*The unique electronic properties of graphene – a one-atom-thick sheet of carbon that was produced*

for the first time just two years ago – make it an ideal testing ground for fundamental physics,

describe Antonio Castro Neto, Francisco Guinea and Nuno Miguel Peres

For example, in quantum electrodynamics (QED),

the strength of electromagnetic interactions between

charged particles is described by the fine-structure constant,

α=e^2/hc, where h– is Planck’s constant divided by

2π and c is the speed of light. With a value of 1 divided

by 137.03599911±0.00000046, this is one of the most

precisely measured physical quantities in nature. Unfortunately,

we have no idea why the fine-structure constant

has this value. Since the effective speed of light

for the Dirac fermions in graphene is 300 times less,

graphene’s fine-structure constant should have a much

larger value of about two, though it has not yet been

measured precisely.

for the first time just two years ago – make it an ideal testing ground for fundamental physics,

describe Antonio Castro Neto, Francisco Guinea and Nuno Miguel Peres

For example, in quantum electrodynamics (QED),

the strength of electromagnetic interactions between

charged particles is described by the fine-structure constant,

α=e^2/hc, where h– is Planck’s constant divided by

2π and c is the speed of light. With a value of 1 divided

by 137.03599911±0.00000046, this is one of the most

precisely measured physical quantities in nature. Unfortunately,

we have no idea why the fine-structure constant

has this value. Since the effective speed of light

for the Dirac fermions in graphene is 300 times less,

graphene’s fine-structure constant should have a much

larger value of about two, though it has not yet been

measured precisely.

Now the above argument is logically the same as mine for gravity.

Above, it is stated that the large-scale collective IR limit effective coupling of electric charge to light is

(index of refraction)(vacuum coupling of electric charge to compensating gauge boson field of light)

in precisely the same way, I claim

(index of refraction)^4(vacuum coupling of gravitational charge to compensating gauge boson field of gravity)

Note that "vacuum coupling" simply means the way the local gauge boson field scatters off virtual off mass shell particles inside the vacuum.

index of refraction = 1 + scattering of the light off the real on mass shell particles outside the vacuum

also note that light is the compensating SPIN 1 gauge boson field from localizing the internal U1 Lie group

Einstein's gravity is the compensating SPIN 1 gauge boson field from localizing the external universal T4 translation Lie group. Here the SPIN 2 graviton is an entangled pair of SPIN 1 gravity "tetrad" quanta.

The fundamental couplings are dimensionless for both light and gravity, hence renormalizable.

"Background independence" = gauge invariance under local T4(x) frame transformations

*At a Glance: Graphene*

Graphene was first isolated by Andre Geim’s team at the University of Manchester (2004) ... using the surprisingly simple technique of ripping layers from a graphite surface using adhesive tape. By repeatedly peeling away thinner layers (left), single-atom-thick sheets were obtained (right), as shown in these scanning electron micrographs.

Graphene was first isolated by Andre Geim’s team at the University of Manchester (2004) ... using the surprisingly simple technique of ripping layers from a graphite surface using adhesive tape. By repeatedly peeling away thinner layers (left), single-atom-thick sheets were obtained (right), as shown in these scanning electron micrographs.

*The trademark behaviour that distinguishes a graphene*

sheet from an ordinary metal, for example, is

the unusual form of the Hall effect.

In the original Hall effect, discovered in 1879, a current flowing

along the surface of a metal in the presence of a transverse

magnetic field causes a drop in potential at right

angles to both the current and the magnetic field. As

the ratio of the potential drop to the current flowing

(called the Hall resistivity) is directly proportional to

the applied magnetic field, the Hall effect is used to

measure magnetic fields.

A century later, Klaus von Klitzing discovered that

in a 2D electron gas at a temperature close to absolute

zero the Hall resistivity becomes quantized, taking only

discrete values of h/ne^2 (where h is Planck’s constant,

n is a positive integer and e is the electric charge). The

quantization is so precise that this “quantum Hall

effect” (QHE) is used as the standard for the measurement

of resistivity.

During a discussion about the discovery of graphene

at a tea party in Boston in early 2005, the present

authors started to wonder whether the QHE would

be different in graphene. We realized that due to a

quantum-mechanical effect called a Berry’s phase, the

Hall resistivity should be quantized in terms of odd

integers only. Graphene has a Berry’s phase of π, meaning

that if you “rotate” the quantum-mechanical wavefunction

of the Dirac fermions in graphene through a

full 360°, the system does not end up in the state that

it started in; instead the wavefunction changes sign.

A similar prediction to ours was made independently

in 2005 by Valery Gusynin at the Bogolyubov Institute

for Theoretical Physics in Kiev, Ukraine, and Sergei

Sharapov at McMaster University in Canada.

Soon after its prediction, this “anomalous integer

QHE” was observed experimentally by both Geim and

Kim, laying to rest any lingering doubts that graphene

had really been isolated. Interestingly, Geim’s group

observed the QHE in graphene at room temperature,

while it is only observed in ordinary metals at very low

temperatures. This is because the magnetic energy of

the electrons, called the cyclotron energy, in graphene

is 1000 times greater than it is in other materials. The

researchers also found that the anomalous integer

QHE is extremely sensitive to the thickness of the sample.

For instance, a sample with two layers of graphene

displays a different effect again – meaning that the

anomalous integer QHE can be used to distinguish

between single-layer graphene and multilayer samples.

