ex13 - x ) N ↓ N e = 1 2 (1-x ) (2) where N ↑ ( ↓ )...

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Solid State Theory Exercise 13 SS 11 Prof. M. Sigrist Exercise 13.1 Critical temperature in the Stoner model We consider three types of dispersion relations: ± k = ± 0 ± ~ 2 k 2 2 m (3D) and ± k = ± 0 + αk (1D). Plot the critical temperature of the Stoner model for fixed interaction strength U depend- ing on the chemical potential μ . Exercise 13.2 Stoner instability In the lecture, it was shown that a system described by the mean-field Hamiltonian H MF = 1 Ω X k ,s ( ± k + Un - s ) c k s c k s - Un n (1) shows an instability towards a magnetically ordered state at N ( ± F ) U C = 1 (note that here, N ( ± ) is the density of states per spin). Show for the case of a parabolic dispersion and T = 0 that there are actually three distinct states: a paramagnetic state: N ( ± F ) U < 1, an imperfect ferromagnetic state: 3 / 2 4 / 3 > N ( ± F ) U > 1 and a perfect ferromagnetic state: N ( ± F ) U > 3 / 2 4 / 3 . Hint: Introduce a variable for the magnitude of the polarization N N e = 1 2 (1 +
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Unformatted text preview: x ) N ↓ N e = 1 2 (1-x ) (2) where N ↑ ( ↓ ) is the total number of up-spins (down-spins) and N e is the total number of electrons. Minimize the total energy of the system with respect to x . Plot the polarization of the system x as a function of N ( ± F ) U . Exercise 13.3 Particle-Hole Excitations in Itinerant Ferromagnets In section 7.3 of the lecture notes the low-energy spectrum of (magnons) spin-waves in itinerant ferromagnets was derived. It is crucial for the existence of well-defined magnons that elementary particle-hole excitations are gapped. Try to explain (without detailed calculations) why there is such a gap and why it is important for the observability of magnons! Office hour: Monday, May 30th, 2011 - 9:00 to 11:00 am HIT K 12.1 Jonathan Buhmann...
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