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| {{other uses2|Fermi}}
| | Some people call me Gabrielle. I am a cashier and I'm doing pretty good financially. As an important girl what I really like is going on to karaoke but I do not have made a dime with it. My husband and I decided upon to reside in Guam but I will attain to move in a year or two. See all that is new on my blog here: http://prometeu.net<br><br>My web blog - [http://prometeu.net hack clash of clans] |
| {{Condensed matter physics|expanded=States of matter}}
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| '''Fermi liquid theory''' (also known as '''Landau–Fermi liquid theory''') is a theoretical model of interacting [[fermion]]s that describes the normal state of most [[metal]]s at sufficiently low temperatures.<ref name=phillips>{{cite book|last=Phillips|first=Philip|title=Advanced Solid State Physics|year=2008|publisher=Perseus Books|isbn=978-81-89938-16-1|pages=224}}</ref> The interaction between the particles of the many-body system does not need to be small. The [[Phenomenology (science)|phenomenological]] theory of Fermi liquids was introduced by the Soviet physicist [[Lev Davidovich Landau]] in 1956, and later developed by [[Alexei Alexeyevich Abrikosov|Alexei Abrikosov]] and [[I. M. Khalatnikov]] using [[Feynman diagrams|diagrammatic]] [[perturbation theory]].<ref name=caltech>{{cite web|last=Cross|first=Michael|title=Fermi Liquid Theory: Principles|url=http://www.pma.caltech.edu/~mcc/Ph127/c/Lecture9.pdf|publisher=California Institute of Technology|accessdate=14 February 2011}}</ref> The theory explains why some of the properties of an interacting fermion system are very similar to those of the [[Fermi gas]] (i.e. non-interacting fermions), and why other properties differ.
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| Important examples of where Fermi liquid theory has been successfully applied are most notably electrons in most metals and Liquid He-3.<ref name=schulz>{{cite journal|last=Schulz|first=H. J.|title=Fermi liquids and non–Fermi liquids|date=March 1995|url=http://arxiv.org/PS_cache/cond-mat/pdf/9503/9503150v2.pdf|accessdate=14 February 2011| arxiv=cond-mat/9503150}}</ref> Liquid [[Helium-3|He-3]] is a Fermi liquid at low temperatures (but not low enough to be in its [[superfluid]] [[phase (matter)|phase]].) He-3 is an [[isotope]] of [[Helium]], with 2 [[proton]]s, 1 [[neutron]] and 2 electrons per atom. Because there is an odd number of fermions inside the atom, the atom itself is also a fermion. The [[electron]]s in a normal (non-[[superconductivity|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 superconductor]]s like [[cuprate]]s.<ref name=wysokinski>{{cite journal|last=Wysokiński|first=Carol|coauthors=et al|title=Spin triplet superconductivity in Sr2RuO4|journal=Physica Status Solidi|year=2003|volume=236|issue=2|doi=10.1002/pssb.200301672|url=http://www.phy.bris.ac.uk/people/annett_jf/papers/physicab.pdf|accessdate=8 April 2012|arxiv = cond-mat/0211199 |bibcode = 2003PSSBR.236..325W }}</ref>
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| ==Description==
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| The key ideas behind Landau's theory are the notion of ''adiabaticity'' and the [[Pauli exclusion principle|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}} (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.
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| 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_F</math> 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 ''[[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 "quasi-particles".<ref name=phillips />
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| Landau quasiparticles are long-lived excitations with a lifetime <math>\tau</math> that satisfies <math>\frac{1}{\tau}\ll\epsilon_p</math> where <math>\epsilon_p</math> is the [[Fermi energy]].
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| For this system, the [[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=0-7506-2636-4}}</ref> (near its poles) in the form
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| <math>G(\omega,p)\approx\frac{Z}{\omega+\mu-\epsilon(p)}</math>
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| where <math>\mu</math> is the [[chemical potential]] and <math>\epsilon(p)</math> is the energy corresponding to the given momentum state.
