We have investigated the behavior of the resistance of graphene at the $n=0$ Landau Level in an intense magnetic field $H$. Employing a low-dissipation technique (with power $P<$3 fW), we find that, at low temperature $T$, the resistance at the Dirac
point $R_0(H)$ undergoes a 1000-fold increase from $sim$10 k$Omega$ to 40 M$Omega$ within a narrow interval of field. The abruptness of the increase suggests that a transition to an insulating, ordered state occurs at the critical field $H_c$. Results from 5 samples show that $H_c$ depends systematically on the disorder, as measured by the offset gate voltage $V_0$. Samples with small $V_0$ display a smaller critical field $H_c$. Empirically, the steep increase in $R_0$ fits acccurately a Kosterlitz-Thouless-type correlation length over 3 decades. The curves of $R_0$ vs. $T$ at fixed $H$ approach the thermal-activation form with a gap $Deltasim$15 K as $Hto H_c^{-}$, consistent with a field-induced insulating state.
The thermal Hall conductivity $kappa_{xy}$ and Hall conductivity $sigma_{xy}$ in CeCoIn$_5$ are used to determine the Lorenz number ${cal L}_H$ at low temperature $T$. This enables the separation of the observed thermal conductivity into its electron
ic and non-electronic parts. We uncover evidence for a charge-neutral, field-dependent thermal conductivity, which we identify with spin excitations. At low $T$, these excitations dominate the scattering of charge carriers. We show that suppression of the spin excitations in high fields leads to a steep enhancement of the electron mean-free-path, which leads to an interesting scaling relation between the magnetoresistance, thermal conductivity and $sigma_{xy}$.
