FIG. 1: Allowed energies of electrons in a crystal in a magnetic field vs magnetic flux. Empty or white

regions are the forbidden energies.

I. HARPER’S EQUATION DESCRIBING ELECTRONS IN A CRYSTAL IN A MAGNETIC

FIELD

In the tight binding approximation, the eigenstates for an electron in a square lattice with flux

φ = Ba2/φ0 are given by Harper’s equation,

ψm+1 + ψm−1 + 2 cos(2πφm− ky)ψm = Eψm (1)

This equation was derived in 1955 by Philip Harper and is known as the Harper’s equation.

Note, again, the two-dimensional problem is reduced to one-dimensional problem.

Important parameter φ

The parameter φ that appears in Harper equation is the pure number φ (“pure” in the sense

that it has no units attached to it), which measures the magnetic field’s strength in an extremely

natural fashion. This pure number, with no units at all, tell us how strong a magnetic field is. It

all hinges on the fact that there is a fundamental amount of magnetic flux — the flux quantum,

equal to hc/e — that emerges intrinsically out of quantum mechanics. This minuscule quantity is

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an inherent fact about our universe, just as are the speed of light and the charge on the electron.

Given that this tiny amount of flux is the natural chunk of flux, it is as if nature had handed us a

measuring stick on a silver platter! This beautiful and generous favor on nature’s part must not be

ignored.

2C1 1 0 0 . . e−ikx

1 2C2 1 0 0 . 0

0 1 2C3 1 0 . 0

. . . . . . .

. . . . . . .

. . . . . . .

eikx . 0 0 0 1 2Cq

ψ1

ψ2

ψ3

.

.

.

ψq

= E

ψ1

ψ2

ψ3

.

.

.

ψq

where Cn = cos(2πnφ − ky). The lower-left and upper-right corner terms e±ikx in this matrix

reflect Bloch’s theorem, which assumes periodic boundary conditions — namely, ψn = eikxψn+q.

In a handful of very simple cases, such as φ = 1, 12, 13, 14, the eigenvalues E can be determined

analytically. However, for q > 4, the above matrix-eigenvalue problem can only be solved

numerically.

Rational vs Irrational Flux

Let φ = pq.

For rational flux, Harper equation describes a system that is periodic with period q. Such

systems can be studied using “ Bloch Theorem”.

Bloch’s theorem

Named after physicist Felix Bloch, Bloch’s theorem states that the energy eigenstates of

an electron moving in a crystal (a periodic potential) can be written in the following form:

Ψ(r) = eik·ru(r) (2)

where u(r) is a periodic function with the same periodicity as that of the underlying potential

— that is, u(r) = u(r + a). The exponential preceding the periodic function u is a kind of helical

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wave, or “corkscrew”, which multiplies the the wave function by a spatially-changing phase that

twists cyclically as one moves through space in a straight line.

Some Analytic Results

(I) Analytic expressions for the energy dispersions E(kx, ky) for a few simple cases

(a) For φ = 1, the energy spectrum consists of a single band given by

E = 2(cos kx + cos ky).

(b) For φ = 1/2, the two bands having energies E+ and E− are given by

E± = ±2√

cos2 kx + cos2 ky.

(c) For φ = 1/3, the three bands having energies E0, E1, and E2 are given by

Ei = 2√

2 cos(θ ± i23π). Here, θ = 1

3arccos[(cos 3kx + cos 3ky)/2

√2].

(d) For φ = 1/4, the four bands having energies E++, E+−, E−+, and E−− are given

by the expression E±± = ±√

4± 2[3 + 12(cos 4kx + cos 4ky)]2.

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• Using analytical expressions in four cases, understand what is meant by a band.

• For every rational flux p/q, there are q bands. In other words, a single band for zero magnetic

field splits into q bands when magnetic flux is a rational number with denominator q.

• For q-even, two bands touch at E=0.

• For irrational value of φ, we have infinity of bands – known as a Cantor set ( A type of

fractal) . It turns out that for each irrational, the sum of the band widths is zero, that is, the

Cantor set has a zero measure. The proof that the set forms a Cantor set has a long history

and is known as Ten Martini Problem.

• Smooth gaps in otherwise fractal structure.

