The symmetry protected Dirac spectrum near the charge neutrality has been a source of exotic physical behaviors observed in graphene. In a disorder-free clean sample, attained by encapsulating graphene in between hexa-boron nitride, we can tune the Fermi energy to be much smaller than any relevant energy scales. In this presentation, we will discuss several examples where extremely small Fermi energy plays a critical role to correlate electrons in graphene. Examples include splitting superconducting Cooper pairs into a pair of electron and hole across the charge neutrality point when the Fermi energy of graphene becomes smaller than superconducting energy gap. Another example is the demonstration of a drastic enhancement of the electronic thermal conductivity at the charge neutrality as the Fermi energy becomes smaller than thermal energy, suggesting the realization of the relativistic hydrodynamic transport.