Fermi liquid theory

Fermi liquid theory (also known as Landau–Fermi liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interactions among the particles of the many-body system do not need to be small. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas (i.e. non-interacting fermions), and why other properties differ. Fermi liquid theory (also known as Landau–Fermi liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interactions among the particles of the many-body system do not need to be small. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas (i.e. non-interacting fermions), and why other properties differ. Important examples of where Fermi liquid theory has been successfully applied are most notably electrons in most metals and Liquid helium-3. Liquid helium-3 is a Fermi liquid at low temperatures (but not low enough to be in its superfluid phase). Helium-3 is an isotope of helium, with 2 protons, 1 neutron and 2 electrons per atom. Because there is an odd number of fermions inside the nucleus, the atom itself is also a fermion. The electrons in a normal (non-superconducting) metal also form a Fermi liquid, as do the nucleons (protons and neutrons) in an atomic nucleus. Strontium ruthenate displays some key properties of Fermi liquids, despite being a strongly correlated material, and is compared with high temperature superconductors like cuprates. The key ideas behind Landau's theory are the notion of adiabaticity and the Pauli exclusion principle. Consider a non-interacting fermion system (a Fermi gas), and suppose we 'turn on' the interaction slowly. Landau argued that in this situation, the ground state of the Fermi gas would adiabatically transform into the ground state of the interacting system. By Pauli's exclusion principle, the ground state Ψ 0 {displaystyle Psi _{0}} of a Fermi gas consists of fermions occupying all momentum states corresponding to momentum p < p F {displaystyle p<p_{ m {F}}} with all higher momentum states unoccupied. As interaction is turned on, the spin, charge and momentum of the fermions corresponding to the occupied states remain unchanged, while their dynamical properties, such as their mass, magnetic moment etc. are renormalized to new values. Thus, there is a one-to-one correspondence between the elementary excitations of a Fermi gas system and a Fermi liquid system. In the context of Fermi liquids, these excitations are called 'quasi-particles'. Landau quasiparticles are long-lived excitations with a lifetime τ {displaystyle au } that satisfies ℏ τ ≪ ϵ p {displaystyle {frac {hbar }{ au }}ll epsilon _{ m {p}}} where ϵ p {displaystyle epsilon _{ m {p}}} is the quasiparticle energy (measured from the Fermi energy). At finite temperature, ϵ p {displaystyle epsilon _{ m {p}}} is on the order of the thermal energy k B T {displaystyle k_{ m {B}}T} , and the condition for Landau quasiparticles can be reformulated as ℏ τ ≪ k B T {displaystyle {frac {hbar }{ au }}ll k_{ m {B}}T} . For this system, the Green's function can be written (near its poles) in the form G ( ω , p ) ≈ Z ω + μ − ϵ ( p ) {displaystyle G(omega ,p)approx {frac {Z}{omega +mu -epsilon (p)}}} where μ {displaystyle mu } is the chemical potential and ϵ ( p ) {displaystyle epsilon (p)} is the energy corresponding to the given momentum state. The value Z {displaystyle Z} is called the quasiparticle residue and is very characteristic of Fermi liquid theory. The spectral function for the system can be directly observed via angle-resolved photoemission spectroscopy (ARPES), and can be written (in the limit of low-lying excitations) in the form: