Why must some insulators conduct at their edges?

Bend a coffee mug into a doughnut and the hole survives. Properties like that, which no smooth change can remove, are the subject of topology, and they turn up in quantum matter as twists in the way electrons' quantum states wind through a material. The twists have a striking consequence: a material can be an insulator inside yet be forced to conduct along its edges or surface. Theorists at the Institute find new kinds of these twists, in materials driven by lasers and in systems with no energy gap at all, and connect them to the quantum field theories of particle physics.

Insulators with conducting skins

The story begins with the quantum Hall effect. In 1980 Klaus von Klitzing discovered that in a strong magnetic field, the Hall resistance of a thin layer of electrons comes in perfectly sharp steps. The sharpness comes from topology: the electrons' quantum states have a kind of twist, counted by a whole number, that cannot change without a drastic event, so impurities and imperfections do not spoil it. The same twist forces current to flow along the edges of the sample, as in the animation above.

In 2005 theorists realized that a related twist could exist in materials that keep time-reversal symmetry, with no magnetic field at all. In 2006 Rahul Roy showed how this kind of topological label, called a Z₂ invariant, extends to three-dimensional crystals; several groups reached closely related results independently in 2006 and 2007. The consequence is striking: a crystal with this twist is an insulator inside, yet its surface must conduct, because the twist cannot simply end at the edge. These materials are now called three-dimensional topological insulators.

Topology on a timer

Topology need not be frozen in place. If a material is driven by a rhythmic push, such as a laser or a pulsed field, its properties repeat in time; physicists call it a Floquet system. In 2016 Roy and Fenner Harper built a "periodic table" for such driven topological insulators, showing that driving enlarges the set of possible topological types compared with static materials. They also showed that interacting driven systems of bosons, fermions or spins can move charge and information around their edges in ways no static system can, a form of order that exists only because the system is being driven.

Phases without gaps

Ryan Thorngren works on the mathematical foundations of topological phases, where condensed matter physics, quantum field theory and mathematics meet. Topological phases were long thought to need an energy gap, a minimum cost to excite the system. With Ashvin Vishwanath and Ruben Verresen, Thorngren identified "intrinsically gapless" topological phases, which have no gapped counterpart and still show protected edge modes.

The Higgs phase, seen as topology

With collaborators, Thorngren has also argued that the Higgs phase of a gauge theory, a concept familiar from particle physics, can be understood as a symmetry-protected topological phase, which predicts edge modes in models where none were expected. His work on symmetry itself, including symmetries that cannot be undone, is told in our Symmetry story.

Why it matters

Because small imperfections cannot change them, topological states are candidates for building blocks of quantum devices that resist noise. They also feed back into fundamental physics: the same mathematics of symmetry and topology connects the electrons in a crystal to the quantum field theories used for particles.