In the past decade the (hard) condensed matter physics community has become obsessed(?) with Dirac cones [graphene, topological insulators, Weyl semimetals, ...]. They occur in the electronic band structure [one-electron spectrum] when two energy bands cross. Here the system has spatial periodicity and the k's are Bloch quantum numbers.

I want to highlight some similarities between conical intersections (CIs) and Dirac cones (DCs) but also highlight some important differences.

First the

**similarities.**

A. Both CIs and DCs give rise to rich (topological) quantum physics associated with

**a geometric phase**and the associated gauge field (a fictitious magnetic field), the Berry curvature, monopoles, ....

B. There are definitive experimental signatures associated with this Berry phase. However, obtaining actual experimental evidence is very nebulous. For example, this paper discusses the problem for extracting the Berry phase from quantum oscillations in a topological insulator. This post discusses the elusive experimental evidence for CIs.

C. A history of under appreciation. Both these concepts could have been elucidated in the 1930s, but were either ignored, or thought to be pathological or highly unlikely. CIs occur in the Jahn-Teller effect (1937) in systems with enough symmetry (e.g. C_s) to produce degenerate electronic states.

However, then people made the mistake of assuming that symmetry was a necessary, rather than a sufficient, condition for a CI. Given that most molecules, particularly large ones have little or no symmetry, it was assumed CIs were unlikely. It was not until the 1980s, with the rise of high-level computational quantum chemistry and femtosecond laser spectroscopy, that people discovered that symmetry was not only unnecessary, but CIs are quite ubiquitous in large molecules. This is facilitated by the large number of nuclear co-ordinates.

DCs have only become all the rage over the past decade because of new materials: graphene, topological insulators, ...

In spite of the similarities above it is important to appreciate some

**significant differences**in the physics associated with these entities.

1. The role of symmetry. As a minimum DCs requires translational symmetry and an infinite system to ensure the existence of a Bloch wave vector. Most require further symmetries, e.g. the sub-lattice in graphene, or something else in a topological insulator. As mentioned, above, CIs don't involve any translational symmetry. One does not even need some local symmetry (e.g. C_3) as observed with some common structural motifs for CIs.

2. Good quantum numbers and quantum evolution. For DCs the Bloch wave vector k is a good quantum number. In the absence of scattering an electron in state k will stay there forever. For CIs R is a classical nuclear co-ordinate. If one starts on a particular surface one will "slide down" the surface and pass through the CI.

3. The role of correlations. DCs are generally associated with a band structure, an essentially one-electron picture. [Strictly, one could look at poles in spectral functions in a many-body system but that is not what one generally does here]. In contrast, CIs are associated with quantum many-body states not single electron states. In particular, although one can in principle have CIs associated with molecular orbital energies and find them with Hartree-Fock methods, in general one usually finds them with multi-reference [i.e. multiple Slater determinant] methods. For a nice clear discussion see this classic paper which explains everything in terms of what physicists would call a two-site extended Hubbard model.

4. Occupation of quantum states. For DCs one is generally dealing with a metal where all the k states below the Fermi energy are occupied. For CIs only one of the R's is "occupied".

The post was stimulated by Ben Levine and Peter Armitage.

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