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Tobias Fritz: on 7/6/11 at 9:24am UTC, wrote Thank you all for the compliments! This second prize was *very* unexpected!...

narsep: on 6/22/11 at 13:26pm UTC, wrote Congratulations for the prize. I wish you a fruitful future and I hope our...

Deepak Vaid: on 6/9/11 at 18:33pm UTC, wrote This essay summarizes the well known work of Gonzalez, Guinea, Vozmediano,...

Georgina Parry: on 6/6/11 at 21:34pm UTC, wrote Congratulations on your second prize.I haven't read your essay, probably...

Vladimir Tamari: on 6/6/11 at 9:05am UTC, wrote Hola Tobias Congratulations for your well-deserved win! Good luck with...

Giacomo D'Ariano: on 3/30/11 at 9:50am UTC, wrote Dear Tobias, it seems to me that electrons flowing in the graphene may...

Tobias Fritz: on 3/28/11 at 18:44pm UTC, wrote I see, so by "circuit" you mean not just the qubits themselves, but the...

Giacomo D'Ariano: on 3/24/11 at 21:26pm UTC, wrote Tobias, in a quantum circuit there are both space and time. If you...


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To build the ultimate artificial mimics of real life systems, we may need to use quantum memory.

October 18, 2017

CATEGORY: Is Reality Digital or Analog? Essay Contest (2010-2011) [back]
TOPIC: Quantum Graphenity by Tobias Fritz [refresh]
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Author Tobias Fritz wrote on Feb. 18, 2011 @ 09:46 GMT
Essay Abstract

One obstacle for further progress on quantum gravity and issues like the discrete vs. continuous debate is the lack of experimental data. To a certain extent, this problem can be overcome by the simulation of models for fundamental physics by other physical systems. Here I focus on graphene as a particularly fascinating example of such a simulator and try to explain how continuous three-dimensional relativistic spacetime emerges from the discrete hexagonal crystal lattice of graphene. The potential of graphene to simulate aspects of fundamental physics is discussed.

Author Bio

As a postdoctoral researcher at the Institute of Photonic Sciences in Barcelona, I mainly work on mathematical methods for quantum foundations and quantum information. I find it enjoyable and challenging to keep a broad scope and to venture into new fields.

Download Essay PDF File

Yuri wrote on Feb. 18, 2011 @ 15:30 GMT
Hello Tobias!

I guess that good opponent for your essay

would be

You can ask him.

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Member Tommaso Bolognesi wrote on Feb. 18, 2011 @ 18:47 GMT

I think I've seen something about this at, but I can't find it right now. Are you from that group?

How would you compare the simulation of fundamental models for physics by other physical systems (as you do) vs. by computer?

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Author Tobias Fritz replied on Feb. 19, 2011 @ 14:21 GMT
see my reply below!

Steve Dufourny wrote on Feb. 18, 2011 @ 19:19 GMT
Hello dear Tobias Fritz ,

Interesting essay.Congratulations.

you say "It seems plausible that the resulting curvature of the graphene sheet

results in a curvature of the Lorentzian metric determining the Dirac equation .If this is indeed the case, then a crumpled graphene sheet naturally models Dirac particles on a crumpled background spacetime, i.e.

a background spacetime with gravity!"

Could you develop please?


The aromatics, the benzens,....are relevant in their stability gravitational.I love the biology and all days I Class or link animals, vegetals , fact I class all and I search the links of mass and the links of evolution.fetr all we are evolved cells.

If we take our stabilities, evolved in time as H2O and its pôle + and- ....or tyrosin and its friends fact the interactions are relevant considering the ionic link, the atomic link , the hydrogen link and the interactions of London, do you know these interactions between H and C London interactions are very very relevant,....or an other beautiful example is the centriol and its tori.Or the adn and its tori of sortings or you know the chlorophyl and the caroten.....HCNO ..they build these spheres dear thinkers....see the cellulose or the pectin or the chitin.... the tetrahedre of spheres indeed( but all rests in 3D and the rotations spinals and orbitals make the rest.

Still an other with biology,Have you already seen the prophase, meta-, ana-, and telophase, Fascinating these entangled spheres implying complementarity of evolution.

What do you think?



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Author Tobias Fritz wrote on Feb. 19, 2011 @ 14:08 GMT
Dear Yuri, Tommaso, Steve,

thank you for your feedback. I am happy to see that there is some interest in my essay. I am also looking forward to read your entries and will do so next week.

