1. Topological Insulators and Topological Band Theory
k=La
k=Lb E
k=La
k=Lb

2. The Quantum Spin Hall Effect and 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 4s
Dirac
Vacuum
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 Haldane Model (PRL 1988) + -k
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. is a topological property of the manifold of occupied states
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
g=0
g=1

7. Edge States Gapless Chiral Fermions : E = v k IQHE state n=1 Vacuum
Gapless states must exist at the interface between different topological phases
IQHE state
n=1
Vacuum
n=0
n=1
n=0 y x
Edge states ~ skipping orbits
Smooth transition : gap must
pass through zero
Gapless Chiral Fermions : E = v k
Band inversion – Dirac Equation E
M>0
Egap
Egap
M<0
Domain wall bound state y0 K’ K ky
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:
|Haldane|2
(conserves Sz) J↑ J↓ E
Bulk energy gap, but gapless edge states
Spin Filtered edge states
Edge band structure ↑ ↓
p/a k
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 B=0 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 0<k<p/a
n=0 : Conventional Insulator
n=1 : Topological Insulator E E
Kramers degenerate at
time reversal
invariant momenta
k* = -k* + G
k*=0
k*=p/a
k*=0
k*=p/a

10. Quantum Spin Hall Insulator in HgTe quantum wells
Theory: Bernevig, Hughes and Zhang, Science 2006
HgTe
HgxCd1-xTe d
Predict inversion of conduction and valence
bands for d>6.3 nm → QSHI
Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007
d< 6.3 nm
normal band order
conventional insulator
Landauer Conductance G=2e2/h ↑ ↓ V I
d> 6.3nm
inverted band order
QSH insulator
G=2e2/h
Measured conductance 2e2/h independent of W for short samples (L<Lin)

11. 3D Topological Insulators
There are 4 surface Dirac Points due to Kramers degeneracy ky L4 L1 L2 L3 E
k=La
k=Lb E
k=La
k=Lb 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. Bi1-xSbx Bi2 Se3 Insulator n0;(n1,n2,n3) = 1;(111) between G and M
Theory: Predict Bi1-xSbx is a topological insulator by exploiting
inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07)
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 ↑,↓
-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
E=0 -D

15. Majorana Fermion Potential Hosts : Current Status : NOT OBSERVED
Particle = Antiparticle : g = g†
Real part of Dirac fermion : g = Y+Y†; Y = g1+i 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 n=5/2
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 = g1+i 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

17. Manipulation of Majorana Fermions
Control phases of S-TI-S Junctions f1 f2 + -
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 |0>.
Braid
A single Majorana can be
moved between junctions.
Allows braiding of multiple
Majoranas
Measure
Fuse a pair of Majoranas.
States |0,1> distinguished by
• presence of quasiparticle.
• supercurrent across line
junction E E E
f-p
f-p
f-p

18. Conclusion
A new electronic phase of matter has been predicted and observed
- 2D : Quantum spin Hall insulator in HgCdTe QW’s
- 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.