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The question is as follow:

Consider a chain of free fermions with $H=\sum_{n}c_n^{\dagger}c_{n+1}+h.c.$. Show that the low-energy excitations at a generic value of the filling are described by the massless Dirac lagrangian in 1+1 dimensions.

Conceptual doubt:

How do you go about relating a fermionic field $\psi$ to this model. And what does the time and spatial derivative, $\partial_t\psi$ and $\partial_x\psi$, correspond to in this case?

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