1. Majorana Fermions and Topological Insulators
Charles L. Kane, University of Pennsylvania
Topological Band Theory
- Integer Quantum Hall Effect
- 2D Quantum Spin Hall Insulator
- 3D Topological Insulator
- Topological Superconductor
Majorana Fermions
- Superconducting Proximity Effect on Topological Insulators
- A route to topological quantum computing?
Thanks to Gene Mele, Liang Fu, Jeffrey Teo

2. 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

3. The Integer Quantum Hall State
2D Cyclotron Motion, Landau Levels E
Hall Conductance sxy = n e2/h
IQHE without Landau Levels (Haldane PRL 1988)
Graphene in a periodic magnetic field B(r)
B(r) = 0
Zero gap,
Dirac point
B(r) ≠ 0
Energy gap
sxy = e2/h
Band structure + - k
Egap

4. 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
The set of occupied Bloch wavefunctions defines a U(N)
vector bundle over the torus.
Classified by the first Chern class (or TKNN invariant) (Thouless et al, 1984)
Berry’s connection
Berry’s curvature
1st Chern class
Trivial Insulator: n = Quantum Hall state: sxy = n e2/h
The TKNN invariant can only change at a phase transition
where the energy gap goes to zero

5. Edge States Bulk – Edge Correspondence : Dn = # Chiral Edge Modes
Gapless states must exist at the interface between different topological phases
IQHE state
n=1
Vacuum
n=0
n=1
n=0 y x
Smooth transition : gap must
pass through zero
Edge states ~ skipping orbits
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
Bulk – Edge Correspondence : Dn = # Chiral Edge Modes

6. time reversal invariant momenta
Time Reversal Invariant 2 Topological Insulator
Time Reversal Symmetry :
Kramers’ Theorem :
All states doubly degenerate
2 topological invariant (n = 0,1) for 2D T-invariant band structures
n=0 : Conventional Insulator
n=1 : Topological Insulator E
k*=0
k*=p/a
Edge States E
k*=0
k*=p/a
Kramers degenerate at
time reversal invariant momenta
k* = -k* + G
n is a property of bulk bandstructure. Easiest to compute if there is extra symmetry:
1. Sz conserved : independent spin Chern integers : (due to time reversal) J↑ J↓ E
Quantum spin Hall Effect :
2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states

7. 2D Quantum Spin Hall Insulator↑
I. Graphene
Kane, Mele PRL ‘05 Eg ↓
Intrinsic spin orbit interaction
 small (~10mK-1K) band gap
Sz conserved : “| Haldane model |2”
Edge states : G = 2 e2/h
p/a
2p/a ↑ ↓ ↑ ↓
II. HgCdTe quantum wells
HgTe
Theory: Bernevig, Hughes and Zhang, Science ’06
Experiement: Konig et al. Science ‘07
HgxCd1-xTe d
HgxCd1-xTe
d < 6.3 nm
Normal band order
d > 6.3 nm:
Inverted band order
G ~ 2e2/h in QSHI E E
Normal
G6 ~ s
G8 ~ p k
Inverted
G8 ~ p
G6 ~ s
Conventional Insulator
QSH Insulator

8. 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 2 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

9. 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

10. Topological Superconductor, Majorana Fermions
BCS mean field theory :
Bogoliubov de Gennes
Hamiltonian
Particle-Hole symmetry :
Quasiparticle redundancy :
1D 2 Topological Superconductor : n = 0,1
(Kitaev, 2000)
Discrete end state spectrum :
END
n=0
“trivial”
n=1
“topological” E
Majorana Fermion
bound state D D E
E=0 -E -D -D
“half a state”

11. Periodic Table of Topological Insulators and Superconductors
Kitaev, 2008
Schnyder, Ryu,
Furusaki, Ludwig 2008
Anti-Unitary Symmetries :
- Time Reversal :
- Particle - Hole :
Unitary (chiral) symmetry :
Complex
K-theory
Altland-
Zirnbauer
Random
Matrix
Classes
Real
K-theory
Bott Periodicity

12. Majorana Fermion : spin 1/2 particle = antiparticle ( g = g† )
Potential Hosts :
Particle Physics :
Neutrino (maybe) Allows neutrinoless double b-decay.
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
Topological Quantum Computing Kitaev, 2003
2 Majorana bound states = 1 fermion bound state
- 2 degenerate states (full/empty) = 1 qubit
2N separated Majoranas = N qubits
Quantum Information is stored non locally
- Immune from local sources of decoherence
Adiabatic Braiding performs unitary operations
- Non Abelian Statistics

13. Proximity effects : Engineering exotic gapped
states on topological insulator surfaces m
Dirac Surface States :
Protected by Symmetry
1. Magnetic : (Broken Time Reversal Symmetry)
Fu,Kane PRL 07
Qi, Hughes, Zhang PRB (08)
Orbital Magnetic field :
Zeeman magnetic field :
Half Integer quantized Hall effect :
M. ↑
T.I.
2. Superconducting : (Broken U(1) Gauge Symmetry)
proximity induced
superconductivity
Fu,Kane PRL 08
S-wave superconductor
Resembles spinless p+ip superconductor
Supports Majorana fermion excitations
S.C.
T.I.

14. Majorana Bound States on Topological Insulators
1. h/2e vortex in 2D superconducting state E D
h/2e SC -D TI
Quasiparticle Bound state at E= Majorana Fermion g0
2. Superconductor-magnet interface at edge of 2D QSHI M
S.C.
m>0
Egap =2|m|
QSHI
m<0
Domain wall bound state g0

15. 1D Majorana Fermions on Topological Insulators
1. 1D Chiral Majorana mode at superconductor-magnet interface E M SC kx TI
: “Half” a 1D chiral Dirac fermion
2. S-TI-S Josephson Junction
f = p f
f  p SC SC TI
Gapless non-chiral Majorana fermion for phase difference f = p

16. 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

17. e e e h A Z2 Interferometer for Majorana Fermions g2 g1 g2 -g2 g1
A Signature for Neutral Majorana Fermions Probed with Charge Transport
N even g2 e e g1
N odd
Chiral electrons on magnetic domain wall split
into a pair of chiral Majorana fermions
“Z2 Aharonov Bohm phase” converts an
electron into a hole
dID/dVs changes sign when N is odd. g2
-g2 e h g1
Fu and Kane, PRL ‘09
Akhmerov, Nilsson, Beenakker, PRL ‘09

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, Bi2Te3
Superconductor/Topological Insulator structures host Majorana Fermions
- A Platform for Topological Quantum Computation
Experimental Challenges
- Charge and Spin 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
- Further manifestations of Majorana fermions and non-Abelian states
- Effects of disorder and interactions on surface states