Lecture 1: Linear Response TheoryResponse Functions and the Interaction RepresentationResponse FunctionsElectron Density Response to an Applied Electric PotentialSanity Check: Free FermionsThe Correlation Function S("017Er, t)Measuring S("017Eq, )MIT OpenCourseWare http://ocw.mit.edu 8.512 Theory of Solids IISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.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 c an 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 1023 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 χH ≡ ∂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 ˆH 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 ˆH. As interesting as that behavior may be, we will now consider that to be a solved problem. That ˆis, we will assume the e xistence of a set of eigenkets {|n�} that diagonalize H with associated eigenvalues (energies) En. In addition to ˆH, we now turn on an external probe potential Vˆ, such that the total Hamil-tonian HT ot satisfies: ˆ ˆHT ot = H + Vˆ(1.1) In particular, we are interested in probe potentials that arise from the coupling of some external scalar or vector field to s ome 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 ρˆ(�r) such that � ˆV = d�r ˆρ (�r) U (�r, t) (1.2) V where the electron density operator ˆρ (�r) is given by N� ˆρ (�r) = δ (�r − �ri) (1.3) i=1 1� �� �� �� ���Response Functions and the Interaction Representation 2 In first quantized language, with r�ibeing the position of electron i the N-electron system. In second quantized notation, recall ρˆ(�r) = Ψ† (�r) Ψ (�r) (1.4) where Ψ† (�r) and Ψ (�r) are the electron field creation and annihilation operators, respectively. The momentum space version of the electron density operator, ˆρ (q�), is related to ρˆ(�r) through the Fourier transforms: i� �ρˆ(�r) = e q· rρˆ(�q) (1.5) q i��rΨ (�r) = e k·c�(1.6)k k such that e−i� �rρˆ(q�) = q·(1.7) r = c†c�(1.8)k−�kq k Equation (1.7) is the first quantized form of ˆρ (�q), and equation (1.8) is the second quantized form with c†the creation ope rator for an electron with momentum1 �k−q� and c�the destruction k−�kq operator for an electron with momentum �k. Returning to equation (1.1), we’d like to think about Vˆas a perturbation on the external field-free system Hamiltonian ˆH as the unperturbed Hamil-H. This leads us naturally to consider ˆˆtonian within the interaction picture representation. Recall that this H is a very complicated beast with all of the electron-electron repulsions included, but for our purposes we just take as a given that there are a set of eigenstates and energies that diagonalize this Hamiltonian. Recall the formulation of the interaction representation: ∂ h∂t|φ (t)� = (ˆi¯ H + Vˆ) φ (t)� (1.9)|ˆWe can “unwind” the natural time dependence due to H from the state ket φ (t)� to form an interaction representation state ket φ˜(t)�I by || ˜Ht|φ (t)�I = e i ˆφ (t)� (1.10)|Htφ˜(t)� (1.11)|φ (t)�I = e−i ˆ| Note that in the absence of Vˆ, these interaction picture state kets are actually the Heisenberg picture state kets of the system. Also, we have now officially set ¯h = 1. After substituting (1.11) into (1.9), we obtain i¯Ht ˆHt˜h∂ = e i ˆV e−i ˆφ (t)� (1.12)∂t | ˆ= VIφ˜(t)� (1.13)| 1 �k and q� are actually wavevectors, which differ from momenta by a factor of ¯ h = 1. h. When in doubt, assume ¯Response Functions and the Interaction Representation 3 where we have set Ht ˆHt VˆI = e i ˆV e−i ˆ(1.14) Thus the interaction picture state ket evolves simply according to the dynamics governed solely by the interaction picture perturbing potential VˆI. More generally, we c an write any observable (operator) in the interaction picture as ˆHt ˆHt AI = e i ˆAe−i ˆ(1.15) We can integrate equation (1.12) with respect to t to get � t ˆdt� VI (t�) φ˜(t�)� (1.16)|φ˜(t)� = |φ0� − i |−∞ At first it seems like we have not done much to benefit ourselves, since all we have done is to convert the ordinary Schrodinger equation, a PDE, into an integral equation. However, if VˆI is small, then we can iterate equation (1.16): � t dt� VˆI (t�) |φ0� + · · · (1.17)|φ˜(t)� ≈ |φ0� − i −∞ The essence of linear response theory is that we focus ourselves on cases where VˆI is suffi-ciently weak that the perturbation series represented by equation (1.17) has essentially converged after including just the first non-trivial term listed ab ove. This term is linear in VˆI. Throughout this discussion, we will be working at T = 0, so φ0� is simply the ground state ˆ|of the non-perturbe d total system Hamiltonian H. Note that we have taken our initial time, i.e. the lower limit of