27 - Elementary excitations: the Bogoliubov–Valatin transformation

Summary

We recall expressions (26.19) for the expectation value of the reduced Hamiltonian

Let us calculate the energy required to add an electron to the system in a state |k↓〉 assuming its companion state | –k↓〉 is empty. In order to do this we must: (i) account for the energy increase due to removing the amplitude of the pair associated with this wave vector k and (ii) add the energy of the lone electron introduced into the state | k〉. When we remove the bound pair from the ground state, according to (27.1) the energy of the system changes by an amount

(the factor 2 in the second term arises because the chosen pair state (k, – k) occurs twice since we have a double sum). Using Eq. (26.23) we may write the form (27.2) as

Adding to (27.3) the energy ξ_k of one (unbound) electron then yields the quasiparticle excitation energy,

where we used Eqs. (26.24) and (26.27). Thus the energy needed to add an electron in state k↓ is σ_k. If we calculate the energy required to remove an electron in a state – k↓ we also obtain σ_k. Note the minimum excitation energy is Ak; i.e., the excitation spectrum has an energy gap.