Week 10: Oct 27, 2016

Preamble to Quantum Hall Effect: Exotic Phenomenon in Quantum Physics

The quantum Hall effect is one of the most remarkable of all quantum-matter phenomena,

quite unanticipated by the physics community at the time of its discovery in 1980. This very

surprising discovery earned von Klitzing the Nobel Prize in physics in 1985.

The basic experimental observation is the quantization of resistance, in two-dimensional

systems, to an extreme precision, irrespective of the sample’s shape and of its degree of purity.

This intriguing phenomenon is a manifestation of quantum mechanics on a macroscopic scale,

and for that reason, it rivals superconductivity and Bose–Einstein condensation in its fundamental

importance. As we will see later in this class, that this effect is an example of Berry phase and

the “quanta” that appear in the quantization of Hall conductivity are the topological integers that

emerge from quantum analog of Gauss-Bonnet theorem. This years Nobel prize to David Thouless

is for explaining this exotic quantization in terms of topology.

It turns out that the classical and quantum anholonomy are described by the same

mathematics. To understand that, we need to introduce the concept of “curvature. As we know,

it is the curved space that leads to anholonomy in classical physics. This allows us to define

curvature in quantum physics. That is, using the concept of anholonomy in quantum physics, we

can define the concept of curvature in quantum physics – that is, in Hilbert space.

Curvature leads us to Topological Invariants that tells us that anholonomy may have its roots

in topology. It will also lead to topological integers, which are the quantum numbers in quantum

Hall effect.

I. TOPOLOGY AND TOPOLOGICAL INVARIANTS:

A. Topological Invariant for Two-Dimensional Surfaces

This index, like most topological invariants in physics, arises as the integral of a geometric

quantity. It is defined as follows:

χ =1

2π

∫S

κ ds (1)

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Here κ is the local curvature of the surface — a concept that we shall now define.

First of all, let us consider curves in a Euclidean plane. If the curve is a circle (the simplest

case), then the reciprocal of its radius, κ = 1/R, is defined to be its local curvature (and the

curvature is the same at every point of the circle). Clearly, the smaller the circle, the greater the

curvature, and vice versa. And a straight line, being a circle of infinite radius, has zero curvature,

which makes perfect sense, since it is not curved at all.

For curves that are not circular, the curvature will vary from point to point. The local

curvature will of course be greater at points where the curve turns more sharply, and lower where

it approaches straightness. To obtain the precise numerical value of the curvature, one finds the

circle that best fits the curve at the point in question, and the inverse radius of that circle is defined

to be the local curvature at that point.

This notion of local curvature can be extended from two-dimensional curves to

three-dimensional manifolds (surfaces in 3-space), such as a tin can, a fruit bowl, an egg,

someone’s cheek, and so forth. The most naıve approach to defining the local curvature of such

an object (that is, the idea based on the simplest possible analogy to the two-dimensional case)

would be, given a point on the object’s surface, to try to find “the best-fitting circle” at that point.

Unfortunately, however, this naıve approach doesn’t work, because in general, many different

circles of different radii will fit snugly against the object at the chosen point, since at each point

on the object’s surface, there are infinitely many different directions that one could choose.

The key idea for defining the Gaussian curvature of a two-dimensional surface is one whose

curvature κmax is greatest and one whose curvature κmin is least. These two circles’ curvatures

are known as the manifold’s principal curvatures at that point. (By the way, some 250 or so years

ago, Euler proved that for any point of any manifold, the extremal circles are always oriented at 90

degrees to each other.) We now define κ, the local Gaussian curvature at any given point, as the

product of the two extremal curvatures: κ = κmaxκmin. For the pipe, since the minimal curvature

is 0, it doesn’t matter what the maximal curvature is, since their product will always be 0. Thus

the local curvature at every point of the pipe equals 0.

