Landau Theory of Fermi Liquids: An In-Depth Explanation

Landau theory, developed by the renowned Russian physicist Lev Davidovich Landau (1908–1968), is a fundamental concept in the field of condensed matter physics. This theory is particularly significant in understanding the behavior of metals at low energies. By analyzing the free energy of the system and its analytical properties, Landau provided a comprehensive framework to describe the excitations in a conductor composed primarily of a gas of free electrons. This article delves into the key aspects of Landau theory and its applications in modern physics.

Introduction to Landau Theory

Landau theory was developed to address the behavior of metals, specifically focusing on the concept of a Fermi liquid. A Fermi liquid is a system of interacting fermions (such as electrons) that behaves similarly to an ideal gas at low energies. This theory is particularly applicable in the context of condensed matter and provides a low energy effective theory for the behavior of electrons in conductors. The theory is based on two basic assumptions: the free energy F of the system must be analytical and must follow the symmetry properties of the Hamiltonian.

The Central Equation of Landau Theory

The central equation of Landau theory can be expressed as:

F(ξ, T) -g1ξ2/2 g2ξ4/4 g4ξ6/6 ...

where the free energy F is a function of the order parameter ξ, and the multiplier parameters gi can be temperature-dependent: gi gi(T).

Key Concepts and Assumptions

The theory relies on several key concepts:

1. Order Parameter

The order parameter ξ is a measure of the deviation from equilibrium. In the context of Landau theory, this parameter captures the deviation from the ideal gas behavior. The higher order terms in the equation represent increasingly complex interactions between the electrons.

2. Symmetry and Analyticity

The symmetry of the Hamiltonian is a crucial aspect of the theory. The free energy must be a symmetric function of the order parameter ξ. Additionally, the function must be analytical, meaning it can be expressed as a power series. This allows for the application of perturbation theory, which is essential for making quantitative predictions.

3. Infinity of Terms

The series in the central equation can be extended to an infinite number of terms, reflecting the increasing complexity of the interactions. Each term in the series corresponds to a different order of interaction and contributes to the overall behavior of the system.

Applications and Implications

Landau theory is significant because it provides a framework for understanding the behavior of metals at low energies. The theory makes numerous quantitative predictions about various properties of metals, such as specific heat, heat capacity, and electrical conductivity.

Conclusion

In conclusion, Landau theory of Fermi liquids offers a fundamental understanding of the behavior of metals at low energies. The theory is based on the analysis of the free energy and its analytical properties, and it provides a powerful tool for making quantitative predictions. For further reading, I recommend the article Effective Field Theory and the Fermi Surface, which delves deeper into the details and applications of Landau theory.