...

Unlike an ordinary metal,

in which any impurities in the crystal scatter electrons

and so lead to energy loss, the electrical resistance in

graphene is independent of the number of impurities.

This means that electrons can travel for many microns

without colliding with any impurities, making graphene

a promising material for a potential high-speed electronic

switching device called a “ballistic transistor”.

? Graphene is a one-atom-thick sheet of carbon that was isolated for the first time in

2004 – a feat long thought to be impossible

? Graphene’s 2D nature and honeycomb atomic structure cause electrons moving in

the material to behave as if they have no mass

? Electrons in graphene move at an effective speed of light 300 times less than the

speed of light in a vacuum, allowing relativistic effects to be observed without

using particle accelerators

? A key experimental signature of graphene is the way it modifies the quantum Hall

effect seen in metals and semiconductors

? The electrons in graphene can travel large distances without being scattered,

making it a promising material for very fast electronic components

sheet from an ordinary metal, for example, is

the unusual form of the Hall effect.

In the original Hall effect, discovered in 1879, a current flowing

along the surface of a metal in the presence of a transverse

magnetic field causes a drop in potential at right

angles to both the current and the magnetic field. As

the ratio of the potential drop to the current flowing

(called the Hall resistivity) is directly proportional to

the applied magnetic field, the Hall effect is used to

measure magnetic fields.

A century later, Klaus von Klitzing discovered that

in a 2D electron gas at a temperature close to absolute

zero the Hall resistivity becomes quantized, taking only

discrete values of h/ne^2 (where h is Planck’s constant,

n is a positive integer and e is the electric charge). The

quantization is so precise that this “quantum Hall

effect” (QHE) is used as the standard for the measurement

of resistivity.

During a discussion about the discovery of graphene

at a tea party in Boston in early 2005, the present

authors started to wonder whether the QHE would

be different in graphene. We realized that due to a

quantum-mechanical effect called a Berry’s phase, the

Hall resistivity should be quantized in terms of odd

integers only. Graphene has a Berry’s phase of π, meaning

that if you “rotate” the quantum-mechanical wavefunction

of the Dirac fermions in graphene through a

full 360°, the system does not end up in the state that

it started in; instead the wavefunction changes sign.

A similar prediction to ours was made independently

in 2005 by Valery Gusynin at the Bogolyubov Institute

for Theoretical Physics in Kiev, Ukraine, and Sergei

Sharapov at McMaster University in Canada.

Soon after its prediction, this “anomalous integer

QHE” was observed experimentally by both Geim and

Kim, laying to rest any lingering doubts that graphene

had really been isolated. Interestingly, Geim’s group

observed the QHE in graphene at room temperature,

while it is only observed in ordinary metals at very low

temperatures. This is because the magnetic energy of

the electrons, called the cyclotron energy, in graphene

is 1000 times greater than it is in other materials. The

researchers also found that the anomalous integer

QHE is extremely sensitive to the thickness of the sample.

For instance, a sample with two layers of graphene

displays a different effect again – meaning that the

anomalous integer QHE can be used to distinguish

between single-layer graphene and multilayer samples.

...

Unlike an ordinary metal,

in which any impurities in the crystal scatter electrons

and so lead to energy loss, the electrical resistance in

graphene is independent of the number of impurities.

This means that electrons can travel for many microns

without colliding with any impurities, making graphene

a promising material for a potential high-speed electronic

switching device called a “ballistic transistor”.

? Graphene is a one-atom-thick sheet of carbon that was isolated for the first time in

2004 – a feat long thought to be impossible

? Graphene’s 2D nature and honeycomb atomic structure cause electrons moving in

the material to behave as if they have no mass

? Electrons in graphene move at an effective speed of light 300 times less than the

speed of light in a vacuum, allowing relativistic effects to be observed without

using particle accelerators

? A key experimental signature of graphene is the way it modifies the quantum Hall

effect seen in metals and semiconductors

? The electrons in graphene can travel large distances without being scattered,

making it a promising material for very fast electronic components

Therefore the effective index of refraction in graphene is 300, which should increase its bending of spacetime by a factor of 300^4 ~ 10^10 - this may be detectable in the lab, though it would still be a tiny effect.

*Massless electrons*

Graphene’s unique properties arise from the collective behaviour of electrons. That in itself is nothing new: as summarized in Philip Anderson’s famous dictum “more is different”, we know that when a large number of particles interact strongly with each other, unexpected collective motions can emerge. In the case of graphene, however, the interaction between electrons and the honeycomb lattice causes the electrons to behave as if they have absolutely no mass (see box on page 35). Because of this, the electrons in graphene are governed by the Dirac equation – the quantum-mechanical description of electrons moving relativistically – and are therefore called Dirac fermions.

Graphene’s unique properties arise from the collective behaviour of electrons. That in itself is nothing new: as summarized in Philip Anderson’s famous dictum “more is different”, we know that when a large number of particles interact strongly with each other, unexpected collective motions can emerge. In the case of graphene, however, the interaction between electrons and the honeycomb lattice causes the electrons to behave as if they have absolutely no mass (see box on page 35). Because of this, the electrons in graphene are governed by the Dirac equation – the quantum-mechanical description of electrons moving relativistically – and are therefore called Dirac fermions.