curvature, then the low-energy composite Fermi surface dynamics are effectively time-reversal symmetric such that $\hat\rho_\text{cf}$, $\hat\alpha_\text{cf}$ and hence also $\hat S$ to be purely diagonal (see Table \[t:summary\] for detailed comparison between HLR and Dirac).

We note that inelastic scattering from gauge fluctuations, whose propagator contains a Chern-Simons term and hence does not respect this fine-tuned spinless time-reversal, and can produce an off-diagonal component to $\sigma_{cf}$ and corresponding non-zero Nernst signal. However, neglecting extrinsic impurity effects, the transport scattering rate due to inelastic gauge scattering rate, $\Gamma_{g,\text{tr}}$, is expected to be suppressed at low temperatures by powers of $T$ (e.g. previous estimates [@Lee92; @HLR] give $\Gamma_{g,\text{tr}}\sim T^{2}$ for unscreened Coloumb interactions and $\Gamma_{g,\text{tr}}\sim T^{4/3}$ with interactions screened by a nearby gate). Then, the resulting gauge-fluctuation induced thermopower $S_g^{xy}\sim T \Gamma_{g,\text{tr}}$ can be distinguished by its temperature dependence from that arising from Berry phase in the composite Dirac liquid, which depends linearly on $T$. Thus, the absence of Berry curvature in the HLR state directly leads to a vanishing ($T$-linear component of) the Nernst signal.

[[**Effect of PHS breaking –**]{}]{} In experimental systems, inter-LL mixing is always present to some degree due to Coulomb interactions. However, at large fields for which 