For a sphere of radius R, κmax = κmin = 1R

, so κ = 1R2 . Consider next a horse’s saddle,

and pick some point near its middle. At that spatial point, if you move backwards or forwards

(i.e., headwards or tailwards, if the saddle is sitting on a horse), the saddle curves upwards,

while if you move sideways, the saddle curves downwards. For this reason, we assign the two

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curvatures opposite signs, so that κmax > 0 and κmin < 0. Thus the Gaussian curvature at that

point, κmaxκmin, will be negative.

it turns out, rather miraculously, that for any two-dimensional closed manifold, the total

Gaussian curvature is always an integer. This is a consequence of a beautiful theorem, the

Gauss–Bonnet theorem in differential geometry. It is named after Carl Friedrich Gauss, who was

aware of a version of the theorem but never published it, and Pierre Ossian Bonnet, who published

a special case of it in 1848.

The Gauss–Bonnet Theorem and Topological Integers

χ =1

2π

∫S

κ ds = 2(1 − g) (2)

In this equation, g is the genus of the object in question — that is, the number of handles it

possesses (always an integer, of course). Both g and the number χ are topological invariants. For

a sphere, which clearly has no handle, g = 0 and hence χ = 2. For a torus, g = 1 and thus χ = 0.

A coffee mug has one handle, and so, topologically speaking, it is a torus; that is to say, coffee

mug = doughnut (at least as viewed by a topologist).

The Gauss–Bonnet theorem states that the integral of the curvature over the whole surface is

“quantized” (restricted to integer values, in this case), and is a topological invariant.

We will next discuss anholonomy, a phenomenon that is rooted in curved space in classical

systems such as Foucault pendulum. It relates directly to the notion of curvature for classical

systems. This phenomena of anholonomy allows us to define the concept of curvature in quantum

systems.

Define Solid Angle

In geometry, a solid angle symbol Ω is the two-dimensional angle in three-dimensional space

that an object subtends at a point. It is a measure of how large the object appears to an observer

looking from that point.

A small object nearby may subtend the same solid angle as a larger object farther away. For

example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed,

as viewed from any point on Earth, both objects have approximately the same solid angle as well

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as apparent size.

A solid angle in steradians equals the area of a segment of a unit sphere in the same way a

planar angle in radians equals the length of an arc of a unit circle. If the circle has a radius R, and

the arc length dl, the angle is dl/R, as ld = Rdθ. Similarly for a sphere of radius R, the solid

angle is dA/R2 where dA is the area of the cap, dA = R2 sin θdθdφ.

Therefore, a cap of area A on the surface of a sphere of radius R will subtend a solid angle

Ω = AR2 .

For a triangle on a sphere as shown in Fig. (2), the solid angle is Ω = θ1 + θ2 + θ3 − π.

The Foucault Pendulum: An Example of Classical Anholonomy

As the earth rotates through an angle of 2π radians (360 degrees), the system’s cyclic pathway

results in the plane of oscillation of the pendulum rotating through a smaller angle. An observer

on Earth witnesses that the orientation of the pendulum — that is, of its plane of oscillation —

slowly rotates during the course of the day, and in general does not return to its original orientation

after 24 hours.

II. GEOMETRIZATION OF THE FOUCAULT PENDULUM

NOTE: Straight lines on a sphere are great circles

When a pendulum is carried around a great circle, the angle between the pendulum’s

swing and the great circle never changes. If, however, the pendulum is taken along a path that is

not a great circle — for example, a path of fixed latitude other than the equator — then the angle

between the pendulum’s swing and the path does change.

Triangular paths on a sphere

Before discussing a pendulum that moves along a circle at a fixed latitude, let us consider a

triangular pathway C on the sphere — that is, a pathway consisting of three great-circle segments,

as shown in Figure 2. Let θ1, θ2, and θ3 be the angles at the vertices of the triangle. We want to

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see how the pendulum’s plane of oscillation changes as it follows the closed triangular pathway.

