Imagine a haunted house of sorts, one with many entrances, each of which eventually leads you to a long, dark corridor. You know that the house consists of something more than empty hallways, because you can hear laughter and see lights that suggest courtyards and magnificent chambers within. You head off in a promising direction, convinced that something more lies ahead, but to no avail. You can try the glass ceilings or staircases that lead up from the underground or one of the many entryways that scale the outer walls, but it is only a matter of time — the walls rearrange themselves, and the floor slides beneath your feet, so that you always find yourself in that same, eerie tunnel. You’re convinced. The conspiracy is *real*.

But you have prestigious company. Certain physicists at Jadwin Hall *also* believe the conspiracy is real. They call their shape-shifting haunted houses topological insulators. What’s their issue? Well, these materials — as their name suggests — are insulators, so they do not generally allow electrons, or electric current, to pass through them. But if you cut them in any way, they always reveal surfaces that conduct electricity. This is a very strange property; no matter how you slice these fickle beasts, you cannot get them to show their inner nature. This point is worthy of emphasis: no matter how clever you are about your cut, no matter how many electrons you displace and how much internal symmetry you distort, you will always find an eerily smooth conducting surface. These devious fiends aren’t really insulators, then, nor are they are conductors. So what *are* they?

It turns out that when Professor M. Zahid Hasan and several other leading physicists discovered these topological insulators (TIs), they uncovered an entirely new “electronic” phase of matter. Not like solid, liquid or gas phase, but another kind of phase, one that earns its own category near the likes of conductors, insulators, semiconductors, superconductors and magnets. TIs are special in another sense. Most materials we know of today can be understood in terms of their potential energy maps (think: a heat scan showing areas of highest energy, lowest energy and everything in between as a gradient of colors). For ordinary materials, we can learn about their inner structure — where the bonds between molecules are, where the weakest links are — just by examining this energy map, a remarkable fact in itself. But the maps of TIs don’t really give us any information. We can in fact distort the geometry of a TI, and not change its fundamental properties — like shifting a wall in the haunted house, but leaving the mysterious structure intact. We have to do a more serious sort of distortion, one that turns a surface inside out, or introduces a hole where there was none before, to see a response in the potential energy landscape. This sort of abuse is called a topological deformation. Understanding the topology of these haunted houses is thus the key to unraveling their mystery.

It turns out that when Professor M. Zahid Hasan and several other leading physicists discovered these topological insulators (TIs), they uncovered an entirely new “electronic” phase of matter.

Whenever you see strange, strange things happening in physics, like particles going through solid walls or electrons feeling uncertain about their whereabouts, you can suspect that quantum mechanics is involved — and nine times out of ten, you’ll be right. Topological insulators, too, are hosts to beautiful quantum effects. It turns out that the electrons inside TIs experience quantum entanglement. Two electrons, say, Alice and Bob, that are entangled may be separated in space, but they have interlocked existences, so that what goes in Alice’s vicinity can affect Bob, and vice versa, without any “information” being sent. The idea of quantum entanglement has been known for many decades now (Einstein did not like it one bit, calling it “spukhafte Fernwirkung*,*” German for “spooky action at a distance”), but it tends to show up in new and unexpected places. It is quantum entanglement that explains why conducting surfaces can form, seemingly spontaneously, when a topological insulator is cut. When the knife approaches Alice, Bob *already* knows and can rearrange himself, so that when he and Alice are finally exposed, the surface they form conducts electricity. It is through this quantum entanglement that topological insulators retain their properties despite geometric deformation.

Topological insulators, it turns out, have many applications in fundamental physics. They could also, however, lead to a potentially lucrative breakthrough in the area of quantum computing, a field that lies at the intersection of applied physics and theoretical computer science. What are quantum computers? Well, in short, while computers today store information as 1s and 0s in electronic bits, quantum computers can store information in many more states, including “fuzzy” levels between 0 and 1, allowing computation at unprecedented speeds and scales. The possibilities are scarcely imaginable. Quantum computers could solve computational problems previously deemed impossible in mathematics, operations research, computer science, genetics, ecommerce and many other disciplines with a fraction of the resources of today’s computers. Some very small quantum computers have already been built in laboratories at universities and the research departments of major companies. Progress has been limited, however, by a fatal effect called decoherence, when information stored in the quantum bits (qubits) becomes horribly mangled. Why do our beloved TIs matter in this futuristic-seeming fantasy? Well, TIs offer a solution to the problem of decoherence. Since they are really tough in a 21^{st} century sort of way, resistant to geometric deformation, they can preserve information that other materials may not be able to. Topological insulators thus have the potential to be used in large-scale quantum computing systems, possibly playing a pivotal role in the inevitable — and imminent — computing revolution.

#### Sources (Einstein’s quote):

-Letter from Einstein to Max Born, 3 March 1947; *The Born-Einstein Letters; Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955*, Walker, New York, 1971. (cited in M. P. Hobson; *et al.*. “Quantum Entanglement and Communication Complexity (1998)”