# How to derive electron number equation of Bogoliubov Hamiltonian using thermodynamic relations.

+ 5 like - 0 dislike
91 views

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
I suggest trying $\partial F/\partial \mu=0$.

This post imported from StackExchange Physics at 2014-08-22 05:03 (UCT), posted by SE-user leongz

Thermodynamic relation $N=-\frac{\partial J}{\partial \mu}$ exactly gives you the particle number equation, wherein $J$ is the macroscopic thermodynamic potential, i.e., the quantity $F$ in your question.
By thermodynamics, $dJ=-SdT+Ydy-Nd\mu$ tells you why the partial derivative equation is valid. And in statistical mechanics, macroscopic ensemble formalism defines $J$ as $e^{-\beta J}\equiv\mathrm{Tr}[e^{-\beta (\hat{H}-\mu\hat{N})}]$. Of course, you can derive the partial derivative equation within this formalism as well.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.