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 3 Properties of t h e Response Function In this lecture we will discuss some general properties of the response functions X and some uselul relations that they satisfy 3 1 General Properties of q w Recall that wit11 n F i E IR and U F i E IR Under Fourier transform this implies n w U f w n q w US gl ww As a result where X is the imaginary part of the response function X gl ww Consider the extension of w to the complex plane We can then rewrite the expression for x q 74 as Without the iq term in the energy denominator there would be siryularities poles on the real axis whenever w is equa 1to the spacing between the ground state and some excited state The presence of iq pushes these poles just into the lower l Bplane emuring that x q7 w is analytic in the entire upper l 2 w plane including the real axis Analyticity of x w in the upper 1 2 plane is needed to build causality into the theory Consider the response function in time t To evaluate this integral we perform a contour integral in the complex w plane For i 0 closing the contour in the upper 1 2 plane ensures that leJiwtl 0 on the curved portion of the Kramers Kronig 2 contour Since we have ensured that is analytic in the upper half plane Cauchy s residue theorem guarantees that the integral over the entire contour is 0 As a result the piece we need i e the integral from to along the real axis must also be 0 Thus t 0 for t 0 which means that the system cannot respond to a perturbation until after the perturbation has occurred What about the t 0 case In this case the contour must be closed in the lower 1 2 plane to prevent the exponential from blowing up However the i in the energy denominator has pushed the singularities into this region of the complex plane Thus the value of the contour integral will be nonzero and the system will respond to perturbations for t 0 3 2 Kramers Kronig Consider the integral d q 0 3 8 As we have just shown in the previous section our de nition of q ensures that q is analytic in the upper 1 2 complex plane Although the integrand here has a pole on the real axis due to the in the denominator by making a hump over this pole we can ensure that the value of the contour integral itself vanishes by Cauchy s residue theorem Assuming that q 0 as 1 0 d q Pr i q 3 9 where the additional term i q is one half of the contribution from the pole at that we picked up by making a hump over the pole Thus for xed q 1 q q Pr d 3 10 q 1 q Pr d 3 11 or equivalently q lim 0 d q i 3 12 The important message from all of this is that the entire response function q can be reconstructed from its imaginary or real part alone
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