In his 18th century research on the fundamental nature of electricity, Benjamin Franklin found that there are two kinds of materials: conductors, in which electric charges flow freely, and insulators, in which charges are immobile. This discovery was of more than just academic interest: it enabled one of the most important inventions of his day: the lightning rod. Fast forward 250 years, and electronic materials like semiconductors, whose electrical conduction can be exquisitely controlled, now form the basis for the technologies that fuel today’s information age.

These capabilities were made possible by the 20th century revolution in our understanding of the structure of matter in terms of quantum mechanics. Insulators insulate through the concept of an energy gap—the barrier an electron must overcome to be dislodged. In a conductor, this energy gap is zero, which allows electrons to move freely. The foundations of the quantum theory of solids were established almost 90 years ago. Surprisingly, this theory contained a hidden treasure, whose significance was not originally fully appreciated. Insulators come in distinct varieties that can be characterized using the mathematical principles of topology. This new understanding has led to the development of new classes of 21st century electronic materials.

The presence of a nonzero energy gap enables a form of theoretical alchemy. In the mind’s eye, one can imagine smoothly changing composition or atomic arrangement to transform one material into another. In this process, the properties of the material will also smoothly change, but as long as the energy gap remains nonzero it will remain an insulator at each step along the way. This notion of smooth transformation introduces the mathematical concept of topology, which posits that geometric objects are essentially the same if they can be smoothly deformed into one another. Remarkably, there exist insulators—called topological insulators—that *cannot* be smoothly transformed into a conventional insulator while maintaining a nonzero energy gap. Somewhere along the way, the gap must vanish, and at that point the material is actually an electrical conductor. Conventional insulators and topological insulators are sharply delineated electronic phases of matter that are in many ways as fundamental as the familiar liquid and solid phases of water.

When a topological insulator contacts a conventional insulator, the interface between them is inevitably conducting. This is simplest to understand if the interface smoothly interpolates between the two materials, so that the gap must vanish somewhere between them. But this interfacial conductor also appears at atomically sharp interfaces and even at the surface of a topological insulator, which can be thought of as the boundary separating a topological insulator from insulating empty space.

This general idea was encountered previously in the study of electrons that are forced to move laterally in a two-dimensional plane sandwiched between two semiconductors in the presence of a strong magnet. This gives rise to a phenomenon called the quantized Hall effect, in which the two-dimensional interior is a kind of topological insulator and its one-dimensional edge is unavoidably a conductor. This edge conductor has magical properties unlike any ordinary conductor. In an ordinary conductor, the motion of the electrons is ultimately limited by collisions. It is analogous to the plight of a person navigating a crowded train station at rush hour: just one collision after another.

The interfacial conductor is more like a one-way moving walkway guiding the commuters smoothly around the confusion. What the interior of the topological insulator does is to effectively create a conductor that is split into spatially separate parts that move in opposite directions. The interior is insulating and acts as a median strip that separates the opposing lanes of traffic. Interestingly, this one-way edge conductor is “impossible” in isolation. The other walkway must inevitably exist, providing a return path.

The rise of interest in topological electronic materials was enabled by the discovery that there are many more versions of this phenomenon where a topological insulating state can split electrically conducting pathways in half, resulting in “impossible” boundary conductors similar to the one-way edge conductor. This can occur in two-dimensional systems without a magnet, which exhibit a different kind of one-dimensional edge conductor. It can also arise in fully three-dimensional materials, which then exhibit conducting surfaces that are each half of an ordinary two-dimensional conductor. This has initiated a bustling subfield of condensed matter physics. These phenomena can be theoretically predicted in specific materials, which can then be synthesized and studied in the laboratory. These materials are not uncommon. A recent survey has turned up approximately 8,000 candidate three-dimensional topological materials.

The “impossible” conducting interface states of topological insulators have novel properties analogous to the one-way edge conductors, which fuels speculation on their possible practical applications. While applications remain in the future, there are medium and long-term goals for developing useful devices using topological insulators. One proposal is to take advantage of the fact that the flow of electrons in the conducting interface states is more organized than in an ordinary conductor. A fundamental problem with packing ever more logic gates into ever smaller computer chips is managing the flow of charge and the dissipation of heat. As conductors are made smaller and smaller, the crowded train station analogy becomes more acute. Perhaps by separating the conductors into separated lanes, it will be possible to organize the flow of charges and to control where energy is dissipated.

A second, more speculative, but also more ambitious, proposal is to use topological insulators and related materials to construct a kind of quantum computer. Ordinary computers perform logic operations on bits: the 0s and 1s that can be combined to encode binary numbers. A quantum bit (or qubit) can be 0 or 1 or *both at the same time*. This combination of 0 and 1 is enabled by a mysterious feature of quantum mechanics known as superposition. This allows a quantum computer—a device that performs operations on qubits—to be much more powerful than an ordinary computer.

The gist of the reason for this is that while a string of N ordinary bits can encode a single N digit binary number, a string of N qubits can encode *all* of the N digit binary numbers at *the same time*, allowing a quantum computer to perform many operations in parallel. It is known that there exist classes of problems that can be solved much more quickly on a quantum computer than on an ordinary computer.

Unfortunately, a quantum computer is very hard to make. Qubits are fragile, and if you measure a qubit, you lose most of the information that it encodes. The difficulty is therefore making sure that the quantum computer doesn’t accidentally measure itself. This fact makes constructing a quantum computer one of the grandest technological challenges for the coming century. One approach is to try to make the qubits very well spatially isolated, so that they don’t get destroyed. There is another approach that takes advantage of a topological electronic phase.

The idea is to *split the qubit*. Qubits are ordinarily localized objects—like atoms—that can exist in two distinct states and exist at a specific location. However, just as a topological insulator can split an electrical conductor into two, there exists a related topological phase called a topological superconductor that can split a qubit into two pieces that reside at the two ends of a one-dimensional material. The beauty of this is that the qubit is shared between the two ends and cannot be measured with any local measurement on one end. The quantum information is thus topologically protected and immune from accidental measurement.

One possible route to creating a topological superconductor is to combine an ordinary superconductor (which can easily be found) with a topological insulator or a related topological material. There is promising experimental evidence that such topological superconductors can be created. Demonstrating that they have the capacity to store quantum information remains a challenge—but one which seems likely to be solved.

The examples outlined above provide two goals to motivate the further development of topological electronic materials. Along the route to those goals there are many good science problems that need to be solved, and it is often the case that new ideas and new applications will emerge precisely from those studies. Fourteen years ago, when we began the work that ultimately led to our recognition with the Breakthrough Prize for Fundamental Physics, we had no idea that the world of topological materials would open so wide. So one lesson that we have learned is to expect the unexpected.