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| The value <math>Z</math> is called the ''quasiparticle residue'' and is very characteristic of Fermi liquid theory. The spectral function for the system can be directly observed via [[ARPES]] experiment, and can be written (in the limit of low-lying excitations) in the form:
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| <math>A(\vec{k},\omega)=Z\delta(\omega-v_Fk_{\|})</math>
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| where <math>v_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 }}</ref>
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| 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 />
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| Another important property of Fermi liquids is related to the scattering cross section for electrons. Suppose we have an electron with energy <math>\epsilon_1</math> above the Fermi surface, and suppose it scatters with a particle in the [[Composite fermion#Fermi sea|Fermi sea]] with energy <math>\epsilon_2</math>. By Pauli's exclusion principle, both the particles after scattering have to lie above the Fermi surface, with energies <math>\epsilon_3,\epsilon_4>\epsilon_F</math> Now, suppose the initial electron has energy very close to the Fermi surface <math>\epsilon\approx\epsilon_F</math> Then, we have that <math>\epsilon_2,\epsilon_3,\epsilon_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 />
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| ==Similarities to Fermi gas==
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| 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 interacting [[quasiparticle]]s, each of which carries the same [[Spin (physics)|spin]], [[Electric charge|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).
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| ==Differences from Fermi gas==
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| The following differences to the non-interacting Fermi gas arise:
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| ===Energy===
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| 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 \epsilon_k</math>, where <math>\epsilon_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 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.
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| ===Specific heat and compressibility===
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| [[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.
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| ===Interactions===
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| 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).
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| ===Structure===
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| 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 <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.
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| ===Distribution===
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| 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>.
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| ===Electrical Resistance===
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| 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 <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.
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| ==Instabilities of the Fermi Liquid==
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| The experimental observation of exotic phases in strongly correlated systems has triggered an enormous effort from the theoretical community to try to understand their microscopic origin. One possible route to detect instabilities of a FL is precisely the analysis done by Pomeranchuk.<ref>I. J. Pomeranchuk, Sov. Phys. JETP 8, 361 (1958)</ref> Due to that, the Pomeranchuk instability has been studied by several authors <ref>Actually, this is a subject of investigation, see for example: http://arxiv.org/abs/0804.4422.</ref> with different techniques in the last few years and in particular, the instability of the FL towards the nematic phase was investigated for several models.
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| ==Non-Fermi liquids==
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| The term '''non-Fermi liquid''' is used to describe a system which displays breakdown of Fermi-liquid behaviour. The simplest example of such a system is the system of interacting fermions in one-dimension, called ''[[Luttinger liquid]]''.<ref name=schulz /> Although Luttinger liquids are physically similar to Fermi liquids, the restriction to one dimension gives rise to several qualitative differences such as the absence of a ''quasiparticle peak'' in the momentum dependent spectral function and the presence of [[spin wave|spin density waves]].
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| Another example of such behaviour is observed at [[quantum critical point]]s of certain second-order [[phase transitions]], such as
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| [[Heavy fermion]] criticality, [[Mott insulator|Mott criticality]] and high-<math>T_c</math> [[High-temperature superconductivity#Cuprates|cuprate]] phase transitions.<ref name=senthil /> The ground state of such transitions is characterized by the presence of a sharp Fermi surface, although there may not be well-defined quasiparticles. That is, on approaching the critical point, it is observed that the quasiparticle residue <math>Z\to0</math>
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| Understanding the behaviour of non-Fermi liquids is an important problem in condensed matter physics. Approaches towards explaining these phenomena include the treatment of ''marginal Fermi liquids''; attempts to understand critical points and derive [[critical scaling|scaling relations]]; and descriptions using ''emergent'' [[gauge theory|gauge theories]] with techniques of [[Holographic principle|holographic]] gauge/gravity duality.<ref name=polchinsky>{{cite arXiv|last=Faulkner|first=Thomas|coauthors=Polchinski, Joseph|title=Semi-Holographic Fermi Liquids|year=2010| eprint=1001.5049}}</ref>
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| ==References==
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| <references/>
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| ==See also==
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| *[[Fermi gas]]
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| *[[Classical fluids]]
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| *[[Fermionic condensate]]
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| *[[Luttinger liquid]]
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| *[[Luttinger's theorem]]
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| {{DEFAULTSORT:Fermi Liquid}}
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| [[Category:Condensed matter physics]]
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| [[Category:Fermions]]
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| [[Category:Electronic band structures]]
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| [[Category:Enrico Fermi|Liquid]]
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Some people call me Gabrielle. I am a cashier and I'm doing pretty good financially. As an important girl what I really like is going on to karaoke but I do not have made a dime with it. My husband and I decided upon to reside in Guam but I will attain to move in a year or two. See all that is new on my blog here: http://prometeu.net
My web blog - hack clash of clans