Tunable orbital susceptibility in $alpha$-${cal T}_3$ tight-binding models

We study the importance of interband effects on the orbital susceptibility of three bands $alpha$-${cal T}_3$ tight-binding models. The particularity of these models is that the coupling between the three energy bands (which is encoded in the wavefunctions properties) can be tuned (by a parameter $alpha$) without any modification of the energy spectrum. Using the gauge-invariant perturbative formalism that we have recently developped, we obtain a generic formula of the orbital susceptibility of $alpha$-${cal T}_3$ tight-binding models. Considering then three characteristic examples that exhibit either Dirac, semi-Dirac or quadratic band touching, we show that by varying the parameter $alpha$ and thus the wavefunctions interband couplings, it is possible to drive a transition from a diamagnetic to a paramagnetic peak of the orbital susceptibility at the band touching. In the presence of a gap separating the dispersive bands, we show that the susceptibility inside the gap exhibits a similar dia to paramagnetic transition.