Topological Insulators and Superconductors
I. Topological Insulators and Band Theory
Unifying theme: bulk – boundary correspondence
- Integer Quantum Hall Effect
- 2D Quantum Spin Hall Insulator
- 3D Topological Insulator
- Topological Superconductivity, Majorana fermions
II. Summary and Outlook
- What we have accomplished
- Challenges for the Future
Thanks to Gene Mele, Liang Fu, Jeffrey Teo
Charles L. Kane, University of Pennsylvania
The Insulating State
The Integer Quantum Hall State
IQHE with zero net magnetic field
Graphene with a periodic magnetic field B(r) (Haldane PRL 1988)Band structure
B(r) = 0
Zero gap,
Dirac point
B(r) ≠ 0
Energy gap
sxy = e2/h
Eg
k
Eg ~ 1 eV
e.g. Silicon
g cE
E2D Cyclotron Motion,
Landau Levels
sxy = e2/h
+ - + - + - +
+ - + - +
+ - + - + - +
Topological Band TheoryThe distinction between a conventional insulator and the quantum Hall state
is a topological property of the band structure
Insulator : n = 0
IQHE state : sxy = n e2/hThe TKNN invariant can only change at a phase
transition where the energy gap goes to zero
( ) :H k2 B lo c h H a m ilto n a n s
B rillo u in zo n e ( = to ru s T ) w ith e n e
rg y g a p
The set of N occupied Bloch wavefunctions defines a U(N)
vector bundle over the Brillouin zone torus.
Classified by the first Chern number (or TKNN invariant) (Thouless et al, 1984)
Closely related to theory of electric polarization
1
( )N
ii
u
k
2
21 1
2 2C T
n d d
A k F k
1
( ) ( ) ( )
N
i i
i
i u u
- kA k k k
( ) ( ) k
F k A k
Berry connection
Berry curvature
1st Chern number
kx
ky
C/a
/a
-/a
-/a
Edge States
Gapless states must exist at the interface between different topological phases
IQHE state
n=1
Egap
Domain wall bound state
Vacuum
n=0
Edge states ~ skipping orbits
n=1 n=0
Dirac Equation :
x
y
M<0
M>0
Gapless Chiral Fermions : E = v k
E
Haldane Model
ky
Egap
KK‟
Smooth transition : band inversion
Jackiw, Rebbi (1976)
Su, Schrieffer, Heeger (1980)
Bulk – Boundary Correspondence : Dn = # Chiral Edge Modes
( )x x y y z
H i M xs s s - + +v
| |
( ') '/
0
x
y
M x d x
ik y
e e
- v
Time Reversal Invariant 2 Topological Insulator
n=0 : Conventional Insulator n=1 : Topological Insulator
Kramers degenerate at
time reversal
invariant momenta
k* = -k* + G
k*=0 k*=/a k*=0 k*=/a
Even number of bands
crossing Fermi energy
Odd number of bands
crossing Fermi energy
Understand via Bulk-Boundary correspondence : Edge States for 0<k</a
2 topological invariant (n = 0,1) for 2D T-invariant band structures
Time Reversal Symmetry :
Kramers‟ Theorem :
1
( )H H-
-k k *y
i s
21 - All states doubly degenerate
2D Quantum Spin Hall Insulator
Theory: Bernevig, Hughes and Zhang, Science ‟06
Experiement: Konig et al. Science „07
HgTe
HgxCd1-xTe
HgxCd1-xTed
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
Normal
Inverted
G ~ 2e2/h in QSHI
G6 ~ s
G8 ~ p
k
E
Conventional Insulator
2( ) 1
n a +
d < 6.3 nm : Normal band order
G6 ~ s
G8 ~ p k
E
QSH Insulator
2( ) 1
n a -
d > 6.3 nm : Inverted band order
band
inversion
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
n0 = 1 : Strong Topological Insulator
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
OR
4
1 2
3
EF
How do the Dirac points connect? Determined
by 4 bulk 2 topological invariants n0 ; (n1n2n3)
kx
ky
Moore & Balents PRB 07
Roy, cond-mat 06
Fu, Kane & Mele PRL 07
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 n0;(n1,n2,n3) = 1;(111)
• 5 surface state bands cross EF
between G and M
ARPES Experiment : Y. Xia et al., Nature Phys. (2009).
Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3
• 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
Surface Quantum Hall Effect
21
2x y
en
hs
+
2
2xy
e
hs
2
2xy
e
hs
B
n=1 chiral edge state
Orbital QHE :
M↑ M↓
E=0 Landau Level for Dirac fermions. “Fractional” IQHE
Anomalous QHE : Induce a surface gap by depositing magnetic material
Chiral Edge State at Domain Wall : DM ↔ -DM
†
0( v )
zMH i s s D - - +
Mass due to Exchange field
Egap = 2|DM|
EF
0
1
2
-2
-1
TI
2
sg n ( )2
xy M
e
hs D
2
2
e
h+
2
2
e
h-
Topological Magnetoelectric Effect
Consider a solid cylinder of TI with a magnetically gapped surface
21
2x y
eJ E n E M
hs
+
Magnetoelectric Polarizability
M E
EJ
M
21
2
en
h
+
The fractional part of the magnetoelectric polarizability is determined
by the bulk, and independent of the surface (provided there is a gap)
Analogous to the electric polarization, P, in 1D.
