Fermi Surfaces and Fermi Liquids

A Fermi surface organizes the low-energy excitations and response of a metal. Landau theory describes long-lived quasiparticles, while microscopic Green functions and renormalization methods reveal when that description becomes unstable.

Enter this chapter

Begin with The Fermi Gas and Fermi-Surface Kinematics to relate particle density to Fermi-surface geometry and phase space.

Helpful background. The Fermi Gas and Fermi-Surface Kinematics and Quasiparticle Poles, Residues, and Lifetimes develop the definitions used by the chapter’s opening pages.

Topics

1. The Fermi Gas and Fermi-Surface Kinematics — Derives $k_F$, density of states, linearization, and particle–hole phase space.
2. Landau Fermi-Liquid Theory — Defines the quasiparticle energy functional, response coefficients, and Pomeranchuk stability inequalities.
3. Microscopic Quasiparticles and Fermi-Liquid Self-Energy — Connects $G^R$ and vertices to $Z$, $v_F^*$, and lifetime scaling.
4. Luttinger’s Theorem, Fermi Volume, and Failure Modes — States the count with its symmetry, unit-cell, analyticity, and topology qualifications.
5. Fermi-Liquid Response and Zero Sound — Derives static response and the collisionless collective pole in the correct order of limits.
6. Fermi-Surface Patch Theory and Low-Energy Scaling — Power-counts the curved surface and identifies exceptional marginal channels.
7. Functional RG for Competing Fermi-Surface Instabilities — Compares many-patch channels while exposing regulator and truncation dependence.
8. Pomeranchuk and Density-Wave Instabilities — Separates $q=0$ distortions from charge- and spin-density waves at finite $q$.

How the ideas fit together

The self-energy determines whether adding a fermion produces a sufficiently sharp excitation. Landau’s energy functional then describes how such excitations interact, connecting quasiparticle kinematics to thermodynamic response and collisionless zero sound Baym and Pethick 1991, chs. 1–3.

A stable pole does not settle every collective question. Forward, pairing, and finite-momentum channels can develop different instabilities, and a Fermi-volume theorem has its own symmetry and topology hypotheses. A growing approximate vertex identifies a tendency that must be checked before assigning an ordered phase.

The comparison below separates the assumptions and tests needed for these different conclusions.

Fermi-liquid claim table

Claim	Necessary input	Convention or limit	Invariant observable	Control or hypothesis	Falsifying signature
Well-defined quasiparticle	Pole of $G^R$ near the surface	Retarded function and momentum normal to the interacting surface	Peak position and width with $\Gamma/\lvert\omega\rvert\to0$	Smooth self-energy and Ward-consistent vertex	Width comparable to energy or loss of the pole without an alternative sharp excitation
Landau thermodynamics	Quasiparticle energy functional	Total or per-species DOS declared; static response limit	$C_V/T$, compressibility, and spin susceptibility	Stable Landau parameters and adiabatic low-energy states	A stability inequality fails or properly translated responses disagree
Zero sound	Collisionless kinetic equation	$q\to0$ at fixed $s=\omega/(qv_F)$ with $\omega\tau\gg1$	Pole outside the particle–hole continuum	Suitable interaction eigenvalue and a sharp fermion pole	Continuum damping or crossover to hydrodynamic first sound
Luttinger volume	Charge conservation, translations, and theorem topology	Physical unit cell, flavor factor, and filled-band equivalence fixed	Density modulo filled bands compared with the singular surface	No unaccounted symmetry enlargement or topological momentum sector	Count fails after including the correct cell and every theorem sector
Pomeranchuk tendency	Forward-scattering eigenvalue	Normalized lattice harmonic at $q=0$	Thermodynamic shape susceptibility	Stable quasiparticles and the appropriate point-group channel	A finite-$q$ peak dominates or there is no thermodynamic size growth
fRG leading channel	Regulated many-patch vertex flow	Regulator, patching, frequency grid, and stopping rule declared	Converged eigenmode and stopping scale	Weak coupling plus regulator, discretization, self-energy, and feedback convergence	Reordering under admissible truncations or pole breakdown before the stop scale

Where to go next

Continue to Correlated Lattice Fermions and Mott Physics to choose the orbitals and interactions of an effective lattice model.

References

• Gordon Baym and Christopher Pethick, Landau Fermi-Liquid Theory: Concepts and Applications, Wiley-VCH (1991), chs. 1–3, doi:10.1002/9783527617159.