Exercises for

Exercises for Solid State Theory Prof. Dr. A. Muramatsu

SS 2013 Sheet 12

Exercise 1 - The Bogoliubov-Valatin Transformation (6 points) We consider the following Hamiltonian:

H −µN =X

kσ

ξ_kn_kσ−X

k

h

∆_kc^†_k↑c^†_−k↓+ ∆^∗_kc_−k↓c_k↑−∆_kb^∗_k i

. (1)

where b^∗_k = −ψ^∗_k of the notes. This Hamiltonian can be diagonalized by a suitable linear transformation to define new Fermi operators α_k, the Bogoliubov-Valatin-Transformation:

c_k↑ =u^∗_kα_k↑+v_kα^†_−k↓

c^†_−k↓ =−v^∗_kα_k↑+u_kα^†_−k↓

α_k↑ =u_kc_k↑−v_kc^†_−k↓

α_−k↓ =v_kc^†_k↑+u_kc_−k↓, with|u_k|²+|v_k|² = 1.

a) We first study this Hamiltonian for non-interacting fermions with ∆ = 0. Demonstrate that the new operators can be interpreted has hole creation operator for k < kf and particle creation operator for k>kf.

b) We now study the superconducting situation with ∆ 6= 0. Calculate explicitly the com- mutation rules for the new operators α_kσ, {α^†_k↑, α_k↑} and {α_k↑, α−k↓}, to get conditions on u_k and v_k.

c) Rewrite the Hamiltonian in terms of the Bogoliubov operators α_kσ.

d) To diagonalize this Hamiltonian we choose uk,vk such that the non diagonal terms van- ishes. For this, regroup the terms proportional to α_−k↓α_k↑ and α^†_k↓α^†_−k↓ in a system of two equations and show that it gives us the condition :

∆^∗_kv_k

u_k =−ξ_k±q

ξ_k²+|∆_k|² ≡E_k−ξ_k, (2) and give the resulting condition onu_k and v_k in function ofξ_kand E_k.

e) Using the previous results show that the Hamiltonian has the following form:

H −µN =X

k

(ξ_k−E_k+ ∆_kb^∗_k) +X

k

E_k(α^†_k↑α_k↑+α^†_−k↓α_−k↓). (3)

Solutions due on: 8th of July, 2013

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