Editing Fermi liquid theory (section)

==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 <math>\delta n_k</math> in occupation of states <math>k</math> contains terms both linear and quadratic in <math>\delta n_k</math> (for the Fermi gas, it would only be linear, <math>\delta n_k \varepsilon_k</math>, where <math>\varepsilon_k</math> 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 parametrized 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.

The renormalization of the mass of a fluid of interacting fermions can be calculated from first principles using many-body computational techniques. For the two-dimensional [[homogeneous electron gas]], [[GW approximation|GW calculations]]<ref>{{cite journal|author1=R. Asgari |author2=B. Tanatar |title= Many-body effective mass and spin susceptibility in a quasi-two-dimensional electron liquid|journal= Physical Review B|volume=74|issue=7 |pages=075301|year=2006|doi= 10.1103/PhysRevB.74.075301|bibcode = 2006PhRvB..74g5301A |url=http://repository.bilkent.edu.tr/bitstream/11693/23741/1/bilkent-research-paper.pdf|hdl=11693/23741 |hdl-access=free}}</ref> and [[quantum Monte Carlo]] methods<ref>{{cite journal|author1=Y. Kwon |author2=D. M. Ceperley |author3=R. M. Martin |title= Quantum Monte Carlo calculation of the Fermi-liquid parameters in the two-dimensional electron gas|journal= Physical Review B|volume= 50
|issue=3 |pages=1684–1694|year=2013|doi= 10.1103/PhysRevB.50.1684|pmid=9976356 |bibcode = 1994PhRvB..50.1684K |arxiv=1307.4009}}</ref><ref>{{cite journal|author1=M. Holzmann |author2=B. Bernu |author3=V. Olevano |author4=R. M. Martin |author5=D. M. Ceperley |title= Renormalization factor and effective mass of the two-dimensional electron gas|journal= Physical Review B|volume= 79 |issue=4 |pages= 041308(R)|year=2009|doi= 10.1103/PhysRevB.79.041308|arxiv = 0810.2450 |bibcode = 2009PhRvB..79d1308H |s2cid=12279058 }}</ref><ref>{{cite journal|author1=N. D. Drummond |author2=R. J. Needs |title= Diffusion quantum Monte Carlo calculation of the quasiparticle effective mass of the two-dimensional homogeneous electron gas|journal= Physical Review B|volume=87 |issue=4 |pages=045131|year=2013|doi= 10.1103/PhysRevB.87.045131|arxiv = 1208.6317 |bibcode = 2013PhRvB..87d5131D |s2cid=53548304 }}</ref> have been used to calculate renormalized quasiparticle effective masses.

===Specific 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, [[Werner Heisenberg|Heisenberg]]'s [[Uncertainty principle|uncertainty relation]] would prevent an accurate definition of the energy).

===Structure===
The structure of the "bare" particles (as opposed to quasiparticle) [[Green's function (many-body theory)|many-body 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 <math>0<Z<1</math>. 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 <math>Z</math>.

===Electrical resistivity===
In a metal the resistivity at low temperatures is dominated by electron–electron scattering in combination with [[umklapp scattering]]. For a Fermi liquid, the resistivity from this mechanism varies as <math>T^2</math>, 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. In certain cases, umklapp scattering is not required. For example, the resistivity of compensated [[semimetal]]s scales as <math>T^2</math> because of mutual scattering of electron and hole. This is known as the Baber mechanism.<ref>{{Cite journal|last=Baber, W. G.|date=1937|title=The Contribution to the Electrical Resistance of Metals from Collisions between Electrons|journal=Proc. R. Soc. Lond. A|volume=158|issue=894|pages=383–396|doi=10.1098/rspa.1937.0027|bibcode=1937RSPSA.158..383B|doi-access=}}</ref>

===Optical response===
Fermi liquid theory predicts that the scattering rate, which governs the optical response of metals, not only depends quadratically on temperature (thus causing the <math>T^2</math> dependence of the DC resistance), but it also depends quadratically on frequency.<ref>{{cite journal|author=R. N. Gurzhi|year=1959|title=MUTUAL ELECTRON CORRELATIONS IN METAL OPTICS|journal=Sov. Phys. JETP|volume=8|pages=673–675}}</ref><ref>{{cite journal|author1=M. Scheffler |author2=K. Schlegel |author3=C. Clauss |author4=D. Hafner |author5=C. Fella |author6=M. Dressel |author7=M. Jourdan |author8=J. Sichelschmidt |author9=C. Krellner |author10=C. Geibel |author11=F. Steglich |year=2013|title=Microwave spectroscopy on heavy-fermion systems: Probing the dynamics of charges and magnetic moments|journal=Phys. Status Solidi B|volume=250|issue=3 |pages=439–449|doi=10.1002/pssb.201200925|arxiv = 1303.5011 |bibcode = 2013PSSBR.250..439S |s2cid=59067473 }}</ref><ref>{{cite journal|author1=C. C. Homes |author2=J. J. Tu |author3=J. Li |author4=G. D. Gu |author5=A. Akrap |year=2013|title=Optical conductivity of nodal metals|journal=Scientific Reports|volume=3|issue=3446 |pages=3446|doi=10.1038/srep03446|pmid=24336241 |pmc=3861800 |arxiv = 1312.4466 |bibcode = 2013NatSR...3.3446H }}</ref> This is in contrast to the [[Drude model|Drude prediction]] for non-interacting metallic electrons, where the scattering rate is a constant as a function of frequency.
One material in which optical Fermi liquid behavior was experimentally observed is the low-temperature metallic phase of [[Sr2RuO4|Sr<sub>2</sub>RuO<sub>4</sub>]].<ref>{{cite journal |author1=D. Stricker |author2=J. Mravlje |author3=C. Berthod |author4=R. Fittipaldi |author5=A. Vecchione |author6=A. Georges |author7=D. van der Marel |year=2014|title=Optical Response of Sr<sub>2</sub>RuO<sub>4</sub> Reveals Universal Fermi-Liquid Scaling and Quasiparticles Beyond Landau Theory |journal= Physical Review Letters|volume=113|issue=8 |pages=087404 |doi=10.1103/PhysRevLett.113.087404 |pmid=25192127 |bibcode=2014PhRvL.113h7404S|arxiv=1403.5445 |s2cid=20176023 }}</ref>