Scientific American - USA (2020-08)

August 2020, ScientificAmerican.com 55

At around the same time, Barry Simon, a mathematical phys-
icist at the California Institute of Technology, noticed that adia-
batic evolution formed a mathematical bridge between the Hall
conductance and the local curvature of the two-dimensional
phase space generated by the two fictitious magnetic fluxes. This
local curvature is called Berry curvature after its discoverer, math-
ematical physicist Michael Berry. In particular, Simon showed
that the Hall conductance was equal to h / 2 π times the local cur-
vature at the origin of that phase space. This was a big deal. A
famous mathematical result from 1848—the Gauss-Bonnet theo-
rem—declared that the total curvature of a geometric shape was
a topological feature, not a geometric one. In other words, the
sum of all the local curvatures of a three-dimensional shape is
the same for all topologically equivalent shapes with the same
surface area. Even more exciting, the total curvature is simply
given by 2π(2 − 2 g ), where g is the number of holes in the shape.

Most important for us, a modern generalization of Gauss-Bon-

net by geometer Shiing-shen Chern showed that the same result

applied for the total Berry curvature of our two-dimensional phase

space describing the quantum Hall effect. The Berry curvature of

that space was now given by 2π C, with C denoting an integer

known as the first Chern number. To show that the Hall conduc-

tance was quantized, Simon and his collaborators looked at the

average of the conductance over the whole phase space, which is

given by h / 2 π times (total curvature) divided by (surface area).

Plugging in 2π C for the total curvature and ( h / e )^2 for the surface

area yielded C × e^2 / h. Et voilà. The average Hall conductance was

an integer multiple of e^2 / h , as Thouless had shown. But for the

first time ever, the integer in front of e^2 / h was identified with a

“topological invariant”—a property that does not change if you

rotate or deform a shape—and therefore the result was impervi-

ous to small perturbations and imperfections in the physical set-

up of the quantum Hall effect. This was a breakthrough insight.

Unfortunately, the beauty of the preceding arguments by Thou-

less and Simon was marred by a serious issue: the Hall conductance

that experimentalists measured corresponded to the local curva-

ture at the origin of the two-dimensional phase space, not the aver-

age curvature over the whole space. To see why the local curvature

of an arbitrary shape is almost never equal to its average curvature,

consider a torus. Gauss-Bonnet implies that the average curvature

of a torus, and of any shape with a single hole in it, is zero. But the

local curvature of a torus is obviously nonzero along most points

on the surface and can take both positive and negative values. Thou-

less and his collaborators actually tried to address this issue, yet

the question remained: Why was the Hall conductance quantized,

if one was not allowed to average over all possible initial conditions

of Laughlin’s pump? Indeed, that was the question I had to answer.

A SENSE OF DESPAIR

my f Irst steps Into the mystery of the quantum Hall effect were sup-

posed to be illuminated by a book written by Thouless himself: Topo-

logical Quantum Numbers in Nonrelativistic Physics. A couple of

weeks after receiving the book from Matt, I determined that I did

not have the background required to understand any of the phys-

ics within. I locked the book inside my desk drawer and put the

key away. Yet the book’s simple existence gave me a sense of despair.

How could I make any progress in solving the problem if I could not

understand the contents of that book? Back then, I was a blank slate.

Of course, I had the option of going to Matt for help. He could

teach me what I needed to know. Heck, we could even work close-

Square

surface area =

2

e

h

( )

Cylinder

surface area =

Join

orange

edges

Join

blue

edges

2

e

(h)

Torus

surface area =

2

e

h

( )

e

h

e

h

Gauss-Bonnet Theorem

Total curvature = 4π (1 − g)

Sphere

Torus

Cube

=

= 4π (1 − 1)

= 0

= 4π (1 − 0)

= 4π

Average curvature = 0

Negative

curvature

Positive

curvature

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