Conformal symmetry of electron-hole puddles in ungated graphene

In this paper the mono-layer graphene at the charge neutrality point is considered whithin Thomas-Fermi-Dirac theory, treating inhomogeneous external potentials and electron-electron interactions on equal footing. We present some general considerations concerning the probability measure of the ground state charge density. The system shows degrees of self-similarity. By analyzing the ground state carrier density profile, we show that although it is not Gaussian, the critical exponents are consistent with Kondev hyper-scaling relations. Using Schramm-Loewner (SLE) evolution we show that the ungated graphene has conformal invariance and the random zero-charge density contours are SLE$_{\kappa}$ with $\kappa=1.8\pm 0.2$.