Quantum Graphenity by Tobias Fritz

Your model is indeed interesting dear Narsep,

Best

Steve

Hi there. I used this opportunity to write out some conditions for emergence of continuous structures, and in particular that of Lorentz invariance, in general based on the classification provided by effective field theory. Graphene famously has such emergent symmetry, but in more complicated models which include all the matter content and structures of the standard model, might be more difficult to achieve. If you are interested the argument (and known loopholes) are here:

http://www.fqxi.org/community/forum/topic/856

I am curious about your thoughts.

Cheers,

Moshe

Dear Tobias,

Trigonal warping: very interesting! I like the idea of modified dispersion relation similar to those of Smolin, Magueijo and Amelino-Camelia. This may provide a way to discover the digital nature of reality!

However, I cannot believe that the massless field has no digital analog, there must be a way! Otherwise we are proving that the world is not digital! In the reply to your thread in my blog I conjectured a possible mechanism to cure the problem with an anisotropic refraction index. I hope it will work, since I believe that "reality" is digital!

Hi Tobias,

I was thinking that it was possible to get single graphene rings and their interference would be "interesting".

Good Luck,

Don Limuti

Tobias,

I'm posting another question about your (let me say it again) very interesting work. I'm very interested in your graphene simulator, since, as you can imagine from my work, I want to understand more Dirac quantum simulation in space-dimensions d>1, e.g. your case d=2. The way in which I do things I have a tripartite gate, which indeed builds up a graphene spatial network, but it corresponds to a Dirac equations with a 3x3 (differential) Hamiltonian matrix, since the gate is tri-partite. I'm still trying to understand if this is the only possibility, but it looks so ... Now, I want to come back to your idea of the tight-binding effective Hamiltonian.

The best way to explain myself, again, is through a figure. By the way, this is part of my talk at the March Meeting next tuesday. As you see, I'm quoting you!

Here's the fugure

Sorry, the figure was too big.

Here it is again...Attachment #1: Dirac2plus1_small.jpg

Dear Tobias,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

http://prl.aps.org/abstract/PRL/v106/i11/e116803

Tobias,

in a quantum circuit there are both space and time. If you consider the graphene as a quantum circuit (namely the gates are the edges of the hexagon, whence they are bipartite), then your computational circuit is 2dl, means there is only a single space dimension! Then my simple circuit simulates the Dirac in 1+1 very well, and graphene would not. I think that you should look at graphene as a the spatial projection (a leaf in the rest-frame foliation) of a 2+1 dim. circuit!

Cheers

Dear Tobias,

it seems to me that electrons flowing in the graphene may simulate vs time the evolution of the Dirac qstate in 2dspace, but it is certainly not the quantum circuit that simulates 2plus1 Dirac.

I also need to read the original paper that you are quoting.

Second: the way in which I see things, the Dirac eq. in 2plus1dims should have a 3x3 Ham matrix since the gate is tri-partite---not a 2x2 as you write in your paper. And my way I automatically get the spin in 3plus1dims.

Reader of this blog: look for figure Dirac2plus1_small.jpg in the next thread.

Cheers

Hola Tobias

Congratulations for your well-deserved win! Good luck with your reserach.

Best wishes from Vladimir

Congratulations on your second prize.I haven't read your essay, probably because the title sounds a bit obscure to me and there were so many of them to choose from. Now given its final placing I feel I must have missed out on something very interesting indeed! Well done and best wishes for the future, Georgina.

This essay summarizes the well known work of Gonzalez, Guinea, Vozmediano, Novoselov, Geim and others according to which the effective long-distance theory of the half-filled Hubbard model on the honeycomb lattice is given by the Dirac equation for a massless quasiparticle, along with a gauge field which can be used to model valley degrees of freedom and the effect of defects and disorder in the lattice on the quasi-particle dynamics. This fact was known as far back as 1992, see arXiv:cond-mat/9208004.

There a few other facts not mentioned in the essay but which are important for this topic:

1. In arXiv:0909.3057, Gonzalez and Herrero have also shown how one can model a Dirac particle in the presence of a wormhole background in a graphene based setup.

2. It is also well known (see for e.g. arXiv:gr-qc/9405070) that gravity in 2+1 dimensions is a purely topological theory with an equivalent description as a theory of a Chern-Simons gauge field. Consequently a quantum hall system - in which the CS theory plays an integral part - can be utilized as a substrate for tests of quantum gravity.

3. The entropy of a black hole horizon also is given by a Chern-Simons theory in the LQG approach (e.g. arXiv:gr-qc/9710007). Thus a quantum hall system could plausibly be used to model an isolated horizon.

4. Kitaev has also done a great deal of work (arXiv:cond-mat/0506438) in showing how non-abelian anyons in a hexagonal lattice can be used for quantum computation.

In this way, three different threads of theoretical physics - Chern-Simons theory describing the quantum hall effect, 2+1 dimensional quantum gravity and quantum computation on a lattice come together quite naturally in an experimentally accessible setup as simple as that of graphene.

The essay itself is a very readable summary of such efforts and a pedagogical explanation of this line of research. In fact, this topic can use all the publicity it can get and I am happy to see that happen. Congratulations on this honor, Tobias!

Congratulations for the prize. I wish you a fruitful future and I hope our earlier correspondence to be as useful as it was for me.

Best ragards,

narsep (Hadjidakis)