Zeeman field is generated by Kondo exchange with the Pr moments, so $H \sim J_K M$, with $M$ the (dimensionless) Pr magnetization, which implies a highly unconventional sublinear dependence of $\sigma_{xy}$ on $M$ for the pristine LAB. If the Fermi level is displaced from the band touching by an amount $\epsilon_F$, then we expect, treating the above power as a square root, $\sigma_{xy} \sim
\frac{e^2}{h} \sqrt{\frac{m H}{\hbar^2}} \mathcal{S}(H/\epsilon_F)$, where $\mathcal{S}$ is a scaling function (see Supp. Mat.). This gives an order of magnitude quantitative estimate $$\label{eq:8}
  \sigma_{xy} \sim 10^3 \Omega^{-1} cm^{-1} \times
  \left\{ \begin{array}{cc} \sqrt{\frac{m}{m_e}}  \sqrt{\frac{J_K}{\rm Ry}}
      \sqrt{M}, & J_K M \gg \epsilon_F \\
      \sqrt{\frac{m}{m_e}} \frac{J_K}{\sqrt{\epsilon_F {\rm Ry}}} M,
      &  J_K M \ll \epsilon_F \end{array}\right. \; ,$$ where Ry=13.6eV is the Rydberg, and $m_e$ is the electron mass. The lower regime gives $\sigma_{xy} \sim 0.1 \Omega^{-1} cm^{-1}$, within an order of magnitude of observations in Pr$_2$Ir$_2$O$_7$,[@machida] for parameters $m \sim 20m_e$ (estimated from the calculations in Ref. ) , $\epsilon_F \sim $10meV, $J_K \sim$100K (estimated from the measured Curie-Weiss temperature), and $M \sim 0.01$.

Another interesting experimental observation in Pr$_2$Ir$_2$O$_7$ is a diverging magnetic Gruneisen number $\Gamma_H = \frac{1}{T}
\left. \frac{\partial T}{\partial H}\right|_S$ in the zero field limit [@si; @gruneisen; 