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I am making different plots for a 2-d non-interacting tight binding Hamiltonian $$ H = - t \sum_{<ij>, \sigma} c_{i \sigma}^{\dagger} c_{j \sigma} + h.c$$

I get the dispersion relation $$\epsilon (k) = -2 t ( \cos(k_{x} a) + \cos (k_{y} a))$$

Plotting the contours of this, I get many k values giving me the same energy $\epsilon = 0$, that contour looks like a rhombus. I know that this has something to do with nesting but I don't understand exactly what is it.

Contours of energy in Brillouin Zone

I'll appreciate referring to good sources on this too. I can't seem to find any that explains this clearly without referring to other things I am not familiar with.

Hoda
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1 Answers1

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"Nesting" refers to a Fermi surface where two points on the Fermi surface are connected by half a reciprocal lattice vector. When this occurres, it usually indicates the system is critical or unstable with respect to an interaction.

If your think about adding an interaction term to the Hamilton via perturbation theory, you'll find that any translationally invariant term only couples states with the same momentum, up to a reciprocal lattice vector. If there's no nesting, the states will have very different energies, and thus be suppressed in perturbation theory. However, if the Fermi surface has nesting, there exists states with the same crystal momentum and same energy, which diverge in perturbation theory.

Jahan Claes
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