## The amazing Graphene

Continuing the interesting talks series from DICE2014, I was
blown away by a talk by

**Alfredo Iorio**:**“What after**CERN ?”. Physics is an experimental science and the lack of experiments forces theoreticians to construct alternative models which most likely have nothing to do with how nature really is.
In high energy physics the experiments are extremely
expensive and the price tag for a new accelerator is in the billions. Why do
people need larger accelerators? Because to probe smaller and smaller regions
of space you need larger and larger energies. Accelerators circulate a beam of
particles in a circle to gain the required energy, and the faster they go
(closer and closer to the speed of light), the heavier the particle become and
they need larger and larger circle radius. To probe at the scales of the string
theory for example, one needs an accelerator the size of the galaxy. So is
there an alternative to this?

It turns out that there are theoretical and experimental
efforts of outstanding value circumventing this brute force approach and Iorio’s
research belongs into this rare breed in physics.

In the past I was blogging at FQXi about an experiment done
by Bill Unruh with a laboratory waterfall which was able in principle to
simulate a black hole and its Hawking radiation. However even more amazing things can be achieved with Graphene

So what is so special about this
material? There are two key properties which makes it extremely interesting.

First, the hexagonal structure
requires two translations to reach any atom.

Given an origin, any atom can be specified

**first**by a linear combination of two vectors: \( (a_1 , a_2 )\): \( x_a = n_1 a_1 + n_2 a_2\) where \( n_1 , n_2\) are positive or negative integers, followed by a**second translation**using the vectors \( s_1 , s_2 , s_3\).
Second, the band structure in graphene is very special: the
conductivity and valence band touch in exactly one point (called the Dirac
point) making the structure a semi-metal:

Graphene Band |

When the excitation energy is small (~ 10 \( \mu \) eV),

**Two of the 4 spinor components come from the Lattice A vs. Lattice B, and the other two come from the up and down bands touching at the Dirac point.**__the quasi-particle excitations respect Dirac’s equation.__
By its very geometrical structure, graphene is an ideal
simulator of spin ½ particles.

Now the hard work begins. How can we use this to obtain
answers about quantum field theory in curved space-time? First we can start
easy and consider defects in the hexagonal pattern. A defect changes the Euler
number and introduces curvature. This is tractable mathematically for all
simple defects using a bit of advanced geometry, but you don’t get very far
except in the description of the phenomena in terms of critical charges and
magnetic fluxes.

But if you can manufacture surfaces of negative curvature:

called Beltrami spheres then the real fireworks begins. Under the right conditions you can simulate
the Unruh effect (http://arxiv.org/pdf/1308.0265.pdf
): an observer in A sees the quantum vacuum in the frame B as a condensate of
A-particles. To observe this the tip of a Scanning Tunneling Microscope is
moved across the graphene surface and probes the graphene quasi-particles.

Of course there are drawbacks/roadblocks too: the defects in
manufacturing which might spoil those effects. It is unclear how accurate are
manufacturing techniques at this time. Also I don’t know if the impurities
effects are properly computed. Much more serious I am skeptical of the ability
to maintain the hexagonal pattern while creating the Beltrami funnel. And if
this is not maintained, in turn it will affect the band structure which can
ruin the validity quasi-particle model of Dirac’s equation.

I brought my concerns to Alfredo and his response put my
mind to ease. To avoid playing telephone, with his permission I am sharing his
answer here:

*“- So, you are perfectly correct when you doubt that the Beltrami shape can be done all with hexagons. In fact, this is not possible, not because of technical inabilities of manufacturers, but because of the Euler theorem of topology.*

- How do we cope with that? Although at the location of the defects the Dirac structure is modified, the hexagonal structure resists in all the other places. When the number of atoms N is big enough, one can safely assume that the overall hexagonal structure dominates (even when the defects start growing, as they do with N, all they do is to distribute curvature more evenly over the surface).