@Yuri: yes, I am indeed going to contact some experts.

@Tommaso: without knowing which group exactly you mean, I can definitely say that I'm not a member of it ;) I am not an expert on graphene and have merely studied its simulation capabilities.

About you to compare the simulation by physical systems with the simulation by computer: great question. For the sake of concreteness, let's consider the Dirac equation, although the ideas are general. Then if we use the charge carrier (quasi-)particles graphene as the simulator, we have a system whose dynamics is precisely given by the Dirac equation. There are actual Dirac particles around. However if we do the simulation on a digital computer, there are no (quasi-)particles actually obeying the Dirac equation. There are only bit strings which let us calculate things about the Dirac equation. So in a certain sense, this is not even a simulation, it is only numerics.

@Steve: you asked first why the crumpled graphene sheets may model Dirac particles on a curved spacetime. I should say about this that I am not absolutely sure that this is correct, since I am not an expert on the subject; I should contact the authors of my reference [CV] to see whether this is true. My idea about it is the following: we know that the dynamics of charge carriers in a flat graphene sheet is governed by the Dirac equation. In this sense, the flat graphene sheet simulates a Minkowski spacetime. Now in the presence of ripples, we may imagine the graphene sheet like a landscape which is not flat, but contains valleys and hills. Geometrically, this landscape is curved: it is impossible to map it to a flat surface while preserving all the distances, similarly as to how it is impossible to fabricate a map of the earth in which all the distances are to scale. What about the wave equation governing the charge carriers in this case? They are going to obey a Dirac equation on a curved 3-dimensional spacetime! But according to Einstein's general relativity, "curvature of spacetime" and "gravity" are two words for the same thing. Hence, the rippled graphene sheet should simulate Dirac particles with a background gravitational field.

About your other comments, I'm not sure what to say. I agree that the subject of studying molecular forces and chemical bondings is fascinating. About graphene: no, it is *not* composed by 3D spheres! Rather, the point of graphene is that is a 2-dimensional structure. If you want to say so, it's composed out of 2D rings. But I don't see what one could possibly mean by saying that "they turn".

Steve Dufourny replied on Feb. 19, 2011 @ 17:40 GMT
Thanks for your posts, it's interesting.Well

I am sorry but you are false! No dear it's a real 3D , only the application is in 2D, for a kind of plan;that's all.And yes it's composed by 3D spheres.It's like that everywhere in the micro meso and macro.

Thus the graphene is composed by points and they do not turn? ok ok

Good luck in this contest,say hello to Han Gueurdes and Joy Christian, no but frankly we are on rational platform of what???

ARE YOU SURE YOU UNDERSTAND THE POLARITY BETWEEN MASS AND LIGHT IN TIME SPACE EVOLUTION ???it is well to insert this graphene due to this nobel prize but please make it correctly and not with point and without rotation....If it's that which is learned at universities, I suggest a new library simply.But it's well.



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narsep replied on Feb. 21, 2011 @ 08:24 GMT
Dear Tobias,

Graphene is considered as a 2D structure in a macro scale. In a micro scale of course it is in a 3D (3 plus 1) as atoms are physical objects. I suppose you take carbon atom as a point and in this case graphene IS a 2D structure.

good luck, marsep (ioannis hatzidakis)

P.S. I will refer to your paper later in a post (for my contribution) that my impressions from this contest will be included.

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Steve Dufourny replied on Feb. 21, 2011 @ 15:18 GMT
All scales are in 3D, the micro, the meso and the macro,of course if we insert the evolution, the 4d space time helps and duration of time is implied by rotations of the entangled spheres.When you compute you can't consider a 2D , that has no sense,all must be in 3D. We just need to see more our scales by a kind of spherical topology, and its evolution.Still one thing, their rotations spinals and orbitals of these entangled spheres are proportional with mass.They have a mommentum these particles....



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Philip Gibbs wrote on Feb. 20, 2011 @ 08:38 GMT
Dear Tobias, thanks for this very tantalising idea for studying discrete quantum gravity using graphene. I hope someone will be able to put it into practice.

Your explanation of how massless Dirac fermions emerge on the graphene sheet is very clearly written. I wonder if some structure with the properties of a blackhole could be created. It might then be possible to study the effects of quantum gravity around a black hole in discrete space.

