the composite Fermi liquid state of the half-filled Landau level in GaAs [@Ying94; @Bayot95]. There, a sizable Seebeck coefficient was observed, whose magnitude agrees reasonably well with that of an ordinary Fermi liquid with the same density of particles as the composite Fermi-surface [@Cooper97]. Given the expectation that particle-hole symmetry results in vanishing thermopower, such observations naively seem to rule out a (even approximately) particle-hole symmetric description of the CFL in GaAs.

Can one reconcile the experimentally observed large thermopower with the naive expectation of ordinary particle-hole symmetry? The key to resolving this apparent contradiction, is to note that the PHS present in the half-filled LL is inherently non-local, involving not only exchange of particles and holes but also filling a LL $|0{\rangle}\rightarrow \prod_{n\in LL} c_n^\dagger|0{\rangle}$, where $c_n^\dagger$ create an electron in the $n^\text{th}$ orbital of the LL. For example this non-local PHS exchanges $\sigma^{xy}\rightarrow -\sigma^{xy}+1\frac{e^2}{h}$ requiring non-vanishing Hall conductance: $\sigma^{xy}=\frac{1}{2}\frac{e^2}{h}$ in a PHS state. Then, non-vanishing thermopower can arise in a system with non-local PHS, due to the non-vanishing combination $S^{xx}=\rho^{xy}\alpha^{xy}$, where $\hat{\rho}\equiv\hat\sigma^{-1}$ is the resistivity tensor. We note that non-local PHS still constrains $\alpha^{xx}=0$, as a filled Landau level makes no contribution to $\hat\alpha$. Given these basic symmetry 