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 1 8 512 Theory of Solids Problem Set 9 Due April 29 2004 Consider a Fermi gas with dispersion k and a repulsive interaction U r Now if N 0 U 1 we nd in mean eld theory the spontaneous appearance of the order parameter U n n and the splitting of the up and down spin bands k k 2 k k 2 1 Show that the transverse spin susceptibility of a system described by the mean eld 2 Hamiltonian is given by 0 B 0 where 0 q f k q f k k q k q i k This is the generalization of the Lindhard function to a spin split band 2 Now include the interaction term in the response to the transverse eld in a self consistent eld approximation Show that q 2B 0 q 1 U 0 q 3 The poles of the numerator in describe the single particle hole excitations Sketch the region in q space where Im 0 due to these excitations 4 The other pole in q occurs when the denominator vanishes Calculate the dispersion of this pole which we identify as the spin wave excitation as follows a Show that at q 0 the denominator vanishes Hint the condition 1 U 0 0 is the same as the self consistency equation for T 2 b Expand 0 q for small q and show that the location of the pole of is given by q Dq 2 Note that unlike the Lindhard function for free fermions the existence of the gap makes the expansion well behaved
View Full Document
Unlocking...