Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a...
Transcript of Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a...
![Page 1: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/1.jpg)
Topological Phases under Strong Magnetic Fields
Mark O. Goerbig
ITAP, Turunc, July 2013
![Page 2: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/2.jpg)
Historical Introduction
What is the common point between
• graphene,
• quantum Hall effects
• and topological insulators?
... and what is it?
![Page 3: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/3.jpg)
The 1920ies: Band Theory
• quantum treatment of (non-interacting) electrons in aperiodic lattice
• bands = energy of the electrons as a function of aquasi-momentum
![Page 4: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/4.jpg)
Chrystal Electrons and Bloch’s Theorem
• discrete translations T = exp(ip ·Rj)represent a symmetry in a Bravais lattice(described by lattice vectors Rj)
• the operator p (generator ofdiscrete translations) plays therole of a momentum(quasi-momentum or latticemoment)
R j
![Page 5: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/5.jpg)
Chrystal Electrons and Bloch’s Theorem
• discrete translations T = exp(ip ·Rj)represent a symmetry in a Bravais lattice(described by lattice vectors Rj)
• the eigenvalues of p are goodquantum numbers: energy bandsǫl(p)
• Bloch functions:
ψ(r) =∑
p
eip·r/~up(r)
![Page 6: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/6.jpg)
Bravais and Arbitrary Lattices
arbitrary lattice=
Bravais lattice+
basis (of N “atoms”)
M. C. Escher (decomposed)
triangular lattice + complicated basis
There are as many electronic bands as atoms per unit cell
![Page 7: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/7.jpg)
Band Structure and Conduction Properties
I II I II
gap
metal (2D)
energy
Fermi
level
momentum
electron metal hole metal
insulator (2D)Fermi
level
semimetal (2D) Fermi
level
Fermilevel
density
of states
energy
![Page 8: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/8.jpg)
1950-70: Many-Body Theory
• Physical System described by an order parameter(a) ∆k = 〈ψ†
−k,↑ψ†k,↓〉 (superconductivity)
(b) Mµ(r) =∑
τ,τ ′〈ψ†τ (r)σ
µτ,τ ′ψτ ′(r)〉 (ferromagnetism)
• Ginzburg-Landau theory of second-order phase transitions(1957)
∆ = 0(disordered)
↔∆ 6= 0
(ordered)
• symmetry breaking(a) broken (gauge) symmetry U(1)(b) broken (rotation) symmetry O(3)
• emergence of (collective) Goldstone modes(a) superfluid mode, with ω ∝ |k|(b) spin waves, with ω ∝ |k|2
![Page 9: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/9.jpg)
The Revolution(s) of the 1980ies
3 essential discoveries:
• integer quantum Hall effect (1980, v. Klitzing, Dorda,Pepper)
• fractional quantum Hall effect (1982, Tsui, Störmer,Gossard)
• high-temperature superconductivity (1986, Bednorz, Müller)
![Page 10: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/10.jpg)
Integer Quantum Hall Effect (I)
8 12 160 4Magnetic Field B (T)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ xy
(h/e
)2
0
0.5
1.0
1.5
2.0
ρΩ
xx(k
)
2/3 3/5
5/9
6/11
7/15
2/53/74/9
5/11
6/13
7/13
8/15
1 2/3 2/
5/7
4/5
3 4/
Vx
VyIx
4/7
5/34/3
8/57/5
123456
Magnetic Field B[T]
[mesurement by J. Smet et al., MPI-Stuttgart]
QHE = plateau in Hall res. & vanishing long. res.
