I've seen the term 'low-energy approximation' several times and also 'low-energy Hamiltonian' but I didn't got the reasoning of these terms.
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It depends on the context: e.g., effective mass approximation used in semiconductor theory is a low-energy approximation: we assume that electrons and holes are not excited too far away from the band extremum, so that we can expand the dispersion relation near these point and neglect the transitions deep into the band or to other bands.
Similar approximations are used in quantum field theory and the theory of critical phenomena. Sometimes it may overlap with adiabatic approximation, long-wavelength approximation, etc. E.g., spin excitations can be approximately described as waves, if they have low energy.
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$\begingroup$ I am reading a paper about bilayer graphene and the hamiltonian written was a low-energy hamiltonian. Why is it called this way and how does this hamiltonian differ from the normal one? $\endgroup$– nouhaCommented Oct 27, 2022 at 10:48
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$\begingroup$ @nouha again, several things can be meant - you could add more contenmxt to the question. Do they use linearized dispersion law rather than the real band structure? $\endgroup$– Roger V.Commented Oct 27, 2022 at 11:22