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.1 8.512 Theory of Solids Problem Set 9 Due April 29, 2004 Consider a Fermi gas with dispersion �k and a repulsive interaction Uδ(r). Now if N(0)U > 1, we find in mean field theory the spontaneous appearance of the order parameter: Δ = U�n ↑ − n ↓� , and the splitting of the up- and down-spin bands, �˜k = �k − Δ/2 ↑ �˜k = �↓ k + Δ/2 . 1. Show that the transverse spin susceptibility of a system describ ed by the mean field Hamiltonian is given by χ0 = µ2Γ0 where⊥ B� f (�˜k+q,) �k,) Γ0(q, ω) = ↑− f (˜↓.ω − �˜k+q+ ˜ + iη↑ �k+qk ↓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 field in a self consistent field approximation. Show that µ2 Γ0(q, ω) χ (q, ω) =B.⊥1 − UΓ0(q, ω) 3. The poles of the numerator in χ describe the single particle-hole excitations. Sketch ⊥the region in (ω, q) space where Imχ = 0 due to these excitations. ⊥4. The other pole in χ (q, ω) 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 = ω = 0, the denominator vanishes. [Hint: the condition 1 − UΓ0 = 0 is the same as the self-consistency equation for Δ(T ).] �2 (b) Expand Γ0(q, ω) for small q, ω and show that the location of the pole of χ⊥ is given by ω(q) = Dq2 . Note that unlike the Lindhard function for free fermions, the existence of the gap Δ makes the expansion well
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