Good luck in the contest

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Steve Dufourny replied on Feb. 20, 2011 @ 11:09 GMT
I have a simple ask, do you think a BH can be created in a lab via the quantyum world, if yes, it's serious, because a BH is a sphere , central of galaxies and others universal rotations.Never we can create a micro BH.That has no sense at my humble and arrogant opinion.



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Author Tobias Fritz replied on Feb. 21, 2011 @ 18:02 GMT
@Phil: thanks for the encouragement! I am happy to see you here, as presumably you are the author of two of my references?

Anyway, relating graphene to quantum gravity is, at least for my understanding, still a little shady at the present time; not that I have mostly been writing about simulating Dirac fermions in (2+1)-dimensional spacetime, with some ideas about how to extend this to curved spacetime. But while obtaining a quantized form of this spacetime may be possible with graphene, to me it seems extremely speculative at the present time. I bet, however, that some people have thought about this, so the speculation is mostly due to my ignorance. In any case, I have to admit that my choice of title was mainly for effect...

@Steve: as I have mentioned in my essay, there have been successful *simulations* of black holes by other physical systems. But note that these are only simulations and not actual black holes with a gravitational singularity. For example, while from behind the horizon of an actual black hole no particle can escape, this holds in the simulation only for certain kinds of quasi-particles. About the creation of actual microscopic black holes, I suppose we will have to wait a few years for the LHC to produce more data before we can decide either way.

Steve Dufourny replied on Feb. 23, 2011 @ 13:28 GMT
seee what is a real BH .....please before pondering these stupidities.Never they exist these micro BH no but we dream in live there.If you simulate that ...wawww I am impressed.A BH is a cosmological sphere.The LHC must be rational and stop the researchs of stupidities as higgs or extradimensions or this and VS rationality .....



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narsep wrote on Feb. 21, 2011 @ 14:19 GMT
Dear Tobias,

I do not resist to respond to one of the excellent essays of this contest. As my essay is straightly related to yours (mine is dealing with 3D (3 plus 1 spacetime)) I think more appropriate leave any extensive detail for a private correspondance. However, it is valuable to emphasize that carbon atom together with oxygen, nitrogen, silicon, phosphorous and sulphur are the ones that consist the more interesting part of nature. The first three plus hydrogen consist the "live" part of nature (vis vitalis) the "spirituality" of which causes the most inexplicable causality in nature. These atoms are able to appear in four discrete forms denoted as hybridization. Carbon atom (C) without any hybridization is in atomic form (no interconnection), at sp3 hybridization C occurs as tetravalent (model of 3D spacetime - see our contribution (uc)), at sp2 hybridization C occurs as trivalent (model of 2D spacetime present contribution and uc), at sp hybridization C occurs as divalent (model of 1D spacetime uc). You say that free electrons "are not participating in any chemical bonding to neighboring atoms" this is not so as these "free electrons" are forming double bonds with neighbors. However, because of hybridization they can move freely in a molecular orbital that is extended to the whole structure of graphene and that is why "they behave AS IF they were free". Your two types of C resemble real and virtual points to our essay. Your sublattice is triangle although the quantum space is a parallelogram. In your fig. 2 each C is connected to three (why you depict only two?) C of the same type. This is not so, because the third one should be in a some short of a different kind and this is not shown in your text. In order to be right the two sublattices should be the ones (two) that form 30 degrees angles with horizontal direction (direction that charge is transferring). The vertical direction is the charge front that is moving from left to right. In chemistry, we call your red ellipses double bonds and the structure that you show in fig. 2 is a conjugation form of graphene that is appropriate for the charge flow from left to right (or the opposite). Please do not take my points as criticism but as minor details that I feel clarify your model.

Best wishes and good luck, narsep (ioannis hadjidakis)

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Author Tobias Fritz replied on Feb. 21, 2011 @ 18:44 GMT
Dear Ioannis,

my knowledge of chemistry is very superficial, while you're apparently an expert. So, do you think that the phenomenon of hybridization is responsible for the important role they play in biological systems?

Sorry if I cannot follow your explanation of the lattice structure and the different 'types' of carbon atoms. What I can say is that the red ellipses are *not* supposed to depict double bonds; pairing the atoms up into pairs and defining each atom to be of a certain 'type' is just a mathematical trick. In principle, each atom can be paired up with any of its neighbors. There is no physical/chemical significance to the red ellipses, and neither to the blue lines. Also, basically there exists no physical/chemical connection at all between different atoms of the same type.