![Page 11: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/11.jpg)
Integer Quantum Hall Effect (II)
Quantised Hall resistance at low temperatures
RH =h
e21
n
h/e2: universal constantn: quantum number (topological invariant)
• result independent of geometric and microscopic details
• quantisation of high precision (> 109)
⇒ resistance standard: RK−90 = 25 812, 807Ω
![Page 12: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/12.jpg)
Fractional Quantum Hall Effect
partially filled Landau level → Coulomb interactions relevant
1983: Laughlin’s N -particle wave function
• no (local) order parameter associated with symmetrybreaking
• no Goldstone modes• quasi-particles with fractional charges and statistics
1990ies : description in terms of topological (Chern-Simons)field theories
![Page 13: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/13.jpg)
The Physics of the New Millenium
• simulation of condensed-matter models with optical lattices(cold atoms)
• 2004 : physics of graphene (2D graphite)
• 2005-07 : topological insulators
![Page 14: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/14.jpg)
Graphene – First 2D Crystal
• honeycomb lattice =two triangular (Barvais) lattices
AB
B
B
e3
e1
2e
band structure
![Page 15: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/15.jpg)
Band Structure and Conduction Properties (Bis)
I II I II
gap
metal (2D)
energy
Fermi
level
momentum
electron metal hole metal
Fermi
levelinsulator (2D)
semimetal (2D) Fermi
level
graphene (undoped)Fermi
level
Fermilevel
energy
density
of states
![Page 16: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/16.jpg)
Topological Insulators
generic form of a two-band Hamiltonian:
H = ǫ0(q)1+∑
j=x,y,z
ǫj(q)σj
• Haldane (1988): anomalous quantum Hall effect → quantumspin Hall effect (QSHE)
• Kane and Mele (2005): graphene with spin-orbit coupling• Bernevig, Hughes, Zhang (2006): prediction of a QSHE in
HgTe/CdTe quantum wells• König et al. (2007): experimental verification of the QSHE
⇒ 3D topological insulators (mostly based on bismuth):surface states ∼ ultra-relativistic massless electrons
![Page 17: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/17.jpg)
Outline of the Classes
Mon 1: Introduction and Landau quantisation
Mon 2: Landau-level degeneracy and disorder/confinementpotential
Tue 1: Issues of the IQHE
Tue 2: Towards the FQHE, Laughlin’s wave function and itsproperties
![Page 18: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/18.jpg)
Further Reading
• D. Yoshioka, The Quantum Hall Effect, Springer, Berlin (2002).
• S. M. Girvin, The Quantum Hall Effect: Novel Excitations and BrokenSymmetries, Les Houches Summer School 1998http://arxiv.org/abs/cond-mat/9907002
• G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101(2003).http://arxiv.org/abs/cond-mat/0205326
• M. O. Goerbig, Quantum Hall Effectshttp://arxiv.org/abs/0909.1998
![Page 19: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/19.jpg)
1. Introduction To the Integer Quantum Hall
Effect and Materials
![Page 20: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/20.jpg)
Classical Hall Effect (1879)
B
I
longitudinal Hallresistance resistance
C1
C4
C2 C3
C5C6
2D electron gas_ _ _ _ _ _
++ + + ++
Quantum Hall system :2D electrons in a B-field
Hal
l res
ista
nce
magnetic field B
RH(b)
Hall resistance:
RH = B/enel
Drude model (classical stationary equation):
dp
dt= −e
(
E+p
m×B
)
−p
τ= 0
![Page 21: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/21.jpg)
Shubnikov-de Haas Effect (1930)H
all r
esis
tanc
e
magnetic field B
long
itudi
nal r
esis
tanc
e
Bc
(a)
Den
sity
of s
tate
s
EnergyEF
hωC
(b)
oscillations in longitudinal resistance→ Einstein relations σ0 ∝ ∂nel/∂µ ∝ ρ(ǫF )→ Landau quantisation (into levels ǫn)
σ0 ∝ ρ(ǫF ) ∝∑
n
f(ǫF − ǫn)
![Page 22: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/22.jpg)
Quantum Hall Effect (QHE)
8 12 160 4Magnetic Field B (T)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ xy
(h/e
)2
0
0.5
1.0
1.5
2.0ρ
Ωxx
(k)
2/3 3/5
5/9
6/11
7/15
2/53/74/9
5/11
6/13
7/13
8/15
1 2/3 2/
5/7
4/5
3 4/
Vx
VyIx
4/7
5/34/3
8/57/5
123456
Magnetic Field B[T]
QHE = plateau in RH & RL = 0
1980 : Integer quantum Hall effect (IQHE)1982 : Fractional quantum Hall effect (FQHE)
![Page 23: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/23.jpg)
Metal-Oxide Field-Effect Transistor (MOSFET)
conductionband
acceptorlevels
valenceband
conductionband
acceptorlevels
valenceband
conductionband
acceptorlevels
valenceband
E
z
FE
z
F
E
z
F
(a)
(b) (c)
VVG
G
metal oxide(insulator)
semiconductor
metal oxide(insulator)
semiconductor metal oxide(insulator)
II
I
VG
z
z
E
E
E
1
0
metaloxide
semiconductor
2D electrons
usually silicon-based materials (Si/SiO2 interfaces)
![Page 24: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/24.jpg)
GaAs/AlGaAs Heterostructure
dopants
AlGaAs
z
EF
GaAs
dopants
AlGaAs
z
EF
GaAs(a) (b)
2D electrons
Impurity levels farther away from 2DEG (as compared toSi/SiO2)
⇒ enhanced mobility (FQHE)
![Page 25: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/25.jpg)
Nobel Prize in Physics 2010 : Graphene
Kostya Novoselov Andre Geim
"for groundbreaking experiments regarding the two-dimensionalmaterial graphene"
![Page 26: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/26.jpg)
What is Graphene?
