

In this paper we study the evolution of the Berry phase ( Φ B ) for cases that exhibit merging of the Dirac cones in models of massless Dirac fermions . The Berry phase captures the band topology and hence facilitates prediction of the topological character of the system. Motivated by such a scenario we consider three different systems, namely, (i) an artificial model for graphene with a constant parameter, tuning of which makes the two Dirac cones merge (Montambaux et al., 2009), (ii) a generic case for graphene where all the three hopping amplitudes (say, t a , t b and t c ) are different and in such systems merging of Dirac cones are demonstrated when the hopping amplitudes satisfy a triangle inequality (Hasegawa et al., 2006), and finally (iii) in a Haldane model where the merging of the cones can be engineered as a function of the band deformation. We find that Φ B discontinuously jumps from a value π to zero at the point where the Dirac cones merge in (i) and (ii), while (iii) exhibits a different scenario where Φ B changes from 2 π to π . The result is surprising, since except for the topological gap that exists in (iii), the band structures for the cases (i) and (ii) display similar features where two of the hopping amplitudes (say t b and t c ) are equal and the third one (say t a ) is continuously varied with respect to other two. Thus the Haldane flux enforces a non-relativistic character to the otherwise relativistic Dirac fermions. Hence there is an interplay between the presence of the topological gap in the Haldane model with the Berry phase. Finally we show the evolution of the topological properties of the Haldane model by computing the Berry curvature and the Chern number where the latter shows the gradual vanishing of the topological regime as long as one of the hopping amplitudes remain smaller than twice of the other two. Since a large number of two dimensional materials exhibit topological character, and quite a few of them undergo topological phase transitions, our work will enable a simple, yet robust technique to study these systems. • Topological aspects and electronic properties of graphene • Haldane model • Chern insulators.