It is very much like in benzene, which is sometimes depicted as having three double bonds and three single bonds, although all of the six bonds are identical, which makes benzene a nicely symmetric molecule.

Maybe we have been saying the same thing in different words. Feel free to email me if you don't deem the discussion appropriate for this forum; I'll be happy to learn more chemistry!

I haven't yet looked at your essay in detail yet (will do so tomorrow) in order to be able to comment on the relation.

narsep replied on Feb. 22, 2011 @ 10:58 GMT
Dear Tobias,

Hybridization is (one of) the most important factor(s) that atoms and molecules' properties are depented of. Chemical orbitals are in a sense the application of quantum theory in chemistry. So, double bond is also a mathematical "trick" - model. A figure is attached in order to help understand what I mean by lattices staff (lattices in green). Regards narsep (ioannis)

attachments: fig.jpg

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Author Tobias Fritz replied on Feb. 22, 2011 @ 17:56 GMT
Hi Ioannis,

ok, good to clear this up, so then we have been saying the same thing! It was the terminology of "double bond" which confused me: it sounds a lot like something which is actually deemed to exist. Thanks also for the figure. Maybe you can allow me to ask, as a final question, what the "charge front" and "charge transfer" mean in the figure?

James Lee Hoover wrote on Feb. 22, 2011 @ 00:19 GMT

Isn't graphene difficult to work with? What is the effect of a graphene flake lying atop an iridium crystal causing new iridium atoms?

Not my area of understanding.

Jim Hoover

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Author Tobias Fritz replied on Feb. 22, 2011 @ 17:57 GMT
Dear Jim,

sorry if I cannot comment on this, as I'm not aware of any relation between graphene and iridium. Can you expand or point to a reference?

And probably yes, I suppose that graphene is difficult to work with in the lab, also because of its unprecedented properties like two-dimensionality and its stiffness, and whatnot. But then anything is difficult to work with when it's new, and simple to work with once the techniques have been established. Obvious examples come to mind, take Bose-Einstein condensation for a recent one which took so long to be realized experimentally, but is now produced on a daily basis in many labs around the world.

Mike Bradley wrote on Feb. 22, 2011 @ 10:22 GMT
Hi Tobias,

Nice essay and a good idea! I completely agree about the importance of graphene in this context and I am continually amazed by the mapping between a 2D flake of graphite and the Dirac equation. In particular if Zitterbewegung can be seen that would extraordinary.

Thanks again for the essay, you've given me some things to think about.


Mike Bradley

PS How do you like ICFO ? I have been there once (more or less by accident) while on holiday in Barcelona; it seems like a nice place.

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Author Tobias Fritz replied on Feb. 22, 2011 @ 17:58 GMT
Dear Mike,

thank you for the kind words, I appreciate it! About ICFO, yes, I think it's a nice place and I'm quite happy here. There is a lot going on, though mostly on the experimental side, but also in the quantum information group we always have something to discuss. Only the location is a bit off; but maybe this is what you came for: the marvelous beaches?

Mike Bradley replied on Feb. 25, 2011 @ 10:50 GMT
Hi Tobias,

Yes exactly, I came down to visit Barcelona with a girlfriend around Xmas and since we were staying in a hotel in Castelldelfels we drove by ICFO to check it out. As I say it looked like a nice place.

What sort of work is your group doing there ? I'm working at the BIPM in Sevres, near Paris, on a superconducting Watt Balance machine for measuring Planck's constant h and also for the planned redefinition of the SI kilogram standard in 2015. I am particularly interested in applying quanutm information ideas in precision measurement applications-- hence my interest in graphene also.


Mike Bradley

Dr. Michael P. Bradley, Ph.D., P.Eng.

Chercheur Associé/Research Fellow

Electricity Department/Watt Balance

Bureau International des Poids et Mesures (BIPM)

Pavillon de Breteuil

F-92312 Sèvres CEDEX


Tél: +33 1 45 07 62 92



Associate Professor

Dept. of Physics and Engineering Physics

University of Saskatchewan

116 Science Place

Saskatoon, SK S7N 5E2 Canada


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Florin Moldoveanu wrote on Feb. 26, 2011 @ 04:19 GMT
Hello Tobias,

Congrats on a good essay. I only wanted to correct one thing. Zitter was experimentally observed. Check out Hestenes' winning essay on the nature of time.

