My question arise from this article: Edge superconducting correlation in the attractive-U Kane-Mele-Hubbard model. I will describe my question in detail so that you might not need to look into that article. What I want to ask for help is an easy way to derive the electron number equation, may be using some thermodynamic relations just as how I derive the gap equation.
The Hamiltonian considered is the mono-layer graphene with intrinsic SO interaction, plus the negative-U Hubbard term. After the mean field approximation with S-wave superconducting order parameter, we obtain the mean-field Hamiltonian:
$$
H=\sum_k\phi_k^\dagger H_k\phi_k+E_0
$$
where $\phi_k$ is the Nambu spinor $\phi_k^\dagger=(a_{k\uparrow}^\dagger, b_{k\uparrow}^\dagger, a_{-k\downarrow}, b_{-k\downarrow})$, $E_0=2N\Delta^2/U$, $N$ is the number of unit cell, and
$$
H_k=\begin{pmatrix}
\lambda_k-\mu & -t\gamma_k & -\Delta & 0 \\
-t\lambda_k^* & -\lambda_k-\mu & 0 & -\Delta \\
-\Delta^* & 0 & -\lambda_k+\mu & t\gamma_k \\
0 & -\Delta^* & t\gamma_k^* & \lambda_k+\mu
\end{pmatrix}
$$
The $\mu$ is the chemical potential in the original Hubbard term, $-t\gamma_k$ is the sum of the graphene hopping integral of nearest neighbors, $\gamma_k$ is from SO term. We can just ignore their physical meaning and regard them as some parameters, they are not very important to my question.
Diagonalizing $H_k$, we have four eigenvalues, $\omega_{ks\alpha}=\alpha\omega_{ks}=\alpha\sqrt{(\epsilon_k+s \mu)^2+\Delta^2}$ with $\epsilon_k=\sqrt{\lambda_k^2+t^2|\gamma_k|^2}$, where $s,\alpha$ are $\pm 1$.
Now we have gap equation and electron number equation:
$$
\frac{1}{U}=\frac{1}{4N}\sum_{ks}\frac{\tanh{(\beta\omega_{ks}/2)}}{\omega_{ks}}
$$
$$
n_e-1=-\frac{1}{N}\sum_{ks}\frac{s\epsilon_k -\mu}{\omega_{ks}}\tanh(\beta \omega_{ks}/2)
$$
where $n_e$ is the average electron number on one sublattice.
The following is how I derive the gap equation, the free energy is:
$$
F=-\frac{1}{\beta}\sum_{ks\alpha}\ln{(1+\mathrm{e}^{-\beta\omega_{ks\alpha}})}+\frac{2N\Delta^2}{U}
$$
the free energy is minimized when $\Delta$ choose to have its true value, i.e. using $\partial F/\partial \Delta=0$ we can derive the gap equation showing above.
How can I derive the electron number equation? I know in principle I can derive it by representing the original electron operators instead of the diagonalized Bogoliubov quasi-particle operators, but this is much too complicated even one trying to derive them using Mathematica.
So just as I said in the beginning of this question:I need your help to get an easy way to derive the electron number equation, may be using some thermodynamic relations just as how I derive the gap equation
This post imported from StackExchange Physics at 2014-08-22 05:03 (UCT), posted by SE-user luming
Q&A (4871)
