Topological Insulators and
Superconductors
Akira Furusaki
2012/2/8
YIPQS Symposium
1
Condensed matter physics
• Diversity of materials
– Understand their properties
– Find new states of matter
“More is different” (P.W. Anderson)
• Emergent behavior of electron systems at low energy
– Spontaneous symmetry breaking
crystal, magnetism, superconductivity, ….
– Fermi liquids (non-Fermi liquids)
(high-Tc) superconductivity, quantum criticality, …
– Insulators
Mott insulators, quantum Hall effect, topological insulators, …
Outline
• Topological insulators: introduction
• Examples:
–
–
–
–
–
Integer quantum Hall effect
Quantum spin Hall effect
3D Z2 topological insulator
Topological superconductor
Classification
• Summary and outlook
Introduction
• Topological insulator
– an insulator with nontrivial topological structure
– massless excitations live at boundaries
bulk: insulating, surface: metallic
• Many ideas from field theory are realized
in condensed matter systems
– anomaly
– domain wall fermions
–…
• Recent reviews:
– Z. Hasan & C.L. Kane, RMP 82, 3045 (2010)
– X.L. Qi & S.C. Zhang, RMP 83, 1057 (2011)
Recent developments 2005• Insulators which are invariant under time reversal
can have topologically nontrivial electronic structure
• 2D: Quantum Spin Hall Effect
– theory
C.L. Kane & E.J. Mele 2005; A. Bernevig, T. Hughes & S.C. Zhang 2006
– experiment
L. Molenkamp’s group (Wurzburg) 2007 HgTe
• 3D: Topological Insulators
in the narrow sense
– theory
L. Fu, C.L. Kane & E.J. Mele 2007; J. Moore & L. Balents 2007; R. Roy 2007
– experiment
Z. Hasan’s group (Princeton) 2008 Bi1-xSbx
Bi2Se3 , Bi2Te3 , Bi2Tl2Se, …..
Topological insulators
• band insulators
in broader sense
free fermions (ignore e-e int.)
• characterized by a topological number (Z or Z2)
Chern #, winding #, …
• gapless excitations at boundaries
stable
Topological non-topological
(vacuum)
insulator
Examples: integer quantum Hall effect, polyacetylen,
quantum spin Hall effect, 3D topological insulator, ….
6
Band insulators
• An electron in a periodic potential (crystal)
2
d2

  x   V  x   x   E  x 
2
2m dx
V  x  a  V  x
• Bloch’s theorem
 k  x   eikxuk  x 
Brillouin zone
uk  x  a   uk  x 


a
k

a
E
empty
band gap
Band insulator
occupied


a
0

a
k
Energy band structure:







a mapping k  H k , En k , or un k
Topological equivalence (adiabatic continuity)
Band structures are equivalent if they can be continuously deformed
into one another without closing the energy gap
8
Topological distinction of ground states
deformed “Hamiltonian”
n empty
bands
0  †
1n
Q k U 
U U  m  n 
U
 0 1m 
Q2  1, Q†  Q, trQ  m  n
 
Q : Brillouin zone (k -space)
m filled
bands
U  m  n  U  m  U  n 
ky
kx
 2 U m  n U mU n  
map from BZ to Grassmannian
IQHE (2 dim.)
homotopy class
9
Berry phase of Bloch wave function



Berry connection
 

 
A k  i u k  k u k
Berry curvature
 
  
F k  k  A k
Berry phase

C  
C


  
A k  dk   Fd 2 k
S
Example: 2-level Hamiltonian (spin ½ in magnetic field)
 
     dz
H k  d k    
 d x  id y

d x  id y 

 d z 
 
 
 

H k u k d k u k
10
Integer QHE
11
Integer quantum Hall effect
(von Klitzing 1980)
 xy   H
h
 25812 .807 
2
e
 xx
Quantization of Hall conductance
e2
 xy  i
h
i : integer
exact, robust against disorder etc.
12
Integer Quantum Hall Effect
 xy   H
B
 xx
y
x
 xy
e2
 C
h
(TKNN: Thouless, Kohmoto, Nightingale & den Nijs 1982)
Chern number
integer valued
1
2
C
d
k  k  A  k x , k y  = number of edge modes crossing EF