Now, if you stay at small \( \lambda \) (large energy E), you see all local effects of having defects, and the lattice structure cannot be efficiently described by a smooth effective metric (essentially, since the \( \lambda \) and E we talk about here are those of the conductivity (or \( \pi \) ) electrons that live on the lattice (they don't make the lattice, that is made by other electrons, belonging to the \( \sigma \) bonds), we realize that when their wavelength is big enough, they cannot see the local structure of the lattice, just like large waves in the sea are insensitive to small rocks. Hence, for those electrons, the defects cannot play a local role, but, of course they keep playing a global, i.e., topological, role, e.g., by giving the intrinsic curvature (as well known, in 2 dimensions the Gauss-Bonnet theorem links topology and geometry: Total Curvature = 2 \( \pi \) Euler Characteristic).

- Thus, if I was good enough at explaining the previous points, you should see that the limit for big \( r \) (that is small curvature \( K = \pm 1/r^2 \)) is going in the right direction, in all respects: 1. the number of atoms N grows; 2. the energy \( E_r \sim 1/r \) (see Fig. Graphene Band) gets small, hence the \( \lambda \) involved gets big, hence 3. the continuous metric \( g_{\mu \nu} \) well describes the membrane; 4. the overall Dirac structure is modified, but not destroyed, and, the deformations are given by a ''gauge field'', that is of the fully geometric kind. Indeed, this gauge field describes deformations of the membrane, as seen by the Dirac quasi-particles. The result is a Dirac field (we are in the continuum) in a curved spacetime (i.e. covariant derivatives of the diffeo kind appear). In arXiv:1308.0265 we discuss all of this in Section 2.

- There is also an extra (lucky!) bonus in going to big \( r \), that is the reaching of some sort of horizon (more precisely, that is a conformal Killing horizon, that, for a Hawking kind of phenomenon, is more than enough). Why so? The issue here brings in the negative curvature. In that case the spacetime (the 2+1 dimensional spacetime!) is conformal (Weyl related) to a spacetimes with an horizon (Rindler, deSitter, BTZ). Something that does not happen for the positive curvature, the sphere, that in graphene is a fullerene-like structure. In fact, the latter spacetime is conformal (Weyl related) to an Anti deSitter, that, notoriously, does not have an intrinsic horizon.

Now, once you learn that, you also learn that surfaces of constant negative Gaussian curvature have to stop somewhere in space (they have boundaries). That is a theorem by Hilbert. For small \( r \) (large curvature) they stop too early to reach the would-be-horizon. For large \( r \), though, they manage to reach the horizon. Fortunately, for that to happen, \( r \) needs not be 1 km (that would not be an impossible Gedanken experiment, but still a tremendous task, and just unfeasible for a computer). The job is done by \( r = 1 \) micron! That is something that made us very happy: the task is within reach. It is still hard for the actual manufacturing of graphene, but, let me say, it turned into a problem at the border between engineering and applied physics, i.e. it is no longer a fundamental problem, like, e.g., the mentioned galaxy-size accelerator.

- We are actively working on the latter, as well. In this respect, we are lucky that these ``wonders’’ are happening (well... predicted to be happening) on a material that is, in its own right, enormously interesting for the condense matter friends, hence there is quite a lot of expertise around on how to manage a variety of cases. Nonetheless, you need someone willing to probe Hawking phenomena on graphene, while the standard cond-mat agenda is of a different kind. Insisting, though, very recently I managed to convince a composite group of condensed matter colleagues, mechanical engineers, and computer simulations wizards, to join me in this enterprise. So, now we are moving the first steps towards having a laboratory that is fully dedicated to capture fundamental physics predictions in an indirect way, i.e. on an analog system.

What we are doing right now, between Prague, Czech Republic (where I am based) and Trento, Italy (where the ``experimentalists`` are sitting), is the following:

First, we use ideal situations, i.e. computer simulations, hence we have no impurities nor substrates here. There no mention is made of any QFT in curved space model. We only tell the system that those are Carbon atoms, use QM to compute the orbitals and all the relevant quantified, perform the tight binding kind of calculations. Thus, the whole machinery here runs without knowing that we want it to behave as a kind of black hole.

What we are first trying is to obtain a clear picture of what happens to a bunch of particles, interacting via a simplified potential, e.g., a Lennard-Jones potential, constrained on the Beltrami. This will tell us a lot of things, because we know (from similar work with the sphere, that goes under the name of generalized Thomson problem, see, e.g., the nice work by Bowick, Nelson and Travesset) that defects will form more and more, and their spatial arrangements are highly non trivial.