There is no change while the pendulum moves along any one segment. There is a change,

however, each time that the pendulum moves from one segment to the next. At those moments,

the angle between the plane of oscillation and the path is changed by the angle between the two

segments at the point where they meet. Therefore, the total change of the pendulum’s plane of

oscillation — the pendulum’s phase shift α(C) — is the sum of these three discrete changes,

minus π. (The π comes from the fact that in a Euclidean plane, where the phase shift is zero, the

sum of the angles of a triangle equals π.) Thus the pendulum’s total phase shift is given by the

following formula:

α(C) = θ1 + θ2 + θ3 − π (3)

We combine this with the definition given above for local curvature. In other words, equation

(3) is the Gauss–Bonnet formula for a geodesic triangle,∫κ dS = θ1 + θ2 + θ3 − π, where the

integration is carried out over the surface of the triangle.

The quantity α(C) is the solid angle subtended at the center of the sphere by the triangle

defined by the path C. The above result is reminiscent of the Berry phase example of a spin in a

magnetic field where Berry phase is the solid angle of the path in parameter space.

Foucault Pendulum Trajectories: Paths of Constant Latitude on the Sphere

Consider now a pendulum moving around a circle C at a fixed latitude θ0. (In the case of a

Foucault pendulum, of course, instead of someone moving the pendulum along the fixed-latitude

circle C, it is simply carried along such a pathway by the Earth’s rotation.) Although C is not a

great circle, we can approximate it by a spherical polygon — that is, the union of a large number of

short segments of great circles, where each such segment runs along the path for a short distance

(see Figure 2, part C). The pendulum keeps a fixed angle with respect to each of the three geodesic

segments. The net phase shift of a Foucault pendulum is thus equal to the sum of the three vertex

angles of the spherical polygon, which, by Gauss–Bonnet, equals the solid angle subtended at the

center of the earth by the cyclic path.

In summary, unlike a cyclic pathway followed on a flat surface, which makes a system always

return to its initial state (a holonomic process), a cyclic pathway on a curved surface results in a

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FIG. 1: The parallel transport of a vector in a flat space and in a curved space.

0

A

B

C

FIG. 2: (A) A spherical triangle, whose three sides are segments of great circles. When a pendulum

(whose oscillation direction is represented by the arrows) is taken along such a triangular path, the angle

of the pendulum’s swing with respect to each great-circle segment remains constant. Thus, only the angles

between two segments contribute to the anholonomy — that is, to the overall angular shift of the pendulum.

(B) The path of the Foucault pendulum at latitude θ0, over the course of one day. (C) A circle of fixed

latitude approximated by segments of great circles tangent to it.

mismatch between the system’s initial and final states.

By describing the classical Foucault pendulum as an oscillator transported

around a cyclic pathway on a surface of a sphere, we “geometrize” it.

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III. BERRY MAGNETISM — EFFECTIVE VECTOR POTENTIAL AND MONOPOLES

The mathematics underlying the Foucault pendulum and the Berry phase are quite similar .

Rather surprisingly, it turns out that fictitious magnetic monopoles — the exotic hypothetical

particles are hidden in the mathematics of anholonomy. The key variable in an anholonomic

system is γ, the shift after the system goes around some cyclic pathway and returns home. For

the Foucault pendulum, for example, γ is the total twist angle, or the precession angle, of the

pendulum after a 24-hour period has elapsed.

γ

2π=

Ω

2π=

1

2π

∫S

~Beff · ds ≡ χ =1

2π

∫S

κ ds (4)

Here ~Beff can be interpreted as the magnetic field of a fictitious monopole as ~Beff = R/R2.

This equation reveals the profound fact that anholonomy can be thought of as a kind of curvature.

We note that χ will be an integer — a topological quantum number — provided that we integrate

over the entire surface S.

In the Berry-phase literature, the quantity κ, which emerges out of the analogies like a rabbit

popping out of a hat, is known as the “Berry curvature”.

Perhaps you will experience an “aha” moment upon seeing how cleverly nature exploits one

single idea but realizes it differently in many highly diverse contexts. Michael Berry himself put it

as follows: “A circuit tracing a closed path in an abstract space can explain both the curious shift

in the wave function of a particle and the apparent rotation of a pendulum’s plane of oscillation.”

In other words, although the classical and the quantum systems that exhibit anholonomy involve

very different physics, they have topological aspects that are captured in one and the same

mathematical language.

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