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."

Figure 4. Applying a magnetic field (B) profoundly affects the configuration of microscopic bar magnets. Because bar magnets favor alignment along the direction of a magnetic field, the region in which the bar magnets rotate around a vortex or antivortex becomes confined to a string. This string turns out to enhance the binding of vortex-antivortex pairs. It was commonly assumed that the enhanced binding would stop these pairs from separating, no matter how hot the system might become. In the figure, + indicates a vortex and * an antivortex.

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.

Figure 5 and Figure 6. An actual snapshot of a simulation for a clean system. + indicates a vortex and * an antivortex. Most of these are bound tightly into pairs. Occasionally, however, one can see a relatively-separated pair (in the box, inset, Fig. 6).

The bar magnets are shown as lines with a dot indicating the south pole of the magnet. The magnets are color coded to indicate how much they deviate from the direction favored by the applied magnetic field. Red magnets point nearly antiparallel to the direction favored by the applied field. The region of red connecting the vortex-antivortex pair is the string of overturned spins tethering them together.

The simulation demonstrates that when the system is heated up, the fluctuations of the string--its entropy--can become great enough to effectively allow pairs of vortices to unbind. The nature of this phase transition is unique in that it does not lead to singularities of the type seen in other phase transitions.

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.

This research is supported by the National Science Foundation.

Team Members
Herb Fertig
Joe Straley