Hartree–Fock Equations and Landau’s Fermi-Liquid Theory
In the previous two chapters, we have considered only non-interacting quantum many-particle systems in obtaining exact results for basic thermodynamic quantities and two-particle correlations. However, in real systems, particles interact, making exact statistical-mechanical calculations impossible except for some low-dimensional solvable models. Thus, we are almost always obliged to introduce some approximation when studying interacting systems. Here, we derive the Hartree–Fock equations, i.e., one of the simplest approximation schemes for studying interaction effects at finite temperatures, based on a variational principle for the grand potential. They are most effective when interactions are weak and repulsive and are crucial in describing molecular-field effects, but may not be applicable to systems with attractive potentials, as will be seen in later chapters. Next, we apply the Hartree–Fock equations to fermions at low temperatures to clarify how the interaction affects thermodynamic properties along the lines of Landau’s Fermi-liquid theory.
KeywordsDensity Matrix Variational Principle Fermi Surface Fermi Energy Spin Susceptibility
- 1.M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965)Google Scholar
- 2.G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists (Academic, New York, 2012)Google Scholar
- 4.I.M. Gelfand, S.V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, 1963)Google Scholar
- 5.L.D. Landau, J. Exp. Theor. Phys. 30, 1058 (1956). (Sov. Phys. JETP 3, 920 (1957))Google Scholar
- 7.D. Vollhardt, P. Wölfle, The Superfluid Phases of Helium 3 (Taylor & Francis, London, 1990), p. 31Google Scholar