# August 10, 2009 {#week277}

I'm very happy to have finished writing a huge paper with Aaron
Lauda... a paper we've been working on for 5 years! It's a chronology
of 20th century math and physics, focused on why people are starting to
use $n$-categories in physics:

1) John Baez and Aaron Lauda, A prehistory of $n$-categorical physics, to
appear in Deep Beauty: Mathematical Innovation and the Search for an
Underlying Intelligibility of the Quantum World, edited by Hans
Halvorson. Available at `http://math.ucr.edu/home/baez/history.pdf`

This is a companion to a paper I talked about in
["Week 261"](#week261):

2) John Baez and Mike Stay, Physics, topology, logic and computation: a
Rosetta Stone, to appear in New Structures in Physics, edited by Bob
Coecke. Available as
[`arXiv:0903.0340`](http://arxiv.org/abs/arXiv:0903.0340).

Both of them are supposed to be gentle explanations of how
"diagrammatic thinking" unifies and clarifies our perspective on many
subjects. I can imagine merging them and expanding them to form a book
someday... but not today. Today I want to take a break from
$n$-categories, and talk about some basic physics.

Let's start with a riddle.

What's a million times thinner than paper, stronger than diamond, a
better conductor than copper, and absorbs exactly

\pi  α \~ 3.14159 / 137.035 \~ 2.29254%

of the light you shine through it?

Hint: α is the "fine structure constant" - a fundamental dimensionless
constant that specifies the strength of the electromagnetic force.

Can't guess? Okay, here's the answer: graphene!

::: {align="center"}
![](graphene.jpg)
:::

Graphene is what you get when you take one slice of a crystal of
graphite. It's a hexagonal array of carbon atoms, each connected to
three neighbors. I mentioned this substance back in
["Week 262"](#week262), when I was visiting the National University
of Singapore, because researchers there were working on graphene as a
possible substitute for silicon chips, which might operate 1000 times
faster. Now a group there led by Barbaros Özyilmaz is trying to use
graphene for storing information:

3) Prachi Patel, A step towards superfast carbon memory, Technology
Review, Wednesday April 1, 2009, available at
`http://www.technologyreview.com/computing/22377/`.

No, this is not an April Fool's joke.

Graphene was only discovered in 2004, but the easiest way to make some
is surprisingly lowtech. You press a chunk of graphite onto some scotch
tape, and hope a thin layer sticks:

4) Making graphene 101, Özyilmaz' group, YouTube video, available at
`http://www.youtube.com/watch?v=rphiCdR68TE`

Graphene has some amazing properties. For starters, it acts like a toy
universe containing massless spin-$1/2$ particles, which zip around at the
speed of light. But in this toy universe, light moves 300 times slower
than in the real universe!

To understand this, you have to start by realizing that any sort of wave
also acts like a particle when you take quantum mechanics into account.
The idea is that when something can wiggle back and forth at some
frequency, it can wiggle a little or a lot - but it can't wiggle just
*any* amount. The amount of wiggling is "quantized" - it comes in
discrete steps. These steps are too small to see in normal life, where
for example you might see a rubber band seem to vibrate with any
amplitude it wants. But these quantized steps are there nonetheless, no
matter what is doing the wiggling - it appears to be a completely
general principle.

When the wiggles are waves that are moving along, these quantized steps
are called "particles". So, for example, when we have a wave of light
of some particular frequency that contains 500 times more energy than
the bare minimum, we say it consists of 500 "particles" called
photons. At first this may sound weird, but it turns out that many
things we normally consider particles - like electrons and protons and
neutrons - really are just quantized wiggles of some sort or another.

In cases like these, the stuff that's doing the wiggling is rather
absract: for photons, we say it's the "electromagnetic field", and
for other particles we say it's various other fields.

But since the principle is completely general, there are also cases
where the stuff doing the wiggling is quite mundane. For example, if you
have a crystal, the crystal's atoms can wiggle. This is of course how
*sound* propagates. But this should mean that sound comes in quantized
packets called "phonons". And indeed, experiments with crystals show
that this is true!