2 i
bulk-edge correspondence
filled band
A  kx , k y   k k k

 k   kx ,  k y

Berry connection
13 eB y
kx 
Lattice model for IQHE
(Haldane 1988)
• Graphene: a single layer of graphite
– Relativistic electrons in a pencil
Geim & Novoselov: Nobel prize 2010
B
A
K
E
K’
K’
K
K’
K
K
Matrix element for hopping
between nearest-neighbor sites: t
3ta
c
vF 

2
300


px
H  vF 



K’
px  ip y
0
 ip y
0
0
0
0
0
0
0
px  ip y
0
py
px
A

0
B
px  ip y  A

0
B
0
Dirac masses
• Staggered site energy
 M , on A sites
(G. Semenoff 1984)
Breaks inversion symmetry
 M , on B sites
• Complex 2nd-nearest-neighbor hopping (Haldane 1988)
t2ei
No net magnetic flux through a unit cell
Breaks time-reversal symmetry
K point: H K  vF  px x  p y y   m z
m  M  3 3t2 sin 
K ' point: H K '  vF  px x  p y y   m ' z
m '  M  3 3t2 sin 
Hall conductivity  xy
e2
 C
h
1
C  sgn  m   sgn  m '  
2
C 0
C  1
C 0
C  1
Chern insulator
C0
Massive Dirac fermion: a minimal model for IQHE
H  iv x x   y  y   m z
parity anomaly
 xy  sgn m 
1
2
(2+1)d Chern-Simons theory for EM
mx 
Domain wall fermion
H  iv x x   y  y  mx z
x
x
1

 1 
 x, y   expiky   y  mx'dx' 
0
v

  i 
E  vk
m0
m0
16
Quantum spin Hall effect
(2D Z2 topological insulator)
17
2D Quantum spin Hall effect
Kane & Mele (2005, 2006); Bernevig & Zhang (2006)
• time-reversal invariant band insulator
• spin-orbit interaction
L  S
• gapless helical edge mode (Kramers’ pair)

B
E
up-spin electrons
conduction band

B
down-spin electrons
valence band
kx
Sz is not conserved in general.
Topological index: Z
Z2
18
Quantum spin Hall insulator
Bulk energy gap & gapless edge states
Helical edge states:
(i) Half an ordinary 1D electron gas
(ii) Protected by time reversal symmetry
19
Kane-Mele model
• Two copies of Haldane’s model (spin up & down)
+ spin-flip term
down-spin electrons
up-spin electrons
HKane-Mele  HHaldane    2  HHaldane    2  Hspin-flip
Hspin-flip  iR  ci† (s  rij ) z c j
s : electron spin
n. n .
• Invariant under time-reversal transformation
i  i s   s
  isy K
2  1
• Spin-flip term breaks U (1) U (1) symmetry
– R  0 two copies of Chern insulators C  1  C
– R  0 C  0 a new topological number: Z2 index   0, 1
Effective Hamiltonian
•
•
•
 :  z  1 A sublattice,  z  1 B sublattice
 :  z  1 K point,  z  1 K' point
s : s z  1 up spin, s z  1 down spin
H 0  i vF  x z  x   y  y 
HSO  SO z z sz
complex 2nd nearest-neighbor hopping (Haldane)
H R  R  x z s y   y sx 
H M  M z
spin-flip hopping
staggered site potential (Semenoff)
Time-reversal symmetry
1H total  H total
  i x sy K
2  1
complex conjugation
Chern # = 0
Z2 index
Kane & Mele (2005); Fu & Kane (2006)
 1
 0
Quantum spin Hall insulator
Trivial insulator
E
E
conduction
band
conduction
band
EF

0
k

 
Time-reversal invariant momenta:  a
4
 1  

a 1
 
 
 
Pf  w  a 

  1
det  w  a 


a
 a  G

0
an even number
of crossing
 
k
 
 
wmn k  um k  un k
un k
Bloch wave of occupied bands
Z2 index
EF
valence band
valence band

an odd number
of crossing

 
 
w a  w a
antisymmetric
Time reversal symmetry
Time reversal operator 
 
 
 
 
 k    * k 








 k    * k 
 
  