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Anonymous replied on Feb. 28, 2011 @ 18:57 GMT
Dear Florin,

that's very interesting! Thanks for the pointer, it would indeed have been worth mentioning in the essay.

On the other hand, from what I gather from skimming Hestenes' essay and his paper, his zitter model is a theory of a classical point particle, and therefore clearly cannot be realistic. For example, how would it reproduce the spectrum of the hydrogen atom? Certainly one needs some sort of quantization somewhere.

Evaluating the merits of the experiment seems much more difficult. So I don't dare saying anything about whether this could be an experimental detection of zitterbewegung or not, but only notice that it's not well-known and has not been published in an 'important' journal, which makes me a bit skeptical.

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Author Tobias Fritz replied on Feb. 28, 2011 @ 18:58 GMT
oops, my login had expired, the previous post is mine!

Florin Moldoveanu replied on Mar. 1, 2011 @ 13:15 GMT
Dear Tobias,

Hestenes' idea is only a speculation, but the resonance was experimentally observed and the root cause of zitterbewegung and Klein paradox is the SU(2)xU(1) gauge symmetry of Dirac's equation (combined with the a spin current conservation to be 100% rigurous). This generates a departure from the simple Berry phase of standard nonrelativistic QM and uncovers new physics unavailable from the simpler U(1) symmetry. By the way, in quantum Hall effect experiments, spin current is not conserved and the full SU(2)xU(1) symmetry is experimentally observed and is explained as spin-orbit coupling.

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Russell Jurgensen wrote on Feb. 28, 2011 @ 01:37 GMT
Dear Tobias,

I enjoyed your essay and couldn't help becoming more curious as it progressed. It does seem very attractive to test quantum theory in a controlled environment as you describe. Massless fermions seem especially interesting for study, especially if graphene allows a certain degree of control of the parameters. From your research, is the test something that can be set up in an average lab?

Kind regards, Russell Jurgensen

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Author Tobias Fritz replied on Feb. 28, 2011 @ 19:25 GMT
Dear Russell,

I'm happy to hear your comments! First of all, I should emphasize again that almost all of what I explained is not 'my' research, but merely a write-up of what I learned from reading the papers.

About the experimental realizations, yes, it seems that one indeed has good control over some of the parameters. For example, one can add hydrogen atoms to the lattice sites, which means that the affected lattice sites are not available for the electron hopping. Also, the two-dimensionality is a big advantage in that one has direct access to each atom, in contrast to three-dimensional crystals.

However, my understanding is that most things can only be observed indirectly. For example, observing a single massless free Dirac particle may be impossible, since how would one isolate a single electron? What has been observed is the correct dependence of the cyclotron mass on the electron filling, as predicted by the Dirac formalism. Or maybe one can use doping to introduce internal electric potentials, of which one might then try to observe the energy levels...? I don't know...

Russell Jurgensen replied on Mar. 1, 2011 @ 06:35 GMT
Dear Tobias,

Thank you for the extra detail. It makes it clear it is an involved test. Very interesting.

Kind regards, Russell Jurgensen

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Vladimir F. Tamari wrote on Mar. 1, 2011 @ 04:14 GMT
Hola Tobias

I enjoyed your concept of analyzing graphene as a basis of simulating aspects of physics. In my current fqxi paper and my earlier 2005 Beautiful Universe theory on which it is based I have proposed an entire universe made up solely of one type of node - much as your 2-D graphene 'universe' is made up of one type of node: a carbon atom. In my proposed Face-Centered Cubic (FCC) lattice the nodes are magnetic dipoles whose axes can be aligned in any direction in 3D. (N-S) attraction and (N-N) or (S-S) repulsion play an important role in the interactions of the lattice. I wonder if carbon has such polarity, and whether induction plays a role in the unusual binding strength of graphene's chemical bonds.

A related question that arose in my theory and other discussions here is that in 3D (for example in buckminster fullerine molecules Carbon 60) Brouwer's theorem states that a vector field on a sphere will always have one vortex. This implies a 'weak' spot on a C60 molecule - if that is, magnetic polarity plays a role there. Such phenomena highlight the limitations of 2D simulations in a 3D world, and I hope you can extend your fascinating analysis into 3D lattices, particularly FCC. Good luck to you.


