Editing Fermi liquid theory (section)

==Description==

The key ideas behind Landau's theory are the notion of [[Adiabatic process|''adiabaticity'']] and the [[Pauli exclusion principle]].<ref name=coleman>{{cite book|last=Coleman|first=Piers|title=Introduction to Many Body Physics|publisher=Rutgers University|pages=143|url=http://www.physics.rutgers.edu/~coleman/620/mbody/pdf/bkx.pdf|access-date=2011-02-14|archive-url=https://web.archive.org/web/20120517093528/http://www.physics.rutgers.edu/~coleman/620/mbody/pdf/bkx.pdf|archive-date=2012-05-17|url-status=dead}} (draft copy)</ref> 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 <math>\Psi_0</math> of a Fermi gas consists of fermions occupying all momentum states corresponding to momentum <math>p<p_{\rm F}</math> with all higher momentum states unoccupied. As the 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 ''[[renormalization|renormalized]]'' to new values.<ref name=coleman />  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 "[[quasiparticle]]s".<ref name=phillips />

Landau quasiparticles are long-lived excitations with a lifetime <math>\tau</math> that satisfies <math>{\hbar}/{\tau}\ll\varepsilon_{\rm p}</math> where <math>\varepsilon_{\rm p}</math> is the quasiparticle energy (measured from the [[Fermi energy]]). At finite temperature, <math>\varepsilon_{\rm p}</math> is on the order of the thermal energy <math>k_{\rm B}T</math>, and the condition for Landau quasiparticles can be reformulated as <math>{\hbar}/{\tau}\ll k_{\rm B}T</math>.

For this system, the [[Green's function (many-body theory)|many-body Green's function]] can be written<ref name=landau>{{cite book|last1=Lifshitz|first1=E. M.|last2=Pitaevskii|first2=L.P.|title=Statistical Physics (Part 2)|series=Landau and Lifshitz|volume=9|year=1980|publisher=Elsevier|isbn=978-0-7506-2636-1}}</ref>  (near its poles) in the form

:<math>G(\omega,\mathbf{p})\approx\frac{Z}{\omega+\mu-\varepsilon(\mathbf{p})}</math>

where <math>\mu</math> is the [[chemical potential]], <math>\varepsilon(\mathbf{p})</math> is the energy corresponding to the given momentum state and <math>Z>0</math> is called the ''quasiparticle residue'' or ''renormalisation constant'' which 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:

:<math>A(\mathbf{k},\omega)=Z\delta(\omega-v_{\rm F}k_{\|})</math>

where <math>v_{\rm F}</math> is the Fermi velocity.<ref name=senthil>{{cite journal|last=Senthil|first=Todadri|title=Critical Fermi surfaces and non-Fermi liquid metals|year=2008|journal=[[Physical Review B]]|volume=78|issue=3|page=035103|doi= 10.1103/PhysRevB.78.035103 | arxiv=0803.4009|bibcode = 2008PhRvB..78c5103S |s2cid=118656854}}</ref>

Physically, we can say that a propagating fermion interacts with its surrounding in such a way that the net effect of the interactions is to make the fermion behave as a "dressed" fermion, altering its effective mass and other dynamical properties. These "dressed" fermions are what we think of as "quasiparticles".<ref name="caltech">{{cite web |last=Cross |first=Michael |title=Fermi Liquid Theory: Principles |url=http://www.pmaweb.caltech.edu/~mcc/Ph127/c/Lecture9.pdf |access-date=2 February 2015 |publisher=California Institute of Technology}}</ref>

Another important property of Fermi liquids is related to the scattering cross section for electrons. Suppose we have an electron with energy <math>\varepsilon_1</math> above the Fermi surface, and suppose it scatters with a particle in the [[Composite fermion#Fermi sea|Fermi sea]] with energy <math>\varepsilon_2</math>. By Pauli's exclusion principle, both the particles after scattering have to lie above the Fermi surface, with energies <math>\varepsilon_3,\varepsilon_4>\varepsilon_{\rm F}</math>. Now, suppose the initial electron has energy very close to the Fermi surface <math>\varepsilon\approx\varepsilon_{\rm F}</math> Then, we have that <math>\varepsilon_2,\varepsilon_3,\varepsilon_4</math> also have to be very close to the Fermi surface. This reduces the [[phase space]] volume of the possible states after scattering, and hence, by [[Fermi's golden rule]], the [[scattering cross section]] goes to zero. Thus we can say that the lifetime of particles at the Fermi surface goes to infinity.<ref name=phillips />