5) Wikipedia, Phonon, `http://en.wikipedia.org/wiki/Phonon`

::: {align="center"}
[![](normal_modes.gif)](http://en.wikipedia.org/wiki/Phonon)
:::

Different vibrational modes of the crystal have different numbers of
wiggles per *distance*, and different numbers of wiggles per *time*;
these give phonons with different *momentum* and *energy*. The relation
is simple:

momentum = h (wiggles per distance)

energy = h (wiggles per time)

where h is Planck's constant, a tiny little constant.

(Why am I talking about sound in crystals, rather than air? It's
because our story applies in its simplest form to vibrational modes that
don't interact much with other modes or get damped out quickly by
friction. These give rise to particles that don't interact much with
other particles, and don't decay very fast - so they're easy to see.)

There are other things about a crystal that can vibrate besides the
atoms. For example, the atoms may have an angular momentum, or "spin",
that can change directions. Often each atom's spin affects its
neighbors through magnetic forces. This allows waves of changed spin -
"spin waves" - to propagate through the crystal. And again, these
waves come in quantized packets called "magnons":

6) Wikipedia, Spin wave, `http://en.wikipedia.org/wiki/Spin_wave`

A crystal can also have a "hole": an atom with a missing electron
These holes can move around, so they too act like particles. This
example is perhaps a bit more obvious than the previous ones... so here
I should probably emphasize that just as sound waves or spin waves come
in discrete particle-like units, these holes act like waves!

Similarly, a crystal can have atom with an *extra* electron. Here it's
even *more* obvious that these electrons act like particles - you
probably want to just say they *are* particles. But at this point it's
crucial to emphasize that this example is really like all the rest. In
particular (pardon the pun), we need to understand these extra electrons
in terms of waves, and we need to compute their energy and momentum
using these formulas:

momentum = h (wiggles per distance)

energy = h (wiggles per time)

so the relation between the energy and momentum depends completely on
the details of the crystal: you can't just use some formula you may
happen to know about a "free" electron in empty space. So, electrons
in crystals can do all sorts of crazy things that you'd never expect,
and indeed this is what makes things like conductors and transistors
possible.

Let's talk about relations between energy and momentum. For a particle
of mass m in empty space, special relativity says that

energy^2 = momentum^2 + mass^2

in units where the speed of light is 1. If we work in units where the
speed of light is some number c, the formula looks messier:

energy^2 = momentum^2 c^2 + mass^2 c^4

If you look at a particle at rest, its momentum is zero so a little
calculation reveals that:

energy = mass c^2

which is a pretty famous formula.

But if we go to the other extreme, and look at "massless" particle,
meaning one with mass = 0, we get

energy = momentum c

which is also famous, but less so. If we apply the formulas

momentum = h (wiggles per distance)

energy = h (wiggles per time)

we get

wiggles per time = c (wiggles per distance)

This fact is easy to believe if we imagine light as a wave moving along,
and c as the speed of this wave. Indeed, this fact was important in
helping guess the relation between momentum and "wiggles per
distance", and energy and "wiggles per time", back when they were
first inventing quantum mechanics.

Now, the above formulas apply to photons in empty space. But my real
point is this: if we have any sort of substance and any sort of wave
that can move around in this substance, obeying these formulas, it *acts
like a massless particle*. So, we can use a lot of the same math and
intuition that we use for photons. But, quite likely the value of c will
be different than the real-world speed of light!

And this in fact is what happens for graphene. The calculation is
pathetically simple by ordinary physics standards, yet complicated
enough that I don't feel like presenting the details here. You can see
it near the beginning of this wonderful paper:

7) Jiannis K. Pachos, Manifestations of topological effects in
graphene, to appear in Contemp. Phys.. Also available as
[`arXiv:0812.1116`](http://arxiv.org/abs/arXiv:0812.1116).

You start with a really simplified model: a planar hexagonal honeycomb
that can either have or lack an electron at each vertex. You ignore the
spin of the electron for some reason. It's a bit surprising you can get
away with that! You assume the energy is a bit less whenever you've got
electrons at both ends of any given edge of your hexagonal honeycomb.

It helps to color the vertices of the honeycomb alternately black and
white. Then there are two kinds of electrons: "black" and "white"
ones. The formula for the energy can be expressed nicely in terms of
these, because it's a sum over pairs of neighboring vertices, one black
and one white. Note that whenever an electron moves, it changes from
black to white or vice versa.

