Emergent particle-hole symmetry in the half-filled Landau level

We provide an effective description of a particle-hole symmetric state of electrons in a half-filled Landau level, starting from the traditional approach pioneered by Halperin, Lee, and Read [ Phys. Rev. B 47, 7312 ( 1993)]. Specifically, we study a system consisting of alternating quasi-one-dimensional strips of composite Fermi liquid ( CFL) and composite hole liquid ( CHL), both of which break particle-hole symmetry. When the CFL and CHL strips are identical in size, the resulting state is manifestly invariant under the combined action of a particle-hole transformation with respect to a single Landau level ( which interchanges the CFL and CHL) and translation by one unit, equal to the strip width, in the direction transverse to the strips. At distances long compared to the strip width, we demonstrate that the system is described by a Dirac fermion coupled to an emergent gauge field, with an antiunitary particle-hole symmetry, as recently proposed by Son [ Phys. Rev. X 5, 031027 ( 2015)].