Qi, Hughes, Zhang ‟08; Essin, Moore, Vanderbilt „09
d=1 : Polarization P
d=3 : Magnetoelectric
poliarizability
[ ]2
T rB Z
e
A
2
2
2[ ]
4 3T r
B Z
ed
h + A A A A A
formula “uncertainty quantum”
e
2/e h
(extra end electron)
(extra surface
quantum Hall layer)
L D E B
DL
E B
P E
topological “q term”
2
2
e
h q
0 2T R s ym . : o r m o d q
Topological Superconductivity
†
†
1
2B d G
H H
k
Intrinsic anti-unitary particle – hole symmetry
1
B d G B d GH H
- -
†
E E E E
- -
0
*
0
B d G
H
H
H
D
D -
Bogoliubov de Gennes
Hamiltonian
BCS mean field theory : † † † † * D
Bloch - BdG Hamiltonians satisfy
Topological classification problem similar to time reversal symmetry
*x
Particle – hole redundancy
21 +
0 1
1 0x
=
E
k
E
k
1( ) ( )
B d G B d GH H
- - -k k
same state
Bulk-Boundary correspondence : Discrete end state spectrum
0
D
-D
E
E
-E0
D
-D
E=0
n=0 “trivial” n=1 “topological”
†
0 0E E
G G
Majorana fermion
bound state
1D 2 Topological Superconductor : n = 0,1
END
†
E E-G G
EG
(Kitaev, 2000)
Majorana Fermion : Particle = Antiparticle
Real part of a Dirac fermion :
“Half a state”
Two Majorana fermions define a single two level system
Zero mode
†
1 1 2
† †
2 1 2
;
( ) ;
i
i i
+ +
- - - , 2i j i j
†
0 1 2 02H i 0empty
occupied
21
i
†
Periodic Table of Topological Insulators and Superconductors
Anti-Unitary Symmetries :
- Time Reversal :
- Particle - Hole :
Unitary (chiral) symmetry :
1( ) ( ) 1
2 ; H H
- - - k k
1( ) ( ) 1
2 ; H H
- + - k k
1( ) ( ) ; H H
- - k k
Real
K-theory
Complex
K-theory
Bott Periodicity
Altland-
Zirnbauer
Random
Matrix
Classes
Schnyder, Ryu,
Furusaki, Ludwig 2008
Kitaev, 2008
Hasan and Kane, Rev Mod Phys 82, 3045 (2010).
Moore, Nature 464, 194 (2010).
Qi and Zhang, Phys. Today 63, 33 (2010).
Ryu, Schnyder, Furusaki and Ludwig, New J. Phys. 12, 065010 (2010).
Qi and Zhang, Rev Mod Phys, to appear, arXiv:1008.2026.
Moore and Hasan, Annual Review of Condensed Matter, 2, 44 (2010).
Further Reading:
Major accomplishments :
Topological band theory of insulators and superconductors
is well understood:
- Topological Invariants and bulk-boundary correspondence
- Robustness to disorder and weak interactions
- Electromagnetic and/or gravitational response
Rapid materials progress:
- Several materials have been identified and characterized
experimentally.
- Even more materials have been predicted, based on band
structure calculations.
- Detailed characterization of topological insulators via transport,
optics and spectroscopy is developing.
Grand Challenges
• Perfect existing and new materials
• Design and implement heterostructure devices
• Find Majorana
• Classify and characterize many body topological phases
• Find applications for technology
Perfect New and Existing Materials
Real 3D topological insulator materials
are not such great insulators. Electrical
conductance is dominated by the bulk.
Challenge for materials theory in conjunction
with experiments.
Success Story : Bi2Te2Se
Xiong, et al (Princeton) „11
Electrical resistivity in Bi2Se3
(Checkelsky et al „09)
Topological insulator devices
• Topological Insulator – Trivial Insulator
- protect the surface states
- control the surface state Fermi energy
(modulation doping)
• Topological Insulator – Magnetic Insulator
- achieve magnetically gapped surface states
- anomalous quantum Hall effect
- topological magnetoelectric effect
• Topological Insulator – Superconductor
- achieve proximity induced superconductivity
in the surface states.
Requires control interfaces between materials.
Challenge for materials theory and experiment
M. ↑
T.I.
S.C.
T.I.
I.
T.I.
Potential Hosts :
Particle Physics : Neutrino (maybe)
- Allows neutrinoless double b-decay.
Condensed matter physics : Possible due to pair condensation
- n=5/2 Fractional quantum Hall effect
- Topological superconductivity
Topological Quantum Computation
Find Majorana
† †0
Ettore Majorana 1906 – 1938?
Observation of a Majorana fermion is among
the great challenges of physics today
1937 : Majorana publishes his modification
of the Dirac equation that allowes spin ½
particles to be their own antiparticle.
1938 : Majorana mysteriously disappears at sea
What is the best way to achieve topological superconductivity?
SC
M
TI
SC
TI
Majorana bound state at a vortex (0D) 1D Chiral Majorana mode at a interface with
a magnetic material
• Exotic superconductors (superfluids)- Surface of 4He
- p+ip superconductor (eg Sr2RuO4)
- CuxBi2Se3 ?
• Ordinary superconductor heterostructures- superconductor – topological insulator
- superconductor – semiconductor (eg InAs wire)
SC InAs
Majorana
end state
What are the most feasible experimental
signatures of Majorana modes ?
Classify and Characterize Interacting Topological States
Topological Insulators are like the
Integer Quantum Hall effect. The
single particle energy gap is correctly
described by non interacting band
theory.
Interacting systems exhibit a much
richer collection of fractional quantum
Hall states. Understanding these was
one of the greatest triumphs of many
body physics.
What is a fractional topological insulator ?
Classify possible states
Characterize quasiparticle excitations and surface states.
Need to develop new techniques:
- Parton construction?
- B-F theory?
- Entanglement spectrum?
What is the generalization of the bulk – boundary
correspondence for interacting systems ?