In fact, this problem is more likely to be a math problem.

When I read a paper(http://arxiv.org/abs/0707.2875), the author includes the characteristic polynomial for a type of matrix $A_k$ with eigenvalues $E_k$.

The characteristic polynomial is expressed as \begin{equation} \rho(E_k) = E_k^4 - (\text{Tr}\{A_k\})E_k^3 + \{\frac{1}{2}[|A_k-I| + |A_k+I|] - 1 - \text{det}A_k\}E_k^2 \notag\\ + \frac{1}{2}[|A_k-I| - |A_k+I|]E_k + \text{det}A_k = 0, \end{equation} while the matrix $A_k$ related here is \begin{equation} A_k= \left( \begin{array}{cc} \xi_k - \vec{\sigma}*(\vec{V}-\vec{g}_k) & i*(d_0+\vec{d}*\vec{\sigma})*\sigma_y\\ [i*(d_0+\vec{d}*\vec{\sigma})*\sigma_y]^\dagger &-\xi_k + \vec{\sigma}^T*(\vec{V}+\vec{g}_k) \end{array} \right). \end{equation} Here, $\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$, $\xi_k=\frac{k^2}{2m}-\mu$, $\sigma_i$ are Pauli matrix. $\vec{g}_k$ is a vector satisfy $\vec{g}_k=-\vec{g}_{-k}$.

Sometime $A_k$ is also called the Bogoliubov-de Gennes(BdG) Hamiltonian.

My question here is how can we deduce the characteristic polynomial $\rho(E_k)$.

  • $\begingroup$ I think you'll likely find that the first equation is simply a version of the general expansion of a fourth degree monic polynomial $\prod\limits_{k=1}^4(\lambda-\lambda_k)$ - for example, the co-efficient of $\lambda^3$ is simply $-\sum\lambda_k$, equal to $\mathrm{Tr}(A)$ and so forth and you're making use of the Cayley-Hamilton theorem. $\endgroup$ Apr 22 '14 at 0:42

The characteristic equation for a $N\times N$ matrix is:

$$\text{det}(A_k-\lambda I)=0$$ It is (more or less) obvious that this can be rewritten as $$\rho(\lambda)=\prod_{k=1}^N (E_k-\lambda)=0$$

where $E_k$ are the eigenvalues of the matrix. Note that your particular notation is a little confusing because $E_k$ seems to imply that there are multiple equations here, while the standard notation (using $\lambda$ for your $E_k$ ) avoids this confusion. The characteristic equation can quite simply be rewritten to the form that you cite.

This is all explained at length in the relevant wikipedia article, which also contains a general formula for $N\times N$ matrices (the $N=4$ case is what you are dealing with). Furthermore, a generic textbook on linear algebra (such as the one cited by the authors of your article) will probably show this too.


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