Drunk Euclid still rules

On random metrics and the KPZ universality

Straight lines and circles, polygons and exact constructions. That is the clear and shiny paradise of Euclid and classical geometry. The paradigm of perfect connection between the mind and the reality. No need for awkward limiting processes, just unrefutable syllogisms. A cathedral of mathematics and, at the same time, the most successful
physical theory ever designed, since it is validated by the experiments millions of times a day. Yet, what happens when Euclid does not goes easy on ouzo? Can we still play such good geometry? Surprisingly, the answer is yes.

When Euclid gets drunk, instead of the classical plane, we get a wiggly surface. Assume that you crumple your sheet of paper on which you have drawn some circles and straight lines. They will now look crumpled to you. But, I hear you say, there is nothing you can say about them any more, since you don’t know how the paper was crumpled!

OK, let us start easier. Let us assume that you have a surface or, for math jedis, a two-dimensional riemannian manifold. The key concept is that of distance (also called metric). Let us assume that you put some smart ants on the surface which can measure distances between pairs of points, and these distances are smooth and blah-blah. The role which circles used to play on the plane belongs now to balls which are the set of points which ants can reach from a fixed one in a given time. Balls can have all kinds of shapes, as you can see in the figure below.

What about straight lines? They are now called geodesics, which are (ahem-ahem) the curves of shortest length between two points. Balls and geodesics fulfill some nice geometric relations, such as orthogonality (i.e. perpendicularity) according to the metric. The next figure shows the set of geodesics emanating from a point, along with the balls surrounding it, for a surface with the shape of an eggcrate.

So, what happens if we crumple the Euclidean plane, randomly (angrily, if needed)? Then the metric on the manifold becomes random. We will assume that wrinkles at one point of the paper sheet are independent of wrinkles at points far away. In formal terms, we say that the metric only has local correlations. Let us now draw balls around a given point.

Now the balls are rather rough. That is reasonable. Rough, but not crazily rough, just a bit. In fact, they look like balls drawn by a slightly drunk mathematician. Moreover, they have a certain appeal to it. Maybe they are fractal curves? Well, they are. And their dimension
is d=3/2, independently of how you crumple exactly your manifold, as long as you only study balls which are much larger than the distance above which wrinkles are correlated.

If you define the roughness of a ball as the width of the ring which can contain it, then you get a pretty nice result: . Again a very nice and simple exponent, which shows that drunk geometry hides many interesting secrets. Also the geodesics have fractal behavior, if you are wondering. But we have just scratched the surface of this ethylic paradise. Suffice it to say that we have found the pervasive (and mysterious) Tracy-Widom distribution when we measured the fluctuations on the local radius of the balls.

Earlier this year we published those results, along with S.N. Santalla, T. LaGatta and R. Cuerno. Don’t miss the paper, which has very nice pictures, and and even nicer introduction video, in which I crumple actual paper in front of the camera!

How come we get such nice results, independently of how we crumple the manifold? The magic lies in the notion of universality. Imagine that you have some theory defined by its small-scale behavior. When you look from very far away, some details are forgotten and some get even more relevant. This procedure of looking from further and further away is called renormalization. Typically, only a few details are relevant when you are really far from the smallest scale. And those few details make up the universality class.

So, balls in random manifolds share universality class with other phenomena… from physics! (Well, if you know of a clear boundary between physics and math, let me know.) Growth of amorphous crystals from a vapour, forest fires, cell colonies… Basically, growth phenomena in noisy conditions are known to belong to the Kardar-Parisi-Zhang (KPZ) class. They share the fractal dimension of the interfaces and the Tracy-Widom distribution for the fluctuations.

Why is Nature so kind to us? Why does she agree to forget most of the mumbo-jumbo from the small scale details, so as we can do physics and math on a large scale that makes any sense? Somehow this connects with one of the deepest questions in science, as posed by Eugene Wigner: “why is mathematics so effective in describing Nature?”

