$\theta_B$ is close to $\pi$): $$\begin{aligned}
\theta_B \approx \frac{S^{xy}}{4\sigma^{xx}S^{xx}}\end{aligned}$$ in a manner that is largely insensitive to the details of non-universal features such as composite Fermi velocity and impurity scattering rates. Here, we remind that all conductivities are written in units ${e^2}/{\hbar}$.

![\[fig:NernstPlot\] [**Field dependence of $S^{xy}$ -** ]{} In addition to whether $S^{xy}$ vanishes at half-filling ($B=B_{1/2}$), the asymmetric field dependence of the transverse thermopower (Nernst) signal, distinguishes between the HLR state with zero Berry phase ($\gamma=0$), and the particle-hole symmetric Dirac state with $\pi$ Berry phase ($\gamma=1$).[]{data-label="fig:NernstField"}](NernstPlot.pdf){width="0.7\columnwidth"}

[[**Field detuning –**]{}]{} We have shown that the Nernst coefficient at half-filling provides a sensitive probe for the composite Fermi surface Berry phase. In addition, the Dirac and HLR states also show distinct Nernst signals upon tuning the field away from half-filling, $B\neq B_{1/2}\equiv 4\pi n_{el}$, where $n_{el}$ is the electron density. For $B\neq B_{1/2}$, an effective gauge-magnetic field $b = B-B_{1/2}$ arises, which induces a change in the CF Hall conductance. For non-zero composite FS Berry phase $\theta_B=\pi\gamma$, the field-detuning alters the CF density by: $n_\text{cf} = \frac{B-(1-\gamma)b}{4\pi}$, and the composite Fermi wavevector is $k_F = \sqrt{4\pi n_\text{cf}}$. Let us work in the limit of small detuning, $b\ll B_{1/2}$, where 