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Physical Review

Particle-Hole Inversion in the Fermi Surface: A Symmetry Operation for Collective Excitations

John C. Fisher

Published Jan 15, 1962• 1 citation
Publisher

Abstract

The operation of transposing particles (or holes) on one side of a Fermi surface into holes (or particles) on the other side is shown to be a symmetry operation of the Bardeen, Cooper, and Schrieffer reduced Hamiltonian, and an approximate symmetry operation of the full two-particle scalar interaction Hamiltonian, insofar as particle-conserving collective excitations are concerned. Calling this operation $f$ inversion, and the conserved quantity associated with it $f$ parity, the zero-rest-energy collective excitations in a Fermi gas with weak attractive interactions prove to have $f\mathrm{parity}=\ensuremath{-}1$.

DOI
10.1103/physrev.125.492

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The operation of transposing particles (or holes) on one side of a Fermi surface into holes (or particles) on the other side is shown to be a symmetry operation of the Bardeen, Cooper, and Schrieffer reduced Hamiltonian, and an approximate symmetry operation of the full two-particle scalar interaction Hamiltonian, insofar as particle-conserving collective excitations are concerned. Calling this operation fff inversion, and the conserved quantity associated with it fff parity, the zero-rest-energy collective excitations in a Fermi gas with weak attractive interactions prove to have fparity=\ensuremath−1f\mathrm{parity}=\ensuremath{-}1fparity=\ensuremath−1.
Abstract In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface as bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.
We prove that the random phase approximation provides an upper bound on the correlation energy of a three-dimensional, weakly interacting, Fermi gas in the thermodynamic limit at high density. More precisely, we consider a mean-field scaling regime where the kinetic and interaction Hamiltonian are scaled by ρ−2/3ρ^{-2/3}ρ−2/3 and ρ−1ρ^{-1}ρ−1, respectively. We use a correlated trial state constructed through a quasi-bosonic Bogoliubov transformation of collective particle--hole excitations associated to patches on the Fermi surface. Key of the proof is to control the position-space localization of the collective excitations. Combined with an iterated Duhamel expansion of the expectation values of (localized) particle number operators we obtain extensive estimates, which justify the reduction to an effective quadratic Hamiltonian and its approximate diagonalization.
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 offer a heuristic argument 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 [Phys. Rev. X 5, 031027 (2015)] 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.
The low-energy modes of a superfluid atomic Fermi gas at zero temperature are investigated. The Bose-Einstein-condensate (BEC) side of the superfluid phase is studied in detail. The atoms are assumed to be in only one internal state, so that for a sufficiently diluted gas the pairing of fermions can be considered effective in the l=1 channel only. In agreement with previous works on p-wave superfluidity in Fermi systems, it is found that the px+ipyp_x+ip_ypx​+ipy​ phase represents the lowest energy state in both the Bardeen-Cooper-Schrieffer (BCS) and BEC sides. Our calculations show that at low energy three branches of collective modes can emerge, with different species of dispersion relations: a phonon-like mode, a single-particle-like mode and a gapped mode. A comparison with the Bogoliubov excitations of the corresponding spinor Bose condensate is made. They reproduce the dispersion relations of the excitation modes of the p-wave superfluid Fermi gas to a high accuracy.
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 offer a heuristic argument 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 [Phys. Rev. X 5, 031027 (2015)] 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.