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Statistical mechanics, structure of matter Condensed matter and the structure of matter Theorem

Luttinger's theorem

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Statement

Consider a translation-invariant system of interacting spin-1/21/2 fermions in ℝd\R^d at zero temperature, with density nn and chemical potential μ\mu, and units where ℏ=1\hbar = 1. Let G(k,ω)G(k, \omega) be its single-particle Green's function, written as G(k,ω)=1/(ω+μ−εk−Σ(k,ω))G(k, \omega) = 1/(\omega + \mu - \varepsilon_k - \Sigma(k, \omega)) with self-energy Σ\Sigma. Assume that the system is connected to the non-interacting system by perturbation theory in the interaction, as in a Fermi liquid. Then n=2∫{k : G(k,0)>0}ddk(2π)d.n = 2 \int_{\set{k \,:\, G(k, 0) > 0}} \frac{d^d k}{(2\pi)^d}. In words, the volume enclosed by the interacting Fermi surface equals the volume of the free Fermi sea at the same density. Here the interacting Fermi surface is where G(k,0)G(k, 0) changes sign.

Remarks

Joaquin Luttinger derived it in 1960, using a functional introduced in joint work with John Ward. The proof writes the density as a frequency integral of GG, and shows to all orders of perturbation theory that the extra term coming from Σ\Sigma is zero. The argument is perturbative, so the theorem is not established for systems that are not adiabatically connected to free fermions.

Example: for free fermions in ℝ3\R^3, Σ=0\Sigma = 0 and G(k,0)>0G(k, 0) > 0 exactly when εk<μ\varepsilon_k < \mu, which is the Fermi ball of radius kFk_F, and the formula gives n=kF3/3π2n = k_F^3 / 3\pi^2. For an interacting Fermi liquid the shape of the Fermi surface can change, but the volume it encloses stays fixed by nn.

Going deeper

For a student

The theorem is the reason one can read the electron density off measured Fermi surfaces, for instance from quantum oscillations. In crystals the integral runs over the Brillouin zone, and it counts electrons modulo filled bands. Its proof uses the Luttinger–Ward functional of many-body perturbation theory. Apparent violations in strongly correlated systems are a sign of physics that is not perturbatively connected to a free Fermi gas.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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  1. Fermi liquid theory concept
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  3. Fermi–Dirac statistics concept
  4. Thermodynamic limit concept
  5. Fermi gas example
  6. Luttinger's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Luttinger%27s_theorem
About this page
TagBJYDT498FX
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Kindstatement, theorem
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MSC82D35, 81V74
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UsesFermi surface, Fermi liquid theory, Fermi gas
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Changed3 November 2026

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