coefficient arises despite the particle-hole symmetry of the composite Dirac liquid.

Moreover, we see that the off-diagonal component, $S^{xy}\approx {S^{xx}}/({k_F\ell})$ and closely related Nernst coefficient $\nu_N = S^{xy}/B$ are non-zero. We will shortly see that the non-vanishing $S^{xy}$ marks a sharp departure from the maximally particle-hole asymmetric HLR state with zero Berry phase, for which $S^{xy}_\text{HLR}=0$. More generally, we will show that the non-vanishing Nernst effect is directly tied to the Berry phase of the composite Fermi surface.

[[**Thermopower with zero Berry phase –**]{}]{} Composite fermions in the PHS breaking state originally described by HLR [@HLR] are spinless electrons bound to $-4\pi$ flux, and hence carry electrical charge (equivalently flux of ${\mathbf{a}}$) as well as vorticity. Then, in contrast to the Dirac liquid, the CF current in the HLR state is equal to the electron current, ${\mathbf{j}}_\text{cf} = {\mathbf{j}}_{el} = \frac{1}{4\pi}\hat{\epsilon}{\left(}{\mathbf{E}}-{\mathbf{e}}{\right)}$. In a thermopower measurement (i.e. in an open circuit configuration), zero electrical current flows, ${\mathbf{j}}_\text{el}=0$, implying ${\mathbf{E}}={\mathbf{e}}$, and ${\mathbf{j}}_\text{cf}=0=\hat\sigma_\text{cf}{\mathbf{e}}-\hat\alpha_\text{cf}\nabla T$, which together give ${\mathbf{E}} = \hat\rho_\text{cf}\hat\alpha_\text{cf}\nabla T$, i.e. the electron and CF thermopower tensors coincide: $\hat{S}_{\text{HLR}} = \hat\rho_\text{cf}\hat\alpha_\text{cf}$. Since the CFs in the HLR state experience no residual orbital magnetic field, and if the composite Fermi-surface has zero Berry 