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VORTICES UNBOUND
by Kathleen Ricker, NCSA Research Editor

By simulating the movements of microscopic bar magnets, a condensed matter physicist at the University of Kentucky is shedding some new light on the nature of a fundamental physical process.

The phrase "loss of order" brings to mind images of a momentous nature. Floods uproot enormous trees. Tornadoes lay waste to entire towns. Fires spectacularly blacken hundreds of acres of dry forest. Disorder goes hand-in- hand with enormous, catastrophic change.

Similarly, it has always been assumed that, on a microscopic level, dramatic change goes hand-in-hand with the disordering of matter--what condensed matter physicists call a "phase transition." During a phase transition, matter, in changing from one state into another, loses its structure. Take, for example, the case of water: after it reaches a temperature of 210 degrees Fahrenheit, it can only be converted to steam, regardless of how high the temperature is raised.

But when and where does this disordering start? Previously, it was believed that a phase transition was always marked by dramatic behavior, such as the transformation of water into steam. In the language of phase transitions, these are called "singularities," and the standard theory assumes that a singularity always marks a change in phase. But Herb Fertig, a condensed matter physicist at the University of Kentucky, argues that researchers may have been, in a sense, looking for a sign where there is not always one to be found.

Using the University of Kentucky's Hewlett-Packard Superdome cluster, an Alliance supercomputing resource, Fertig studies the behavior of vortices in thin film ferromagnets: tiny whirlpools within a layer of magnetic material so flat that it is nearly two-dimensional. These systems are very analogous to many other systems--including superfluids, superconducting materials, and thin crystalline solids, all of which have measurable properties determined by the state of the vortices or similar objects. Thus, the study provides an interesting window into how many other systems work.

The vortices are minuscule. Contrary to the conventional wisdom of condensed matter physics, however, their phase transitions may be nearly imperceptible.

Magnets within magnets

Imagine a magnet so thin that it is very nearly two-dimensional. This magnet actually consists of trillions of atoms, which themselves behave like tiny bar magnets. For certain atoms--iron, nickel, or cobalt, for example--these bar magnets favor an ordered state at absolute zero (273 degrees Celsius), in which all the magnets are aligned. In two dimensions, when the temperature is above absolute zero but not too high, there is always some disordering, although not very strong. Condensed matter physicists say such states possess "quasi-long range order". The system only becomes truly disordered when vortices make their presence felt.

In most thin-film magnets, vortices come in two varieties. A vortex consists of a magnet rotating in a horizontal plane around a fixed core in a clockwise direction. By contrast, an antivortex consists of a magnet rotating in a counterclockwise direction. The two objects are, in a way, like bar magnets themselves, oriented perpendicular to the plane of the real magnet. When the north pole of such an imaginary magnet is above the plane, there is a vortex; when it is below the plane there is an antivortex. Like simple magnets, vortex "magnets" that are all pointed in the same direction tend to repel one another, to move as far from one another as possible. Thus,a field full of them will take on a lattice pattern.

However, like two simple magnets positioned so that north and south poles are side-by-side, vortices that point in different directions will pair up, or bind. This is where things get really interesting, because the behavior of vortices is strikingly similar to that of elementary particles. When the temperature is high enough, the vortices are forced apart, or deconfined, despite their natural tendency to bind. It is only then that the thin-film magnet becomes truly disordered. This process was discovered in the early 1970s by a pair of researchers named Kosterlitz and Thouless, and it is accompanied by a weak but definite singularity--evidence of a phase transition.

But what happens when a magnetic (or "symmetry-breaking") field is turned on? "Its effect is to try to make the little bar magnets line up along a particular direction," says Fertig. As the system attempts to force all the magnets to point in this direction, it conflicts with the natural tendency of the bar magnets in the film to rotate around the centers of individual vortices. "They have to rotate in a circle," he explains, "but because of the extra magnetic field, the system doesn't want to do that, and the way it compromises is by confining the rotation to as small an area as possible." The result is what Fertig calls a string-a line in the film across which the bar magnets rotate in spite of the force exerted by the magnetic field. The strings connect vortices to antivortices and enhance their binding. And because of this, researchers realized that the Kosterlitz-Thouless mechanism for vortex unbinding would not work in a symmetry-breaking field.

Elemental changes

If vortex pair unbinding in a magnetic field is a phase transition in the usual sense, it should be accompanied by a singularity--quantities with peaks or cusps when plotted as a function of temperature or magnetic field. "People were looking for those singularities," says Fertig. But they were never found, and strong arguments were developed that said they could not be present. But if unbinding were always accompanied by singularities, this would mean vortices would always be paired. "No matter how hot you make the magnet, you can't make the vortices get to the completely disordered state. This seemed wrong to me."

So Fertig created a physical model in which he could simulate the conditions that would lead to the phase transition, the state at which the vortices would unbind. "I found that this unbinding occurs, but that it has a very different character from [previous studies]-[it's] very subtle," he explains. "The vortices do unbind, but you don't see singularities."

Fertig realized that if this was true--if there really was a previously unknown unbound vortex state--that it could have important implications not only for the study of magnetic vortices but for that of many other phenomena that undergo phase transitions. "It opens up the question of what is meant by a phase transition," says Fertig, "if you don't have to go through a singularity."

If no singularity occurs, the only way to understand what is happening is to look closely at the vortices themselves--a difficult task in the laboratory, where the particles are on the order of nanometers and can be seen only by scanning tunneling microscopes. So, to test his hypothesis, Fertig has created a simulation in which a lattice of bar magnets is subjected to a magnetic field. The simulation allows him to easily track the vortices. "As the simulation goes on, the configuration of the magnets is changing, and at any moment I can...identify where the vortices are."

Fertig has developed a measure of how far apart the vortices are at any moment in the simulation. "The idea is that because of the way the simulation is constructed, there's actually a maximum distance that the vortices can be apart. We run the simulation and we ask how many times during the course of the simulation we see vortices [separated by] the maximum possible distance." Keeping track of the fraction of configurations with this maximum separation allows Fertig to identify when a phase transition might be taking place. If the fraction decreases to zero as the system size is increased, the vortices are in a bound state. However, if the fraction reaches a finite number, he can identify an unbound state.

In his simulation on the Superdome at the University of Kentucky Fertig is able to show three phases. With a low-intensity magnetic field and a reasonably high temperature, the vortices can be unbound. He has also identified two phases in which a high-intensity magnetic field causes the vortices to bind together in two different ways.

As a result, Fertig is getting closer to identifying the boundaries between the three phases. "We have two 'knobs' that we can turn," he says, magnetic field and temperature. "If you make a graph of temperature on one axis and magnetic field on the other, there will be a line that separates the unbound vortex phase from the bound vortex phases, and we would like to know where that line is."

Fertig is currently conducting these same simulations on a much larger scale, trying to map out the phase boundary. To achieve this, and to confirm the correlation between the formation of strings and the unbinding of vortices, Fertig predicts that he will use 120,000 hours of computing time.

It requires a lot of computing power to study something so infinitesimal that it went unnoticed for twenty-five years.