introduction to fractional quantum hall effect milica v. milovanović institute of physics belgrade...
TRANSCRIPT
![Page 1: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/1.jpg)
Introduction to fractional quantum Hall effect
Milica V. Milovanović
Institute of Physics BelgradeScientific Computing Laboratory
(Talk at Physics Faculty, Belgrade, 2010)
![Page 2: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/2.jpg)
Hall experiment:
![Page 3: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/3.jpg)
Classical picture in 2D:
)cen(
BjEj 00
0
0
nec/B
nec/Bˆ
jˆE
00 /1
![Page 4: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/4.jpg)
T= 85mK N/NePlateaus ! Rigidity ! filling factor =
J.P.Eisenstein and H.L.Stormer, Science 248,1461(1990)
![Page 5: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/5.jpg)
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
![Page 6: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/6.jpg)
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
![Page 7: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/7.jpg)
Summary:
m,n|seigenvalue:RR,
guiding center coordinate )Z,Z(R
cyclotronvelocity
![Page 8: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/8.jpg)
Degeneracy of Landau level:
2
pL
2
LL
ip,xiR,R
yx
ijji
N
ehcBS
l2
LL2B
yx
unit of flux“flux quantum”
![Page 9: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/9.jpg)
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
![Page 10: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/10.jpg)
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)
![Page 11: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/11.jpg)
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?
![Page 12: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/12.jpg)
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)
![Page 13: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/13.jpg)
W. Pan et al.,PRL 83, 3530 ,1999.
FQHE at 5/2 !
R. Willet et al., PRL 59, 1776, 1987
![Page 14: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/14.jpg)
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)
![Page 15: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/15.jpg)
)zz(
1
)zz(
1
)zz(
1
)zz(
1
)zz(
1
)zz(
1
324142314321
Pfaffian for 4 particles:
![Page 16: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/16.jpg)
Pfaffian (p-wave superconductor)
otherwise we would have
Fermi-liquid-like state
![Page 17: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/17.jpg)
(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)
![Page 18: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/18.jpg)
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)
![Page 19: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/19.jpg)
theory (mathematical identity)
Pf331)(A
two-component:
)wz()ww()zz( qqp
p3
llk
k3
jji
i331
![Page 20: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/20.jpg)
Pf state can lead to a first topological quantum computer!
We want to know how to make Pfaffian!
?Pf)t(tunneling
331
![Page 21: Introduction to fractional quantum Hall effect Milica V. Milovanović Institute of Physics Belgrade Scientific Computing Laboratory (Talk at Physics Faculty,](https://reader036.vdocument.in/reader036/viewer/2022062314/56649e895503460f94b8de64/html5/thumbnails/21.jpg)
The quest for Pfaffian begins!