Luttinger's theorem
Encyclopedia
In condensed matter physics, Luttinger's theorem 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.
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
or 
where
is the single-particle Green function in terms of frequency and momentum. Then Luttinger's theorem can be recast into the form
,
where
is as above and 
is the differential volume of 
-space in 
dimensions.
Joaquin Mazdak Luttinger
Joaquin Mazdak Luttinger was an American physicist well-known for his contributions to the theory of interacting electrons in one-dimensional metals and the Fermi-liquid theory...
and J. C. Ward
John Clive Ward
John Clive Ward , was a British-Australian physicist. His most famous creation was the Ward-Takahashi identity, originally known as "Ward Identity" . This celebrated result, in quantum electrodynamics, was inspired by a conjecture of Dyson and was disclosed in a one-half page letter typical of...
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 surfaceFermi surface
In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline...
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
or where
is the single-particle Green function in terms of frequency and momentum. Then Luttinger's theorem can be recast into the form,where
is as above and is the differential volume of -space in dimensions.