Then you do a little calculating: you do a Fourier transform here, you
diagonalize a 2\times2 matrix there, and... presto!

There turn out to be two different kinds of waves, each consisting of a
certain linear combination of black and white electrons. The energy of
these waves (or if you prefer, their number of wiggles per time) is
related to their momentum (or wiggles per distance) by a complicated
formula.

To write down this formula, we have to treat momentum as a *vector*,
since the energy of a wave depends on which direction it's going: the
hexagonal honeycomb means that not all direction behave the same. If you
plot the energy E as a function of the momentum vector p = (p~x~,p~y~),
you get a graph like this:

::: {align="center"}
![](graphene_dispersion_relation.png)
:::

It's pretty fancy! But you can see it has hexagonal symmetry, as you'd
expect. And it's also periodic. This comes from the periodicity of the
crystal lattice - but in sneaky way: the crystal lattice in actual space
gives rise to a "dual lattice" in the space of momentum vectors, and
the energy stays the same if you take the momentum and add any vector in
this dual lattice. This is no big deal, it happens for any crystal:

8) Wikipedia, Reciprocal lattice,
`http://en.wikipedia.org/wiki/Reciprocal_lattice`

In crystal jargon, the original crystal lattice is called the "Bravais
lattice" and its dual is called the "reciprocal lattice".

The big deal is that if you look carefully at certain points in the
graph, it looks almost like a cone.

::: {align="center"}
![](graphene_dispersion_relation_2.jpg)
:::

If we move the tip of this cone to the origin, the formula for it is:

E^2 = (p~x~^2 + p~y~^2) c^2

or in other words:

energy^2 = momentum^2 c^2

for some constant c. And this is just the relation we had for a massless
particle!

A digression: when I wrote this relation before, I took a square root
and got

energy = momentum c

But the above graph includes negative energies as well. What do the
negative energies mean, physically? That's an interesting puzzle, which
people had to confront when they were first trying to combine special
relativity and quantum mechanics. I don't want to talk about it here,
but if you're curious, see this:

9) M. I. Katnelson, K. S. Novoselov, A. K. Geim, Chiral tunnelling and
the Klein paradox in graphene, Nature Physics 2 (2006), 620.

Anyway, the upshot is that near one of these special points, graphene
acts like it has two kinds of massless particles in it: "black" and
"white". They are approximately described by the usual equation for
massless spin-$1/2$ particles, the Dirac equation - but in a universe with
just 2 dimensions of space! The speed c is about 1/300th of the
real-world speed of light. Also, a spin-$1/2$ particle in 2d space usually
comes in two states: it can rotate clockwise or counterclockwise. But
here those two states are certain linear combinations of "black" and
"white".

Furthermore, because the formula

energy^2 = momentum^2 c^2

remains unchanged when you rescale both distances and times by the same
factor, graphene reacts exactly the same way to light of different
wavelengths - at least within some range where all the approximations
we've made are good. And, it turns out to absorb \pi α of the light you
shine through it:

10) A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel,
Universal infrared conductance of graphite, Phys. Rev. Lett. 100 (2008),
117401.

11) R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J.
Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, A. K., Fine structure
constant defines visual transparency of graphene, Science 320 (2008),
1308. Also available at
`http://onnes.ph.man.ac.uk/nano/Publications/Science_2008fsc.pdf`

So, graphene is an amazing physical system, but so far I've just
scratched the surface. Graphene also has "topological defects", and
"anyons", and there are wonderful applications of Euler's theorem and
its big brother, the Atiyah-Singer index theorem. All this in a physical
system you can make by pressing graphite against a piece of tape!

I should say more about all this, but I need to go to bed and then catch
an early morning plane. So, I'll quit here - but I highly recomend
Pachos' paper "Manifestations of topological effects in graphene".

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Addenda: For more discussion visit the [$n$-Category Café](http://golem.ph.utexas.edu/category/2009/08/this_weeks_finds_in_mathematic_38.html.html).

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*Error and doubt no longer encumber us with mist;\
We are now admitted to the banquets of the Gods;\
We may deal with laws of heaven above; and we now have\
The secret keys to unlock the obscure earth.* - Halley