# Two-orbital system w/ tight-binding and Coulomb interaction

1. Feb 12, 2014

### phys_student1

We have a system of two orbitals 'a' and 'b'. The Hamiltonian is given as:

$$H=\sum_{\sigma=\uparrow,\downarrow}(\epsilon_a A_\sigma^\dagger A_\sigma+\epsilon_b B_\sigma^\dagger B_\sigma) -t\sum_{\sigma=\uparrow,\downarrow}(A_\sigma^\dagger B_\sigma+B_\sigma^\dagger A_\sigma)+U B_\uparrow^\dagger B_\uparrow B_\downarrow^\dagger B_\downarrow$$
The first term represent the energy of both electrons (will be ignored, it only shifts the zero-level), second term represent the hopping while third one the Coulomb interaction between electrons in orbital 'b'. A's and B's are the creation and annihilation operators. Initially we have 1 spin-up and 1 spin-down electrons in the system. In the limit U>>t, we are to find the ground state and its energy.

I have represented H as the following matrix:

$$H=\begin{pmatrix} 0 & 0 & -t & -t \\ 0 & 0 & +t & +t \\ -t & +t & U & 0 \\ -t & +t & 0 & 0 \end{pmatrix}$$
with basis vector:
$$\begin{pmatrix} \left|\uparrow,\downarrow\right> \\ \left|\downarrow,\uparrow\right> \\ \left|\uparrow\downarrow,.\right> \\ \left|.,\uparrow\downarrow\right> \end{pmatrix}$$
This system can be diagonalized and solved exactly, although the solution is cumbersome. Alternatively, in U>>t limit, we can 'downfold' the Hamiltonian and solve as:
$$H_{eff}=\begin{pmatrix} -t & -t \\ t & t \end{pmatrix} \begin{pmatrix} -1/U & 0 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} -t & t \\ -t & t \end{pmatrix} = -t^2/U \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$$
Which gives the solution as: $$E_1=-2t^2/U, E_2=0$$. Eigen-functions can be obtained straightforwardly.

Now, if I am to use second quantization, then the Hamiltonian becomes:
$$H_{eff}=-2t^2/U(A_\uparrow^\dagger B_\downarrow^\dagger B_\downarrow A_\uparrow - A_\downarrow^\dagger B_\uparrow^\dagger B_\downarrow A_\uparrow - A_\uparrow^\dagger B_\downarrow^\dagger B_\uparrow A_\downarrow + A_\downarrow^\dagger B_\uparrow^\dagger B_\uparrow A_\downarrow )$$
I have tried to write it down in terms of the spin operators, which gave me a compact representation:
$$H_{eff}=2t^2/U(S_a.S_b - n_a.n_b/4)$$
Then how would one normally proceed to get the eigenvalues / functions?

Last edited: Feb 12, 2014
2. Feb 13, 2014

### phys_student1

Another point...If you have your perturbing Hamiltonian in 2nd quantization representation, and your wave functions are also in 2nd Q. rep. Then how would one find the 2nd order perturbation theory, since you will have some thing like <c*c...|c*c...|c*c...> , I mean, everything is in terms of creation and annihilation operators and I do not see how one can proceed in this representation?

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