Solving the BCS Hamiltonian via the Bogoliubov Transformation

Question

I was doing a calculation in Giamarchi's Introduction to Many Body Physics, chapter 3, on BCS theory and second quantization, and ran into some confusion with the BCS Hamiltonian. The pdf is here for your reference: http://dpmc.unige.ch/gr_giamarchi/Solides/Files/many-body.pdf

The main confusion comes with eqn. 3.154. Here, the BCS Hamiltonian is given by

$$ H=\sum_k \left( A(k)(\beta_k^\dagger \beta_k-\alpha_k^\dagger \alpha_k)+\Delta(\alpha_k^\dagger \beta_k+\beta_k^\dagger \alpha_k)\right)+ \sum_k\xi(k) $$

Where $\xi(k)$ and $A(k)$ are functions of $k$ and $\alpha_k$ and $\beta_k$ are fermionic operators. Now, I know that the tight-binding Hamiltonian with a periodic potential is given by eqn. 3.128: $$ H=\sum_{k}\bigg( A(k)(\beta_k^\dagger \beta_k-\alpha_k^\dagger \alpha_k)+V(\alpha_k^\dagger \beta_k+\beta_k^\dagger \alpha_k)\bigg) $$ The solution to this is easy to solve with the Bogoliubov transformation, and is given by

$$ E(k)=\sqrt{\xi(k)^2+V^2} $$ I was able to derive this without any problems. However, my question is this: would the solution of the BCS Hamiltonian then be

$$ E_{BCS}(k)=\sqrt{\xi(k)^2+V^2}+\xi(k) $$

Or would it be identical to the tight-binding Hamiltonian? Would the eigenvectors as given by the Bogoliubov transformation change as well?

Answer 1

As is stated in Giamarchi's document, "This is, up to a simple constant, exactly the Hamiltonian we already examined, and it thus can be solved by exactly the same transformations."

Indeed, $\sum_k\xi(k)$ is a constant: it does not involve any of the operators $\alpha_k$ or $\beta_k$, and may be written $E_0 = \sum_k\xi(k)$

Therefore, the resolution of the BCS Hamiltonian would lead to $$ E_{BCS}(k)=\sqrt{\xi(k)^2+V^2}+ \sum_{k'}\xi(k')=\sqrt{\xi(k)^2+V^2}+ E_o $$

which is just the same energies as TB case with potential above, shifted by a fixed amount, and would lead to the same eigenvectors.

Answer 2

What do you mean by "solution of the BCS Hamiltonian"? In quantum mechanics, a solution is a observable quantity you want to calculate. Notwithstanding, I usually call the partition function (the grand canonical one for many-body problems) a "solution", because it contains most information of the system.

Answering your question assuming your question is about the quasi-particle spectrum (excitation energies): The quasi-particle energies are $E_k=\sqrt{\xi_k^2+V^2}$.

The total "energy" (we are actually dealing with the grand-canonical hamiltonian, so $\xi_ \textbf{k}$ are energies only referred from the chemical potential) after a quasi-particle excitation is $E_\textbf{k}=\sqrt{\xi_\textbf{k}^2+V^2}+\sum_\textbf{k} \xi_ \textbf{k}$ plus some other energy constants you may have droped on the way (sometimes the constants are necessary, but not in general).