In 2016, three physicists, David Thouless, Duncan Haldane, and Mike Kosterlitz, won the Nobel prize in physics for their groundbreaking discoveries of topology in matter/materials. Although many people have heard of topology before, likely associating it with the more familiar term “topography,” the fundamental nature of topology remains relatively opaque to those outside of the fields of math and certain applied sciences. **So what exactly is topology, and in what sense can we talk about topology existing in materials?**

Mathematically speaking, topology is the study of the properties of an object that are generally unaffected by certain kinds of shape change or continuous deformation, such as twisting, pressing, or stretching. The emphasis on continuous deformation excludes changes that involve tearing or “gluing” parts together. A classic example of two topologically equivalent objects is that of a donut and a coffee mug. The hole in the donut becomes the handle for the coffee cup, and the remaining changes are made via a series of hypothetical deformations.

On the other hand, a donut and a baseball are topologically different for the simple reason that a donut has a hole and a baseball does not. One can imagine what it might look like to make the transition from baseball to donut; a hole must first be made through the baseball, tearing apart its internal structure and thus constituting what can be called a strictly “discontinuous” deformation. In the language of topology, we say that the donut and the coffee mug have the same topological index (number of holes, in this case), while the donut and the baseball do not and are, thus, topologically distinct. By acknowledging that two objects are topologically equivalent, we can say that there is a certain property shared by both that is in some sense immutable, as long as no discontinuous deformations are made. In this example, the shared property that remains unchanged is simply the number of holes; there is no “higher-level” physical or electrical property, for example, shared by the mug and donut; they just both possess one hole. It is now understood, however, that in some cases topologically equivalent objects may share more properties than previously thought, and that the once purely mathematical world of topology now has wide-ranging relevance for materials.

Indeed, for some solid-state materials, certain physical properties are *defined* by their topological indices (or topological invariants) instead of their fine details. These are known as topological materials. A famous example includes so-called topological insulators (TIs), materials that are electrically insulating in bulk and resistant to the flow of electrons, but have metallic conducting boundaries no matter their geometric shape. This permits the transmission of an electric current, originating from its topological invariants. A useful heuristic for understanding this mechanism comes from representing a trivial insulator (any non-topological insulating substance, such as air) and a TI as a baseball and a donut respectively, continuing the example from above. In solid-state materials, like TIs and trivial insulators, electrical properties are described by something called electronic bands; basically, the energies of the electrons as a function of their momenta. Both the trivial insulators and the interior of the TIs have an energy gap in their electronic bands, leading to an insulating behavior by prohibiting the transmission of conduction electrons. When the energy gap closes, conduction electrons are able to travel freely, and the material behaves like a conductor. The gap, however, has opposite signs in the two insulators — positive signs in the trivial insulators, and negative signs in the bulk of the TIs. When moving from a trivial insulator (i.e., positive gap) to a TI (i.e., negative gap), the energy gap must vanish at their interface: the surface of the TI. (This transition from a positive to a negative energy gap requires a discontinuous deformation.) As a result, the surface of the TI is conductive, even though its interior is insulating. Thus, the energy gap closing at the surface of the topological insulator is analogous to the tearing process for the case of transforming from a baseball to a donut. The negative energy gap necessitating the surface states of topological insulators is one example of how a material’s topological invariants define its properties. As long as the TI is deformed “continuously,” the material topology guarantees that it will retain its conductive surface states, a kind of “property insurance.”

Because of this topological protection, the electronic properties of topological materials are often much more robust than similar non-topological alternatives which are affected by small perturbations (e.g., disorder, defects, or impurities). This makes TI materials potentially useful in several electronic and technological applications. In a TI, for example, electrons traveling on the two-dimensional surface will experience less resistance than in a similarly-sized trivial insulator, which is one of many reasons why TIs often come up in discussions of future electronic devices and integrated circuits. More generally, topological materials, their properties often being impervious to outside interference, make for promising candidates in quantum computer storage. In classical computing, a bit is used to represent a basic unit of information stored within the computer. Qubits, the quantum computing analog of classical bits, retain information in quantum states which are extremely sensitive to their environment, leading to a loss of stored information when disturbed. Topological materials have the potential to reduce the impact of environmental degradation. In addition, topological materials are known for their especially strong spin-orbit coupling, a property currently taken advantage of in computer information storage that could be made to be even more efficient with the implementation of these kinds of materials.

Topological materials are strangely fascinating, exceedingly useful, and not altogether too uncommon. You probably have yet to notice them, but topological materials are poised to continue their integration within the technology and electronics of the future.

**Acknowledgments:
**The author would like to thank Drs. Zhang and Fertig for their suggestions and guidance and to acknowledge the National Science Foundation for funding support.

**References:
**[1]https://physicsworld.com/a/what-are-the-potential-applications-of-topological-insulators/

[2]https://www.jyi.org/2019-july/2019/7/1/topological-insulators-and-applications-to-quantum-computing

[3]https://www.nanowerk.com/nanotechnology-news2/newsid=50931.php

Edited by Chloe Holden and Evan Leake