rne

them 6x

Lx

Notice that X ke nµI e a Tn l

In the nth Landau level there are

Ie

N I Ly Jd LILY217 e

O

AIB Ad2M He

N Aff 7 Not

For u levels N UNE

Dynamics

VGtl L.SE dganCky4i heiHtket

wave packet

4 crit 12 4Gt

semiclassical motion

tocyclotron motion of wavepacket

Current flowing in single Landau level state 4h

Warm up problem for the case with an

electric field

Heuristically Ig e Vy eDEEI

O

Microscopically

eits

efer x key

Ouest Landa Level

eBx

Tychy em 4nF x y f ih0gteaf4nhcx.usLx

Ig JC Jg x y DX

O

Efx keyI met

d e Kk eBx

eBfx keytg

c f eahRe Ty Jdxe

Efx keyIg e we 1 x hee 0

As expected

Current in a Field

Fix a Ii

Y ExU x e Ex U C e y e Ex

H t iz moi x ke'T eEx

Pj Imai x 2k e'xx e'T t Ex

En Imai E zfee eE.yx ifeeeE.fi

Imf key free Eu

En tmuifx xij iimuifxi xi.itxiTEx.EE TE.fE IIznuEpxiitEuii E

eExi TIMED

H PEI Imai x Xi eEXk f mu

C kn true htt t ee Xt t I mu

Vy In fz et ke Ee Fz T

Physically

Iy neighg

Lx TTejT

Microscopicallye B x Xn te

Ig em yef dx e If the EB x

em fnei.gg dx e f the es x Kia

ester EEy

Is f E I I

Total current in a Landau level is then

Iy dTE JdiedEkn Eh Mr Me

e Yu e Vr Vi r Ms

Iy EU Vr oxy Ivh

Oxy O

Sxy hervEEE

e

Effin

F MREEF t.EE

Edge states run along

equipotentialB

LAUGHLIN's FLUX ARGUMENT

B

FL edue oh

µ5 A IoT

f III.x.meO

gA nfe1fribbon NO CHANCE TO ENERGIES OF

BULK

4nF e x Xo

Xo f time Eg end NocuanoeTfBuckENE EIXo 7 Xo t A TRANSFERING CHARGE

B FROM LEFT TO RIGHT

0h needy

of HeIn Ife 0V nejat

INTEGER Oxy

PERCOCATIONPICTURET

Disorder is essential for the 7 QUE

Require a gapin the excitation spectrum

fnI ED

i FIXED

equipment

Q

f Raise EI

00 tI

ftp.dPERCOLATLON

Percolation will occur 0

µF nth two

When the shoreline percolates the orbital

period diverges t the excitation gapvanisher

so that dissipation takes place at the

transition between plateaux

x

5 FRACTIONAL QUANTUM HALL EFFECT

Early exptsshoved that for v e 1 oxy oxy

vanished as expected from the perspective of the IQUE

However with the inverhoi of modulation doping in GaAs

quantum wells much higher quality sampleswere possible with

much less disorder

A Wigner crystal was expected forlou enough e concentrations

in pure oumplerPinned

by disorder itwould be insulating

1982 Toni Stormer Gossard discovered a Clk plateau

at v Yg wth ai 70 oxy 43 e 1h

Tsui joked that it might be quarksThe effect did

not involve quarks but incredibly theelectrons in the r g

FQH state have indeed condensed into a state uh fractional

charge exahahoii.gg 4ze

Many more fractions observed

Because 0 70 this is a dittipahrinless state

with agap premmeaby

driver by the e e

Coulomb interaction

The new ideas developed explore understandThese

new phases of matter their tpological properties have

had profound implications for physics both in the lab

and the cosmos

5 1 PRELIMINARIES MechanicalMomentum.tGuidingCerter

Recall µ Etzel TI2cm

where MECHANICALLYTT Matty p t e ATT momentum

This is a gauge invariant quantitybut unlike the canonical

momentum Mx Ny do not commute

Th my fpx eAy3 ifeAx py1e f ihdx AyIf ihdy Ax

hxiyt eH I i