Electron Interactions and Nanotube Fluorescence Spectroscopy

1
Topological Insulatorsand Topological Band Theory
2
The Quantum Spin Hall Effectand Topological Band
Theory

• I. Introduction
• - Topological band theory
• II. Two Dimensions Quantum Spin Hall
  Insulator
• - Time reversal symmetry Edge States
• - Experiment Transport in HgCdTe
  quantum wells
• III. Three Dimensions Topological Insulator
• - Topological Insulator Surface States
• - Experiment Photoemission on
  BixSb1-x and Bi2Se3
• IV. Superconducting proximity effect
• - Majorana fermion bound states
• - A platform for topological quantum
  computing?

Thanks to Gene Mele, Liang Fu, Jeffrey Teo,
Zahid Hasan group (expt)
3
The Insulating State
Characterized by energy gap absence of low
energy electronic excitations
Covalent Insulator
Atomic Insulator
The vacuum
e.g. intrinsic semiconductor
e.g. solid Ar
electron
Dirac Vacuum
4s
Egap 10 eV
Egap 2 mec2 106 eV
3p
Egap 1 eV
positron hole
Silicon
4
The Integer Quantum Hall State
2D Cyclotron Motion, Landau Levels
E
Energy gap, but NOT an insulator
Quantized Hall conductivity
Jy
Ex
B
Integer accurate to 10-9
5
Graphene
E
k
www.univie.ac.at
Novoselov et al. 05
Low energy electronic structure Two Massless
Dirac Fermions
Haldane Model (PRL 1988)

• Add a periodic magnetic field B(r)
• Band theory still applies
• Introduces energy gap
• Leads to Integer quantum Hall state

The band structure of the IQHE state looks just
like an ordinary insulator.
6
Topological Band Theory
The distinction between a conventional insulator
and the quantum Hall state is a topological
property of the manifold of occupied states
Classified by the Chern (or TKNN) topological
invariant (Thouless et al, 1982)
The TKNN invariant can only change at a quantum
phase transition where the energy gap goes to zero
Insulator n 0 IQHE state
sxy n e2/h
Analogy Genus of a surface g holes
g0
g1
7
Edge States
Gapless states must exist at the interface
between different topological phases
IQHE state n1
Vacuum n0
n1
n0
y
x
Smooth transition gap must pass through zero
Edge states skipping orbits
Gapless Chiral Fermions E v k
Band inversion Dirac Equation
E
Mgt0
Egap
Egap
Mlt0
Domain wall bound state y0
ky
K
K
Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980)
Haldane Model
8
Quantum Spin Hall Effect in Graphene
Kane and Mele PRL 2005
The intrinsic spin orbit interaction leads to a
small (10mK-1K) energy gap
Simplest model Haldane2 (conserves Sz)
J?
J?
E
Bulk energy gap, but gapless edge states
Spin Filtered edge states
vacuum
?
?
QSH Insulator

• Edge states form a unique 1D electronic conductor
• HALF an ordinary 1D electron gas
• Protected by Time Reversal Symmetry
• Elastic Backscattering is forbidden. No 1D
  Anderson localization

9
Topological Insulator A New B0 Phase
There are 2 classes of 2D time reversal invariant
band structures Z2 topological invariant n
0,1 n is a property of bulk bandstructure, but
can be understood by considering the edge states
Edge States for 0ltkltp/a
n1 Topological Insulator
n0 Conventional Insulator
E
E
Kramers degenerate at time reversal invariant
momenta k -k G
k0
kp/a
k0
kp/a
10
Quantum Spin Hall Insulator in HgTe quantum wells
Theory Bernevig, Hughes and Zhang, Science 2006
Predict inversion of conduction and valence
bands for dgt6.3 nm ? QSHI
Expt Konig, Wiedmann, Brune, Roth, Buhmann,
Molenkamp, Qi, Zhang Science 2007
dlt 6.3 nm normal band order conventional insulator
Landauer Conductance G2e2/h
dgt 6.3nm inverted band order QSH insulator
G2e2/h
Measured conductance 2e2/h independent of W for
short samples (LltLin)
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3D Topological Insulators
There are 4 surface Dirac Points due to Kramers
degeneracy
ky
kx
OR
2D Dirac Point
How do the Dirac points connect? Determined by
4 bulk Z2 topological invariants n0 (n1n2n3)
Surface Brillouin Zone
n0 1 Strong Topological Insulator
EF
Fermi circle encloses odd number of Dirac
points Topological Metal 1/4 graphene
Robust to disorder impossible to localize
n0 0 Weak Topological Insulator
Fermi circle encloses even number of Dirac
points Related to layered 2D QSHI
12
Theory Predict Bi1-xSbx is a topological
insulator by exploiting
inversion symmetry of pure Bi, Sb (Fu,Kane
PRL07) Experiment ARPES (Hsieh et al. Nature
08)
Bi1-xSbx

