How do Dirac fermions arise in graphene, and, what significance (if any) does this have for high-energy physics?

Question

Graphene has a honeycomb lattice (in the absence of defects and impurities). By considering the low-energy limit of the half-filled Hubbard model used to model the strongly interacting electron gas we find that the low-energy quasiparticles obey the dispersion relation for massless fermions. These details are all covered very nicely in a paper by Gonzalez, Guniea and Vozmediano (reference) among others.

It might seem like I'm answering the question. I'm following this line of exposition because I don't want to assume that this is a topic something commonly known or understood outside the condensed matter community. Any answers which elaborated on these basics would be very useful as they would help make the discussion more broadly accessible.

My primary question is more about the implications this fact has for high-energy physics, in particular the question of emergent-matter in theories of quantum gravity. The case of graphene is a canonical example in that regard where one obtains relativistic, massless excitations in the low-energy corner of an otherwise non-relativistic system - the 2D electron gas (2DEG).

Obviously I have my own beliefs in this regard and I will try to outline them in an answer. But I also want to solicit the communities views in this regard.

Answer 1

The emergence of global symmetry at low energies is a familiar phenomena, for example Baryon number emerges in the context of the standard model as "accidental" symmetry. Meaning at low energies it is approximately valid, but at high energies it is not.

The reason this is the case is that it so happens that the lowest dimension operator you can write, with the matter content and symmetries of the standard model, is dimension 5. The effect is then suppressed by one power of some high energy scale - it is an irrelevant operator. This is a model independent way to characterize the possibility of the emergence of global symmetries at low energies.

We can then ask about Lorentz invariance - what are the possible violations of Lorentz invariance at low energies, and what is the dimensions of the corresponding operators. This depends on the matter content and symmetries - for the system describing graphene, there is such emergence. For anything containing the matter content of the standard model, there are lots and lots of relevant operators*, whose effect is enhanced at low energies - meaning that Lorentz violating effects, even small ones at high energies, get magnified as opposed to suppressed at observable energies.

Of course, once we include gravity Lorentz invariance is now a gauge symmetry, which makes its violation not just phenomenologically unpleasant, but also theoretically unsound. It will lead to all the inconsistencies which necessitates the introduction of gauge freedom to start with, negative norm states and violations of unitarity etc. etc.

• At least 46, which were written down by Coleman and Glashow (Phys.Rev. D59, 116008). Relaxing their assumptions you can find even more. Each one of them would correspond to a new fine-tuning problem (like the cosmological constant problem, or the hierarchy problem).

Answer 2

The answer you'll get from most high-energy physicists is that there are no implications whatsoever. Lorentz invariance is extraordinarily well-tested: see, e.g., http://arxiv.org/abs/0801.0287. In particular, there are many relevant operators in the Standard Model that one would expect to be generated if physics at a high scale is not Lorentz-invariant. Even some irrelevant operators that one might naively expect to appear with order-one Planck-suppressed coefficients are constrained to have smaller coefficients. Adding in gravity only makes the problem worse. For instance, most attempts to generate emergent GR from nonrelativistic theories will have an extra scalar mode and run into massive phenomenological difficulties, because they aren't really gauging the full diffeomorphism group.

To be slightly more clear: there are cases (and the free relativistic fermion emerging in the long-distance limit of graphene is one of them) where lattice symmetries can forbid dangerous relevant operators. This shouldn't happen for the full Standard Model (I assume someone has written down a careful argument for this somewhere, but I don't know a reference offhand). Still, even for the graphene case there are irrelevant operators, and we have bounds on those too. Furthermore, once you start thinking about gravity you're more or less forced to give up the hope of an underlying highly symmetric lattice that forbids all the dangerous operators.

One more half-joking comment: this argument also tells you the correct answer to the FQXi essay contest "Is Reality Digital or Analog?," so if someone fleshes it out carefully they could possibly make up to $10k from it.

(It is a good question, by the way; there's an obvious conventional wisdom from effective field theory that explains why you don't see high-energy theorists pursuing this sort of thing much, but from the outside it might not be so clear why such ideas don't generate much interest.)