Fourier transform of t-V model for t=0 case

AI Thread Summary
The discussion focuses on computing the Fourier transform of the t-V model when t=0, emphasizing the Hamiltonian's potential term expressed in momentum space. The electron annihilation operator is transformed using a Fourier series, leading to the expression for the electron number operator in momentum space. The Hamiltonian is reformulated to include momentum space operators and their interactions. Participants question the possibility of further simplification and whether energy eigenvalues can be computed similarly to the V=0 case, which yields a specific energy expression. The conversation centers on the mathematical treatment of the Hamiltonian and the implications for energy calculations.
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Homework Statement
I am trying to compute the Fourier transform of the 2D ##t-V## model for the case ##t=0##.
Relevant Equations
$$\hat H = -t \displaystyle \sum_{\langle i,j\rangle} ( \hat c_i^{\dagger} \hat c_j + \hat c_j^{\dagger} \hat c_i) + V \sum_{\langle i, j \rangle} \hat n_i \hat n_j$$
To compute the Fourier transform of the ##t-V## model for the case where ##t = 0##, we start by expressing the Hamiltonian in momentum space. Given that the hopping term ##t## vanishes, we only need to consider the potential term:

$$\hat{H} = V \sum_{\langle i, j \rangle} \hat{n}_i \hat{n}_j$$

Let's express this in terms of creation and annihilation operators in momentum space. The Fourier transform of the electron annihilation operator ##\hat{c}_i## is given by:

$$\hat{c}_i = \frac{1}{\sqrt{N}} \sum_k e^{-ikr_i} \hat{c}_k$$

where ##N## is the total number of lattice sites, ##k## is the wavevector, and ##r_i## is the position of lattice site ##i##.

Therefore, the electron number operator ##\hat{n}_i## can be expressed as:

$$\hat{n}_i = \hat{c}_i^\dagger \hat{c}_i = \frac{1}{N} \sum_{k,k'} e^{i(k'-k)r_i} \hat{c}_{k'}^\dagger \hat{c}_k$$

Hence, the Hamiltonian in momentum space becomes:

$$\hat{H} = \frac{V}{N^2} \sum_{\langle i, j \rangle} \sum_{k,k'} e^{i(k'-k)(r_j-r_i)} \hat{c}_{k'}^\dagger \hat{c}_k \hat{c}_{k}^\dagger \hat{c}_{k'}$$


I wonder if we can further simplify, is my attempt correct? Is it possible to compute the energy eigenvalues like in the case of ##V=0## where the solution corresponds ##e_k = -2t(cos(kx)+cos(ky))##.
 
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