29. Why is there a problem with one-electron band theory?

General coincidences

  • Overuse and misinterpretation of the terms 'breakdown in band theory', 'failure of the one-electron approximation' and 'the need to go beyond mean-field theory' when discussing strongly correlated materials (see any text book).

  • The unfortunate multiplicity of meanings of the word 'correlation', which confuses some text book writers.

  • A coincidental failure of the LDA in many strongly correlated materials for a somewhat unrelated mathemetical reason.

  • Failure of the LDA (and hence 'band theory') to treat most of the interesting proplems that have arisen in strongly correlated materials (e.g. superconducting cuprates) over the last decade, leading to intense derision in the many body theory community. But the LDA is not band theory.

Being in love with the LDA

  • A failure of imagination on the part of electronic structure theorists to countenance the possibility that there could ever be anything seriously wrong with the LDA.

  • An associated belief that because DFT is 'in principle exact' and includes 'correlation' it is automatically superior to methods like Hartree-Fock for everything.

NB: Hartree-Fock theory is in general a poor approximation, but it is very useful to highlight precisely why the LDA breaks down.



Why do people think band theory or one-electron calculations break down for strongly correlated materials? Because they do, if you use LDA/GGA calculations etc.. But the LDA is not band theory. And the LDA does not break down because we don't include some mysterious magic Coulomb interaction which spontaneously arises in some new state of matter as is sometimes claimed. It's because we do include something spurious which isn't really there, which would be removed if only we treated the non-local exchange properly.. If you do a Hartree-Fock band calculation, then the bands split in just the way you expect from a Hubbard type consideration of the on-site interactions, as we've seen. So its not really correct to say that one-electron bands have no meaning in strongly correlated materials.

Why else do people believe this? Because most text books say so. Many authors get confused between the words 'correlation' in the quantum chemistry sense and the phrase 'strongly correlated', particularly if they have a background in electronic structure theory. They think strong correlation is related to everything that is not in the Hartree-Fock Hamiltonian - but as we've seen that kind of correlation in Mott insulators is just a short range screening effect which will change the numerical results, but will not change the qualitative features of the ground state that we can calculate.

As a final note, I should stress that the point of this talk is not to promote the use of Hartree-Fock theory for any reason at all. I have merely used it as the simplest method available to illustrate what is wrong with DFT calculations of strongly correlated materials.

That said, I believe it is highly likely that orbitals from HF or hybrid functional DFT approaches are likely to provide better trial wave functions for QMC calculations.