- Fermi liquid
Fermi liquid is a generic term for a quantum mechanical
liquid offermion s that arises under certain physical conditions when thetemperature is sufficiently low. The interaction between the particles of the many-body system does not need to be small (see e.g. electrons in a metal). The phenomenological theory of Fermi liquids, which was introduced by the Soviet physicistLev Davidovich Landau in1956 , explains why some of the properties of an interacting fermion system are very similar to those of theFermi gas (i.e. non-interacting fermions), and why other properties differ.Liquid He-3 is a Fermi liquid at low temperatures (but not low enough to be in its
superfluid phase.) He-3 is anisotope ofHelium , with 2proton s, 1neutron and 2 electrons per atom; because there is an odd number of fermions inside the atom, the atom itself is also a fermion. Theelectron s in a normal (non-superconducting)metal also form a Fermi liquid.imilarities to Fermi gas
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 the non-interacting fermions with so-calledquasiparticle s, each of which carries the same spin, charge andmomentum 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).Differences from Fermi gas
The following differences to the non-interacting Fermi gas arise:
Energy
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.pecific heat and compressibility
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.Interactions
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).tructure
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. 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. Distribution
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.
Resistance
In a metal the resistance at low temperatures is dominated by electron-electron scattering in combination with
Umklapp scattering . 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.Instabilities of the Fermi Liquid
The experimental observation of exotic phases instrongly correlated systems has triggered an enormouseffort from the theoretical community to try to understandtheir microscopical origin. One possible route todetect instabilities of a FL is precisely the analysis doneby Pomeranchuk [ I. J. Pomeranchuk, Sov. Phys. JETP 8, 361 (1958)] .. Due to that, the Pomeranchuk instabilityhas been studied by several authors [ Actually, this is a subject of investigation, see for example: [http://arxiv.org/abs/0804.4422] .] with differenttechniques in the last few years and in particular,the instability of the FL towards the nematic phasewas investigated for several models
References
ee also
*
Classical fluids
*Luttinger liquid
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