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 field 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 field ( 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 satisfies: 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 field to some sort of “density” in the sample. For example, the external field can be an electric potential U ( r, t ), which couples to the electronic charge density ρ ˆ( r ) such that ˆ V = d� r 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 first 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 field creation and annihilation operators, respectively.
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