the result to zero as demanded by the equations of motion. Note that, here and throughout the following discussion, unless explicitly stated otherise, we fix the magnetic field to be in the $+z$ direction. Expressions for the oppositely directed field can be obtained from our formulas by changing ${\epsilon}\rightarrow -{\epsilon}$.

Just as electrons see Dirac CF’s as $4\pi$ flux, Dirac CF’s see electrons as $-4\pi$ flux (see Fig. \[fig:Duality\] inset), and repeating the above arguments for a flow of electrons rather than CF’s shows that an electrical current produces a transverse emergent electric field ${\mathbf{e}}=4\pi \hat{\epsilon}{\mathbf{j}}_{el}$. In addition, due to the half-integer Hall conductance ($AdA$ term in Eq. (\[eq:Leff\])) ${\mathbf{j}}_{el}$ also induces an physical electric field ${\mathbf{E}}=-4\pi\hat{\epsilon}{\mathbf{j}}_{el}$. Together these imply: $$\begin{aligned}
{\mathbf{j}}_{el} = \frac{1}{4\pi}\hat{\epsilon}{\left(}{\mathbf{E}}-{\mathbf{e}}{\right)}\label{eq:je}\end{aligned}$$ Since the electron system has both longitudinal and Hall conductance, we expect this electric field along $y$ to be accompanied by one along $x$, indicating non-vanishing thermopower coefficients, $\alpha_{xx},\alpha_{xy}$.

To see how this works in detail we must understand how composite Dirac fermion currents ${\mathbf{j}}_\text{D}$ are generated. In the linear-response regime, the Fermi-surface of CF’s responds to a emergent ${\mathbf{e}}$-fields and temperature gradients as: $$\begin{aligned}
{\mathbf{j}}_\text{D} = \hat\sigma_\text{D} {\mathbf{e}}-\hat\alpha_\text{D}\nabla T
\label{eq:cfresponse}\end{aligned}$$ We note in passing that 