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WORLD SCIENTIFIC eBooks

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

Daniel Charles Mattis, Élliott H. Lieb

Published Dec 1, 1994• 3 citations
Publisher
DOI
10.1142/9789812812650_0008

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

Daniel Charles Mattis, Élliott H. Lieb

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 [p(p), p(−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 p(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 , and find: (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.

Series on directions in condensed matter physics•Oct 24, 2013•0 citations

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

Daniel Charles Mattis, Élliott H. Lieb

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.

Journal of Mathematical Physics•Feb 1, 1965•975 citations
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 [p(p), p(−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 p(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 , and find: (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.
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.