What is a topological insulator?

What is a topo­log­ical insu­lator? This is what I have written at the top of my notes from a con­ver­sa­tion a couple weeks ago with the­o­ret­ical con­densed matter physics pro­fessor Arun Bansil, who has just pub­lished two new arti­cles about topo­log­ical insu­la­tors in Nature Mag­a­zine. But it took about three pages of fran­ti­cally scrib­bled notes before the answer finally began to surface.

I will spare you the wind up of our con­ver­sa­tion, which mostly had to do with the fact that I haven’t taken a physics class in over a decade, and get right to it:

A topo­log­ical insu­lator is a mate­rial whose sur­face is con­duc­tive but whose inside is not. While that may sound pretty straight­for­ward, guess again. There’s no obvious reason why a mate­rial of uni­form struc­ture should dis­play such dif­ferent prop­er­ties in dif­ferent parts. “It’s all about topology and one of the most basic sym­me­tries of nature—the time reversal sym­metry,” said Bansil.

These two key con­cepts – topology and time reversal sym­metry – were totally new to me. In fact, I had no idea there were even mul­tiple kinds of sym­me­tries at all. I’m familiar with mirror sym­metry, as you prob­ably are, too. But time reversal sym­metry is a bit more mind-​​blowing. It refers to the basic prop­erty of phys­ical laws which guar­an­tees that a ball going up fol­lows the same laws as a ball going down. If you record a video of a ball going up and down con­tin­u­ously, you can’t tell if the movie is in play mode or rewind.

Topology at first seems com­pletely unre­lated to sym­metry, but appar­ently it is inti­mately con­nected. Basi­cally, topology describes the unique prop­er­ties of an object’s sur­face and fol­lows the rule that cer­tain types of topolo­gies cannot be exchanged with others. Com­pare, for example, a soccer ball with a doughnut. Without punching a hole in the ball, there’s no way to turn it into a doughnut. There is some­thing fun­da­men­tally dif­ferent about the nature of these two objects. But with a bit of imag­i­na­tion, you can see how a han­dled cup might flu­idly turn into a doughnut.

In topo­log­ical insu­la­tors, the inter­face between the con­duc­tive sur­face and the non-​​conducting bulk mate­rial behaves as if a soccer ball were spon­ta­neously trans­forming into a doughnut, so to speak, and this gives birth to the new types of con­duc­tive states on the surface.

In a reg­ular old insu­lator (or most mate­rials for that matter), elec­trons are bop­ping back and forth between one “spin” direc­tion and another. But in a topo­log­ical insu­lator, time reversal sym­metry pro­tects the spin of each state and the elec­trons cannot hop between dif­ferent spins.

Bansil’s team’s first Nature article explores the fact that mag­netic impu­ri­ties in the mate­rial will break the time reversal sym­metry and cause spins to flip. Strate­gi­cally placed impu­ri­ties could turn topo­log­ical insu­la­tors into very resilient on-​​off switches. In quantum com­puting, these mate­rials could one day replace the zeros and ones of tra­di­tional computers.

Another appli­ca­tion for these weird babies is spin­tronics, an emerging tech­nology that exploits elec­tron spins in elec­tronic devices rather than the charge of the elec­tron, which is the case in much of the cur­rent elec­tronic tech­nology. In normal mate­rials, the number of up and down spins is roughly equal. In topo­log­ical insu­la­tors there is a nat­ural ten­dency for the spins to be sep­a­rated and to be pro­tected from flip­ping by the time reversal sym­metry, allowing the pos­si­bility of novel spin­tronics applications.

The second Nature article explores how in some mate­rials con­ducting states on the sur­face could be pro­tected via mirror sym­metry of the system rather than time reversal sym­metry, and thus lead to  yet another new kind of insu­lator in which the sur­face states could be more easily manip­u­lated for var­ious appli­ca­tions. Bansil’s team pre­dicts tin tel­luride to be one such new type of insulator.

For more arti­cles about Bansil’s work with topo­log­ical insu­la­tors, see Sci­ence, Nature Physics, and Phys­ical Review Let­ters.