When this is clear, we want to get to a point where we tell the machine that we have N points, and she (the machine) plots the Beltrami of those points. i.e. it finds the minimum, the defects, etc. This would be the end of what we are calling: Simulation Box 1 (SB1).

When SB1 is up and running, we fix a N that is of the order of 3000, take away points interacting with Lennard Jones, and substitute them with Carbon atom, i.e. we stick in the data of Carbon, the interaction potential among them, and then let a Density Functional computation go. The latter is highly demanding, computer-time wise, but doable. With this we shall refine various details of the theory, look into the structure of the electronic local density of states (LDOS), although the \( r \) we can get with N = 3000 is still too small for any Hawking anything. That is the first half of SB2.

The work of SB1 and first half of SB2, can be done with existing machines and well tested algorithms. But we need to go further, towards a big \( r \) (the 1 micron at least... although I would be happier with 1 mm, but don't tell my experimentalist friends, they would kill me!). This is possible, but we are going into the realm of non tested algorithms, of dedicated machines (i.e. large supercomputers, etc). Nonetheless, figures of the order of N = 100K (and even whispered N = 1 million) are in the air. That would be second half of SB2, i.e. when the Hawking should be visible.

That is the road I can take with the current group of people involved. I don't give up though the idea of getting someone to actually do the real graphene thing. But this would only mean a handle of a very large number of points, to the expense of more impurities, substrates, etc. Indeed, the SB2 (the computer simulations of true Carbon interactions) would be so accurate, that myself (and, most importantly, the cond-mat community) would take those results as good (if not better, because `fully clean`) as the experiments.”

- How do we cope with that? Although at the location of the defects the Dirac structure is modified, the hexagonal structure resists in all the other places. When the number of atoms N is big enough, one can safely assume that the overall hexagonal structure dominates (even when the defects start growing, as they do with N, all they do is to distribute curvature more evenly over the surface).

Now, if you stay at small \( \lambda \) (large energy E), you see all local effects of having defects, and the lattice structure cannot be efficiently described by a smooth effective metric (essentially, since the \( \lambda \) and E we talk about here are those of the conductivity (or \( \pi \) ) electrons that live on the lattice (they don't make the lattice, that is made by other electrons, belonging to the \( \sigma \) bonds), we realize that when their wavelength is big enough, they cannot see the local structure of the lattice, just like large waves in the sea are insensitive to small rocks. Hence, for those electrons, the defects cannot play a local role, but, of course they keep playing a global, i.e., topological, role, e.g., by giving the intrinsic curvature (as well known, in 2 dimensions the Gauss-Bonnet theorem links topology and geometry: Total Curvature = 2 \( \pi \) Euler Characteristic).

- Thus, if I was good enough at explaining the previous points, you should see that the limit for big \( r \) (that is small curvature \( K = \pm 1/r^2 \)) is going in the right direction, in all respects: 1. the number of atoms N grows; 2. the energy \( E_r \sim 1/r \) (see Fig. Graphene Band) gets small, hence the \( \lambda \) involved gets big, hence 3. the continuous metric \( g_{\mu \nu} \) well describes the membrane; 4. the overall Dirac structure is modified, but not destroyed, and, the deformations are given by a ''gauge field'', that is of the fully geometric kind. Indeed, this gauge field describes deformations of the membrane, as seen by the Dirac quasi-particles. The result is a Dirac field (we are in the continuum) in a curved spacetime (i.e. covariant derivatives of the diffeo kind appear). In arXiv:1308.0265 we discuss all of this in Section 2.

- There is also an extra (lucky!) bonus in going to big \( r \), that is the reaching of some sort of horizon (more precisely, that is a conformal Killing horizon, that, for a Hawking kind of phenomenon, is more than enough). Why so? The issue here brings in the negative curvature. In that case the spacetime (the 2+1 dimensional spacetime!) is conformal (Weyl related) to a spacetimes with an horizon (Rindler, deSitter, BTZ). Something that does not happen for the positive curvature, the sphere, that in graphene is a fullerene-like structure. In fact, the latter spacetime is conformal (Weyl related) to an Anti deSitter, that, notoriously, does not have an intrinsic horizon.

