  The term "particle-hole symmetry" is beset with conflicting meanings in
contemporary physics. Conceived and written from a condensed-matter standpoint,
the present paper aims to clarify and sharpen the terminology. In that vein, we
propose to define the operation of "particle-hole conjugation" as the
tautological algebra automorphism that simply swaps single-fermion creation and
annihilation operators, and we construct its invariant lift to the Fock space.
Particle-hole symmetries then arise for gapful or gapless free-fermion systems
at half filling, as the concatenation of particle-hole conjugation with one or
another involution that reverses the sign of the first-quantized Hamiltonian.
We illustrate that construction principle with a series of examples including
the Su-Schrieffer-Heeger model and the Kitaev-Majorana chain. For an enhanced
perspective, we contrast particle-hole symmetries with the charge-conjugation
symmetry of relativistic Dirac fermions. We go on to present two major
applications in the realm of interacting electrons. For one, we argue that the
celebrated Haldane phase of antiferromagnetic quantum spin chains is
adiabatically connected to a free-fermion topological phase protected by a
particle-hole symmetry. For another, we review the recent proposal by Son for a
particle-hole conjugation symmetric effective field theory of the half-filled
lowest Landau level, and we comment on the emerging microscopic picture of the
composite fermion.
