Landau Fermi-Liquid Theory

Landau theory assumes a one-to-one adiabatic correspondence between low-energy states of an interacting normal fluid and occupations of long-lived quasiparticles. Interactions then enter through an energy functional of those occupations. This phenomenology predicts thermodynamics and response without claiming a microscopic proof for every interacting fermion system.

Required background. Use Fermi-surface kinematics and the general quasiparticle pole criteria.

Quasiparticle energy functional

For small deviations from the ground-state distribution,

$$\delta E=\sum_{\mathbf p\sigma}\epsilon^*_{\mathbf p}\,\delta n_{\mathbf p\sigma} +\frac{1}{2V}\sum_{\mathbf p\sigma,\mathbf p'\sigma'} f_{\sigma\sigma'}(\mathbf p,\mathbf p') \delta n_{\mathbf p\sigma}\delta n_{\mathbf p'\sigma'}.$$

In an isotropic spin-$1/2$ liquid, decompose the forward interaction into spin-symmetric and antisymmetric Legendre harmonics. With $N(0)$ the total density of states,

$$f^{s,a}(\cos\theta)=\frac{1}{N(0)}\sum_{\ell\ge0}F_\ell^{s,a}P_\ell(\cos\theta).$$

The normalization is conventional; observable formulas must be translated with it. In this convention, positivity of every quadratic deformation requires

$$1+\frac{F_\ell^{s,a}}{2\ell+1}>0.$$

Violation signals a Pomeranchuk instability in that angular and spin channel, not merely a large interaction.

Thermodynamics and mass

For an isotropic three-dimensional Galilean-invariant liquid,

$$\kappa=\frac{N(0)}{n^2(1+F_0^s)}, \qquad \chi_s=\frac{\mu_*^2N(0)}{1+F_0^a}, \qquad \frac{m^*}{m}=1+\frac{F_1^s}{3}.$$

Here $\mu_*$ denotes the appropriate quasiparticle magnetic moment; material $g$ factors require separate matching. The last identity follows from Galilean boost invariance and does not hold unchanged on a lattice. The heat-capacity coefficient measures $N(0)\propto m^*$, while the microscopic pole residue $Z$ cancels from equilibrium state counting. Hence $Z$ and $m^*/m$ are not synonyms.

Baym and Pethick derive these relations and the backflow required by conservation in Baym and Pethick 1991, chs. 1–2.

Validity ceiling

The quasiparticle width must vanish faster than its excitation energy as the surface is approached. The state must remain normal, and the response channel must be below the scale where pairing, density waves, or other orders intervene. Landau parameters summarize low-energy forward scattering; they do not determine high-energy spectra or prove adiabatic continuity.

The microscopic self-energy page connects this functional to poles and Ward identities, while zero sound tests the dynamic response.

Exercises

In the convention above, determine the stability bound in the $\ell=2$ symmetric channel.

Solution

The quadratic coefficient is proportional to $1+F_2^s/5$. Stability therefore requires $F_2^s>-5$. At equality the $q=0$ quadrupolar shape susceptibility diverges in the idealized normal state.

References

• Gordon Baym and Christopher Pethick, Landau Fermi-Liquid Theory: Concepts and Applications, Wiley-VCH (1991), chs. 1–2, doi:10.1002/9783527617159.