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# Fermi liquid theory

Fermi liquid theory describes the generic physical properties which a collection of interacting fermions assumes (under certain conditions). In particular, it explains why the behaviour of interacting fermions may be (qualitatively) very similar to the behaviour of a Fermi gas (i.e. non-interacting fermions), in contrast to naive expectation. It also explains the quantitative differences. It was introduced by the russian physicist Lev Davidovich Landau in 1956. In particular, the electrons in a normal metal form a Fermi liquid, as well as the atoms of liquid He-3 (He-3 is an isotope of Helium, with 2 protons, 1 neutron and 2 electrons per atom. This makes an odd number of fermions inside the atom, such that the complete atom itself is also a fermion).

Loosely speaking, the following conditions are usually to be fulfilled in order to have a Fermi liquid: The system of interacting fermions is cooled to low-enough temperatures, the interaction between them is not too strong and the system is translationally invariant (such that momentum is conserved).

The Fermi liquid is qualitatively analogous to the non-interacting Fermi-gas, in the following sense: The system's dynamics and thermodynamics at low excitation energies and temperatures may be described by substituting for the non-interacting fermions so-called "quasi-particles", each of which carries the same spin, charge and momentum as the original particles. Physically these may be thought of as being particles whose motion is disturbed by the surrounding particles and which themselves perturb the particles in their vicinity. Each many-particle excited state of the interacting system may be described by listing all occupied momentum states, just as in the non-interacting system. As a consequence, quantities such as the heat capacity of the Fermi liquid behave qualitatively in the same way as in the Fermi gas (e.g. the heat capacity rises linearly with temperature).

However, the following differences to the non-interacting Fermi gas arise:

• The energy of a many-particle state is not simply a sum of the single-particle energies of all occupied states. Instead, the change in energy for a given change $\delta n_k$ in occupation of states $k$ contains terms both linear and quadratic in $\delta n_k$ (for the Fermi gas, it would only be linear, $\delta n_k \epsilon_k$, where $\epsilon_k$ denotes the single-particle energies). The linear contribution corresponds to renormalized single-particle energies, which involve, e.g., a change in the effective mass of particles. The quadratic terms correspond to a sort of "mean-field" interaction between quasiparticles, which is parameterized by so-called Landau Fermi liquid parameters and determines the behaviour of density oscillations (and spin-density oscillations) in the Fermi liquid. Still, these mean-field interactions do not lead to a scattering of quasi-particles with a transfer of particles between different momentum states.

• Specific heat, compressibility, spin-susceptibility and other quantities show the same qualitative behaviour (e.g. dependence on temperature) as in the Fermi gas, but the magnitude is (sometimes strongly) changed.

• In addition to the mean-field interactions, some weak interactions between quasiparticles remain, which lead to scattering of quasiparticles off each other. Therefore, quasiparticles acquire a finite lifetime. However, at low enough energies above the Fermi surface, this lifetime becomes very long, such that the product of excitation energy (expressed in frequency) and lifetime is much larger than one. In this sense, the quasiparticle energy is still well-defined (in the opposite limit, Heisenberg's uncertainty relation would prevent an accurate definition of the energy).

• Green's function and momentum distribution of quasiparticles behave as for the fermions in the Fermi gas (apart from the broadening of the delta peak in the Green's function by the finite lifetime).

• The structure of the "bare" particle's (as opposed to quasiparticle) Green's function is similar to that in the Fermi gas (where, for a given momentum, the Green's function in frequency space is a delta peak at the respective single-particle energy). The delta peak in the density-of-states is broadened (with a width given by the quasiparticle lifetime). In addition (and in contrast to the quasiparticle Green's function), its weight (integral over frequency) is suppressed by a quasiparticle weight factor $0<Z<1$. The remainder of the total weight is in a broad "incoherent background", corresponding to the strong effects of interactions on the fermions at short time-scales.

• The distribution of particles (as opposed to quasiparticles) over momentum states at zero temperature still shows a discontinuous jump at the Fermi surface (as in the Fermi gas), but it does not drop from 1 to 0: the step is only of size $Z$.

• In a metal the resistance at low temperatures is dominated by electron-electron scattering in combination with umklapp scattering (where momentum is transferred to the crystal lattice). For a Fermi liquid, the resistance from this mechanism varies as $T^2$, which is often taken as an experimental check for Fermi liquid behaviour (in addition to the linear temperature-dependence of the specific heat), although it only arises in combination with the lattice.

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