The quasiparticle description makes sense only if the relaxation rate of the quasiparticles is much smaller than their energy. The relaxation of quasiparticle excitations is due to their mutual collisions and consequent decay into several others. Physically, the effect of the Pauli exclusion principle is felt only when the occupation probability of a given quantum state is of the order of unity. In the Maxwell-Boltzmann case, the probability of occupation of any quantum state is always small and hence Pauli's principle is not important. A natural question arises as to how the properties of a degenerate Fermi gas would change if the fermions interacted mutually. In the excited state, the momentum distribution of the particles will be different. The properties of a degenerate Fermi gas were first studied by Wilson-Sommerfeld in 1927. They turn out to be very different from a classical ideal gas.
The quasiparticle description makes sense only if the relaxation rate of the quasiparticles is much smaller than their energy. The relaxation of quasiparticle excitations is due to their mutual collisions and consequent decay into several others. Physically, the effect of the Pauli exclusion principle is felt only when the occupation probability of a given quantum state is of the order of unity. In the Maxwell-Boltzmann case, the probability of occupation of any quantum state is always small and hence Pauli's principle is not important. A natural question arises as to how the properties of a degenerate Fermi gas would change if the fermions interacted mutually. In the excited state, the momentum distribution of the particles will be different. The properties of a degenerate Fermi gas were first studied by Wilson-Sommerfeld in 1927. They turn out to be very different from a classical ideal gas.
The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A ; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1. The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.
We have produced an interacting quantum degenerate Fermi gas of atoms composed of two spin states of magnetically trapped 40K. The relative Fermi energies are adjusted by controlling the population in each spin state. Thermodynamic measurements reveal a resulting imbalance in the mean energy per particle between the two species, which is a factor of 1.4 at our lowest temperature. This imbalance of energy comes from a suppression of collisions between atoms in the gas due to the Pauli exclusion principle. Through measurements of the thermal relaxation rate we have directly observed this Pauli blocking as a factor of 2 reduction in the effective collision cross section in the quantum degenerate regime.
The Fermi liquid theory may provide a good description of the thermodynamic properties of an interacting particle system when the interaction between the particles contributes to the total energy of the system with a quantity which may depend on the total particle number, but does not depend on the temperature. In such a situation, the ideal part of the Hamiltonian, i.e. the energy of the system without the interaction energy, also provides a good description of the system's thermodynamics. If the total interaction energy of the system, being a complicated function of the particle populations, is temperature dependent, then the Landau's quasiparticle gas cannot describe accurately the thermodynamics of the system. A general solution to this problem is presented in this paper, in which the quasiparticle energies are redefined in such a way that the total energy of the system is identical to the sum of the energies of the quasiparticles. This implies also that the thermodynamic properties of the system and those of the quasiparticle gas are identical. By choosing a perspective in which the quasiparticle energies are fixed while the density of states along the quasiparticle axis vary, we transform our quasiparticle system into an ideal gas which obey fractional exclusion statistics.
Low-lying excitations in a liquid or a solid are considered to have a long lifetime and to obey the Bose or Fermi statistics. They will be called "quasiparticles" in general. The low-lying energy levels of the system are considered to be characterized by a set of occupation numbers of the quasiparticles. The expressions for the grand partition function and the distribution function (the average occupation number) of the quasiparticles have the same structure as that for the classical lattice gas, under a nonuniform external field and with two-, three-,..., body interactions. Therefore the virial expansion formula for the classical lattice gas can be applied to the system of quasiparticles. The result is simplified by assuming that the energy correction due to the simultaneous excitations of two, three,..., n quasiparticles are of $O({V}^{\ensuremath{-}1}),O({V}^{\ensuremath{-}2}),\ensuremath{\cdots},O({V}^{\ensuremath{-}n+1})$, respectively, where V is the volume of the system. The resulting expression for the distribution function of the quasiparticles is either a Fermi or a Bose distribution function with an effective energy. The expression for the entropy in terms of the distribution function is the same as that for an ideal gas. This shows that Landau's formula for the distribution function of quasiparticles for a Fermi liquid is valid under conditions which are more general and more practical than those given by Landau. Finally the situation in the Husimi-Temperley model is discussed in comparison with that in the Ising model from the point of view of the present formulation.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.
A quasiparticle description of various condensed media is a very popular tool in study of their transport and thermodynamic properties. I present here a microscopic theory for the description of diffusion processes in two-component gas of quasiparticles with arbitrary dispersion law and statistics. Particularly, I analyze the role of interaction within each subsystem (i.e. between identical quasiparticles) in relaxation of the whole system. The approach for solution of such kinetic problem allows to study the most important limiting cases and to clarify their physical sense. Classical results for diffusion coefficient of light particles in a massive gas (Lorentz model) and of massive particles in a light gas (Rayleigh model) are obtained directly from the general solution without using artificial approaches, as it was done earlier. This particularly provide a possibility to generalize these popular models on quasiparticle systems.