# Luttinger's theorem

Luttinger's theorem relates a Fermi liquid's particle density to the volume of its Fermi surface.

In condensed matter physics, Luttinger's theorem[1][2] is a result derived by J. M. Luttinger and J. C. Ward in 1960 that has broad implications in the field of electron transport. It arises frequently in theoretical models of correlated electrons, such as the high-temperature superconductors, and in photoemission, where a metal's Fermi surface can be directly observed.

## Definition

Luttinger's theorem states that the volume enclosed by a material's Fermi surface is directly proportional to the particle density.

While the theorem is an immediate result of the Pauli exclusion principle in the case of noninteracting particles, it remains true even as interactions between particles are taken into consideration provided that the appropriate definitions of Fermi surface and particle density are adopted. Specifically, in the interacting case the Fermi surface must be defined according to the criteria that

${\displaystyle G(\omega =0,\,p)\to 0}$ or ${\displaystyle \infty ,}$

where ${\displaystyle G}$ is the single-particle Green function in terms of frequency and momentum. Then Luttinger's theorem can be recast into the form[3]

${\displaystyle n=2\int _{G(\omega =0,p)>0}{\frac {d^{D}k}{(2\pi )^{D}}}}$,

where ${\displaystyle G}$ is as above and ${\displaystyle d^{D}k}$ is the differential volume of ${\displaystyle k}$-space in ${\displaystyle D}$ dimensions.

## References

### Inline

1. ^ Luttinger, J. M.; Ward, J. C. (1960). "Ground-State Energy of a Many-Fermion System. II". Physical Review. 118 (5): 1417–1427. Bibcode:1960PhRv..118.1417L. doi:10.1103/PhysRev.118.1417.
2. ^ Luttinger, J. M. (1960). "Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermions". Physical Review. 119 (4): 1153–1163. Bibcode:1960PhRv..119.1153L. doi:10.1103/PhysRev.119.1153.
3. ^ Alexei M. Tsvelik (2003). Quantum Field Theory in Condensed Matter Physics (2nd ed.). Cambridge University Press. p. 327. ISBN 0-521-82284-X.

### General

• Behnam Farid (2007). "On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions". arXiv: [cond-mat.str-el].
• Behnam Farid; Tsvelik (2009). "Comment on "Breakdown of the Luttinger sum rule within the Mott-Hubbard insulator", by J. Kokalj and P. Prelovšek, Phys. Rev. B 78, 153103 (2008)". arXiv: [cond-mat.str-el].
• Behnam Farid (2009). "Comment on "Violation of the Luttinger sum rule within the Hubbard model on a triangular lattice", by J. Kokalj and P. Prelovšek, Eur. Phys. J. B 63, 431 (2008)". arXiv: [cond-mat.str-el].
• Kiaran B. Dave; Philip W. Phillips; Charles L. Kane (2012). "Absence of Luttinger's theorem". arXiv: [cond-mat.str-el].
• Behnam Farid (2012). "Comment on "Absence of Luttinger's theorem", by Kiaran B. Dave, Philip W. Phillips and Charles L. Kane, arXiv:1207.4201". arXiv: [cond-mat.str-el].
• M. Oshikawa (2000). "Topological Approach to Luttinger's Theorem and the Fermi Surface of a Kondo Lattice". Physical Review Letters. 84 (15): 3370–3373. Bibcode:2000PhRvL..84.3370O. PMID 11019092. arXiv:. doi:10.1103/PhysRevLett.84.3370.
• Mastropietro, Vieri; Mattis, Daniel C. (2013). Luttinger Model: The First 50 Years and Some New Directions. World Scientific. ISBN 978-981-4520-71-3.
• F. D. M. Haldane (2005). "Luttinger's Theorem and Bosonization of the Fermi Surface". In R. A. Broglia and J. R. Schrieffer. Proceedings of the International School of Physics "Enrico Fermi", Course CXXI "Perspectives in Many-Particle Physics". North-Holland. pp. 5–29. arXiv:.