Minimal conductivity and signatures of quantum criticality in ballistic graphene bilayer

We study the ballistic conductivity of graphene bilayer in the presence of next-nearest neighbor hoppings between the layers. An undoped and unbiased system was found in Ref. [1] to show a nonuniversal (length-dependent) conductivity $\sigma(L)$, approaching the value of $\sigma_\star=3/\pi\simeq{}0.95$ for large $L$. Here we demonstrate one-parameter scaling and determine the scaling function $\beta(\sigma)=d\ln{}\!\sigma/d\ln{}\!L$. The scaling flow has an attractive fixed point [$\,\beta(\sigma_\star)=0$, $\beta'(\sigma_\star)<0\,$] reproducing the scenario predicted for random impurity scattering of Dirac fermions with Coulomb repulsion, albeit the system considered is perfectly ballistic and interactions are not taken into account. The role of electrostatic bias between the layers is also briefly discussed.