• As we will discuss later, the integers are the Berry phases in units of 2π . That is the

anholonomy or total curvature as we integrate over (kx, ky) plane – the Brillouin zone. It

turns out that these integers are the quantum numbers of Hall conductivity.

Recall: the concept of Brillouin zone

This notion was developed by the French physicist Leon Brillouin (1889–1969). For

any crystal lattice in three-dimensional physical space, there is a “dual lattice” called the

reciprocal lattice, which exists in an abstract space whose three dimensions are inverse

lengths. This space lends itself extremely naturally to the analysis of phenomena involving

wave vectors (because their dimensions are inverse lengths). If we limit ourselves to

crystals whose lattices are perfectly rectangular (as has generally been done in this book),

then given a lattice whose unit cell has dimensions a × b × c, the reciprocal lattice’s unit

cell will have dimensions 1a× 1

b× 1

c. This cell is called the first Brillouin zone. The various

locations in the Brillouin zone — wave vectors — act as indices labeling the different Bloch

states.

In two-dimension, the Brillouin zone is a torus.

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II. SIMPLEST MODEL OF QUANTUM HALL: FLUX - 1/2

A square lattice immersed in a magnetic field, where electrons can only tunnel to its

neighboring sites is described by Harper’s equation:

ψm+1 + ψm−1 + 2 cos(2πφm− ky)ψm = Eψm (3)

In the above model, all rational flux-values except for φ = 1/2 support a quantum Hall state

due to the presence of a gap.

However, flux-1/2 is the simplest case as the Hamiltonian is a 2 × 2 matrix and hence

describes a spin-1/2 model in a magnetic field.

Mapping to Spin-1/2 in a Magnetic Field

Let us denote hopping along x and y directions as Jx and Jy. For the square lattice described

above, we have set Jx = Jy = 1.

H(kx, ky) = −~h · ~σ, (4)

hx = 2Jx cos kx; hy = 2Jy cos ky; hz = 0 (5)

With hz = 0, we have θ = 0. This means there is no Berry phase in the problem.

Generalized Harper Model

Consider a generalization of the above model where electrons can also tunnel along the

diagonal. This opens a gap at φ = 1/2 as shown in Fig. (2). This also maps to a spin in a

magnetic field problem.

This simple case of flux-value 12, a generalization of Harper’s equation in which the crystal

electrons can hop along diagonals (in addition to nearest-neighbor hopping) turns out to be the

ideal model to describe a quantum Hall system. Such a system maps exactly onto a spin-12

situation

in a magnetic field, as described above. With the diagonal hopping, which we will denote by Jd,

the Hamiltonian of this system can be transformed as follows :

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H(kx, ky) = −~h · ~σ, (6)

hx = 2Jx cos kx; hy = 2Jy cos ky; hz = 4Jd sin kx sin ky (7)

Unlike the spin-12

situation described above, where the parameter space can be realized as a

sphere, the parameter space here is the space of Bloch wave-number vectors (kx, ky). This space

— a Brillouin zone — is not a sphere but a torus. If we are dealing with a rational magnetic

flux-value p/q, then the edges of the Brillouin zone will be defined by ky lying within the interval

[−π/q,+π/q] on one axis (as can be seen from the periodicity of the cosine in Harper equation ,

and kx lying within the interval [−π,+π] along the perpendicular axis.

Although the problem is thereby mapped to a spin-12

problem in a magnetic field, we note

that there is no point in the Brillouin zone where h = 0 (which is the location of the monopole).

Therefore, the fictitious monopole in the quantum Hall problem exists outside the reciprocal space

. It can be seen if we consider samples with edges where from the crossing-point of the two energy

eigenvalues of the edge modes, we can pinpoint the monopole inside the gap.

FIG. 2: Butterfly plot when electrons are allowed to tunnel to the diagonal.

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III. QUANTIZATION OF HALL CONDUCTIVITY

• Calculate Berry phase using the generalized model. Note that the parameter space is

(kx, ky) which is a torus. If you integrate over all values of the parameter, you get an integer

Recall: Gauss-Bonnet Theorem: Week9.pdf...

• Where is the monopole ???

• Thouless and his collaborators showed that Hall conductivity is given by the same formula

as the Berry phase in units of 2π, multiplied by e2/h.

