Fermi liquid theory

The key ideas behind Landau's theory are the notion of adiabaticity and the Pauli exclusion principle.[5] 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 ${\displaystyle \Psi _{0}}$ of a Fermi gas consists of fermions occupying all momentum states corresponding to momentum ${\displaystyle p<p_{\rm {F}}}$ 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 renormalized to new values.[5] 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".[1]

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

For this system, the Green's function can be written[6] (near its poles) in the form

${\displaystyle G(\omega ,p)\approx {\frac {Z}{\omega +\mu -\epsilon (p)}}}$

where ${\displaystyle \mu }$ is the chemical potential and ${\displaystyle \epsilon (p)}$ is the energy corresponding to the given momentum state.

The value ${\displaystyle Z}$ 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 (ARPES), and can be written (in the limit of low-lying excitations) in the form:

${\displaystyle A(\mathbf {k} ,\omega )=Z\delta (\omega -v_{\rm {F}}k_{\|})}$

where ${\displaystyle v_{\rm {F}}}$ is the Fermi velocity.[7]

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".[2]

Another important property of Fermi liquids is related to the scattering cross section for electrons. Suppose we have an electron with energy ${\displaystyle \epsilon _{1}}$ above the Fermi surface, and suppose it scatters with a particle in the Fermi sea with energy ${\displaystyle \epsilon _{2}}$. By Pauli's exclusion principle, both the particles after scattering have to lie above the Fermi surface, with energies ${\displaystyle \epsilon _{3},\epsilon _{4}>\epsilon _{\rm {F}}}$. Now, suppose the initial electron has energy very close to the Fermi surface ${\displaystyle \epsilon \approx \epsilon _{\rm {F}}}$ Then, we have that ${\displaystyle \epsilon _{2},\epsilon _{3},\epsilon _{4}}$ 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.[1]

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 quasiparticles, each of which carries the same spin, 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).

The following differences to the non-interacting Fermi gas arise:

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.[17] Due to that, the Pomeranchuk instability has been studied by several authors [18] 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.

The term non-Fermi liquid, also known as "strange metal",[19] 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.[3] 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).[20]

Another example of such behaviour is observed at quantum critical points of certain second-order phase transitions, such as heavy fermion criticality, Mott criticality and high-${\displaystyle T_{\rm {c}}}$ cuprate phase transitions.[7] 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 ${\displaystyle Z\to 0}$

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 with techniques of holographic gauge/gravity duality.[21]

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