Fermi fluid theory

The Fermi liquid theory or Fermi-Landau liquid theory (after Enrico Fermi and Lew Landau ) is a theory for weakly interacting fermionic many-body systems, so-called Fermi liquids .

Landau developed the theory to describe the properties of liquid ³ He . The broadest use of the theory today is the description of conduction electrons in conventional metals in those cases in which the interactions between the electrons must be taken into account and thus the model of a (non-interacting) Fermi gas does not apply.

Basics and requirements

We consider a many-body system of weakly interacting fermions. Surface effects are not taken into account.

For non-interacting fermions, one obtains Bloch states as eigen-states of the Hamiltonian (see Bloch theorem ). Since they are eigenstates of the Hamiltonian, they do not decay, so they have an infinite lifetime. If there is a Coulomb interaction between the fermions, there are excited states with a finite lifetime because of the scattering processes between the fermions.
It only makes sense to speak of “particles” that we can observe, if they are so long-lived that they cannot escape our observation. So we demand that the decay rate (with lifetime ) is sufficiently small compared to the excitation energy (with the Fermi energy ). Hence our demand follows: . We then talk about quasiparticles. Note: If , then, the excited state lies above the Fermi edge and we can imagine an occupied state in k-space outside the Fermi sphere, which we have in a non-interacting fermionic system at absolute zero. If this excitation energy is less than zero, we can imagine a hole in the Fermi sphere and, strictly speaking, one would have to speak of a quasi-hole. However, we want to save ourselves this paperwork within this theory and speak of quasi- particles as a substitute for the suggestions above and below the Fermi edge . ${\ displaystyle 1 / \ tau}$ ${\ displaystyle \ tau}$ ${\ displaystyle E-E_ {F}}$ ${\ displaystyle E_ {F}}$ ${\ displaystyle | E-E_ {F} | \ gg \ hbar / \ tau}$ ${\ displaystyle E-E_ {F}> 0}$

The interaction must not be singular. However, the Coulomb interaction, as it exists between two electrons, has a singularity with a vanishing distance between the electrons. However, the interaction in the many-body system is shielded by the surrounding charges, so that this singularity does not occur with us. We use the results of the Lindhard theory for this. This is a perturbation theory and is therefore based on weak interaction potentials. We can therefore describe only weakly interacting systems: .

{\ displaystyle T \ ll T_ {F}, | {\ vec {k}} - {\ vec {k}} _ {F} | \ ll k_ {F}, \ hbar \ omega \ ll E_ {F}}

Central statements

The rate of decay of the excitations can be calculated using Fermi's golden rule . For excitations close to the Fermi edge at low temperatures, the following estimate of the decay rate is obtained: where the chemical potential is (at is ), i.e. H. Excitations near the Fermi edge at low temperatures decay so slowly (= are so long-lived) that one can speak of "particles" as a good approximation. This is supported by the following point:

{\ displaystyle 1 / \ tau \ propto (E- \ mu) ^ {2} + (\ pi k _ {\ mathrm {B}} T) ^ {2},}

{\ displaystyle \ mu}

{\ displaystyle T = 0}

{\ displaystyle \ mu = E_ {F}}

There is a "one-to-one correspondence" between the excitations of the interacting system near the Fermi edge (quasiparticles, no eigenstates of the Hamiltonian with interaction) and the eigenstates of the non-interacting systems (Fermigas). H. one can sort of assign the states to one another. Essentially, this means the following: We only know the exact solution of the Schrödinger equation for the non-interacting many-body system, which is effectively a single-particle system (the many-body wave function is a product of single-particle wave functions, see Slater determinant ), and not the solution for the weakly interacting one System. However, in the interactive system we can find similarities to the non-interactive system; the “nature of the states” that we are looking at here (not eigenstates) is not completely destroyed by switching on a weak interaction. The one-particle image is not completely destroyed by the weak interaction. Nevertheless, in order to emphasize the difference between an interactive and a non-interactive system, one speaks of a Fermi liquid and no longer of a Fermigas .

The interaction changes variables such as the Fermi velocity, the density of states at the Fermi edge and specific heat compared to the results that the Fermi gas provides for these variables. These differences are bundled into a phenomenological parameter, the effective mass. For this reason, the Fermi fluid theory is a phenomenological theory, the effective mass cannot be determined within the theory.

If you want to observe the Fermi liquid in a one-dimensional system, you fail: In spatial dimensions there is no Fermi liquid, but a so-called Luttinger liquid. This is because here an integral that appeared when calculating the decay rate using Fermi's Golden Rule diverges and the lifetime then approaches zero. Basically, it is because the shielding by surrounding charges becomes more effective in higher spatial dimensions, simply because more particles are available for shielding in the higher dimensions. This is shown in Lindhard's theory by a decrease in the charge density induced by an interfering charge with a distance dependence of .

{\ displaystyle d = 1}

{\ displaystyle 1 / r ^ {d}}

A jump in the distribution function at the Fermi edge that is less than 1 is characteristic of the Fermi fluid. The Luttinger fluid has no crack here. This jump in the Fermi fluid is not to be confused with the jump in the Fermi-Dirac distribution function at absolute zero, where the distribution function simply corresponds to a step function that suddenly falls from 1 to 0 when the Fermi energy is exceeded. The distribution function for the Fermi liquid also drops slowly at absolute zero before the Fermi edge and is not yet zero above the Fermi edge. This simply reflects the existence of the quasiparticles: Some electrons are in states above the Fermi edge, so that the population probability below the Fermi edge is less than 1 and above the Fermi edge is greater than 0. The decisive factor is the discontinuity at the Fermi edge, the size of which can be calculated using the method of Green's functions. ${\ displaystyle T = 0}$

Individual evidence

↑ Neil W. Ashcroft and David N. Mermin: Solid State Physics . 3. Edition. Oldenbourg, Munich 2007, ISBN 978-3-486-58273-4 , pp. 437-446 .
^ Giuliani, Gabriele, and Giovanni Vignale. Quantum theory of the electron liquid . Cambridge university press, 2005.

literature

Giuliani, Gabriele, and Giovanni Vignale. Quantum theory of the electron liquid . Cambridge university press, 2005.
Ashcroft, Neil W., N. David Mermin, and Jochen Gress. Solid state physics . Vol. 3. Oldenbourg, 2013.

[1] Neil W. Ashcroft and David N. Mermin: Solid State Physics . 3. Edition. Oldenbourg, Munich 2007, ISBN 978-3-486-58273-4 , pp. 437-446 .

[2] Giuliani, Gabriele, and Giovanni Vignale. Quantum theory of the electron liquid . Cambridge university press, 2005.