2s
2p 2p 2p 2px y z zsp spsp
Hybridation sp 2
120o
graphene = 2D carbon crystal(honeycomb)
sp hybridisation2
![Page 27: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/27.jpg)
Graphene and its Family
2D
3D 1D 0D
![Page 28: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/28.jpg)
From Graphite to Graphene
(strong) covalent bondsin the planes
(weak) van der Waalsbonds between the planes
![Page 29: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/29.jpg)
How to Make Graphene: Recipe (1)
put thin graphite chip on scotch−tape
![Page 30: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/30.jpg)
How to Make Graphene: Recipe (2)
~10 : fold scotch−tape on graphite chip and undo :
![Page 31: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/31.jpg)
How to Make Graphene: Recipe (3)
2glue (dirty) scotch−tape on substrate (SiO )
![Page 32: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/32.jpg)
How to Make Graphene: Recipe (4)
lift carefully scotch−tape from substrate
![Page 33: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/33.jpg)
How to Make Graphene: Recipe (5)
place substrate under optical microscope
orientationmarks
thick graphite
graphene ?
less thickgraphite
![Page 34: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/34.jpg)
How to Make Graphene: Recipe (6)
zoom in region where there could be graphene
![Page 35: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/35.jpg)
Electronic Mesurement of Graphene
SiO
Si dopé
V
2
g
Novoselov et al., Science 306,p. 666 (2004)
![Page 36: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/36.jpg)
2. Landau Quantisation and Integer
Quantum Hall Effect
![Page 37: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/37.jpg)
Infrared Transmission Spectroscopy
10 20 30 40 50 60 70 80
0.96
0.98
1.00
B
E
2L3L
2L
3L
0L
1L
Be2cE1 ~1L
1E
1E
A
B
C
D
B
E
2L3L
2L
3L
0L
1L
Be2cE1 ~1L
1E
1E
A
B
C
D
(D)(C)
(B)
Rel
ativ
e tra
nsm
issi
on
Energy (meV)
(A)
0.4 T1.9 K
0.0 0.5 1.0 1.5 2.00
10
20
30
40
50
60
70
80 )(32 DLL )(23 DLL
)(12 CLL )(21 CLL
)(01 BLL )(10 BLL
)(21 ALL
Tran
sitio
n en
ergy
(meV
)
sqrt(B)
10 20 30 40 50 60 70 80 900.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
Rel
ativ
e tra
nsm
issi
on
Energy (meV)
1 T
0.4T
2T4T
10 20 30 40 50 60 70 80 90
0.99
1.00
0.7T
0.2T
0.3T
0.5T
Grenoble high−field group: Sadowski et al., PRL 97, 266405 (2007)
transition C
transition B
rela
tive
tran
smis
sion
rela
tive
tran
smis
sion
Energy [meV]
Energy [meV]
Tra
nsm
issi
on e
nerg
y [m
eV]
Sqrt[B]
selectionrules :
λ, n→ λ′, n±1
![Page 38: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/38.jpg)
Edge States
ymaxn+1
ν = n ν =
n−1
yymaxymaxn n−1
n+1
n
n−1
(a)
(b)
y
xν = n+1
µ
LLs bended upwards atthe edges (confinementpotential)
chiral edge states⇒ only forward scattering
ν= n+1 ν= n ν= n−1
![Page 39: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/39.jpg)
Four-terminal Resistance Measurement
I I
R ~
56
2 3
41
R ~ µ − µ = µ − µ
3µ − µ = 02
5
L
H
µ = µµ = µ2 LL3
µ = µ = µ6 5 R
3 R L
: hot spots [Klass et al, Z. Phys. B:Cond. Matt. 82, 351 (1991)]
![Page 40: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/40.jpg)
IQHE – One-Particle Localisation
n
ε
(n+1)
ν
NL
(a)
density of states
RxyxxR
B=n
h/e n2
FE
![Page 41: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/41.jpg)
IQHE – One-Particle Localisation
n
ε
n
ε(b)
(n+1)
ν
NL
(a)
density of statesdensity of states
RxyxxR
B
EF
RxyxxR
B=n
h/e n2
FE
![Page 42: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/42.jpg)
IQHE – One-Particle Localisation
n
ε
n
ε
n
ε(b) (c)
(n+1)
ν
NL
(a)
density of states density of states density of states
extended states
localised states
RxyxxR
B
EF
Rxy
B
xx
EF
R
h/e (n+1)
h/e n2
2
RxyxxR
B=n
h/e n2
FE
![Page 43: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/43.jpg)
IQHE in Graphene Novoselov et al., Nature 438, 197 (2005)
Zhang et al., Nature 438, 201 (2005)
V =15V
Density of states
B=9T
T=30mK
T=1.6K
∼ ν
∼ 1/ν
Graphene IQHE:
R = h/e
at = 2(2n+1)
at = 2n
ν
ν
H ν2
(no Zeeman)
Usual IQHE:
g
![Page 44: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/44.jpg)
Percolation Model – STS Measurement
2DEG on n-InSb surface Hashimoto et al., PRL 101, 256802 (2008)
(a)-(g) dI/dV for different values of sample potentials (lower spinbranch of LL n = 0)
(i) calculated LDOS for a given disorder potential in LL n = 0
(j) dI/dV in upper spin branch of LL n = 0
![Page 45: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/45.jpg)
Fractional Quantum Hall Effect beyond Laughlin
8 12 160 4Magnetic Field B (T)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ xy
(h/e
)2
0
0.5
1.0
1.5
2.0ρ
Ωxx
(k)
2/3 3/5
5/9
6/11
7/15
2/53/74/9
5/11
6/13
7/13
8/15
1 2/3 2/
5/7
4/5
3 4/
Vx
VyIx
4/7
5/34/3
8/57/5
123456
Magnetic Field B[T]
QHE = plateau in RH & RL = 0
FQHE series: ν = p/(2sp+ 1) = 1/3, 2/5, 3/7, 4/9, ...
