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152 CHAPTER 10. FERMI LIQUID THEORY will evolve in time following the Schrödinger prescription ψ k (r, t) = ψ k (r)e −iɛ kt/h¯ . (10.3) Here ψ k is the Bloch wavefunction satisfying the time-independent Schrödinger equation, and the time-dependent solution oscillates in time with a single frequency, given by h¯ω = ɛ k , the band energy. If we look at this in Fourier space, we should say that ψ k (r, ω) = 2πψ k (r)δ(ω − ɛ k ) (10.4) so that the wavefunction has spectral weight only at ω = ɛ k /h¯. 1 We can say that the probability amplitude of finding an electronic state with energy ω and momentum k is just A(k, ω) = δ(ω − ɛ k ) , (10.5) where the quantity A(k, ω) is usually called the electron spectral function. What about an interacting system In this case, (i) interactions modify the precise form of the dispersion relation, so we should replace ɛ k by a renormalised ˜ɛ k . This latter is often referred to as a mass renormalisation: m ∗ /m = ɛ k / ˜ɛ k . Moreover, (ii), if we add a quasiparticle, it will scatter from other quasiparticles or by creating particle-hole pairs. Because of this, the probability amplitude of finding a quasiparticle of momentum k at time t will decay exponentially at a rate given by the quasiparticle scattering rate Γ k . These two effects change the time-dependence of the state to e (iɛ k/h¯−Γ k )t and after a Fourier transform lead to an ansatz for the spectral function in an interacting system of the form A(k, ω) = − 1 [ ] π I 1 (10.6) ω − ˜ɛ k /h¯ + iΓ(k) Notice that if the inverse lifetime Γ k → 0, the spectral function reduces to the noninteracting (10.5). (10.6) describes quasiparticles with a dispersion curve ˜ɛ k and a decay rate (inverse lifetime) Γ k . We must be careful about the chemical potential. If we are in equilibrium (and at T=0), we cannot add fermionic excitation at an energy ω < µ. So we shall infer that for ω > µ, (10.6) is the spectral function for particle-like excitations, whereas for ω < µ it is the spectral function for holes. If Γ k is small, then the quasiparticles are long-lived and have some real meaning. More precisely: a quasiparticle resonance at energy h¯ω = ˜ɛ k can be resolved, if the peak width in the spectral function is less than the peak’s centre position, or, borrowing the terminology developed for damped harmonic oscillators, if the quality factor (ratio of peak position over peak width) is larger than 1. For an interacting system (right panel in Fig. 10.7), A has width = 2 × Γ k . The phase-space argument made in section 10.3.1 showed that Γ k ∝ ɛ 2 . The quality factor ˜ɛ k /Γ ∝ ˜ɛ k /˜ɛ 2 k therefore diverges for ɛ → 0. We find that quasiparticles are overdamped at high energies, but well-defined for E → E F . 1 Getting the sign here requires one to adopt a sign convention for Fourier transforms that is opposite to the one often used in maths books. We have also set h¯ = 1.
10.3. TOTAL ENERGY EXPANSION FOR LANDAU FERMI LIQUID 153 Figure 10.7: Electron spectral functions for Fermi gas (left) and Fermi liquid (right) This shows that the quasiparticle concept is self-consistent: it is possible to have quasiparticles, which scatter so rarely that their lifetime is sufficiently long to observe them. However, this does not guarantee that the Fermi liquid state always exists in every metal. The conditions under which Fermi liquids exist or not is an active field of both experimental and theoretical research. 10.3.4 Tunability of the quasiparticle interaction ∑ The interaction term 1 2 kk f(k, k ′ )n(k)n(k ′ ) in the Landau expansion for the total energy ′ causes various quasiparticle properties to be changed w.r.t. the free electron value. Most importantly, the effective mass m ∗ can be orders of magnitude larger than the bare electron mass. The quasiparticle interaction function f(k, k ′ ) is completely different from the Coulomb repulsion, which acts on the underlying electrons. In particular, f(k, k ′ ) can be spin-dependent. This is an example of the tunability of correlated electron systems: although the underlying Coulomb interaction is fixed, the effective interaction in the low energy model depends strongly on details of the system, and can therefore be tuned over a wide range by changing, for example, magnetic field, pressure or doping.
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