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Majorana Fermions and Topological Insulators
I. Topological Band Theory
- Integer Quantum Hall Effect - 2D Quantum Spin Hall Insulator - 3D Topological Insulator - Topological Superconductor
II. Majorana Fermions
- Superconducting Proximity Effect on Topological Insulators - A route to topological quantum computing?
Thanks to Gene Mele, Liang Fu, Jeffrey Teo
Charles L. Kane, University of Pennsylvania
The Insulating State
Covalent Insulator
Characterized by energy gap: absence of low energy electronic excitations
The vacuumAtomic Insulator
e.g. solid Ar
Dirac Vacuum
Egap ~ 10 eV
e.g. intrinsic semiconductor
Egap ~ 1 eV3p
4s
Silicon
Egap = 2 mec2 ~ 106 eV
electron
positron ~ hole
The Integer Quantum Hall State2D Cyclotron Motion, Landau Levels
IQHE without Landau Levels (Haldane PRL 1988)
Graphene in a periodic magnetic field B(r)
gap cE E
Hall Conductance xy = n e2/h
Band structureB(r) = 0Zero gap, Dirac point
B(r) ≠ 0Energy gapxy = e2/h
Egap
k
Topological Band TheoryThe distinction between a conventional insulator and the quantum Hall state
is a topological property of the manifold of occupied states
2( ) : ( ) Bloch Hamiltonans Brillouinwith
energy gap
zone H Tk
Trivial Insulator: n = 0 Quantum Hall state: xy = n e2/h
The TKNN invariant can only change at a phase transition where the energy gap goes to zero
Classified by the first Chern class (or TKNN invariant) (Thouless et al, 1984)
2Tr
2 T
in
F
The set of occupied Bloch wavefunctions defines a U(N)vector bundle over the torus.
1
( )N
i iu
k
ij i jA u du
d F A A A
Berry’s connection
Berry’s curvature
1st Chern class
Edge States
Gapless states must exist at the interface between different topological phases
IQHE staten=1
Egap
Domain wall bound state 0
Vacuumn=0
Edge states ~ skipping orbits
n=1 n=0
Band inversion – Dirac Equation
x
y
M<0
M>0
Gapless Chiral Fermions : E = v k
E
Haldane Modelky
Egap
KK’
Smooth transition : gap must pass through zero
Jackiw, Rebbi (1976)Su, Schrieffer, Heeger (1980)
Bulk – Edge Correspondence : n = # Chiral Edge Modes
Time Reversal Invariant 2 Topological Insulator
2 topological invariant (= 0,1) for 2D T-invariant band structures
=0 : Conventional Insulator =1 : Topological Insulator
Kramers degenerate at
time reversal invariant momenta
k* = k* + G
E
k*=0 k*=/a
E
k*=0 k*=/a
Edge States
1. Sz conserved : independent spin Chern integers : (due to time reversal) n n
is a property of bulk bandstructure. Easiest to compute if there is extra symmetry:
, mod 2n
2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states
Quantum spin Hall Effect :J↑ J↓
E
Time Reversal Symmetry :
Kramers’ Theorem :
1 ( )H H k k *yi 2 1 All states doubly degenerate
2D Quantum Spin Hall Insulator
Theory: Bernevig, Hughes and Zhang, Science ’06Experiement: Konig et al. Science ‘07
HgTe
HgxCd1-xTe
HgxCd1-xTed
d < 6.3 nmNormal band order
d > 6.3 nm: Inverted band order
Conventional Insulator QSH Insulator
E
k
E
II. HgCdTe quantum wells
I. Graphene
• Intrinsic spin orbit interaction small (~10mK-1K) band gap
• Sz conserved : “| Haldane model |2”
• Edge states : G = 2 e2/h
Kane, Mele PRL ‘05
0 /a 2/a
↑
↓
↑
↑ ↓
↓
Eg
6 ~ s
8 ~ p 6 ~ s
8 ~ p Normal
Inverted
G ~ 2e2/h in QSHI
3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracy
Surface Brillouin Zone
2D Dirac Point
E
k=a k=b
E
k=a k=b
0 = 1 : Strong Topological Insulator
Fermi circle encloses odd number of Dirac points
Topological Metal :
1/4 graphene
Robust to disorder: impossible to localize
0 = 0 : Weak Topological Insulator
Fermi circle encloses even number of Dirac pointsRelated to layered 2D QSHI
OR
4
1 2
3
EF
How do the Dirac points connect? Determined by 4 bulk 2 topological invariants 0 ; (123)
