  Single-layer carbon, or graphene, demonstrates amazing transport properties,
such as the minimum conductivity near $\frac{4e^2}{h}$ independent of shapes
and mobility of samples. This indicates there exist some unusual effects due to
specific Dirac dispersion relation of fermion in two dimensions. By deriving
fermion-lattice interaction Hamiltonian we show that Berry phases can be
produced in fermion states around two Dirac points by relative rotations of two
sublattices. The Berry phases in turn remove the degeneracies of energies for
states near the Fermi surface, leading to a dynamical instability of the
lattice with respect to the rotations. By considering the Berry phases emerging
in an uncertain way on fermion wavefunctions in vicinities of the Fermi
surface, the conductivity is calculated by using the Landauer-B\"{u}tticker
formula together with the transfer-matrix technique, verifying $\sim
\frac{4e^2}{h}$ quantized minimum conductivity as observed in experiments
independent of shapes and sizes. The relationship between the chaotic structure
of fermions due to the Berry phases and the classical transport properties are
discussed. The physical meaning is profound as this relationship provides an
excellent example to elucidate the mechanism of quantum-classical transition.