• Bi1-x Sbx is a Strong Topological
• Insulator n0(n1,n2,n3) 1(111)
• 5 surface state bands cross EF
• between G and M

Bi2 Se3
ARPES Experiment Y. Xia et al., Nature
Phys. (2009). Band Theory H.
Zhang et. al, Nature Phys. (2009).

• n0(n1,n2,n3) 1(000) Band inversion at G
• Energy gap D .3 eV A room temperature
• topological insulator
• Simple surface state structure
• Similar to graphene, except
• only a single Dirac point

EF
Control EF on surface by exposing to NO2
13
Superconducting Proximity Effect
Fu, Kane PRL 08
Surface states acquire superconducting gap D due
to Cooper pair tunneling
s wave superconductor
Topological insulator
-k?
BCS Superconductor
(s-wave, singlet pairing)
k?
Superconducting surface states
-k
?
Dirac point
?
?
(s-wave, singlet pairing)
?
Half an ordinary superconductor Highly nontrivial
ground state
k
14
Majorana Fermion at a vortex
Ordinary Superconductor Andreev bound
states in vortex core
E
D
Bogoliubov Quasi Particle-Hole redundancy
E ?,?
0
-E ?,?
-D
Surface Superconductor Topological zero
mode in core of h/2e vortex
E

• Majorana fermion
• Particle Anti-Particle
• Half a state
• Two separated vortices define one zero energy
• fermion state (occupied or empty)

D
0
E0
-D
15
Majorana Fermion

• Particle Antiparticle g g
• 
  
• Real part of Dirac fermion g YY Y g1i
  g2 half an ordinary fermion
• Mod 2 number conservation ? Z2 Gauge symmetry
  g ? g

• Potential Hosts
• Particle Physics
• Neutrino (maybe)
• - Allows neutrinoless double b-decay.
• - Sudbury Neutrino Observatory
• Condensed matter physics Possible due to pair
  condensation
• Quasiparticles in fractional Quantum Hall
  effect at n5/2
• h/4e vortices in p-wave superconductor Sr2RuO4
• s-wave superconductor/ Topological Insulator
• among others....
• Current Status NOT OBSERVED

16
Majorana Fermions and Topological Quantum
Computation
Kitaev, 2003

• 2 separated Majoranas 1 fermion Y g1i
  g2
• 2 degenerate states (full or empty)
• 1 qubit
• 2N separated Majoranas N qubits
• Quantum information stored non locally
• Immune to local sources decoherence
• Adiabatic braiding performs unitary operations

Non-Abelian Statistics
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Manipulation of Majorana Fermions
Control phases of S-TI-S Junctions
Majorana present
Tri-Junction A storage register for
Majoranas
Create A pair of Majorana
bound states can be created from the vacuum in a
well defined state 0gt.
Braid A single Majorana can be moved
between junctions. Allows braiding of
multiple Majoranas
Measure Fuse a pair of
Majoranas. States 0,1gt distinguished by
presence of quasiparticle. supercurrent across
line junction
E
E
E
0
0
0
f-p
f-p
f-p
0
0
0
18
Conclusion

• A new electronic phase of matter has been
  predicted and observed
• - 2D Quantum spin Hall insulator
  in HgCdTe QWs
• - 3D Strong topological insulator
  in Bi1-xSbx , Bi2Se3 and Bi2Te3
• Superconductor/Topological Insulator structures
  host Majorana Fermions
• - A Platform for Topological Quantum
  Computation
• Experimental Challenges
• - Transport Measurements on
  topological insulators
• - Superconducting structures
• - Create, Detect Majorana
  bound states
• - Magnetic structures
• - Create chiral edge states,
  chiral Majorana edge states
• - Majorana interferometer
  
• Theoretical Challenges
• - Effects of disorder on surface
  states and critical phenomena
• - Protocols for manipulating and
  measureing Majorana fermions.