- #1
gonadas91
- 80
- 5
Hey guys this
is my first question, I am trying to solve a first approach of the interacting resonant level model (IRLM), where we suppose that there is only one energy level acting as a quantum dot between the leads. The hamiltonian is of the form
\begin{eqnarray}
H=\sum_{kL}\epsilon_{kL} c_{kL}^{\dagger}c_{kL} + \sum_{kR}\epsilon_{kR} c_{kR}^{\dagger}c_{kR} - t \sum_{kL,kR}\phi_{kL(0)}\phi_{kR(0)}^* c_{kL}^{\dagger}c_{kR} + \phi_{kL(0)}^* \phi_{kR(0)} c_{kR}^{\dagger}c_{kL}
\end{eqnarray}
where the $c_{ks}^{\dagger} , c_{ks}$ with $s=L,R$ refers to the fermionic creation/annihilation operators n the left and right lead respectively. The $ \phi_{ks} (0)$ are the eigenfunctions of the single particle hamiltonian when we have a delta function as a barrier ($\delta (x)$)at $x=0$, that is the single particle equivalent problem.\\
I would like to express the hamiltonian in another basis, where it is diagonalised and that also had fermionic operators, but I have just tried a Bogoliubov transformation for fermions and realized that I cant, because the u and v parameters become sines and cosines and in some time I get a divergence with the anticommutation relations. Does anyone know how to solve this? Thank you!
is my first question, I am trying to solve a first approach of the interacting resonant level model (IRLM), where we suppose that there is only one energy level acting as a quantum dot between the leads. The hamiltonian is of the form
\begin{eqnarray}
H=\sum_{kL}\epsilon_{kL} c_{kL}^{\dagger}c_{kL} + \sum_{kR}\epsilon_{kR} c_{kR}^{\dagger}c_{kR} - t \sum_{kL,kR}\phi_{kL(0)}\phi_{kR(0)}^* c_{kL}^{\dagger}c_{kR} + \phi_{kL(0)}^* \phi_{kR(0)} c_{kR}^{\dagger}c_{kL}
\end{eqnarray}
where the $c_{ks}^{\dagger} , c_{ks}$ with $s=L,R$ refers to the fermionic creation/annihilation operators n the left and right lead respectively. The $ \phi_{ks} (0)$ are the eigenfunctions of the single particle hamiltonian when we have a delta function as a barrier ($\delta (x)$)at $x=0$, that is the single particle equivalent problem.\\
I would like to express the hamiltonian in another basis, where it is diagonalised and that also had fermionic operators, but I have just tried a Bogoliubov transformation for fermions and realized that I cant, because the u and v parameters become sines and cosines and in some time I get a divergence with the anticommutation relations. Does anyone know how to solve this? Thank you!