look for more sensitive tests. To this end, we investigate the thermoelectric response for composite Fermi-liquid phases for the half-filled Landau level.

Whereas longitudinal thermopower typically vanishes in a system with local PHS, we find that, due to the non-local realization of PHS in the half-filled LL, the Seebeck coefficient obtains a similar non-zero value in both particle-hole symmetric composite Dirac liquid and the particle-hole asymmetric HLR state, and does not qualitatively distinguish these two states. In contrast, we find that the transverse Seebeck coefficient (or closely related Nernst coefficient), provides a more sensitive probe for the degree of particle-hole asymmetry. Namely, in clean systems, this quantity is sensitive to the Berry phase of the Fermi-surface, vanishes in the maximally particle-hole asymmetric CFL state without Berry phase, and achieves a maximum value for the particle-hole-symmetric composite Dirac liquid. We show that a non-zero Nernst coefficient directly results from the composite Fermi-surface Berry phase and represents a qualitative signature of the composite Fermi surface topology. Moreover, we show how the Nernst effect in combination with conductivity and thermopower measurements can be used to quantitatively extract the composite Fermi-surface Berry phase.

[[**Thermoelectric coefficients and PH Symmetry –**]{}]{} We begin by briefly recounting 