University of Minnesota
School of Physics & Astronomy


Topological Edge States

Tobias Gulden
Tobias Gulden

Researchers at the University of Minnesota have closely examined the evolution of topological edge states, a phenomenon that occurs in materials physics. Tobias Gulden, a graduate student working with Professor Alexander Kamenev of the Fine Theoretical Physics Institute, studies the problem of topological phase transitions to determine how the transition from bulk to edge state occurs.

A topological state appears at the edge of a system (see graphic) and is disconnected from the bulk states by topological protection, which provides it with the unique property of zero energy. Gulden says this phenomenon has been a hot topic in physics since 2007 after topological edge states were predicted by condensed matter theorists. He says although the first examples were observed in materials physics as early as the 1980s, people did not link the observations to topological protection until much later. Topological edge states play an important role in spintronics, a branch of materials physics which hopes to use the spin of an electron to store information, and quantum computation where the qubit, the quantum analog of the classical bit, may be realized with such states. Understanding the detailed physics of topological edge states could potentially help physicists create more efficient devices.
Gulden says that topological phase transitions have typically been studied with conformal field theory which allows physicists to study properties at the exact point at which the transition occurs. Another approach commonly used has been to study systems far away from the transition, where edge states are either present or absent. This leads to a skewed view of the phenomenon because these states cannot just appear or disappear out of nowhere. Gulden says the received wisdom was that the topological edge state is a zero energy state and is localized at the edge of the system. “Strictly speaking this holds only for an infinite sized system. But everything in nature is finite so every transition has to be smooth, so we asked ‘how did this evolve from a bulk state to an edge state?’” Gulden used a technique called scaling, frequently used for phase transitions, to calculate what was happening close to the transition. He was able to show the evolution occurs smooth and gradual, and he can predict from properties at the transition whether an edge state will appear.
“Our next question was: there are several different classes of topological states, do the transitions follow similar behavior? And we found that the behavior is not only similar, but exactly the same,” Gulden adds. It was known that the transition point has universal properties, “but it’s quite remarkable that this universality also applies to the gradual evolution of a topological state, qualitatively and quantitatively.”
The motivation behind this work for Gulden was to show that for any finite size system, every state is a continuous function of the physical parameters. “It’s just mathematical theory and that means it’s impossible to have the zero energy state suddenly appear. There must be something happening in between. How does that state actually evolve to become the edge state? This was an unknown piece of the puzzle. People didn’t really look at it.”