![Page 46: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/46.jpg)
Jain’s Wavefunctions (1989)
Idea: “reinterpretation” of Laughlin’s wavefunction
ψLs (zj) =
∏
i<j(zi − zj)2sχp=1(zj)
∏
i<j(zi − zj)2s: “vortex” factor (2s flux quanta per vortex)
χp=1(zj) =∏
i<j(zi − zj):wavefunction at ν∗ = 1
![Page 47: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/47.jpg)
Jain’s Wavefunctions (1989)
Idea: “reinterpretation” of Laughlin’s wavefunction
ψLs (zj) =
∏
i<j(zi − zj)2sχp=1(zj)
∏
i<j(zi − zj)2s: “vortex” factor (2s flux quanta per vortex)
χp=1(zj) =∏
i<j(zi − zj):wavefunction at ν∗ = 1
Generalisation to integer ν∗ = p
ψJs,p(zj) = PLLL
∏
i<j(zi − zj)2sχp(zj, zj)
χp(zj, zj): wavefunction for p completely filled levelsPLLL: projector on lowest LL (→ analyticity)
![Page 48: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/48.jpg)
Physical Picture: Composite Fermions
CF = electron+”vortex” (carrying 2s flux quanta)with renormalised field coupling eB → (eB)∗
ν = 1/3
pseudo−vortex
electronic filling 1/3theory
CF
1 filled CF level
electron
"free" flux quantum
(with 2 flux quanta)
composite fermion (CF)
![Page 49: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/49.jpg)
Physical Picture: Composite Fermions
CF = electron+”vortex” (carrying 2s flux quanta)with renormalised field coupling eB → (eB)∗
ν = 1/3
ν = 2/5
pseudo−vortex
theory
CF
2 filled CF levels
electron
"free" flux quantum
(with 2 flux quanta)
composite fermion (CF)
electronic filling 1/3 1 filled CF level
At ν = p/(2ps+ 1) ↔ ν∗ = nel/n∗B = p, with n∗
B = (eB)∗/h:FQHE of electrons = IQHE of CFs
![Page 50: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/50.jpg)
Generalisation: FQHE at half-filling
• 1987: Obervation of a FQHE at ν = 5/2, 7/2 (evendenominator)
• 1991: Proposal of a Pfaffian wave function (Moore & Read;Greiter, Wilzcek & Wen)
ψMR(zj) = Pf
(
1
zi − zj
)
∏
i<j
(zi − zj)2
⇒ quasiparticle charge e∗ = e/4 with non-Abelian statistics
• Further generalisations to ν = K/(K + 2): Read & Rezayi
![Page 51: Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a two-band Hamiltonian: H= ǫ0(q) 1 + X j=x,y,z ǫj(q)σj • Haldane (1988): anomalous quantum](https://reader033.fdocuments.net/reader033/viewer/2022053004/5f07deec7e708231d41f2b04/html5/thumbnails/51.jpg)
Multicomponent SystemsLa
ndau
leve
ls
|+>
|−>d
ν = 1/2
ν = 1/2 ν = ν + ν = 1
+
+ −− T
: A sublattice : B sublattice
τ
τ
2
3
1
2
e 1e
e
spin + isospin : SU(4)
A physical spin: SU(2)
two−fold valley degeneracy
B bilayer: SU(2) isospin
SU(2) isospin
C graphene (2D graphite)
(doubling of LLs)
exciton