Physics , Condensed Matter Homework

The key insight of Bogoliubov is that although the mean-field Hamiltonian (1) is clearly not diagonal in the space of states created by the operators ck,↑/↓, it may be diagonal in a basis of states created by operators related to these via a simple SU2 transformation. Indeed, it may be the case that ‘electrons’ and ‘holes’ are the wrong degrees of freedom to consider; we should at least look to see if the Hamiltonian is simpler in terms of any collective degrees of freedom. Along these lines, we define the rotated, ‘collective’ or ‘Bogoliubov’ creation and annihilation operators bk,↑/↓ and b † k,↑/↓ given by