Luttinger's theorem

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

:$G(\backslash omega=0,\backslash ,p)\; \backslash to\; 0$ or $\backslash infty,$

where $G$ is the single-particle Green function in terms of frequency and momentum. Then Luttinger's theorem can be recast into the form

{{cite book

| author = Abrikosov, A. A., Gorkov, L. P. and Dzyaloshinsky, I. E.

| year = 1963

| title = Methods of Quantum Field Theory in Statistical Physics

| edition=revised | page = 168

| publisher = Dover, New York

}}

:$n\; =\; 2\; \backslash int\_\{\backslash mathcal\{Re\backslash ,\}G(\backslash omega=0,k)>0\}\backslash frac\{d^D\; k\}\{(2\backslash pi)^D\}$,

where $\backslash mathcal\{Re\backslash ,\}G$ is the real part of the above Green function and $d^D\; k$ is the differential volume of $k$-space in $D$ dimensions.

• Fermi liquid
• Fermi surface
• Luttinger–Ward functional

{{Reflist}}

• {{cite arXiv |author1=Behnam Farid |title=On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions |class=cond-mat.str-el |year=2007 |eprint=0711.0952 }}
• {{cite arXiv |author1=Behnam Farid |author2= Tsvelik |title=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) |class=cond-mat.str-el |year=2009 |eprint=0909.2886 }}
• {{cite arXiv |author1=Behnam Farid |title= 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) |class=cond-mat.str-el |year=2009 |eprint=0909.2887 }}
• {{cite journal |arxiv=1207.4201 |author1=Kiaran B. Dave|author2=Philip W. Phillips|author3=Charles L. Kane |title=Absence of Luttinger's theorem |year=2013 |doi=10.1103/PhysRevLett.110.090403 |pmid=23496693|volume=110 |issue=9 |pages=090403|journal=Physical Review Letters |bibcode=2013PhRvL.110i0403D|s2cid=1134967 }}
• {{cite journal |author=M. Oshikawa |year=2000 |title=Topological Approach to Luttinger's Theorem and the Fermi Surface of a Kondo Lattice |journal=Physical Review Letters |volume=84 |issue=15 |pages=3370–3373 |doi=10.1103/PhysRevLett.84.3370|arxiv = cond-mat/0002392 |bibcode = 2000PhRvL..84.3370O |pmid=11019092|s2cid=9806160 }}
• {{Cite book | title=Luttinger Model: The First 50 Years and Some New Directions |series=Series on Directions in Condensed Matter Physics |volume = 20| year=2013 | last1 = Mastropietro | first1 = Vieri | last2 = Mattis | first2 = Daniel C. | isbn = 978-981-4520-71-3|doi = 10.1142/8875|bibcode = 2013SDCMP..20.....M |publisher=World Scientific}}
• {{cite conference |author=F. D. M. Haldane |year=2005 |title=Luttinger's Theorem and Bosonization of the Fermi Surface |book-title=Proceedings of the International School of Physics "Enrico Fermi", Course CXXI "Perspectives in Many-Particle Physics" |editor=R. A. Broglia |editor2=J. R. Schrieffer |editor2-link=J. R. Schrieffer |publisher=North-Holland |pages=5–29 |arxiv=cond-mat/0505529|bibcode=2005cond.mat..5529H }}

Category:Condensed matter physics

Category:Eponymous theorems of physics

Category:Fermions