Majorana Fermions and Topological Insulators

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
Dirac Vacuum
4s
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)
Band structure
B(r) 0 Zero gap, Dirac point B(r) ? 0 Energy
gap sxy e2/h
k
Egap
4
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)
Berrys connection
Berrys curvature
1st Chern class
Trivial Insulator n 0 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
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
Mgt0
E
Egap
Egap
Mlt0
Domain wall bound state y0
ky
K
K
Jackiw, Rebbi (1976) Su, Schrieffer, Heeger (1980)
Haldane Model
Bulk Edge Correspondence Dn Chiral
Edge Modes
6
Time Reversal Invariant ?2 Topological Insulator
Time Reversal Symmetry
All states doubly degenerate
Kramers Theorem
?2 topological invariant (n 0,1) for 2D
T-invariant band structures
n1 Topological Insulator
n0 Conventional Insulator
Edge States
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)
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

0
p/a
2p/a
?
?
?
?
II. HgCdTe quantum wells
HgTe
HgxCd1-xTe
Theory Bernevig, Hughes and Zhang, Science
06 Experiement Konig et al. Science 07
d
HgxCd1-xTe
d lt 6.3 nm Normal band order
d gt 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
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
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
10
Topological Superconductor, Majorana Fermions
BCS mean field theory
Bogoliubov de Gennes Hamiltonian
Particle-Hole symmetry Quasiparticle
redundancy
(Kitaev, 2000)
1D ?2 Topological Superconductor n 0,1
Discrete end state spectrum
END
n0 trivial
n1 topological
E
Majorana Fermion bound state
D
D
E
E0
0
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 n5/2
• 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 pip 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
0
SC
-D
TI
Quasiparticle Bound state at E0
Majorana Fermion g0
2. Superconductor-magnet interface at edge of 2D
QSHI
S.C.
M
mgt0
Egap 2m
QSHI
mlt0
Domain wall bound state g0
15
1D Majorana Fermions on Topological Insulators
1. 1D Chiral Majorana mode at superconductor-magne
t interface
E
M
SC
kx
TI
Half a 1D chiral Dirac fermion
2. S-TI-S Josephson Junction
f p
f
0
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
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
17
A Z2 Interferometer for Majorana Fermions 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 QWs
• - 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