PH 364 Topologicalphases

s.FI iiTouw uomemrna I8IEEt god1 A particle of change e and mass m T

confined in a ring of radius as.in ringthe x y f Cane There is a magneticthx OI threaded along 7 direction q q q qa show that its energy gig fluxspectrumisshifted by

En z th t n O Il IzTwhee fo It he is the quantum of magnetic fluxCt c D

b One can disposeof the gauge potential in theSchrodingerb equation of the ware fume in VIII byagauge transformation that changes the wave function

tovice yay diestacos do2

G aisle Cosshow that the effect of the external flux can be

enforced by the conditions that the Petrelsphase9 Ct oft new wavefunction satisfies a twisted

boundary condition do 00C Io2 TO 7as the particle encloses the ring

2 A humor BohmEIBfitment w.it Ta mqg netie monopoleAiring a Dirac envisioned a infinitesimally

thin and infinitely long solenoid ITINrEee figs

The end ol such a solenoid ycan be thought of a magnetic 1 X

mini L it ae tenfmtfiEiEusolenoid gocenoid

Suppose a point monopole with magnetic t

charge g sits at the end of the solenoid F D so thatits magnetic field is

B g IzShow that the existence of a magnetic monopgle demands

that the vectorpotential II is singular find the locusofsingularity Hint vise spherical co ordinatesDeeign aithought.ABelectron interference experimentsick that an electron of charge e completes a

closed path enclosing the solenoid it's wavefunctionremains

single valued If the election's waketuition remainshigh values this is as good as it does not see the

existence of the solenoidShow that the single valued condition is satisfied

if the magnetic charge g is quantized in unit ofelectric charge e od t l

Show that the existence of the magnetic monopoleof any arbitrary magnetic charge g impliesquantization of electric charge ie

3 For a 2b Italia bar with fixed the field B

and fixed electric field E stitch the regions ofclassical Sd Hr A quantum oscillation and integer

quantum Hall IAH stales as a function of

mobility In encl m All formulas and definitionsof E e nm are given in the class 100

4 Derive a tight bonding model for a Hof'stenderlattice with two sub lattices or D and magneticflux of fShow that the tight bindingmodel has chiral

symmetry5 In a RD Hall bar the magnetic field B is ailing

2 direction and electric field as along y directionFor this case the bulk Landau levels become dispersiveFind the dispersion relation and its drift velocity

for the same setup approximate the boundary potential

upto its linear order in position Then obtain itsdispersion melatonin and drift velocity torte skip herorbits

Solution

OI star B sea AoA OI on

TEA J P f Iitaody

Im t EFI Tn EnTn

Imf E Fort LEE i fun a

DII ri al Ea date out o

c EE EE'To E Intoy n e nm seq n e

na integerdub En 1 fine Ezo2mA

singleraheed w

f

whichmeans y

orbitalangularmomentumis

conserved

OrbitalAngular momentum Le J CftEAT check8 Lisconsened

tsrJ0dIeoddaIrfdfoffe.Faided do

Enif Aadt toousing Leibnitz formula

2 B e g II Age 8C c coedTwicefruit W gauge

L Ao 8457,2A B phone as e Ardor with anothergame

41cg 2am n integerb

18 722a rifff e t I

FA B A

k Eet ei x

re La b area IIIa sina.ci

Ignominy II diagonalpart halfofatuniteell

It octet'te k

on dx todyde It cosKx dy Sin Kx

ozHTk ot HC so cry is the chiral operator

I

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