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Author Tobias Fritz replied on Mar. 1, 2011 @ 18:59 GMT
Dear Vladimir,

thank you for the excellent questions! As far as I understand, a carbon atom does not a priori have any magnetic moment. However, one can introduce magnetic moments by adding additional atoms of other elements "above" or "below" the carbon atoms; see for example this paper:

Moreover, it seems that there is something like an emergent ferromagnetism going on, a phenomenon poorly understood. See here:

Finally, about extending the analysis to higher dimensions: Yes, I am indeed working on that! The goal is to see whether there are any higher-dimensional lattices which also have the property of the emergence of fermions. This is unlikely to be the case in 3D, since otherwise this would be well-known. I think that one would need a "non-rectangular" symmetry of the lattice. I'm not sure what exactly I mean by that, by the hexagonal graphene lattice certainly satisfies it with its ternary rotational symmetry, whereas an FCC lattice doesn't.

But I'm looking forward to study the 4D case, since this is the one relevant for actual physics in spacetime! (I'm thinking of things like lattice simulations of Euclidean Quantum Field Theory.)

Steev Dufourny replied on Mar. 2, 2011 @ 12:50 GMT
Hello to both of you,

What about the magneton of Borh and the nuclear and atomic magnetic's always proportional in factn with the spinning and orbiting spheres.......partition fuction of a sub systems of entangled spheres ...the number is specific and presice for all gravitational stabilities.The entropy is correlated.

The particles inside the system have a spins thus a magnetism nuclear.We can take several quatum numbers which differenciate the different spins and orbitals.If the volumes are inserted also with the biggest volume for th cneter.....we can also differenciate the velocities of these rotations correlated with mass with the magneton of Borh as system of gauge eh/4pimc....we can subtitute the mass of different volumes. You can use the parallelization of Christian also in a deterministic road showing the rationalities of the magnetic momment, all has a momment,only the space hasn't rotations thus mass thus no momment also.It's relevant if the real serie is inserted for the different quantum numbers......the conservation of the parity seems essential as our proprtionalities.



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narsep replied on Mar. 5, 2011 @ 18:21 GMT
Dear Tobias and others,

have you a look to my essay (concerning 3+1 D)? This model is based on C(sp3 hybridization.

Ioannis Hadjidakis

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Vladimir F. Tamari wrote on Mar. 2, 2011 @ 15:19 GMT
Hello Tobias

I am glad my comments made some sense to you. There is so much in physics and mathematics I do not know - for example Euclidean Quantum Field Theory or what is ternary rotational symmetry and why it is important in fermion structure. In my (BU) theory the FCC arrangement was suggested almost ad-hoc and because other researchers such as Norman Cook (Models of the Atomic Nucleus (Springer)) was able to simulate nucleon structure using it. The magnetic lattice nodes I proposed self-assemble into some sort of configuration. I wonder if this process can easily be simulated and with what software.- it is outside my proficiency (all I know is the old BASIC !).

Hello Steve I hope you fine. Good luck to us all.


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Steve Dufourny replied on Mar. 2, 2011 @ 16:22 GMT
Hi Vladimir,

That goes thanks it's nice.Hope you also. Yes indeed ,you are right.good luck to all, this year, the essays are numerous ,it's well for the 3ème year.And furthermore there are many many relevances in several essays.

Best Regards


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Giacomo Mauro D'Ariano wrote on Mar. 6, 2011 @ 15:59 GMT
Dear Tobias,

compliments for your very interesting paper! I'm especially interested in the simulation of Dirac equation by graphene, since this is connected to my work. Can you really consider the carbon atom as a gate in a quantum-computational simulation of Dirac in 2+1 dimensions? In such case I'm indeed very curious about the unitary transformation of the gate!

In the meanwhile I discovered that I didn't see your reply to my answer in my thread, and I answered to it. It seems that you are right. In order to recover an isotropic velocity of light in the analog coordinate system, one needs another way, maybe the thickness of events?


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Author Tobias Fritz replied on Mar. 9, 2011 @ 17:44 GMT
Dear Mauro,

excellent question! I think something like this is indeed possible. In a second quantized formalism, the electrons in the tight-binding approximation to graphene would be modeled as follows: take one qubit for each lattice site, i.e. at each carbon atom. The Hamiltonian is given by

where the sum runs over all pairs of adjacent atoms and the a_i are fermionic annihilation operators. Intuitively, this says that the particles hop from one atom to a neighboring one. So for small timesteps, this interaction funtions like a partial SWAP gate with a small swapping angle. If we approximate the continuous time by discrete steps, we therefore obtain a three-dimensional network of partial SWAP gates, and these simulate the massless Dirac equation.