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Qubism

Scientists tend to be very visual people. We love to understand through pictures. About one year ago, we had one of those ideas which remind you why it’s so fun to be a theoretical physicist… Simple and deep. The idea was about how to represent quantum many-body wavefunctions in pictures. Speaking very coarsely, the high complexity of the wavefunction maps into fractality of the final image.

So, more slowly. As you know, bit can take only two values: 0 and 1. A qubit is a quantum bit, which can be in any linear combination of 0 and 1, like Schrödinger’s cat, which we denote by and . In other terms: a qubit is represented by two complex numbers: . If you have two qubits, the basic states are four: 00, 01, 10 and 11, so we get

If you add one qubit, the number of parameters doubles. For N qubits, you need parameters in order to specify completely the state! The task of representing those values in a picture in a meaningful way seems hopeless… Our idea is to start with a square and divide it in four quadrants. Each quadrant will be filled with a color associated with the corresponding parameter.

What if we get a second pair of qubits? Then we move to “level-2”: we split each quadrant into four parts, again, and label them according to the values of the new qubits. We can go as deeply as we want. The thermodynamical limit corresponds to the continuum limit.

The full description of the algorithm is in this paper from arXiv, and we have launched a webpage to publish the source code to generate the qubistic images. So, the rest of this blog entry will be just a collection of pictures with some random comments…

This is the ground state of the Heisenberg hamiltonian for qubits. It is an antiferromagnetic system, which favours neighbouring qubits to be opposite (0-1 or 1-0). The main diagonal structures are linked to what we call a spin liquid.

These four pics correspond to the so-called half-filling Dicke states: systems in which half the qubits are 0 and the other half 1… but you do not know which are which! The four pics show the sequence as you increase the number of qubits: 8, 10, 12 and 14.

This one is the AKLT state for N=10 qu-trits (each can be in three states: -1, 0 or 1). It has some nice hidden order, known as the Haldane phase. The order shows itself quite nicely in its self-similarity.

This one is the Ising model in a transverse field undergoing a quantum phase transition… but the careful reader must have realized that it is not fitting in a square any more! Indeed, it is plotted using a different technique, mapping into triangles. Cute, ein?

But I have not mentioned its most amazing properties. The mysterious quantum entanglement can be visualized from the figures. This property of quantum systems is a strong form of correlation, much stronger than any classical system might achieve.

So, if you want to learn more, browse the paper or visit this webpage, although it is still under construction…

With warm acknowledgments to my coauthors: Piotr Midgał, Maciej Lewenstein (ICFO), Miguel I. Berganza and Germán Sierra (IFT), and also to Silvia N. Santalla and Daniel Peralta.

Rough is beautiful (sometimes)

No posts for three weeks… you know, we’ve been revolting in Spain, and there are times in which one has to care for politics. But physics is a jealous lover… :)

So, we have published a paper on kinetic roughening. What does it mean? OK, imagine that, while your mind is roaming through some the intricacies of a physics problem, the corner of your napkin falls into your coffee cup. You see how the liquid climbs up, and the interface which separates the dry and wet parts of the napkin becomes rough. Other examples: surface gowth, biological growth (also tumors), ice growing on your window, a forest fire propagating… Rough interfaces appear in many different contexts.

We have developed a model for those phenomena, and simulated it on a computer. Basically, the interface at any point is a curve. It grows always in the normal direction, and the growth rate is random. The growth, also, is faster in the concavities, and slower in the convex regions. After a while, the interfaces develop fractal morphology. I will show you a couple of videos, one in which the interface starts out flat, and another one in which it starts as a circle. The first looks more like the flames of hell, the second more like a tumor.

The fractal properties of those interfaces are very interesting… but also a bit hard to explain, so I promise to come back to them in a (near) future.

The work has been done with Silvia Santalla and Rodolfo Cuerno, from Universidad Carlos III de Madrid. Silvia has presented it at FisEs’11, in Barcelona, a couple of hours ago, so I got permission at last to upload the videos… ;) The paper is published in JSTAT and the ArXiv (free to read).