Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,

Introduction to fractional quantum Hall effect

Milica V. Milovanović

Institute of Physics BelgradeScientific Computing Laboratory

(Talk at Physics Faculty, Belgrade, 2010)

Hall experiment:

Classical picture in 2D:

)cen(

BjEj 00

0

0

nec/B

nec/Bˆ

jˆE

00 /1

T= 85mK N/NePlateaus ! Rigidity ! filling factor =

J.P.Eisenstein and H.L.Stormer, Science 248,1461(1990)

Quantum Mechanics: a particle in 2D in perpendicular B

2

Ac

qp

m2

1H

Velocity: Ac

qp

0qB,B

c

qi, ijji

yxyx ii

Bq2

cb

Bq2

cb

1b,b

2

1bbH c

mc

eBc

Landau levels

0H,Ac

qp

qB

cl0H,R

lerR B

2B

z

1l10R, B

2

iRRZ

2

iRRZ yxyx

1Z,Z

Z,Z act in a fixed Landau level

Summary:

m,n|seigenvalue:RR,

guiding center coordinate )Z,Z(R

cyclotronvelocity

Degeneracy of Landau level:

2

pL

2

LL

ip,xiR,R

yx

ijji

N

ehcBS

l2

LL2B

yx

unit of flux“flux quantum”

quantum Hall effect:

x~p.e.iiy,x

i]p,y[ip,x

y

yx

onedimensional problem

tconfinemenB

disorder

x

localizedstates inbulk

no backscattering!0xx

localizedstates

jˆldEld

t

BSd

c

1

dt

d

c

1

Insertion of one fluxquantum does notchange spectrum! andexpect transfer of integer(i) number of electrons

eiq,ce

h,qc H

2H ei

h

integer quantum Hall effect (Laughlin argument)

In rotationally symmetric gauge in two dimensions: iyxz

Single particle wave functions:

2|z|4

1m ez

1N,,0m

Orbits at radius: m2r2

Imagine that we are at the middle of the plateau at 1/3 -

How the ground state of the system would look like?

Laughlin answer:

mj

jii )zz(

2i |z|

4

1

e

3m 3

1

N

Ne

antisymmetry

and in the cases of other “hierarchical constructions” odd denominator expected!

R.B. Laughlin, PRL 50, 1995 (1983)

W. Pan et al.,PRL 83, 3530 ,1999.

FQHE at 5/2 !

R. Willet et al., PRL 59, 1776, 1987

Moore-Read answer:

2j

jii )zz(

)zz(

1

)zz(

1A

ee N1N21

Pfaffian

Pfaffian part describes a pairing amongparticles as in a superconductor =

BCS pairing of spinless fermions

G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)

)zz(

1

)zz(

1

)zz(

1

)zz(

1

)zz(

1

)zz(

1

324142314321

Pfaffian for 4 particles:

Pfaffian (p-wave superconductor)

otherwise we would have

Fermi-liquid-like state

(a) 5/2 : numerics favorable for Pfaffian in 2nd LL Pfaffian is the most simple ansatz if not only explanation of plateau R.H. Morf, PRL 80, 1505 (1998), E.H. Rezayi and F.D.M. Haldane, PRL 84, 4685 (2000)

(b) 1/2 : exps. and numerics find Fermi-liquid-like phase (no plateau) E. Rezayi and N. Read, PRL 72, 900 (1994)

at 1/2 (1/4) in WQWs (wide quantum wells):

signatures of FQHE – minima in !xx

likely nature of these states is multi-component (two-component)

J. Shabani et al., Phys. Rev. Lett. 103, 256802 (2009)

theory (mathematical identity)

Pf331)(A

two-component:

)wz()ww()zz( qqp

p3

llk

k3

jji

i331

Pf state can lead to a first topological quantum computer!

We want to know how to make Pfaffian!

?Pf)t(tunneling

331

The quest for Pfaffian begins!