Concerning the other issue, see my reply in your essay's forum.

Member Giacomo Mauro D\'Ariano replied on Mar. 10, 2011 @ 11:06 GMT
Dear Tobias,

I really love your idea of the graphene Dirac simulator. The second quantization in the tight-binding approximation to graphene that you give is really juicy! I will iimmediately explore this.

We are currently communicating in parallel on our two blogs, and some of the ideas that I'm posting here are also reported in my reply to your last post on my blog.

My problem is to prove that it is possible to simulate the Dirac equation by a quantum computer with a periodic topology of gate connections. This is also your problem, if your Graphene can be regarded as such a kind of a quantum computer (as you seem to assert in your answer). As you saw in my essay, I showed that this is possible in 1 plus 1 dimensions (with a mass-dependent renormalization of the speed of light). I'm trying now to prove it in 3 plus 1 dimensions (here it seems that a 5-simplex geometry is needed for each gate). Now, the problem is the following. In my blog you are mentioning a simple proof that a regular lattice will never give an isotropic propagation speed. How can you reconcile this with the covariance of the Dirac equation that you are simulating by the regular-lattice quantum-simulator graphene? I'm very intrigued and very curious.

Let me compliment again on your work!



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Author Tobias Fritz replied on Mar. 14, 2011 @ 19:15 GMT
See the answer in your essay's forum! Essentially, the main point is that the emergence of the massless Dirac equation only holds for small particle momenta... For higher momenta, anisotropies appear also in graphene, and this is known as trigonal warping. These higher order contributions are suppressed by additional factors of the lattice spacing constant. So the momentum scale at which the anisotropies appear depends on the lattice spacing.

Probably it should also be mentioned that this is why any extension of this to the massive Dirac equation is pointless. If the massless Dirac equation holds only for small momenta, then one can also approximate the system up to the same order by the Pauli equation, i.e. the Schrödinger equation with spin. No relativistic spacetime emerges in this case.

Sridattadev wrote on Mar. 7, 2011 @ 19:47 GMT
Dear Tobias,

Wisdom is more important than imagination is more important than knowledge for all the we know is just an imagination chosen wisely.

Please read Theory of everything at your convenience posted by me in this contest.

Who am I? I am virtual reality, I is absolute truth.



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John Benavides wrote on Mar. 9, 2011 @ 07:52 GMT
Dear Tobias

Your essay is very interesting, I think the idea of emergence have not been study very seriously until now but models like the one you expose or others based on a discrete computational basis, show that this is the central point. On my essay, I try to explain this emergence from a different perspective but I think there is a very closed connection with your ideas. I will like to hear what do you think about it.


J .Benavides

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Lawrence B. Crowell wrote on Mar. 9, 2011 @ 22:38 GMT

Your essay is one of the better essays in the lot here. It was an enjoyable reading.

Cheers LC

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Steve Dufourny replied on Mar. 11, 2011 @ 15:42 GMT
He is skilling indeed Tobias.A little of 3D harmonized with 4D and a little of rationality about the entanglement and it's very relevant.

Now of course for a quantum computer , the realism is deterministic in the pure road of real numbers.The graphene is a step, a weak step.but it's well , they try to converge with the reality, it's the most important.

Tobias....... operators hamiltonians and Laplacians more green and stokes more the rotational operators ....and if you insert the real number.....but perhaps an irrotational vectorial field is prefered U=-1/4INTdivVdv/r....poisson helping and the serie respected...and of course the harmonious function...a real puzzle all that ....fourier always is of course the volumes of entangled spheres is essential.....and what about the theory of big number and the probabilities and the errors also...Laplace where are you and Bernouilli....and the law of repartition of maxwell ...and pi always which smiles.....errors...moy. simple,moy. quadratic ,probable and precise...n=1/rac(pih)....DETERMINISM AND FINITE SERIE .....Pierce helping and Wolfram hihihi

Spherically yours.


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Don Limuti wrote on Mar. 14, 2011 @ 03:51 GMT
Hi Tobias,

Thanks for the introduction to graphene.

Question, do you think graphene will show interference patterns similar to C60.

I like your essay and think it is one of the best, but would encourage you to venture a little more into speculation. I think physics is at a local peak and it is going to be hard to get off it into something more productive without some leaps of faith.

