MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For 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 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 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 H M H which is a measure of the response of the net magnetization M of a system to an applied magnetic eld 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 eld 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 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 H As interesting as that behavior may be we will now consider that to be a solved problem That with associated is we will assume the existence of a set of eigenkets n that diagonalize H eigenvalues energies En In addition to H we now turn on an external probe potential V such that the total Hamil tonian HT ot satis es T ot H V H 1 1 In particular we are interested in probe potentials that arise from the coupling of some external scalar or vector eld to some sort of density in the sample For example the external eld can be an electric potential U r t which couples to the electronic charge density r such that d r r U r t 1 2 V V where the electron density operator r is given by r N r r i 1 3 i 1 1 Response Functions and the Interaction Representation 2 In rst quantized language with r ibeing the position of electron i the N electron system In second quantized notation recall r r r 1 4 where r and r are the electron eld creation and annihilation operators respectively The momentum space version of the electron density operator q is related to r through the Fourier transforms r ei q r q 1 5 q r eik r c k 1 6 k such that q e i q r 1 7 c 1 8 r ck k q k Equation 1 7 is the rst quantized form of q and equation 1 8 is the second quantized form with c the creation operator for an electron with momentum1 k q and c k the destruction k q operator for an electron with momentum k Returning to equation 1 1 we d like to think about V as a perturbation on the external This leads us naturally to consider H as the unperturbed Hamil eld free system Hamiltonian H is a very complicated tonian within the interaction picture representation Recall that this H 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 i h V t t H t 1 9 from the state ket t to form an We can unwind the natural time dependence due to H interaction representation state ket t I by t I eiHt t iHt t I e t 1 10 1 11 Note that in the absence of V these interaction picture state kets are actually the Heisenberg picture state kets of the system Also we have now o cially set h 1 After substituting 1 11 into 1 9 we obtain i h 1 eiHt V e iHt t t V I t 1 12 1 13 k and q are actually wavevectors which di er from momenta by a factor of h When in doubt assume h 1 Response Functions and the Interaction Representation 3 where we have set V I eiHt V e iHt 1 14 Thus the interaction picture state ket evolves simply according to the dynamics governed solely by the interaction picture perturbing potential V I More generally we can write any observable operator in the interaction picture as iHt A I eiHt Ae We can integrate equation 1 12 with respect to t to get t t 0 i dt V I t t 1 15 1 16 At rst it seems like we have not done much to bene t ourselves since all we have done is to convert the ordinary Schrodinger equation a PDE into an integral equation However if V I is small then we can iterate equation 1 16 t t 0 i dt V I t 0 1 17 The essence of linear response theory is that we focus ourselves on cases where V I is su ciently weak that the perturbation series represented by equation 1 17 has essentially converged after including just the rst non trivial term listed above This term is linear in V I Throughout this discussion we will be working at T 0 so 0 is simply the ground state Note that we have taken our initial time of the non perturbed total system Hamiltonian H i e the lower limit of integration in equation 1 16 to be This is because we want to imagine turning on the probing potential V adiabatically that is so slowly that the system tracks the ground state for all nite times If we were to turn on the probe sharply the system would exhibit complicated ringing behavior that we are not interested in We now return to our model experiment for studying the properties of our system After applying some probe via the external potential V we want to measure the response of some We characterize this response through the expectation value of A observable of the system A A t A t A t A I t 1 18 1 19 The key now is to substitute in the approximation for t given by equation 1 17 into equation 1 19 Since we have only kept terms up to linear order in V I we must be careful only to keep terms to this order After performing …