3
$\begingroup$

I know that by using Jordan-Wigner transform(JWT), we can transform spin-$1/2$ systems into fermions. My problem is, for example, after JWT, we have a hamiltonian of form $$\epsilon\left(c_{1}^{\dagger} c_{1}+c_{2}^{\dagger} c_{2}\right)+\lambda\left(c_{1}^{\dagger} c_{2}^{\dagger}+c_{2} c_{1}\right)\tag{1}$$ with $\lambda$ and $\epsilon\in \mathbb R$. If I directly write it as quadratic form $$\left( \begin{matrix} c_{1}^{\dagger}& c_2& c_{2}^{\dagger}& c_1\\ \end{matrix} \right) \left( \begin{matrix} \epsilon& \lambda& 0& 0\\ \lambda& 0& 0& 0\\ 0& 0& \epsilon& 0\\ 0& 0& 0& 0\\ \end{matrix} \right) \left( \begin{array}{c} c_1\\ c_{2}^{\dagger}\\ c_2\\ c_{1}^{\dagger}\\ \end{array} \right) $$ the eigenvalues of the middle hermitian matrix will be $$0,\epsilon ,\frac{\epsilon \pm \sqrt{\epsilon ^2+4\lambda ^2}}{2}$$

But if I change eq.(1) into another form with canonical commutation relation for fermions as follows: $$ \begin{align} &\epsilon \left( c_{1}^{\dagger}c_1+c_{2}^{\dagger}c_2 \right) +\lambda \left( c_{1}^{\dagger}c_{2}^{\dagger}+c_2c_1 \right) \\ &=\epsilon \left( c_{1}^{\dagger}c_1+c_{2}^{\dagger}c_2 \right) +\frac{\lambda}{2}\left( c_{1}^{\dagger}c_{2}^{\dagger}+c_2c_1 \right) -\frac{\lambda}{2}\left( c_{2}^{\dagger}c_{1}^{\dagger}+c_1c_2 \right) \\ &=\frac{\epsilon}{2}\left( c_{1}^{\dagger}c_1+c_{2}^{\dagger}c_2 \right) +\frac{\epsilon}{2}\left[ \left( 1-c_1c_{1}^{\dagger} \right) +\left( 1-c_2c_{2}^{\dagger} \right) \right] +\frac{\lambda}{2}\left( c_{1}^{\dagger}c_{2}^{\dagger}+c_2c_1 \right) -\frac{\lambda}{2}\left( c_{2}^{\dagger}c_{1}^{\dagger}+c_1c_2 \right) \\ &=\frac{\epsilon}{2}\left( c_{1}^{\dagger}c_1+c_{2}^{\dagger}c_2 \right) -\frac{\epsilon}{2}\left[ c_1c_{1}^{\dagger}+c_2c_{2}^{\dagger} \right] +\frac{\lambda}{2}\left( c_{1}^{\dagger}c_{2}^{\dagger}+c_2c_1 \right) -\frac{\lambda}{2}\left( c_{2}^{\dagger}c_{1}^{\dagger}+c_1c_2 \right) +\epsilon \end{align} $$

The same reason, we can write it as quadratic form $$\mathcal{H}=\frac{1}{2}\left(\begin{array}{cccc} c_{1}^{\dagger} & c_{2} & c_{2}^{\dagger} & c_{1} \end{array}\right)\left(\begin{array}{cccc} \epsilon & \lambda & 0 & 0 \\ \lambda & -\epsilon & 0 & 0 \\ 0 & 0 & \epsilon & -\lambda \\ 0 & 0 & -\lambda & -\epsilon \end{array}\right)\left(\begin{array}{c} c_{1} \\ c_{2}^{\dagger} \\ c_{2} \\ c_{1}^{\dagger} \end{array}\right)+\epsilon$$

But this time, eigenvalues become $$\pm \sqrt{\epsilon ^2+\lambda ^2}$$

I think this might be the reason that the unitary used to diagonalize the middle hermitian matrix cannot keep the canonical commutation relation for fermions.

So my problem is, after JWT, is there some routine that we can write the right hermitian matrix so the eigenvalue of the hermitian matrix is the answer we want?

