Tight binding in graphene (complex energy?)

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SUMMARY

The discussion focuses on calculating the band structure of graphene using the tight binding model, specifically addressing the complexity of energy values. The user highlights that while the energy is a physical observable, the off-diagonal elements of the Hamiltonian matrix can be complex due to the non-equivalence of nearest neighbors in different sub-lattices. The tutor clarifies that despite the complex nature of these elements, the eigenvalues remain real because the Hamiltonian is Hermitian. This insight resolves the confusion regarding the expectation value of energy in this context.

PREREQUISITES
  • Tight binding model for band structure calculations
  • Understanding of Hermitian matrices in quantum mechanics
  • Knowledge of graphene's lattice structure and sub-lattices
  • Familiarity with eigenvalues and eigenvectors in linear algebra
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  • Study the tight binding model in detail, focusing on its application to graphene
  • Learn about Hermitian operators and their significance in quantum mechanics
  • Explore the concept of sub-lattices in two-dimensional materials
  • Investigate the implications of complex numbers in quantum mechanics and their physical interpretations
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Physics undergraduates, materials scientists, and researchers interested in condensed matter physics, particularly those studying the electronic properties of graphene.

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Hi, I'm a 4th-year physics undergrad and I have a question about calculating the band structure of graphene using tight binding. Following the calculation here https://wiki.physics.udel.edu/phys8...rgy_quasiparticles,_Berry_phase,_and_all_that , the E(k) is a complex number, and the modulus must be taken to get its expectation value. I asked my tutor about this, since energy is a physical observable, how can it be a complex number? And he said that it's because here the nearest neighbours to any atom are not equivalent to that atom (they are on different sub-lattices) and to do the calculation properly, you'd need to treat the basis like a C-C dimer and consider the transfer integral between nearest-neighbour dimers. He also said there may be some subtle reason why it's acceptable to do the calculation like in the link above, and take the modulus of the complex energy, but he doesn't know what that reason might be.

Can anyone point me in the right direction?

I'm not sure if I should be putting this in the homework help forum, but it's not that I can't do the calculation, just that I can't understand it - apologies if this post shouldn't be here!
 
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The energies are real, what is complex are the off-diagonal elements of the hamiltonian matrix.
However, as this matrix is hermitian, the eigenvalues are real, automatically.
 
It seems that both my tutor and I got bogged down in irrelevant details and missed the obvious - thank you so much, Dr Du!
 

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