54 Scientific American, August 2020need for a master conductor and, even more astonishingly, were
impervious to the principles of physics that, for billions of years,
had guarded the quantum realm from macroscopic interlopers.
A door to the quantum realm was opened that day—a macro-
scopic door that many thought did not exist. In 1985, five years
after the discovery, von Klitzing was awarded the Nobel Prize in
Physics. His finding would lead to further breakthroughs, with
three more Nobel Prizes awarded to two experimentalists (Horst
Störmer and Daniel Tsui) and a theorist (Robert Laughlin) in 1998,
for discovering that electrons acting together in strong magnet-
ic fields can form new types of “particles,” with charges that are
mere fractions of electron charges, a phenomenon now known
as the fractional quantum Hall effect.LAUGHLIN’S QUANTUM PUMP
laughlIn was one of the fIrst physIcIsts to attempt an explana-
tion of the quantum Hall effect. In 1981 he came up with a bril-
liant thought experiment—an idealized simulation of the original
experiment that provided a mathematical metaphor to under-
stand it. Laughlin imagined electrons traveling along a conduct-
ing loop with a flat edge, like a wedding band. A magnetic field
ran perpendicular to the surface of the band, but Laughlin added
a fictitious magnetic field line—called a magnetic flux—threading
through the middle of the loop like a finger through the ring.
Increasing the fictional flux induced a current running around the
loop, thus introducing the longitudinal current present in the clas-
sical Hall effect. The process, named Laughlin’s quantum pump,
would complete one cycle every time the fictional magnetic flux
increased by one “flux quantum”—an amount defined as h / e, where
h is Planck’s constant and e is the electron’s charge.
After each cycle, the quantum system would return to its origi-
nal state as the result of a phenomenon known as gauge invariance.
Laughlin argued that this reset implied that the Hall conductance
was quantized in whole numbers equal to the number of electrons
moved by the quantum pump. Great! Alas, there was an issue. The
Hall conductance was experimentally measured (and averaged) over
many cycles of the pump. Because Laughlin assumed (correctly)
that the system was described by quantum mechanics, there was
no guarantee that each cycle would transfer the same number of
electrons. As Avron and Seiler would write later with their collab-
orator Daniel Osadchy: “Only in classical mechanics does an exact
reproduction of a prior state guarantee reproduction of the prior
measured result. In quantum mechanics, reproducing the state of
the system does not necessarily reproduce the measurement out-
come. So one cannot conclude from gauge invariance alone that the
same number of electrons is transferred in every cycle of the pump.”
Physicists needed a new set of ideas to show that the average num-
ber of electrons transferred over several cycles was also an integer.
Inspired by Laughlin’s argument, the next attempts at explain-
ing the quantization of the Hall conductance relied heavily on
the concept of adiabatic evolution. Adiabatic evolution is a phys-
ical process that aims to capture the evolution of a system that
remains in its lowest-energy state at all times while some exter-
nal parameter varies. When the system’s spectral gap—the ener-
gy required for it to jump to an excited state—becomes small, adi-
abatic evolution slows down to prevent the system from crossing
over to an excited state. Laughlin’s original argument used this
notion to mathematically model the quantum Hall effect as the
adiabatic evolution of the electronic state of a quantum Hall sys-
tem under the increase of a fictitious magnetic flux.UNBREAKABLE PLAY-DOH
to study the quantum hall effect more deeply, physicists turned
to a branch of mathematics called topology. Topology is a way of
thinking about the fundamental essence of shapes—the proper-
ties that do not change even as they are continuously deformed.
Think of a kind of Play-Doh that is unbreakable and impossible
to glue onto itself. You can turn a cube of this substance into a
ball by rounding out its sharp edges and corners, but you cannot
turn it into a doughnut. The latter transformation would require
either poking a hole through the cube or stretching and gluing it
onto itself. In that sense, cubes and doughnuts are topologically
distinct shapes, but cubes and balls are topologically the same
(although they are all geometrically different). Topology was for-
malized in 1895 but had rarely interacted with physics until the
1950s and 1960s.
The initial efforts to understand the role of topology in the quan-
tum Hall effect were considered so significant, in fact, that in 2016
theoretical physicists David Thouless and F. Duncan M. Haldane
won a Nobel Prize for this work. Thouless and his collaborators, in
particular, extended Laughlin’s argument by showing that the Hall
conductance was quantized on average. Because one fictitious flux
was not enough to prove quantization, they proposed a second fic-
titious flux. In the new thought experiment, one flux induced the
electric current across a semiconductor, and the other detected
changes in the current between pump cycles. This scenario simu-
lated cycles of Laughlin’s pump under distinct initial conditions.
The adiabatic evolution generated by the extra fictitious flux played
the role of averaging over many cycles of Laughlin’s pump and
showed that the average Hall conductance was quantized.Laughlin’s PumpFictional magnetic fluxDeflected chargesMagnetic field
perpendicular
to loopOriginal currentMagnetic flux
loopsFlux driving
current across
semiconductorFlux detecting
changes in
Hall current© 2020 Scientific American