Published 13 June 2019 by Benjamin Skuse

# Unentangling a Quantum Enigma

2019 is a festival of science anniversaries – 50 years since the first moon landing, 100 since Eddington made Einstein world-famous with his general relativity-testing eclipse expedition, 150 since Mendeleev’s first periodic table and 500 since Magellan circumnavigated the globe.

Hidden among these celebrations is a more obscure 140^{th} anniversary: the discovery of the Hall effect. Though most people outside physics will not have even heard of it, the Hall effect was regarded by Lord Kelvin as the “greatest discovery…since the times of Faraday” when it was announced. And like a gift that keeps on giving, it continues to fascinate.

While experimenting with electric and magnetic fields on gold leaf in 1879, Johns Hopkins University graduate student Edwin Hall discovered a tiny discrepancy from conventional physics of the time – a small voltage fired out at right angles to the electric current and magnetic field.

In the years that followed, learning the Hall effect’s secrets exposed many fundamental principles regarding the nature of charge carriers. For instance, it was used to offer the first real proof that currents in metals are carried by negatively charged electrons and not positively charged protons.

### Quantum Mysteries

By the mid-20^{th} century the Hall effect was a well-established and understood phenomenon. But in 1980, history repeated itself when it once again defied conventional wisdom. German experimental physicist – and 2019 Lindau Meeting attendee – Klaus von Klitzing showed that if you reduce the temperature and increase the magnetic field strength, a thin semiconductor strip’s conductance (the ease with which an electric current passes) doesn’t grow linearly. Instead, it jumps, which is the simple way of saying it is ‘quantised’. These jumps are extremely precise, regardless of impurities or the details of the system. And each jump is always an integer multiple of a fixed value (known as the fine structure constant).

Showing the world a macroscopic quantum phenomenon for the first time, von Klitzing received the Nobel Prize for discovering the quantum Hall effect. But how and why it happened was a mystery.

This mystery deepened when in 1982 the fractional quantum Hall effect was discovered. The fractional version is a surprising phenomenon with new quantum steps, some of which have Hall values that are fractions above and between the integers. Its discovery and explanation as a new form of quantum fluid led to three more Nobel Prizes for Horst Störmer, Daniel Tsui and Robert Laughlin.

While Nobel Laureate Klaus von Klitzing (left) has already attended 17 Lindau Meetings, Michael Kosterlitz and Duncan Haldane will participate in this year’s meeting for the first time. © Peter Badge/typos 1 in coop. with Lindau Nobel Laureate Meetings

### Topology to the Rescue

Seeing exciting discoveries being made that had no reasonable physical explanation, theoretical physicists were immediately attracted to the problem. One of these researchers – David Thouless – soon realised that aspects of his work with Lindau attendee Michael Kosterlitz were key to explaining the mysterious results. He began to paint a new picture of the quantum Hall effect, a picture that involves something called topology.

Topology ignores the precise shape of an object to just look at its most fundamental properties. This means a topologist sees no difference between a bagel and a mug, as both have just one hole. Drill a hole in a bagel though and it is a completely different object to a topologist. Objects can be defined topologically by properties that are called topological invariants, which are often integers like their number of holes.

Kosterlitz and Thouless had previously used topology to show that thin layers of materials could exhibit superconductivity, superfluidity and other exotic phenomena by switching between states that can be described as having different topologies.

In 1982, Thouless again used topology to treat the electrons in the quantum Hall effect as an individual object, a continuous quantum sea. This allowed him to explain the sudden jumps in conductance as topological phase transitions, like going from a bagel to a shirt button to a pretzel. It also exposed why the effect was precise and robust even when the experiment wasn’t – just as bending or twisting a bagel doesn’t change the fact it is still a bagel unless you rip it or punch a hole in it.

Just a few years later in 1988, another theoretical physicist – Duncan Haldane – realised that the quantum Hall effect could happen in the absence of an external magnetic field. The ‘anomalous quantum Hall effect’ Haldane described is the first topological insulator; a bizarre class of materials that behave like insulators in their interior but whose surfaces conduct.

For their pioneering work to understand exotic phases of matter using topology, Thouless, Kosterlitz and Haldane were awarded the 2016 Nobel Prize in Physics. This takes the current count of Nobel Prizes directly related to the Hall effect to seven. But this is not where the story ends.

### A New Era of Quantum Computing?

Mathematicians and physicists continue to refine scientific understanding of how topology governs the quantum Hall effect. And they are wielding this ever-increasing knowledge in the real world. The most exciting potential application is quantum computers.

A quantum computer could rapidly solve certain types of problems that regular supercomputers would find impossible. But a useful quantum computer has not yet been made, because quantum bits (qubits) are easily destroyed. This is why the likes of Microsoft are exploring a new type of qubit, a topological qubit. In fact, the first quantum computer may end up running on qubits that rely on manipulating the fractional quantum Hall effect.

To think that the next breakthrough in computing, which could reshape our understanding of physics and the universe, can be traced back to a graduate student noticing a tiny odd voltage in his experiment 140 years ago is staggering. But at least one person may have seen it coming.

Soon after Hall detected his eponymous effect, his thesis supervisor Henry Rowland wrote: “The recent discovery by Mr. Hall… opens a wide field for the mathematician”. How prescient those words were.

*Additional information: Klaus von Klitzing, Michael Kosterlitz and Duncan Haldane will all hold lectures at the 69 ^{th} Lindau Nobel Laureate Meeting.*