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Some materials have various competing and/or synergistic orders, such as charge density waves and superconducting pairing.

Are there signatures of such an interplay between orders that can be seen in the results of ab initio calculations?

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In the specific case of charge density wave and superconductivity, both are phenomena that can be calculated using density functional theory:

  1. Charge density wave. It usually manifests as an imaginary phonon in the reference structure that drives the charge density wave formation.
  2. Superconductivity. Phonon-mediated superconductivity can be calculated almost routinely in the context of density functional theory, the basic quantity being the electron-phonon matrix element.

A nice example of this interplay occurs in TiSe$_2$, and density functional theory is capable of describing both phenomena as explained in this paper.

I do not really have a good general answer, but I would say that the usual approach in a first principles context (e.g. DFT) would be to separately evaluate the energy of the system in the competing phases and then compare the results.

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