the inter-LL spacing $\omega_c\approx \frac{eB}{m^*c}$ is much larger than the interaction energy $E_C\approx \frac{e^2 k_F}{{\epsilon}_r}$, one expects only weak inter-LL mixing and hence an approximate PHS. In this case, the composite Fermi-surface Berry phase is not quantized precisely to $\pi$, but should still deviate from $\pi$ only by a small amount $\approx \frac{E_C}{\omega_c}\ll 1$. Given this approximate PHS, it is natural to expect that the composite Fermi surface in 1/2-filled GaAs is approximately described by the composite Dirac theory, and may have appreciable Berry phase near $\pi$.

To compare the thermoelectric responses of the particle-hole symmetric composite Dirac liquid, obtained above, to those of a particle-hole asymmetric state, we can include a particle-hole symmetry breaking Dirac mass ($m\neq 0$) in Eq. (\[eq:Leff\]), which produces a half-integer Hall conductance for the gauge field $a$. Physically, this PH breaking mass arises from inter-LL mixing, and we expect $m\approx \frac{E_C}{\omega_c}$.

For non-zero PHS breaking mass, $m\neq 0$, the completely filled valence band of the composite Dirac cone contributes 1/2-integer quantized Hall conductance $\sigma_{D,v}^{xy}=\frac{\text{sgn}(m)}{4\pi}$, and the non-trivial Berry curvature enclosed by the conduction Fermi surface (FS) gives anomalous CF Hall conductance, $\sigma^{xy}_{D,\text{FS}}=\int_{k<k_F}\frac{d^2k}{(2\pi)^2}\Omega_k  = -\frac{\text{sgn}(m)}{4\pi}\gamma$, where $\Omega_k = \frac{m}{2{\left[}(vk)^2+m^2{\right]}^{3/2}}$ is the conduction band Berry curvature, 