Topological Matter in Optical Lattices

Illustration by Kamran Samimi This month’s Learnin’ Corner is an explanation of topological matter in optical lattices by University of Pittsburgh associate professor of physics W. Vincent Liu, whose paper “Topological Matter in Optical Lattices” first interested us in the subject of topological matter in optical lattices. Cold atomic gases are a new form of quantum matter, which is currently at the forefront of many-body physics research. The main step in creating this form is laser cooling. Think about the laser as a bunch of photons—the photons are tiny particles, and the atom is a giant particle. It’s like there’s a big ship in the sea, and in the beginning the ship is moving fast, but if it’s surrounded by millions of particles, which are like fish, those fish will take away the kinetic energy and slow down the motion of this big ship. The second step in this process is cooling by evaporation. What you do is you have a kind of cup that can hold the atoms inside so the colder atoms cannot fly away, and the hotter particles will escape from the cup, and eventually the system will cool down. And this evaporation will further cool down the particles to the nano-Kelvin scale, which is the lowest temperature human beings have achieved so far. So why do the atoms remain as a vapor or a gas instead of a solid? The atoms remain a gas because the density is so low. The density actually is lower than the density of the air that is surrounding us. Now, the term “semi-metal” has been used to refer to several different things. We used the term “topological semi-metal” to refer to a novel state, which is neither a metal nor an insulator, and this state is topologically protected. A topological insulator has to have something that sets its atoms rotating in a single direction. Normally in a quantum Hall state of electrons, you use a magnetic field to create this effect, but here we put particles into a double-well super-lattice and the interaction between them forces them to line up, in a sense, the whole system develops a spontaneous rotation. All the particles are rotating, and this orientation lowers the total energy. This is known in physics as spontaneous symmetry breaking. It plays the role of a magnetic field. It’s remarkable to see. This is a very special material whose bulk is an insulator (no currents can flow inside the system—for electron systems, it is electric current), but at the same time, its edge is metallic, along which currents can flow. For example, if we consider a two-dimensional electron-based topological insulator (which is a thin sheet), electronic currents cannot flow anywhere inside this sheet. Nevertheless, the edge of this sheet is like a wire, and currents can flow along the edge. In many cases, the edge current can only flow in one direction, known as chiral current, much like cars flowing on a one-way road or on one lane of the interstate highway. Another interesting, and quite unexpected, property of topological insulators lies in the fact that the conductivity of the whole system is very sensitive to the shape (topology). For an ordinary insulator/metal, it is always an insulator/metal no matter what shape we made it into. However, for a topological insulator, if we make the sheet into a sphere, because the sphere has no edges, the system will turn into a true insulator, where currents cannot flow anywhere. But if we make the sheet into a disk, which has an edge, the system can conduct electronic currents due to the metallic edge. The particle flow in one direction is a really remarkable phenomenon. Usually, you have particles in a system that move randomly in both directions—think about a one-dimensional wire—but on the edge of our material, everything moves in the same direction and there are no collisions between particles or random jiggling around back and forth. So this kind of system has very different properties from ordinary wires. It could have huge implications, this phenomenon.