Fermi Surface

In quantum mechanics, the Fermi surface arises due to the Pauli exclusion principle, which dictates that no two fermions (electrons) can occupy the same quantum state. As a result, electrons fill energy bands starting from the lowest available energy. The Fermi surface represents the boundary in momentum space that separates occupied and unoccupied electronic states at absolute zero temperature.

Studying systems with a Fermi surface is crucial as they provide the foundation for understanding a wide range of physical phenomena in condensed matter physics, from the behavior of metals and superconductors to exotic quantum states in strongly correlated systems. Complex states of matter, such as superconductivity, magnetism, and density waves, often emerge from instabilities or interactions involving the Fermi surface. Notably, systems with a Fermi surface exhibit a unique mixing of UV and IR scales, distinguishing them fundamentally from simpler systems like scalar field theories and giving rise to rich and unconventional scaling behavior.

Interacting Fermi Liquid

The Fermi liquid phase is a state of matter describing a system of interacting fermions (e.g., electrons in metals) that behaves similarly to a gas of non-interacting particles but with renormalized properties due to interactions. Its physical properties include

Specific heat: Linear in temperature at low T, due to the density of states at the Fermi surface.

Electrical resistivity: At low temperatures, ~ T^2

Landau Fermi liquid theory:

The low-energy excitations of the system are described as quasiparticles, which are fermions with the same quantum numbers (spin, charge) as the original particles but with renormalized properties such as effective mass, lifetime.
The interactions between quasiparticles are characterized by Landau parameters, which quantify how interactions affect properties like compressibility and magnetic susceptibility.
The interacting system can be continuously connected to the free Fermi gas by "turning on" interactions, without a phase transition.

Particle-hole channel
Forward scattering

Particle-particle channel
BCS scattering

Non-Fermi Liquid

A non-Fermi liquid (NFL) is a phase of matter in which the behavior of interacting fermions deviates from the predictions of Landau Fermi liquid theory. It emerges in systems where strong interactions or quantum fluctuations significantly disrupt the quasiparticle picture.

In NFLs, the concept of long-lived quasiparticles fails, particularly near the Fermi surface, due to strong interactions or scattering. An NFL can also arise when the Fermi surface is coupled to a critical boson. This bosonic mode is often associated with fluctuations of an order parameter, such as in magnetic, charge, or nematic phase transitions. Near the Fermi surface, the lifetime of fermionic excitations becomes comparable to or even shorter than their energy scale, resulting in non-Fermi-liquid behavior characterized by unconventional scaling laws.

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