Landau Theory of Fermi Liquids : Equilibrium Properties

Transcription

1 Quantum Liquids LECTURE I-II Landau Theory of Fermi Liquids : Phenomenology and Microscopic Foundations LECTURE III Superfluidity. Bogoliubov theory. Bose-Einstein condensation. LECTURE IV Luttinger Liquids. G-ology model and spin-charge separation LECTURE V Renormalization Group (RG) for interacting electrons. RG in one-dimension and overview in 2D LECTURE VI Transport properties: conductivity, persistent currents, quantum Hall effect, disorder effect

2 Landau Theory of Fermi Liquids : Equilibrium Properties Effective Mass m*/m=(1+f 1 /3) Specific Heat FERMI GAS Cv=1/3m*k F k B2 T Compressibility T χ 1 =ρk F 2 /(3m*) (1+F 0 )/(3+F 1 ) Magnetic susceptibility χ Μ =γ 2 k F m*/[4π 2 (1+Z 0 )]

3 Landau Theory of Fermi Liquids : Non-equilibrium Properties BIBLIOGRAPHY David Pines and Philippe Nozieres The Theory of Quantum Liquids Part I L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, Menlo Park, 1962). G. Baym and C. Pethick, in The Physics of Liquid and Solid Helium, edited by K. H.Bennemann and J. B. Ketterson (Wiley, New York, 1976), Vol. 2. Dieter Vollhardt Peter Wolfe The Superfluid Phases of Helium 3

4 Basic assumptions Deviations from equilibrium are described by a Boltzmann equation for a space and time dependent quasiparticle distribution function n(k, r, t), which describes the density of quasi particles of momentum and spin (k, σ) at point r and time t. (Constraint on the spatial-temporal variation ) Because of the r dependent n, the quasiparticle energy is itself r dependent. One then assumes the following quasi classical equations of motion

5 Basic consequence Linearizing the Boltzmann equation for a space and time dependent quasiparticle distribution function n(k, r, t), in the collisionless regime there exists a collective mode corresponding to the phase fluctuation of the Fermi surface (zero sound) Thermodynamic sound cannot exists and requires a collisional integral different from zero. The zero sound is a purely quantum mechanical effect.

6 Preliminarities Let us consider a state characterized by a distribution function The energy can be expanded in powers of np: Assume spatial translation and localized interactions then: The local excitation energy of a quasiparticle is equal to

7 Balance flow in the phase space: Boltzmann Equation B. Equation in the collisionless regime Linearization: Keeping only terms of first order in np Adding collisions contributions and external forces, the transport eq. is: Drift term Collision integral: rate of change of the distribution

8 Collective Modes We consider excitations which are periodic in space and time, with wave vector q: The transport equation in the collisionless regime gives: In terms of the normal displacement u:

9 Collective Modes: continued Introducing the polar coordinates: Ratio between wave velocity and Fermi velocity Zero sound Neglecting the spin dependence in the interaction, a solution is: After the integration:

10 Zero sound: continued Other modes: spin waves

11 Properties of degenerate 3 He Normal Fermi liquid between 0.003K and 0.1K Superfluid at 0.003K

12 Superfluidity of 3 He- 4 He 4 He Superfluidity is a consequence of Bose-Einstein condensation 3 He

13 Order Parameter and Mean Field Theory For BEC Mean field theory formulated by Bogoliubov (1947) In the Fock space: Boson annihilation operator BEC occurs when:

14 States are equivalent thus: C-number Small perturbation Generalization to time-dependent configuration Condensate density=>brokengauge symmetry Classical field, or order parameter OFF-DIAGONAL LONG-RANGE ORDER

15 IN A FINITE SIZE SYSTEM Is the eigenfunction of the one-body density matrix with the highest eigenvalue EQUATION FOR THE ORDER PARAMETER Gross-Pitaevskii Equation (1961)

16 EQUIVALENT DERIVATION GROUND STATE Chemical potential Non-linear Schroedinger Equation

17 GROUND STATE:HARMONIC POTENTIAL