Fermi liquid theory
Introduction
Fermi liquid theory is a theoretical framework used to describe the normal state of most metals at low temperatures. It provides a comprehensive understanding of the behavior of interacting fermions, which are particles that follow the Pauli exclusion principle. This theory was developed by the physicist Lev Landau in the mid-20th century and has since become a cornerstone in the field of condensed matter physics. Fermi liquid theory is particularly significant because it explains how the properties of a system of interacting fermions can be described in terms of quasiparticles, which behave like non-interacting particles.
Historical Background
The development of Fermi liquid theory was motivated by the need to understand the properties of electrons in metals. Prior to Landau's work, the Drude model and the Sommerfeld model provided a basic understanding of electron behavior, but they failed to account for interactions between electrons. Landau's insight was to treat the low-energy excitations of a Fermi system as quasiparticles, which are long-lived and have well-defined properties such as energy and momentum. This approach allowed for the successful description of the thermodynamic and transport properties of metals.
Fundamental Concepts
Quasiparticles
In Fermi liquid theory, the concept of quasiparticles is central. Quasiparticles are excitations that behave like particles with modified properties due to interactions. They have a one-to-one correspondence with the states of non-interacting fermions, but their effective mass and lifetime are altered by interactions. The quasiparticle picture is valid at low temperatures, where the interactions between particles are weak compared to their kinetic energy.
Landau Parameters
Landau introduced a set of parameters, known as Landau parameters, to characterize the interactions between quasiparticles. These parameters are dimensionless and describe the strength and nature of the interactions. They play a crucial role in determining the physical properties of the Fermi liquid, such as its specific heat, magnetic susceptibility, and compressibility.
Fermi Surface
The Fermi surface is a key concept in Fermi liquid theory. It is the surface in momentum space that separates occupied from unoccupied states at zero temperature. The shape and size of the Fermi surface are determined by the density of particles and the interactions between them. In a Fermi liquid, the Fermi surface remains well-defined, and its properties can be used to predict various physical phenomena.
Mathematical Formulation
The mathematical formulation of Fermi liquid theory involves the use of Green's functions and perturbation theory. The many-body problem is tackled by introducing the concept of a self-energy, which accounts for the interactions between particles. The self-energy modifies the properties of the quasiparticles, leading to the renormalization of their mass and lifetime.
Green's Functions
Green's functions are a powerful tool in the study of many-body systems. They provide a way to calculate the response of a system to external perturbations and are used to derive various physical properties. In Fermi liquid theory, the retarded Green's function is particularly important, as it describes the propagation of quasiparticles in the system.
Self-Energy and Renormalization
The self-energy is a complex function that encapsulates the effects of interactions on the quasiparticles. It is used to calculate the quasiparticle residue, which measures the overlap between the quasiparticle state and the non-interacting state. The self-energy also leads to the renormalization of the quasiparticle mass, which is different from the bare mass of the particles.
Physical Properties
Fermi liquid theory provides a framework for understanding a wide range of physical properties of metals and other systems of interacting fermions. These properties include:
Specific Heat
The specific heat of a Fermi liquid is linear in temperature at low temperatures. This behavior is a direct consequence of the linear density of states near the Fermi surface. The specific heat coefficient is related to the effective mass of the quasiparticles and can be used to extract information about the interactions in the system.
Electrical Conductivity
The electrical conductivity of a Fermi liquid is determined by the scattering of quasiparticles. At low temperatures, the scattering rate is proportional to the square of the temperature, leading to a characteristic T-squared dependence of the resistivity. This behavior is a hallmark of Fermi liquid behavior and is observed in many metals.
Magnetic Susceptibility
The magnetic susceptibility of a Fermi liquid is enhanced compared to that of a non-interacting system. This enhancement is due to the interactions between quasiparticles and is characterized by the Landau parameter related to spin interactions. The susceptibility provides insights into the magnetic properties of the system and can be used to study ferromagnetism and antiferromagnetism.
Extensions and Limitations
Fermi liquid theory has been successfully applied to a wide range of systems, but it also has limitations. It is not applicable to systems with strong correlations, such as the high-temperature superconductors and heavy fermion compounds. In these systems, the quasiparticle picture breaks down, and new theoretical approaches are needed.
Non-Fermi Liquids
Non-Fermi liquids are systems that do not conform to the predictions of Fermi liquid theory. They exhibit anomalous properties, such as non-linear specific heat and unusual resistivity behavior. Understanding non-Fermi liquids is an active area of research, with implications for the study of quantum criticality and exotic phases of matter.
Quantum Criticality
Quantum criticality refers to the behavior of a system near a quantum phase transition, where quantum fluctuations play a dominant role. Near a quantum critical point, the properties of the system can deviate significantly from those predicted by Fermi liquid theory. This deviation is often associated with the emergence of non-Fermi liquid behavior.