Kramers’ theorem
2  1
time-reversal pair
All states are doubly degenerate.
23
Z2: stability of gapless edge states
(1) A single Kramers doublet
E
E
Kramers’ theorem
stable
k
k
(2) Two Kramers doublets
E
E
k
k
Two pairs of edge states are unstable against perturbations that respect TRS.
24
Experiment
HgTe/(Hg,Cd)Te quantum wells
CdTe
HgCdTe
CdTe
Konig et al. [Science 318, 766 (2007)]
Trivial Ins.
QSHI
25
Z2 topological insulator
in 3 spatial dimensions
26
3 dimensional Topological insulator
• Band insulator
Z2 topologically nontrivial
• Metallic surface: massless Dirac fermions
(Weyl fermions)
E
y
ky
x
Theoretical Predictions made by:
Fu, Kane, & Mele (2007)
Moore & Balents (2007)
Roy (2007)
kx
an odd number of Dirac cones/surface
Surface Dirac fermions
topological
insulator
• “1/4” of graphene
K
E
K’
K’
K
K
K’
ky
kx
Hsurface  i y  x  i x y
• An odd number of Dirac fermions in 2 dimensions
cf. Nielsen-Ninomiya’s no-go theorem
28
Experimental confirmation
• Bi1-xSbx
0.09<x<0.18
theory: Fu & Kane (PRL 2007)
exp: Angle Resolved Photo Emission Spectroscopy
Princeton group (Hsieh et al., Nature 2008)
5 surface bands cross Fermi energy
• Bi2Se3
ARPES exp.: Xia et al., Nature Phys. 2009
photon
p, E
a single Dirac cone
Other topological insulators:
Bi2Te3, Bi2Te2Se, …
Response to external EM field
Qi, Hughes & Zhang, 2008
Essin, Moore & Vanderbilt 2009
    ieA
Integrate out electron fields to obtain effective action for the external EM field
Seff
 e2
 e2
3 
3

dtdx

F
F

dtdx
EB
 
2
2


32 c
4 c
    2
  
0
 

axion electrodynamics (Wilczek, …)
time reversal
trivial insulators
topological insulators
vacuum   0
 
topological insulator
FF  d  AdA
(2+1)d Chern-Simons theory

surface
 xy
dtdx 2  A  A
e2

2h
Topological magnetoelectric effect
Seff
2
 e2

e
3 
3

dtdx

F
F

dtdx
EB
 
2
2


32 c
4 c
Magnetization induced by electric field
 S  e2
M

E
 B  2hc
Polarization induced by magnetic field
 S  e2
P

B
 E  2hc
Topological superconductors
Topological superconductors
•
•
•
•
BCS superconductors
Quasiparticles are massive inside the superconductor
Topological numbers
Majorana (Weyl) fermions at the boundaries
stable
topological
superconductor
vacuum
(topologically trivial)
Examples: p+ip superconductor, fractional QHE at  
5 3
, He
2
Majorana fermion
Ettore Majorana
mysteriously disappeared in 1938
• Particle that is its own anti-particle
• Neutrino ?
• In superconductors:
condensation of Cooper pairs
nothing (vacuum)
particle
hole
Quasiparticle operator
  uc  vc†
   † if u  v
This happens at E=0.
2D p+ip superconductor
• (px+ipy)-wave Cooper pairing
(similar to IQHE)
angular momentum =
• Hamiltonian for Nambu spinor  c p  (spinless case)

p


2m
Hp  
 
px  ip y 


 pF
2
 † 
 c p 


px  ip y  
S2

pF
  d  p  
ˆd d
d
px , p y
2

p


2m 
wrapping # = 1
 x H* p x  H p


• Majorana Weyl fermion along the edge
px+ipy
E
2
px-ipy
 k   †k
k
  x 
 e
ikx
k 0
 †  x
 k  eikx †k dk
S2
Majorana zeromode in a quantum vortex
Zero-energy Majorana bound state
vortex

