Chern-Simons gauge field that breaks particle-hole symmetry (PHS) – which we will henceforth refer to as the “HLR state". While it is in principle possible that the experimental system chooses to spontaneously break PHS and form the HLR state, this option does not appear to be energetically favored in numerical simulations [@Geraedts15].

It was recently realized [@Son15; @Wang15short; @Metlitski15; @Wang15rev; @MetlitskiSDuality; @Mross15] that the HLR picture could be modified to produce a manifestly particle-hole symmetric candidate phase for the 1/2-filled Landau level – the composite Dirac liquid (CDL). Like the HLR state, the CDL has a Fermi-surface of flux-like composite fermion excitations. However unlike the original HLR description in terms of spinless electrons bound to flux, these composite fermions are neutral Dirac-like particles that possess a pseudospin-1/2 degree of freedom, whose $x,y$ components are physically interpreted as an electrical dipole moment. This pseudospin is rigidly locked perpendicular to the CF momentum in close analogy to the Dirac surface state of a topological insulator (TI) [@Son15; @Wang15short; @Metlitski15; @Wang15rev]. The resulting pseudospin winding implies a Berry phase $\theta_{\rm B} = \pi$ whose value is fixed by PHS. On the face of it, the appearance of a single-Dirac cone in a purely 