and $\gamma \equiv \theta_B/\pi=1-\frac{m}{\sqrt{(vk_F)^2+m^2}}$ is a convenient measure of the composite FS Berry phase, $\theta_B$, which interpolates between $0$ (HLR) and $1$ (Dirac). The non-zero Berry curvature also induces anomalous CF thermoelectric Hall conductivity [@Takehito11], generically related to the CF number conductivity by the Mott formula: $\alpha_\text{D}^{ij} \approx \frac{\footnotesize\L T}{v_F} \frac{{\partial}\sigma^{ij}}{{\partial}k_F}$[@AHERMP; @Xiao06; @Qin11].

Hence, for non-zero PHS breaking mass, $m$, $\sigma_D$ in Eq. (\[eq:cfresponse\]) becomes: $$\begin{aligned}
\hat\sigma_{D}(m\neq 0) = \sigma_0\hat 1 +\text{sgn}(m)\frac{(1-\gamma)}{4\pi}\hat{\epsilon}\end{aligned}$$ Then, proceeding as before by eliminating the internal electric field ${\mathbf{e}}$ using Eqs. (\[eq:jD\]),(\[eq:cfresponse\]), we find the electron conductivity and thermopower tensors for $m\neq 0$: $$\begin{aligned}
\sigma &= {\left[}{\left(}\hat\sigma_\text{D}-\frac{\hat{\epsilon}}{4\pi}{\right)}^{-1}-4\pi\hat{\epsilon}{\right]}^{-1}\nonumber\\
S &= {\left(}\hat\sigma_\text{D}-\frac{\hat{\epsilon}}{4\pi}{\right)}^{-1}\hat\alpha_\text{D}
\label{eq:PHbreakingresponse}\end{aligned}$$ The original HLR theory is recovered in the limit of zero Berry phase ($m\rightarrow \infty$). As previously remarked, in this limit the CF sector has a fine-tuned “spinless" time-reversal symmetry, such that $\sigma_\text{cf}^{xy}=0$ and $\alpha_\text{cf}^{xy}\approx \frac{\scriptsize{\L} T}{v_F}\frac{{\partial}\sigma_\text{cf}^{xy}}{{\partial}k_F}=0$, resulting in vanishing Nernst coefficient.

For clean systems, $k_F\ell\gg 1$, $\sigma_0\gg \frac{1}{4\pi}$, the Hall conductivity and Nernst coefficient for generic composite Fermi surface Berry phase at half-filling are: $$\begin{aligned}
\sigma^{xy} &\approx \frac{1}{4\pi}{\left(}1-\text{sgn}(m)\frac{1-\gamma}{{\left(}4\pi\sigma_0{\right)}^2}{\right)}\nonumber\\
S^{xy} &\approx \L T\frac{k_F}{4\pi \sigma_0 v_F}{\left[}\frac{{\partial}\ln\sigma_0}{{\partial}\ln k_F}+\text{sgn}(m)\frac{{\partial}\ln\gamma}{{\partial}\ln k_F}{\right]}\times \gamma
\label{eq:Nernst}\end{aligned}$$ where, for the massive Dirac model of Eq. (\[eq:Leff\]), $\frac{{\partial}\ln\gamma}{{\partial}\ln k_F} = (1-\gamma)(2-\gamma)$. Therefore, we see 