Don Limuti

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Author Tobias Fritz replied on Mar. 14, 2011 @ 18:52 GMT
Hi Don,

thank you for the feedback! Indeed I could have ventured more into speculation, but unfortunately at the present time I do not have interesting and original speculations worth mentioning...

About the interference patterns: I'm not sure what you mean. C60 is a pretty small molecule which one can shoot at a diffraction grating and observe an interference pattern. At least in theory, this is not specific to a molecule of carbon atoms; it should work with anything of small enough size. As you probably know, the next step in this kind of experiments is to do it with a virus, which doesn't have anything to do with C60 or graphene. On the other hand, a graphene sheet can be quite large. So maybe you can elaborate on your question a little more?

Don Limuti replied on Mar. 15, 2011 @ 12:30 GMT
Hi Tobias,

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

Good Luck,

Don Limuti

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T H Ray wrote on Mar. 14, 2011 @ 13:42 GMT

Wow. I regret not reading your essay until now, but was happy I was able to rate it high before the close of the contest.

Masterful job of bringing the essence of simulation and modeling theory down to Earth. Very nice explanation of the relation of the Dirac equation to spacetime values.

I think you're overly modest about the significance of your explanation of continuous vs. discrete as relates to leading edge research. There are a number of important unsolved problems -- protein folding comes immediately to mind -- in which a continuous and random time dependent walk contrasts with the discrete lowest energy state. Classical computing hasn't been much help so far, that I know of; a simulation from another system could be a breakthrough.

(I like the clever distinction between "quantum graphenity" and PI's "guantum graphity.")

Hope you get a chance to read my essay. I, too, chose to survey the subject rather than dwell on research results. I think that your research program and mine have much in common mathematically, however.

All best,


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Member Moshe Rozali wrote on Mar. 14, 2011 @ 19:15 GMT
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:

I am curious about your thoughts.



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Member Giacomo Mauro D\'Ariano wrote on Mar. 18, 2011 @ 19:08 GMT

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!

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Author Tobias Fritz replied on Mar. 21, 2011 @ 10:47 GMT
First of all, I should point out again that the simulation of the Dirac equation by graphene has not been 'my' idea! So you may want to quote either Wallace, who first considered the tight-binding approximation in a two-dimensional hexagonal lattice, or Semenoff, who considered the simulation aspect already in 1984.

Then I have to admit that I don't understand the correspondence between your model and the graphene lattice. In the latter, the gates are the edges of the hexagons, so they are bipartite. Anyway, I'm sure you have thought this over well, and I will understand the details in due time. Enjoy the Meeting :)

Member Giacomo Mauro D\'Ariano replied on Mar. 24, 2011 @ 21:26 GMT

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!


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Author Tobias Fritz replied on Mar. 28, 2011 @ 18:44 GMT
I see, so by "circuit" you mean not just the qubits themselves, but the qubits together with the gates as a circuit in spacetime. Yes, this makes sense, given that one usually draws a quantum circuit as a two-dimensional figure with one direction being space and the other direction being time.

Am I understanding correctly that the hexagonal lattice is split up into two layers, corresponding to adjacent spatial slices? I am a bit confused about this point, since a qubit at every instant of time and have a 1-dimensional worldline, shouldn't it?

Member Giacomo Mauro D\'Ariano wrote on Mar. 18, 2011 @ 19:12 GMT
Here's the fugure

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Member Giacomo Mauro D\'Ariano wrote on Mar. 18, 2011 @ 20:14 GMT
Sorry, the figure was too big.

Here it is again...

attachments: Dirac2plus1_small.jpg

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Alan Lowey wrote on Mar. 19, 2011 @ 11:06 GMT
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,


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Author Yuri Danoyan+ wrote on Mar. 21, 2011 @ 13:25 GMT

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Vladimir F. Tamari wrote on Jun. 6, 2011 @ 09:05 GMT
Hola Tobias

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

Best wishes from Vladimir

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Georgina Parry wrote on Jun. 6, 2011 @ 21:34 GMT
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.

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Deepak Vaid wrote on Jun. 9, 2011 @ 18:33 GMT
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!

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narsep wrote on Jun. 22, 2011 @ 13:26 GMT
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)

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Author Tobias Fritz wrote on Jul. 6, 2011 @ 09:24 GMT
Thank you all for the compliments! This second prize was *very* unexpected!

Also, we have had many interesting discussions here and some of them have spawned new collaborations and work going beyond the essays, which is more important than the prizes. So, thanks to all!

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