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I am starting to do some work on free-fermionic models, but I am having some problems understanding some things. My professor led me know that the hamiltonian for free fermions without mass in a lattice with $N$ sites is $$ H = -\sum_{i=1}^{N-1} c_i^\dagger c_{i+1} + \text{h.c.}\ , $$ with $c_i^\dagger$ and $c_i$ the creation and anhilation operators on site $i$. On the other hand, the hamiltonian on the case of non-zero mass is $$ H = -\sum_{i=1}^{N-1} (1 + (-1)^i \delta) c_i^\dagger c_{i+1} + \text{h.c.}\ . $$

My first question is about the parameter $\delta$. I supposed it is the mass of the fermions, since it is the unique new parameter in this model with respect to the first one, and making $\delta = 0$ recovers the former. Am I right on this assumption?

Second, in the first model, are the coefficients of $c_i^\dagger c_j$ a way of saying that the fermions have a probability of hopping between neightbour sites on the lattice? If this is the case, why are coefficients for $c_i^\dagger c_i$ missing, meaning that there is a probability for a fermion to remain on its site? Regarding these questions, why the second model alternates between $1-\delta$ and $1 + \delta$?

I think I'm having trubles understanding these models because I am not sure what is their origin and derivation. I wondered about the existance of a lagrangian density $\mathcal{L}$ that describes the fields for these fermions, such that the corresponding hamiltonian density $\mathcal{H}$ and its later integration over space leads to the given hamiltonians $H$. Is that the case or am I completely wrong? I am new to many-body quantum mechanics and quantum field theories, so any help would be appreciated.

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  • $\begingroup$ A tip on reading many-body Hamiltonians: They only involve terms with an energy associated with them. The lack of a $c_i^\dagger c_i$ term, i.e. the number operator, does not mean that it is impossible for the particle to remain on its site, it just means the model does not account for the energy associated with being at that site. It only accounts for the energy of hopping between neighboring sites. $\endgroup$ Commented Mar 12, 2022 at 22:21

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Are you familiar with the single particle band theory for each of these models? Just write out the corresponding matrix and diagonalize it. The first is just the simplest one-band tight-binding model with band energy $E_k = -2 \cos k$. The second is the SSH (Su-Schrieffer-Heeger) model for Polyacelylene (poly-CH$_n$) which has two bands. $\delta$ controls the gap between them.

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