August 2020, ScientificAmerican.com 57even though the dynamics describing the system returned to their
original state, the quantum state of the system itself may have
changed significantly. If that were the case, then a key element of
Laughlin’s and Thouless’s arguments would go up in smoke.
To overcome this obstacle, I needed to introduce two more fic-
titious magnetic fluxes in addition to the original two (for a grand
total of four), which allowed me to transform the evolution under
QAC into one that guaranteed a safe return to the original ground
state at the end of a cycle. This trick, borrowed from Matt’s LSM
paper, forced the state of the system to maintain the exact same
energy throughout the modified evolution around the boundary
of the 2-D phase space, even when that energy no longer corre-
sponded to the lowest possible energy of the system. In other
words, to guarantee the return of the system to its initial state, all
one ever needed to know was that the two states had the same
energy. The fact that the ground state of the system was unique-
ly specified by that energy took care of the rest. Adiabatic evolu-
tion’s insistence on keeping the system in its lowest-energy state
throughout the evolution was overkill. More important, as I came
to realize later, the insistence on using adiabatic evolution to
quantize the Hall conductance was also the reason progress had
stalled for nearly two decades.
By now I felt exhausted. But the main hurdle was finally in view.
Everything I had accomplished up to this point was a fancy way of
showing what Thouless, Simon and their collaborators had already
proved: that the averaged Hall conductance was indeed quantized
in integer multiples of e^2 / h. It would seem that I had made no prog-
ress in removing the averaging assumption plaguing every effort
to explain the mystery of the integer quantum Hall effect. Except
for one minor detail: the two-dimensional phase space generated
by QAC had near-perfect uniform Berry curvature. In other words,
the real Hall conductance, the one corresponding to the Berry cur-
vature of a tiny patch near the origin of the 2-D phase space, was
equal to the average curvature over the total flux space. Because the
latter was famously quantized, it followed that the actual Hall con-
ductance was also quantized. Quod erat demonstrandum —QED.
This final theoretical hurdle took many months of restless days
and sleepless nights to cross over. I nearly gave up several times
before reaching my goal. During a particularly dark time, I told
my mom that I was not sure I wanted to wake up the next morn-
ing. In typical Greek fashion, she responded, “If you do anything
stupid, I will fly out there and strangle you with my own two
hands.” Lost in a world of hyperanalytical thinking, I needed such
an absurd statement to snap me out of it. I finished the proof in
November 2009, shared it with Matt, who quickly added a section
on how the result could be extended to also explain the fraction-
al quantum Hall effect, and then posted it online. It would take us
five more years before getting the result published and another
four years before the mathematical physics community could ful-
ly digest it. On February 25, 2018, I opened an e-mail from Michael
Aizenman—a letter I had waited for eight years to receive. It read:Dear Matt and Spiros,
The Open Problems in Mathematical Physics web page
was now updated with the statement that the IQHE ques-
tion, which was posted by Yosi Avron and Ruedi Seiler, was
solved in your joint work.
I thank you here for your contribution, and also con-
gratulate you on it. It is a pleasure to note that in each of
the two problems on which progress is reported there, the
advance came through deep novel insights and new tools.
The list of solvers is a veritable honor roll.The fundamental mystery we started with was the question of
why a microscopic, quantum phenomenon was showing up on a
macroscopic scale. Instead what we found was that one of the most
fundamental constants of nature was the reflection of global order
beyond our finite grasp—the infinite communing with the infini-
tesimal. And although we have focused on the theory behind the
quantum Hall effect, the experimental efforts it has inspired over
the past three decades have been equally, if not more, exciting.
Research on topological phases of matter beyond two-dimension-
al quantum Hall systems is paving the way toward technologies
such as large-scale, fault-tolerant quantum computing. Impres-
sive results coming out of labs such as Ana Maria Rey’s at the Uni-
versity of Colorado Boulder are even tackling fundamental ques-
tions about the very nature of time.
This experience also taught me a valuable lesson: my self-
worth is not tied to my success in life. The fateful call with my
mom took place three months before I put the finishing touches
on the solution. I did not turn into a mathematical genius with-
in the span of a few months. But I made progress by breaking the
problem down into simple parts I could understand. To do that,
I needed to be okay with feeling incompetent most of the time.
Without the faith of my parents in me as a person, whether I was
good enough to solve the problem or not, I would have given up
right before the finish line. Had I done that, the problem may still
be unsolved and Marvel’s Avengers would have had to find an
even more scientifically implausible way to save the universe than
to jump into the quantum realm via a macroscopic portal.“Upside-down” phase space Uniform curvatureEach of the additional fluxes generated an upside-down version
of the phase space so that the new space has uniform curvature.Original phase space curvatureMORE TO EXPLORE
Quantization of Hall Conductance for Interacting Electrons on a Torus. Matthew B.
Hastings and Spyridon Michalakis in Communications in Mathematical Physics, Vol. 334,
No. 1, pages 433–471; February 2015. https://arxiv.org/abs/0911.4706
Open Problems in Mathematical Physics: http://web.math.princeton.edu/~aizenman/
OpenProblems_MathPhys/index.html
FROM OUR ARCHIVES
The Un(solv)able Problem. Toby S. Cubitt, David Pérez-García and Michael Wolf;
October 2018.
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