• Note that integrating over all values of (kx, ky) means that the band is full and we have

an insulator. If more than more band is full, we need to sum over all the filled bands.

The gap after the last filled band has an integer labeling that gives the quantization of Hall

conductivity.

IV. SPIN AND STATISTICS

“Spin is a mysterious beast, and yet its practical effects prevails over the whole of science.

The existence of spin, and the statistics associated with it, is the most subtle and ingenious design

of nature– without it the whole universe would collapse.”... Sin-itiro Tomonaga

Another weird aspect of spin is that although spin character of particles and relation between

spin and statics exists in non-relativistic quantum mechanics, theory of relativistic quantum

mechanics is essential in understanding it. According to Richard Feynman, “It appears to be one

of the few places in physics where there is a rule which can be stated very simply, but for which

no one has found a simple and easy explanation. The explanation is down deep in relativistic

quantum mechanics. This probably means that we do not have a complete understanding of the

fundamental principle involved”.

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V. BERRY PHASE AND GAUGE THEORIES

Recently, ideas of gauge theories have emerged in the study of geometric phases. This is due

to the emergence of gauge theories of interactions of elementary particles.

Many gauge theoretical ideas appear in geometric phase and conversely, ideas associated

with geometric phases clarify some subtle issues in quantum field theory.

Gauge theories are important as the successful field theories explaining the dynamics of

elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry

group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being

the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group

U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons

and eight gluons.

Gauge theories are also important in explaining gravitation in the theory of general relativity.

Its case is somewhat unique in that the gauge field is a tensor, the Lanczos tensor. Theories

of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a

gauge boson known as the graviton. Historically, these ideas were first stated in the context of

classical electromagnetism and later in general relativity. However, the modern importance of

gauge symmetries appeared first in the relativistic quantum mechanics of electrons quantum

electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter,

nuclear and high energy physics among other subfields.

History

• The earliest field theory having a gauge symmetry was Maxwell’s formulation, in 1864-65,

of electrodynamics . The importance of this symmetry remained unnoticed in the earliest

formulations.

• Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the

invariance of the action under a general coordinate transformation. Later Hermann Weyl, in

an attempt to unify general relativity and electromagnetism, conjectured that Eichinvarianz

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or invariance under the change of scale (or ”gauge”) might also be a local symmetry of

general relativity.

• After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London

modified gauge by replacing the scale factor with a complex quantity and turned the scale

transformation into a change of phase, which is a U(1) gauge symmetry. This explained the

electromagnetic field effect on the wave function of a charged quantum mechanical particle.

This was the first widely recognised gauge theory, popularised by Pauli in the 1940s.

• In 1954, attempting to resolve some of the great confusion in elementary particle physics,

Chen Ning Yang and Robert Mills introduced non-abelian gauge theories as models to

understand the strong interaction holding together nucleons in atomic nuclei. (Ronald

Shaw, working under Abdus Salam, independently introduced the same notion in his

doctoral thesis.) Generalizing the gauge invariance of electromagnetism, they attempted

to construct a theory based on the action of the (non-abelian) SU(2) symmetry group on

the isospin doublet of protons and neutrons. This is similar to the action of the U(1) group

on the spinor fields of quantum electrodynamics. In particle physics the emphasis was on

using quantized gauge theories.

• This idea later found application in the quantum field theory of the weak force, and

its unification with electromagnetism in the electroweak theory. Gauge theories became

even more attractive when it was realized that non-abelian gauge theories reproduced a

feature called asymptotic freedom. Asymptotic freedom was believed to be an important

characteristic of strong interactions. This motivated searching for a strong force gauge

theory. This theory, now known as quantum chromodynamics, is a gauge theory with the

action of the SU(3) group on the color triplet of quarks. The Standard Model unifies the

description of electromagnetism, weak interactions and strong interactions in the language

of gauge theory.

• The importance of gauge theories in physics is exemplified in the tremendous success of

the mathematical formalism in providing a unified framework to describe the quantum field

theories of electromagnetism, the weak force and the strong force. This theory, known as the

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Standard Model, accurately describes experimental predictions regarding three of the four

fundamental forces of nature, and is a gauge theory with the gauge group SU(3)×SU(2)×

U(1) . Modern theories like string theory, as well as general relativity, are, in one way or

another, gauge theories.

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