Non-Fermi Liquid
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Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s that describes the normal state of most
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typi ...
s at sufficiently low temperatures. The interactions among the particles of the many-body system do not need to be small. The
phenomenological Phenomenology may refer to: Art * Phenomenology (architecture), based on the experience of building materials and their sensory properties Philosophy * Phenomenology (philosophy), a branch of philosophy which studies subjective experiences and a ...
theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and
Isaak Khalatnikov Isaak Markovych Khalatnykov ( uk, Ісаа́к Ма́ркович Хала́тников; 17 October 1919 – 9 January 2021) was a leading Soviet theoretical physicist who has made significant contributions to many areas of theoretical physics, ...
using diagrammatic
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal
Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer s ...
(i.e. non-interacting fermions), and why other properties differ. Important examples of where Fermi liquid theory has been successfully applied are most notably electrons in most metals and
liquid helium Liquid helium is a physical state of helium at very low temperatures at standard atmospheric pressures. Liquid helium may show superfluidity. At standard pressure, the chemical element helium exists in a liquid form only at the extremely low t ...
-3. Liquid
helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is th ...
is a Fermi liquid at low temperatures (but not low enough to be in its
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
phase). Helium-3 is an
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers ( mass number ...
of
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
, with 2 protons, 1
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...
and 2 electrons per atom. Because there is an odd number of fermions inside the nucleus, the atom itself is also a fermion. The
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...
s in a normal (non-
superconducting Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
) metal also form a Fermi liquid, as do the
nucleons In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons w ...
(protons and neutrons) in an
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden experiments, Geiger–Marsden gold foil experiment. After th ...
. Strontium ruthenate displays some key properties of Fermi liquids, despite being a
strongly correlated material Strongly correlated materials are a wide class of compounds that include insulators and electronic materials, and show unusual (often technologically useful) electronic and magnetic properties, such as metal-insulator transitions, heavy fermi ...
, and is compared with high temperature superconductors like cuprates.


Description

The key ideas behind Landau's theory are the notion of ''adiabaticity'' and the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
. (draft copy) Consider a non-interacting fermion system (a
Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer s ...
), 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 \Psi_0 of a Fermi gas consists of fermions occupying all momentum states corresponding to momentum p 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 '' renormalized'' to new values. 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". Landau quasiparticles are long-lived excitations with a lifetime \tau that satisfies \frac\ll\epsilon_ where \epsilon_ is the quasiparticle energy (measured from the
Fermi energy The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi ga ...
). At finite temperature, \epsilon_ is on the order of the thermal energy k_T, and the condition for Landau quasiparticles can be reformulated as \frac\ll k_T. For this system, the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
can be written (near its poles) in the form :G(\omega,p)\approx\frac where \mu is the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
and \epsilon(p) is the energy corresponding to the given momentum state. The value Z is called the ''quasiparticle residue'' and is very characteristic of Fermi liquid theory. The spectral function for the system can be directly observed via
angle-resolved photoemission spectroscopy Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoele ...
(ARPES), and can be written (in the limit of low-lying excitations) in the form: :A(\mathbf,\omega)=Z\delta(\omega-v_k_) where v_ is the Fermi velocity. 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". Another important property of Fermi liquids is related to the scattering cross section for electrons. Suppose we have an electron with energy \epsilon_1 above the Fermi surface, and suppose it scatters with a particle in the Fermi sea with energy \epsilon_2. By Pauli's exclusion principle, both the particles after scattering have to lie above the Fermi surface, with energies \epsilon_3,\epsilon_4>\epsilon_. Now, suppose the initial electron has energy very close to the Fermi surface \epsilon\approx\epsilon_ Then, we have that \epsilon_2,\epsilon_3,\epsilon_4 also have to be very close to the Fermi surface. This reduces the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
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.


Similarities to Fermi gas

The Fermi liquid is qualitatively analogous to the non-interacting
Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer s ...
, 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 In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For exa ...
s, each of which carries the same spin, charge and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
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 In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
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 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 Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed i ...
, GW calculations and quantum Monte Carlo methods have been used to calculate renormalized quasiparticle effective masses.


Specific heat and compressibility

Specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of he ...
,
compressibility In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a ...
, 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 Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
's uncertainty relation would prevent an accurate definition of the energy).


Structure

The structure of the "bare" particle's (as opposed to quasiparticle)
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
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.


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 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. In certain cases, umklapp scattering is not required. For example, the resistivity of compensated
semimetal A semimetal is a material with a very small overlap between the bottom of the conduction band and the top of the valence band. According to electronic band theory, solids can be classified as insulators, semiconductors, semimetals, or metal ...
s scales as T^2 because of mutual scattering of electron and hole. This is known as the Baber mechanism.


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 T^2 dependence of the DC resistance), but it also depends quadratically on frequency. This is in contrast to the 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.


Instabilities

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 Fermi liquid is precisely the analysis done by
Isaak Pomeranchuk Isaak Yakovlevich Pomeranchuk (russian: Исаа́к Я́ковлевич Померанчу́к (Polish spelling: Isaak Jakowliewicz Pomieranczuk); 20 May 1913, Warsaw, Russian Empire – 14 December 1966, Moscow, USSR) was a Soviet physicist ...
. Due to that, the Pomeranchuk instability has been studied by several authors with different techniques in the last few years and in particular, the instability of the Fermi liquid towards the nematic phase was investigated for several models.


Non-Fermi liquids

The term non-Fermi liquid, also known as "strange metal", 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 the Luttinger liquid. 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, spin-charge separation, and the presence of spin density waves. One cannot ignore the existence of interactions in one-dimension and has to describe the problem with a non-Fermi theory, where Luttinger liquid is one of them. At small finite spin-temperatures in one-dimension the ground-state of the system is described by spin-incoherent Luttinger liquid (SILL). Another example of such behaviour is observed at
quantum critical point A quantum critical point is a point in the phase diagram of a material where a continuous phase transition takes place at absolute zero. A quantum critical point is typically achieved by a continuous suppression of a nonzero temperature phase t ...
s of certain second-order
phase transitions In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
, such as heavy fermion criticality, Mott criticality and high-T_ cuprate phase transitions. 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 Z\to0. 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 scaling relations; and descriptions using ''emergent''
gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
with techniques of
holographic Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, it ...
gauge/gravity duality.


See also

* Classical fluid * Fermionic condensate * Luttinger liquid * Luttinger's theorem * Strongly correlated quantum spin liquid


References

{{DEFAULTSORT:Fermi Liquid Condensed matter physics Fermions Electronic band structures Lev Landau