$\endgroup$
4
  • 1
    $\begingroup$ I don't understand. Shouldn't a unitary transformation of the single-particle basis preserve the (anti-) commutation relations of the creation and annihilation operators?! $\endgroup$ Aug 17 at 9:13
  • $\begingroup$ @JasonFunderberker Em.. Then I don't know why the eigenvalues of the two hermitian matrix will differ. $\endgroup$
    – narip
    Aug 17 at 9:36
  • 1
    $\begingroup$ Have you made a mistake? Check for example section 1.9 and 2 of these lecture notes. $\endgroup$ Aug 17 at 9:40
  • 1
    $\begingroup$ @JasonFunderberker I don't think I made a mistake. But anyway, I've given another example in your link to show my problem. At the same time, I will try to solve it myself. Thank you very much for your help : ) $\endgroup$
    – narip
    Aug 17 at 11:09

1 Answer 1

2
$\begingroup$

The example you find has a mistake. When you diagonalize the matrix $\mathcal{H}$, you are applying some unitary transformation $U$, which is a $4 \times 4$ matrix, to the vector $(c_1,c^{\dagger}_2,c_2,c^{\dagger}_1)^T$. Let the transformed vector be $(\tilde{c}_1,\tilde{c}^{\dagger}_2,\tilde{c}_2,\tilde{c}^{\dagger}_1)^T$. We must have $\tilde{c}^{\dagger}_1=(\tilde{c}_1)^{\dagger}$ and $\tilde{c}^{\dagger}_2=(\tilde{c}_2)^{\dagger}$, but a $4 \times 4$ matrix $U$ gives extra degrees of freedoms to the transformation, and the relation generally no longer holds. Therefore, you will have different eigenvalues for the same Hamiltonian as invalid results.

In your previous edition of the post, you mentioned a link about Jordan-Wigner transformation and its use in $1$-dimensional quantum Ising model. The Hamiltonian after Jordan-Wigner transformation is $$H=\sum_{k}{\big(-c^{\dagger}_kc_k(2\cos{k})-(c^{\dagger}_kc^{\dagger}_{-k}e^{ik}+c_{-k}c_ke^{ik})+2h_zc^{\dagger}_kc_k\big)}$$ You argue that we can write the Hamiltonian as $$H=\sum_{k} \begin{pmatrix} c^{\dagger}_k & c_{-k} \end{pmatrix} \begin{pmatrix} -2\cos{k}+2h_z & -e^{ik} \\ -e^{-ik} & 0 \end{pmatrix} \begin{pmatrix} c_k \\ c^{\dagger}_{-k} \end{pmatrix}$$ This is true, but diagonalizing the $2 \times 2$ matrix in the above equation does not give you superposition of $c_k$ and $c^{\dagger}_{-k}$ which can diagonalize the Hamiltonian. There is the other $-k$ term as contribution in the sum, meaning \begin{align} H & = \sum_{k>0} \begin{pmatrix} c^{\dagger}_k & c_{-k} \end{pmatrix} \begin{pmatrix} -2\cos{k}+2h_z & -e^{ik} \\ -e^{-ik} & 0 \end{pmatrix} \begin{pmatrix} c_k \\ c^{\dagger}_{-k} \end{pmatrix} \\ & + \sum_{-k,k \geq 0} \begin{pmatrix} c^{\dagger}_{-k} & c_k \end{pmatrix} \begin{pmatrix} -2\cos{k}+2h_z & -e^{-ik} \\ -e^{ik} & 0 \end{pmatrix} \begin{pmatrix} c_{-k} \\ c^{\dagger}_k \end{pmatrix} \end{align} Therefore, you can see the diagonalization of $\begin{pmatrix} -2\cos{k}+2h_z & -e^{ik} \\ -e^{-ik} & 0 \end{pmatrix}$ only diagonalizes the first part of the Hamiltonian. As written in the link, the correct way to do this is to sum up the two parts, and have $$H=\sum_{k} \begin{pmatrix} c^{\dagger}_k & c_{-k} \end{pmatrix} \begin{pmatrix} -\cos{k}+h_z & -i\sin{k} \\ i\sin{k} & \cos{k}-h_z \end{pmatrix} \begin{pmatrix} c_k \\ c^{\dagger}_{-k} \end{pmatrix}$$ The diagonalization of the matrix gives you the superposition of $c_k$ and $c^{\dagger}$ which diagonalizes the Hamiltonian.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.