Fermi surfaces and Fermi liquids

For a homogeneous, noninteracting three-dimensional gas, take two spin states with equal populations and dispersion $\varepsilon(k)=\hbar^2k^2/(2m)$. At zero temperature, all states inside a sphere of radius $k_F$ are occupied. Periodic boundary conditions put one momentum state per volume $(2\pi)^3/V$ per spin. Counting the sphere therefore relates number density to $k_F$. The Fermi energy is the band energy at that radius measured from the chosen band minimum. This counting uses a continuum gas; a crystal requires its actual bands and Brillouin zone.

The Fermi surface is a surface in momentum space, not an outer layer of electrons in real space. In a general band it is the set where the energy equals the chemical potential, and it can contain several sheets rather than one sphere. At nonzero temperature, the occupation step is smoothed over an energy range of order $k_BT$. The chemical potential also adjusts to maintain the prescribed density. A sharp zero-temperature surface remains a useful reference when that thermal scale is small compared with the relevant electronic scales.

An electron deep below the zero-temperature chemical potential cannot respond to an arbitrarily small added energy by moving to a nearby occupied state. The destination is already filled. Particle-hole excitations at low energy instead remove an electron just below the surface and place it just above. This phase-space restriction explains why the electronic heat capacity of an ordinary degenerate metal is proportional to temperature at sufficiently low temperature, rather than approaching the classical constant per electron. The proportionality coefficient depends on the density of available low-energy states.

For interacting systems, Landau’s construction associates low-energy excitations with those of a reference Fermi sea as the interaction is turned on without crossing a phase transition. A quasiparticle carries the electron’s conserved quantum numbers but includes a rearrangement of the surrounding medium. Its energy and effective mass can differ from bare values, and quasiparticles interact with one another. This association is a physical low-energy hypothesis, not an assertion that the exact many-body wavefunction is a product of independently dressed orbitals. Collective response requires interaction parameters as well as a dispersion.

In a conventional Fermi liquid with regular short-range interactions, decay is restricted by energy conservation and Pauli blocking. Near the surface, a characteristic decay rate scales quadratically with excitation energy or temperature, up to dimension-dependent and interaction-dependent corrections. In three dimensions the familiar leading behavior is proportional to $\xi^2+(\pi k_BT)^2$, divided by an appropriate microscopic energy and by $\hbar$. As the excitation energy tends to zero, its width becomes small compared with that energy, so the excitation remains a useful propagating degree of freedom.

One-dimensional interacting fermions generically require Luttinger-liquid ideas; critical fluctuations, disorder, or pairing can also change the description. Resistance alone does not settle quasiparticle behavior: momentum relaxation and single-particle decay are different observables. The quadratic lab is only a one-dimensional energy-versus-coordinate analogy: with coefficient positive it shows a parabola and its slope. It does not populate a three-dimensional Fermi sphere, calculate occupations, or predict interacting lifetimes. Compare the local slope with velocity, then return to state counting.