subtracting bound magnetization current contributions, see Appendix \[app:PartonResponse\]).

[lCCC CCC]{} & $\sigma^{xx}$ & $\sigma^{xy}$& $\alpha^{xx}$ & $\alpha^{xy}$& $S^{xx}$ & $S^{xy}$\
HLR & $\dfrac{\sigma_0}{1+{\left(}4\pi\sigma_0{\right)}^2}$ &$\dfrac{1}{4\pi}\dfrac{(4\pi\sigma_0)^2}{1+(4\pi\sigma_0)^2}$ & $\dfrac{\alpha_0}{1+(4\pi\sigma_0)^2}$&$\dfrac{4\pi\sigma_0\alpha_0}{1+(4\pi\sigma_0)^2}$&$\sigma_0^{-1}\alpha_0$&$0$\
Dirac& $\dfrac{1}{{\left(}4\pi{\right)}^2\sigma_0}$ & $\dfrac{1}{4\pi}$ & $0$ & $\dfrac{\alpha_0}{4\pi \sigma_0}$ & $\dfrac{(4\pi)^2\sigma_0\alpha_0}{1+(4\pi\sigma_0)^2}$ & $ \dfrac{4\pi\alpha_0}{1+(4\pi\sigma_0)^2}$\

Eliminating the emergent electric field, ${\mathbf{e}}$ and ${\mathbf{j}}_\text{cf}$, from Eq. (\[eq:je\]) using Eqs. (\[eq:jD\]),(\[eq:cfresponse\]), we find: ${\mathbf{j}}_{el} = {\left[}\frac{1}{4\pi}\hat{\epsilon}+\frac{\rho_0}{{\left(}4\pi{\right)}^2}{\right]}{\mathbf{E}}-{\left[}\frac{1}{4\pi}\rho_0\alpha_0\hat{\epsilon}\nabla T{\right]}$, from which we can identify the electronic conductivity and thermoelectric susceptibility for the composite Dirac liquid: $$\begin{aligned}
\hat\sigma &= \frac{\rho_0}{{\left(}4\pi{\right)}^2}\hat{1}
+\frac{1}{4\pi}\hat{\epsilon}\hspace{0.25in}
\hat\alpha = \frac{\rho_0\alpha_0}{4\pi}\hat{\epsilon}\label{eq:PHSresponse}\end{aligned}$$ where $\rho_0 = \sigma_0^{-1}$ is the resistivity of the composite Fermi surface, $i,j\in\{x,y\}$.

Rather than observing $\hat\alpha$ directly, one typically measures the electric field, ${\mathbf{E}}=\hat S\nabla T$, induced by a thermal gradient in a geometry where no electric current is permitted to flow: ${\mathbf{j}}_{el} = \hat\sigma {\mathbf{E}}-\hat\alpha\nabla T=0$, which defines the Seebeck tensor: $$\begin{aligned}
\hat S &= \hat\sigma^{-1}\hat\alpha = \frac{\rho_0\alpha_0}{1+{\left(}\rho_0/4\pi{\right)}^2}{\left(}\hat 1+\frac{\rho_0}{4\pi}\hat{\epsilon}{\right)}\end{aligned}$$ For a clean system ($k_F\ell \gg 1$), ${\rho_\text{0}}/({4\pi})\ll 1$, and the diagonal Seebeck coefficient $S^{xx}\approx \rho_0\alpha_0$, is essentially equal to the effective Seebeck coefficient that would be obtained by viewing the composite fermions as electrons, in reasonable quantitative agreement with previous analysis of thermopower measurements in GaAs [@Ying94; @Bayot95; @Cooper97]. Note that this large Seebeck 