kx
ky
Bi1-xSbx
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-x Sbx is a Strong Topological Insulator 0;(1,2,3) = 1;(111)
• 5 surface state bands cross EF
between and M
ARPES Experiment : Y. Xia et al., Nature Phys. (2009).Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3
• 0;(1,2,3) = 1;(000) : Band inversion at
• Energy gap: ~ .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 byexposing to NO2
Topological Superconductor, Majorana Fermions
††( ) k
k k BdGk k
cH c c H k
c
Particle-Hole symmetry :
Quasiparticle redundancy :
Discrete end state spectrum :
1( ) ( )BdG BdGH k H k † E E E E
0
*0
BdG
HH
H
0
E
E
-E0
E=0
Bogoliubov de GennesHamiltonian
=0“trivial”
=1“topological”
†0 0
Majorana Fermionbound state
“half a state”
1D 2 Topological Superconductor : = 0,1
END
†E E
E
(Kitaev, 2000)
BCS mean field theory : † † † † *
Periodic Table of Topological Insulators and Superconductors
Anti-Unitary Symmetries : - Time Reversal : - Particle - Hole :
Unitary (chiral) symmetry :
1( ) ( ) 12 ; H H k k
1( ) ( ) 12 ; H H k k
1( ) ( ) ; H H k k
RealK-theory
ComplexK-theory
Bott Periodicity
Altland-ZirnbauerRandom MatrixClasses
Kitaev, 2008Schnyder, Ryu, Furusaki, Ludwig 2008
Majorana Fermion : spin 1/2 particle = antiparticle ( † )
Particle Physics : • Neutrino (maybe) Allows neutrinoless double -decay.
Condensed matter physics : Possible due to pair condensation
• Quasiparticles in fractional Quantum Hall effect at =5/2• h/4e vortices in p-wave superconductor Sr2RuO4
• s-wave superconductor/ Topological Insulator ... among others
Current Status : NOT OBSERVED
† † 0
• 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 a ab bU
1 2i
Potential Hosts :
Topological Quantum Computing Kitaev, 2003
Proximity effects : Engineering exotic gappedstates on topological insulator surfaces
†0 ( v )H i
† † *
Dirac Surface States :
Protected by Symmetry
1. Magnetic : (Broken Time Reversal Symmetry)
• Orbital Magnetic field :
• Zeeman magnetic field :
• Half Integer quantized Hall effect :
2. Superconducting : (Broken U(1) Gauge Symmetry)
proximity induced superconductivity
p p eA †
zM 21
( )2xy
en
h
• S-wave superconductor
• Resembles spinless p+ip superconductor
• Supports Majorana fermion excitations
Fu,Kane PRL 07Qi, Hughes, Zhang PRB (08)
Fu,Kane PRL 08
M. ↑
T.I.
S.C.
T.I.
Majorana Bound States on Topological Insulators
SC
h/2e
1. h/2e vortex in 2D superconducting state
2. Superconductor-magnet interface at edge of 2D QSHI
Quasiparticle Bound state at E=0 Majorana Fermion
TI
0
E
†0 0
S.C.M
QSHIEgap =2|m|
Domain wall bound state 0
m<0
m>0
| | | |S Mm
1D Majorana Fermions on Topological Insulators
2. S-TI-S Josephson Junction
SC
TI
SC
SCM
1. 1D Chiral Majorana mode at superconductor-magnet interface
TI
kx
E
†k k : “Half” a 1D chiral Dirac fermion Fv xH i
0
Gapless non-chiral Majorana fermion for phase difference
cos( / 2)Fv L x L R x R L RH i i
Manipulation of Majorana FermionsControl phases of S-TI-S Junctions
12
0
Majorana present
Tri-Junction : A storage register for Majoranas
CreateA pair of Majorana boundstates can be created from the vacuum in a well definedstate |0>.
BraidA single Majorana can bemoved between junctions.Allows braiding of multipleMajoranas
MeasureFuse a pair of Majoranas.
States |0,1> distinguished by• presence of quasiparticle.• supercurrent across line junction
E
00
1E
E
0
0
0
0
0
0
0
A Z2 Interferometer for Majorana Fermions
A Signature for Neutral Majorana Fermions Probed with Charge Transport
2
hNe
e e
e h
1
2
1
2 2
N even
N odd
†1 2
1 2
c i
c i
• 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.
Fu and Kane, PRL ‘09Akhmerov, Nilsson, Beenakker, PRL ‘09
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