Now, once you learn that, you also learn that surfaces of constant negative Gaussian curvature have to stop somewhere in space (they have boundaries). That is a theorem by Hilbert. For small \( r \) (large curvature) they stop too early to reach the would-be-horizon. For large \( r \), though, they manage to reach the horizon. Fortunately, for that to happen, \( r \) needs not be 1 km (that would not be an impossible Gedanken experiment, but still a tremendous task, and just unfeasible for a computer). The job is done by \( r = 1 \) micron! That is something that made us very happy: the task is within reach. It is still hard for the actual manufacturing of graphene, but, let me say, it turned into a problem at the border between engineering and applied physics, i.e. it is no longer a fundamental problem, like, e.g., the mentioned galaxy-size accelerator.

- We are actively working on the latter, as well. In this respect, we are lucky that these ``wonders’’ are happening (well... predicted to be happening) on a material that is, in its own right, enormously interesting for the condense matter friends, hence there is quite a lot of expertise around on how to manage a variety of cases. Nonetheless, you need someone willing to probe Hawking phenomena on graphene, while the standard cond-mat agenda is of a different kind. Insisting, though, very recently I managed to convince a composite group of condensed matter colleagues, mechanical engineers, and computer simulations wizards, to join me in this enterprise. So, now we are moving the first steps towards having a laboratory that is fully dedicated to capture fundamental physics predictions in an indirect way, i.e. on an analog system.

What we are doing right now, between Prague, Czech Republic (where I am based) and Trento, Italy (where the ``experimentalists`` are sitting), is the following:

First, we use ideal situations, i.e. computer simulations, hence we have no impurities nor substrates here. There no mention is made of any QFT in curved space model. We only tell the system that those are Carbon atoms, use QM to compute the orbitals and all the relevant quantified, perform the tight binding kind of calculations. Thus, the whole machinery here runs without knowing that we want it to behave as a kind of black hole.

What we are first trying is to obtain a clear picture of what happens to a bunch of particles, interacting via a simplified potential, e.g., a Lennard-Jones potential, constrained on the Beltrami. This will tell us a lot of things, because we know (from similar work with the sphere, that goes under the name of generalized Thomson problem, see, e.g., the nice work by Bowick, Nelson and Travesset) that defects will form more and more, and their spatial arrangements are highly non trivial.

When this is clear, we want to get to a point where we tell the machine that we have N points, and she (the machine) plots the Beltrami of those points. i.e. it finds the minimum, the defects, etc. This would be the end of what we are calling: Simulation Box 1 (SB1).

When SB1 is up and running, we fix a N that is of the order of 3000, take away points interacting with Lennard Jones, and substitute them with Carbon atom, i.e. we stick in the data of Carbon, the interaction potential among them, and then let a Density Functional computation go. The latter is highly demanding, computer-time wise, but doable. With this we shall refine various details of the theory, look into the structure of the electronic local density of states (LDOS), although the \( r \) we can get with N = 3000 is still too small for any Hawking anything. That is the first half of SB2.

The work of SB1 and first half of SB2, can be done with existing machines and well tested algorithms. But we need to go further, towards a big \( r \) (the 1 micron at least... although I would be happier with 1 mm, but don't tell my experimentalist friends, they would kill me!). This is possible, but we are going into the realm of non tested algorithms, of dedicated machines (i.e. large supercomputers, etc). Nonetheless, figures of the order of N = 100K (and even whispered N = 1 million) are in the air. That would be second half of SB2, i.e. when the Hawking should be visible.

That is the road I can take with the current group of people involved. I don't give up though the idea of getting someone to actually do the real graphene thing. But this would only mean a handle of a very large number of points, to the expense of more impurities, substrates, etc. Indeed, the SB2 (the computer simulations of true Carbon interactions) would be so accurate, that myself (and, most importantly, the cond-mat community) would take those results as good (if not better, because `fully clean`) as the experiments.”

In conclusion this is an extremely exciting research direction.