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How do we solve a Hamiltonian written in second quantization by using quantum statistical formalism? For example, the following Hamiltonian

$$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$

I have solved few examples in which $H$ had some function form but I am not sure how to find ground-state properties of Hamiltonian wirrten in second quantization. Especially, how do we solve this kind of Hamiltonian in computer? i.e. by using exact diagonalization (for small system sizes)

I am sorry of this question seems stupid. I couldn't find any article which explain things like this.

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You should find the answer to your question in any basic reference which deals with the tight-binding model. You introduce the Fourier transforms of the annihilation and creation operaors. This diagonalizes the Hamiltonian in $k$-space. The energy associated to $k$ is the Fourier transform of the hoppings (1 in your case, in 1D, so just a cosine). See similar question here Dispersion relation in tight binding model with even indices only

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  • $\begingroup$ thank you for your reply, but this does not answer my question. I want to understand the formalism of quantum statistical mechanics. How do we find the partition function and average values of various operators? $\endgroup$
    – Sana Ullah
    Apr 21, 2019 at 9:38
  • $\begingroup$ @SanaUllah After you find the eigenvalues of the Hamiltonian, partition function could be evaluated according to the principles of Statitical Mechanics. They are completely general and cover all the Hamiltonians. $\endgroup$
    – GiorgioP
    Apr 22, 2019 at 9:44

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