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lecture1 notes - Lecture 1 Linear Response Theory Last...

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Lecture 1: Linear Response Theory Last semester in 8.511, we discussed linear response theory in the context of charge screening and the free-fermion polarization function. This theory can be extended to a much wider range of areas, however, and is a very useful tool in solid state physics. We’ll begin this semester by going back and studying linear response theory again with a more formal approach, and then returning to this like superconductivity a bit later. 1.1 Response Functions and the Interaction Representation In solid state physics, we ordinarily think about many-body systems, with something on the order of 10 23 particles. With so many particles, it is usually impossible to even think about a wave function for the whole system. As a result, it is often more useful for us to think in terms of the macroscopic observable behaviors of systems rather than their particular microscopic states. One example of such a macroscopic property is the magnetic susceptibility χ H ∂M , which ∂H is a measure of the response of the net magnetization M of a system to an applied magnetic Feld H ( r, t ). This is the type of behavior we will be thinking about: we can mathematically probe the system with some perturbing external probe or Feld ( e.g. H ( r, t )), and try to predict what the system’s response will be in terms of the expectation values of some observable quantities. Let ˆ H be the full many-body Hamiltonian for some isolated system that we are interested in. We spent most of 8.511 thinking about how to solve for the behavior of a system governed by ˆ H . As interesting as that behavior may be, we will now consider that to be a solved problem. That ˆ is, we will assume the existence of a set of eigenkets {| n } that diagonalize H with associated eigenvalues (energies) E n . In addition to ˆ H , we now turn on an external probe potential V ˆ , such that the total Hamil- tonian H T ot satisFes: ˆ ˆ H T ot = H + V ˆ (1.1) In particular, we are interested in probe potentials that arise from the coupling of some external scalar or vector Feld to some sort of “density” in the sample. ±or example, the external Feld can be an electric potential U ( r, t ), which couples to the electronic charge density ρ ˆ( r ) such that ˆ V = dr ˆ ρ ( r ) U ( r, t ) (1.2) V where the electron density operator ˆ ρ ( r ) is given by N ± ˆ ρ ( r ) = δ ( r r i ) (1.3) i =1 1
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Response Functions and the Interaction Representation 2 In frst quantized language, with r i being the position oF electron i the N -electron system. In second quantized notation, recall ρ ˆ( r ) = Ψ ( r ) Ψ ( r ) (1.4) where Ψ ( r ) and Ψ ( r ) are the electron feld creation and annihilation operators, respectively. The momentum space version oF the electron density operator, ˆ ρ ( q ), is related to ρ r ) through the ±ourier transForms: i ρ r ) = e q · r ρ q ) (1.5) q i r Ψ ( r ) = e k · c (1.6) k k such that e i r ρ q ) = q · (1.7) r = c c (1.8) k k q k Equation (1.7) is the frst quantized Form oF ˆ ρ ( q ), and equation (1.8) is the second quantized Form with c the creation operator For an electron with momentum 1 k q
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