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phys_student1
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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?
$$H=\sum_{\sigma=\uparrow,\downarrow}(\epsilon_a A_\sigma^\dagger A_\sigma+\epsilon_b B_\sigma^\dagger B_\sigma) -t\sum_{\sigma=\uparrow,\downarrow}(A_\sigma^\dagger B_\sigma+B_\sigma^\dagger A_\sigma)+U B_\uparrow^\dagger B_\uparrow B_\downarrow^\dagger B_\downarrow$$
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?
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