**Condensed-Matter-2 Homeworks **

**Keivan Esfarjani, Sharif
University of Technology**

__Saturdays and Mondays 14:00-16:00 __

__Series 1: The Stoner-Wohlfarth model__

Consider the following energy functional for a ferromagnetic grain in an external field and with anisotropy:

E= -M B cos(q-j) - K cos^{2}
q

q represents the angle of M with the easy axis B and (q-j) its angle with the magnetic field. Compute the ground state configurations by minimizing the energy for each magnetic field and plot the hysteresis curves for the 3 cases j=0, p/4 and p/2.

__Series 2: __**Magnons**

Calculate the magnon dispersion relation for spins located on atomic sites of graphene: a bipartite lattice with antiferromagnetic exchange coupling.

__Series 3: Magnon-phonon interaction__

Consider the Heisenberg Hamiltonian where the exchange coupling is a function of interatomic distances. Expand J in powers of the atomic displacements and deduce the magnon-phonon interaction Hamiltonian. Compute to second order in perturbation theory the shift in magnon frequencies due to this interaction.

__Series 4: Phase transitions__

Calculate the critical temperature from ferromagnetic to paramagnetic phase for the Hubbard model in one dimension within the mean field approximation.

__Series 5: Fluctuation-dissipation theorem__

Relate the imaginary part of the retarded response function
c_{AB} (w) to the correlation functions S_{AB}
(w) and S_{BA} (w) defined as

S_{AB }(w)= ∫ <A(t)B(0)> e^{iwt}
dt.

__Series 6: Semiclassical Transport__

Derive the four matrix elements L_{ij }relating the electrical
and thermal currents to the applied electrochemical and temperature gradient
fields. Do the explicit calculation for metals and doped semiconductors with one
type of carriers.