56 Scientific American, August 2020ly on the problem together. But about a month or two after I arrived
at Los Alamos, Matt told me he was leaving the lab. With job inter-
views now taking up most of his time, I barely saw him. A few
months later, when he was offered a position at Microsoft’s Station Q
in Santa Barbara, Calif., my interactions with him all but ended. The
few times we did meet, I became convinced that Matt had made a
serious mistake in giving me a postdoc at Los Alamos. He would
speak, and all I could retain were a few word combinations here
and there. One of the phrases he repeated was “quasi-adiabatic con-
tinuation,” a notion I was unfamiliar with. To my further dismay,
this term did not seem to appear anywhere in the immense litera-
ture devoted to the quantum Hall effect up to that point.
Without much else to go on, I did what every young scientist of
my generation would do and googled “quantum Hall effect” and
“quasi-adiabatic continuation” (QAC). The first phrase returned
hundreds of research papers, but I had as much luck reading
through any of them as with the book by Thouless. The one thing
I did get out of that search, however, was a word that kept coming
up in relation to the quantum Hall effect: topological. When I add-
ed that word to my search, the first thing that popped up was an
article by Avron, Osadchy and Seiler entitled “A Topological Look
at the Quantum Hall Effect.” The piece, which appeared in Phys-
ics Today in August 2003, was meant for nonexpert physicists. This
article was so clearly written that it formed the foundation on
which I would build my understanding of the quantum Hall effect.
In contrast to the hundreds of articles on the quantum Hall
effect, my search on quasi-adiabatic continuation returned just two
results, both by Matt. The first paper, co-authored with theoretical
physicist Xiao-Gang Wen, was an introduction to QAC. The second
paper contained, among other applications, a brief section on using
QAC to compute a version of the Berry curvature relevant to the
fractional quantum Hall effect. This was the first and only published
attempt to apply QAC to any type of Berry curvature. I was excit-
ed to study Matt’s argument inside and out. But I still needed to
understand what QAC was about and how it was connected to adi-
abatic evolution. So I delved into the first paper, and after a month
of poring over it, I felt that I had a good grasp of the technique.
QAC was proposed as an evolution of a quantum system designed
to preserve certain topological properties of its quantum state. In
contrast, adiabatic evolution was better suited for local, geomet-
ric properties, such as the Berry curvature mentioned earlier.
The next task was to figure out how to compute the Berry cur-
vature using QAC. To my dismay, I could not parse Matt’s brief argu-
ment on how the two concepts could be bridged. I decided to re-cre-
ate that bridge (or my version of it, at least) from scratch. The idea
was to follow Simon’s argument connecting adiabatic evolution to
Berry curvature, while sneaking in QAC in place of adiabatic evo-
lution. Substituting one evolution for the other worked out beau-
tifully for one simple reason: I could show that QAC was exactly thesame as adiabatic evolution under the following special condition:
throughout the evolution of the system, the gap in energy between
the ground state and the first excited state had to remain above a
fixed positive value, independent of the size of the system. As luck
would have it, this special condition was satisfied precisely near
the origin of the 2-D phase space. In fact, if that condition was vio-
lated, I could show that the Hall conductance was not quantized.
After going through the exercise of connecting QAC to the Ber-
ry curvature and, hence, to the Hall conductance, I turned my
sights toward the next big hurdle: re-creating Simon’s argument,
which computed the averaged Hall conductance as an unchang-
ing topological quantity that yields the first Chern number. This
was no small feat. As I have mentioned, to get over the initial
problem of simulating adiabatic evolution with QAC, I took
advantage of the fact that QAC tracked adiabatic evolution exact-
ly, as long as there was a big enough spectral
gap between the ground and excited states
of the system. Unfortunately, this assump-
tion about the spectral gap went out the win-
dow the moment I started exploring deeper
into the 2-D phase space, whose total curva-
ture I needed to compute. In fact, this as -
sumption was so powerful that all attempts
to quantize Hall conductance up to that
point had used it. In other words, nobody
thought it was possible to prove quantization without making
that extra assumption. And neither did I. When I finally reached
out to Matt in late spring of 2009 with a solution that made use
of that key assumption, he said to me: “Nice job. But I think you
should be able to prove quantization without it.” Matt pointed
me toward a seemingly unrelated paper of his entitled “Lieb-
Schultz-Mattis in Higher Dimensions” (LSM), where he had laid
the foundations for removing this assumption.
As I began to read through LSM, I had the same sinking feel-
ing as when I had tried to parse Matt’s attempt at connecting QAC
to the Berry curvature. Deciphering it in isolation was going to be
a long and arduous journey. But in a second twist of fate, my Ph.D.
adviser, Bruno Nachtergaele, working with one of his postdocs at
the time, Robert Sims, had published what some considered a math-
ematically rigorous version of Matt’s LSM paper. Although most
of the brilliant insights were already in Matt’s original paper, Bru-
no’s version was so well written and thorough that within a month
I had a clear view of how to proceed. I now knew how to adapt ele-
ments of the LSM argument to overcome the second hurdle: to
show that the averaged Hall conductance computed using QAC,
instead of adiabatic evolution, was still an integer multiple of e^2 / h.
The original Laughlin’s pump argument, which used adiabat-
ic evolution and gauge invariance to deduce a return to the orig-
inal state of the system after one cycle, did not work with QAC.
The main problem was that under QAC, after a flux quantum was
inserted, there was no longer any guarantee that the system would
end up in the same quantum state at the end of a cycle. Adiabat-
ic evolution accomplished such a feat by forbidding the lowest-
energy state of the system from ever getting excited. QAC, on the
other hand, had a mind of its own. If the spectral gap ever dropped
below a critical value as scientists inserted more and more mag-
netic flux, QAC would happily allow the system to jump to a new,
excited quantum state, leaving behind its low-energy past. Unfor-
tunately for me, that meant that at the end of a Laughlin cycle,I made progress by breaking
the problem down into simple
parts I could understand.
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