Exact Solution of a Many-Fermion System and Its Associated Boson Field

1965/02/01 by Daniel C. Mattis, Élliott H. Lieb · 1 voice · 13 citations
Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Quantum and electron transport phenomena #Topological Materials and Phenomena

paper · doi:10.1063/1.1704281

openalex publication_date 1965/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26

Abstract

Luttinger's exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [ρ(p), ρ(−p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators ρ(p) define a boson field which is ipso facto associated with the Fermi-Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter n̄k, and find: ∂n̄k/∂k|kF = ∞ (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.

Citations

• A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas

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• Rigorous construction of the Thirring model: Ward-Takahashi Identities,\n Schwinger-Dyson Equations and New Anomalies
• Bosonizing one-dimensional cold atomic gases

Discussions

• The non-zero commutator arises from the Dirac sea. It is a non-zero $c$ -number precsely because all the deep-lying negative states are full and the high postive energy states are empty. It is worth r [stackexchange, 5 points]

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