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I have a $1$D and one band lattice model with hopping constants $J_R $ (to the right) and $J_L$ (to the left) and under open boundary condition. It has the following Hamiltonian :

$$H = \sum_{n} (J_R c^{\dagger}_{n+1} c_n +J_L c^{\dagger}_n c_{n+1} ) $$ (1)

where $J_L , J_R \in R$ and $c^{\dagger}_n , c_n$ are creation and annihilation operators.

I want to write the matrix representation of this Hamiltonian. I've tried to rewrite this Hamiltonian without using second quantized operators. It looks like this:

$$H = \sum_n (J_R | n+1\rangle\langle n |+ J_L |n+1\rangle\langle n |)$$ (2)

I am not sure if this is correct. I've just used the n+1 and n's. How can i write the matrix representation of this Hamiltonian using the (1) Hamiltonian? Thanks in advance.

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1 Answer 1

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In the associated one-particle hamiltonian has the infinite matrix such that each row has 0 on the digaonal and $J_L$ in the entry to the left and $J_R$ in the entry to the right. Obviously it is not hermitian unless $J_L=J_R^*$. This one-particle Hamiltonian matrix is the one you write down in yoiur equation 2.

The many-body hamiltonian in your first equation acts the much larger many-particle Fock space (Bose or Fermi) and is rather hard to display, so they are not the same thing.

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