## Physics in a nut shell

Physics in a nut shell provides intuitive pictures about various physics ideas for physics students and non-experts. You are welcomed to suggestion topics that you would like to hear about.#### Topological states of matter

**What is topology and what are topological states of matter?**

**[The figure on the left shows a cylinder, while the right one is a Möbius stripe. These two objects has different topology. One way to see that is to realize that a cylinder has two sides, interior and exterior, while the Möbius stripe only has one. (Click the images for larger figures)]**

The idea of topology comes geometry. For example, if we think about a 2D membrane made by rubber, we can stretch it and change its shape, known as "adiabatic deformations". However, we also know that not all membranes can be turned into each other by stretching. One example is offered in the figures shown above, which show a cylinder and a Möbius stripe. These two objects can never be deformed to each other unless we cut the membrane and then glue it back, and cutting is not allowed for adiabatic deformations. If two geometric objects cannot be turned into each other via adiabatic deformations, we say that they have different topology. This line of thinking enables us to classify different geometric objects based on their topology, i.e., whether or not they can be deformed into one another adiabatically.

In the quantum world, quantum states are described by quantum wavefunctions, which has some strong analogy to membranes discussed above. For example, for an electron moving in two dimensions, its wavefunction can be written as Ψ(x,y), where Ψ is a complex number and the square of its absolute value |Ψ|^{2} is the probability of finding this electron at the coordinate (x,y). For a 2D membrane, we can describe it using a similar 2D function z(x,y), which z tells us that if we draw a vertical line in our 3D space by fixing the x and y coordinates, this vertical line intersects with the membrane at the height z. By comparing the function z(x,y) and the quantum wavefunction Ψ(x,y), both of which are 2D functions, the analogy between geometry and quantum wavefunctions is transparent. In other words, we can think of a wavefunction as a geometric object defined in an abstract space, and therefore it is not unexpected that quantum wavefunctions may also have different topology similar to membranes, and this picture offers us a method to distinguish different quantum states of matter based on the topology of their ground state wavefunctions. Of course, it must also be emphasized that because this abstract space, in which a quantum wavefunction lives, is fundamentally different from a 3D Euclidian space, in which a 2D membrane discussed above is embedded, the analogy above is not expected to survive at the quantitative level.

**What are the physics consequences of a nontrivial topology?**

In physics, a wavefunction with nontrivial topology often (but not always) results in interesting behaviors on the surface/edge of the system. For example, in a topological insulator, although the bulk of the material is an insulator, the surface is a conductor. This is exactly the opposite to electric wires used in our home, whose interior is a conductor (i.e., the metal wire) but the outside is an insulator (the plastic/rubber cover).

This conducting surface is a direct consequence of the nontrivial topology. When we experimentally study an insulating sample, inside our sample the system is an insulator, and outside the sample, the air or vacuum (depending on the experimental environment) is also an insulator. So we are in fact dealing with two insulators: the sample and the environment. If the quantum wavefunction of the sample shares the same topology as the environment, we call it a conventional insulator. Most of insulators that we know belong to this family, e.g. glass, rubber, plastic, the air and vacuum. In addition to these conventional insulators, there also exist some other insulators whose quantum wavefunctions are topologically different from the environment (say vacuum). If this is the case, we call the material a topological insulator. A topological insulator typically has a conducting surface (although not always). One way to understand the existence of such a conducting surface is to consider a path with one end inside the sample and the other end outside. Because the topology of the quantum wavefunction inside is different from outside, somewhere along this path (usually at the surface of the sample), the quantum wavefunction shall change its topology. However, as shown above, topology is not something that can be changed adiabatically. For example, we cannot deform a cylinder into a Möbius stripe. The only way to achieve this objective is to destroy the cylinder first (cut it) and then glue it back into a Möbius stripe. For insulators, if we want to change the topology of the quantum wavefunction, the same needs to be done. We have to destroy the insulator first, just as we cut the cylinder. What do we mean by destroying an insulator? Well, we know that if something is not an insulator, it is a probably a conductor. In other words, when we destroy a insulator, we shall expect an conductor. Coming back to a topological insulator, when we move from inside the sample to outside, the topology changes at the interface. We also know that when topology is changed, the insulator needs to be destroyed, which results in a conductor. If we combine all the information together, it tells us that at the surface of the sample, where topology changes, a conductive layer shall arise.

The conducting surface of a topological insulator often (but not always) results in certain quantization effect. The first example of this type is the quantum Hall effect (Nobel Prize in physics for 1985 and 1998). There, the quantization allows us to measure a very important physics constant, the fine structure constant, with extremely high accuracy (error bar in the order of one in a billion). The quantum Hall effect is in fact the second most accurate way to measure the fine structure constant at this moment. The most accurate one is from particle physics by measuring the anomalous magnetic dipole moment(known as g-2). That measurement relies on theoretical input from quantum electrodynamics (known as QED). Using the quantum Hall effect, one can measure the same quantity independent of the theory of quantum electrodynamics. The fact that the g-2 experiment and the quantum Hall effect give the same fine structure constant (within error bar) is a direct verification of our quantum theory of electrodynamics.

Thanks to topology. By working on an insulator, we can prove one of the most important theory in high-energy physics.