hc
e
E
(p+ip) superconductor
0
zero mode
0  0
 0   0
Majorana fermion

energy spectrum
near a vortex
If there are 2N vortices, then the ground-state degeneracy = 2N.
interchanging vortices
i
i+1
braid groups, non-Abelian statistics
 i   i 1
 i 1   i
(p+ip) superconductor
D.A. Ivanov, PRL (2001)
topological quantum computing ?
Majorana zeromode is insensitive to external disturbance (long coherence time).
Engineering topological superconductors
• 3D topological insulator + s-wave superconductor (Fu & Kane, 2008)
s-SC
S-wave SC
Dirac mass for the (2+1)d surface Dirac fermion
Similar to a spinless p+ip superconductor
Z2 TPI
Majorana zeromode in a vortex core (cf. Jakiw & Rossi 1981)
• Quantum wire with strong spin-orbit coupling + B field + s-SC
(Das Sarma et al, Alicea, von Oppen, Oreg, … Sato-Fujimoto-Takahashi, ….)
InAs, InSb wire
B
px2
H0 
   gpx s y  Bsx
2m
s-SC
• Race is on for the search of elusive Majorana!
Classification of topological
insulators and superconductors
Q: How many classes of topological
insulators/superconductors exist in nature?
A: There are 5 classes of TPIs or TPSCs
in each spatial dimension.
Generic Symmetries:
time reversal
charge conjugation (particle hole) SC
40
Classification of free-fermion Hamiltonian fi  Hij f j
in terms of generic discrete symmetries
• Time-reversal symmetry (TRS)  0
anti-unitary
no TRS

TRS   1 TRS with 2  1 spin 0
 1 TRS with 2  1 spin 1/2

  T
TH *T 1  H
• Particle-hole symmetry (PHS)
BdG Hamiltonian
  P
1
PH P   H
*
anti-unitary
0 no PHS

PHS  1 PHS with  2  1
 1 PHS with  2  1

triplet
singlet
• TRS PHS = Chiral symmetry (CS)
TPHTP  H
1
T RS  PHS  0, Ch  1
3  3  1  10
41
10 random matrix ensembles
(symmetric spaces) Altland & Zirnbauer (1997)
TRS
PHS Ch
time evolution operator exp  iHt 
IQHE
WignerDyson
Z2 TPI
chiral
px+ipy
superconductor
• Wigner-Dyson (1951-1963): “three-fold way” complex nuclei
• Verbaarschot & others (1992-1993)
chiral phase transition in QCD
42
• Altland-Zirnbauer (1997): “ten-fold way”
mesoscopic SC systems
10 random matrix ensembles
(symmetric spaces) Altland & Zirnbauer (1997)
TRS
PHS Ch
time evolution operator exp  iHt 
IQHE
WignerDyson
Z2 TPI
chiral
px+ipy
superconductor
“Complex” cases: A & AIII
“Real” cases: the remaining 8 classes
43
TH *T 1  H or PH * P 1   H
How to classify topological insulators and SCs
• Gapless boundary modes are topologically protected.
• They are stable against any local perturbation.
(respecting discrete symmetries)
• They should never be Anderson localized by disorder.
Nonlinear sigma models for Anderson localization
of gapless boundary modes
S   d d 1r tr Q  + topological term (with no adjustable parameter)
2
QM
bulk: d dimensions
boundary: d -1 dimensions
Z2 top. term
 d 1 M   Z 2
WZW term
 d M   Z
 -term
44
NLSM topological terms
Z2: Z2 topological term can exist in d dimensions
Z: WZW term can exist in d-1 dimensions
 d G H 
d+1 dim. TI/TSC
d dim. TI/TSC
Classification of topological insulators/superconductors
Standard
(Wigner-Dyson)
Chiral
BdG
TRS
PHS
CS
d=1
d=2
A (unitary)
0
0
0
--
Z
AI (orthogonal)
+1
0
0
--
-QSHE --
AII (symplectic)
1
0
0
--
Z2
Z2 Z2TPI
--
Z
AIII (chiral unitary)
0
0
1
Z
BDI (chiral orthogonal)
+1
+1
1
Z
CII (chiral symplectic)
1
1
1
Z
D (p-wave SC)
0
+1
C (d-wave SC)
0
1
DIII (p-wave TRS SC)
1
+1
CI (d-wave TRS SC)
+1
1
-Z
0
Z
1
Z2
IQHE
--
polyacetylene
(SSH)
---
0 p SC Z2
--
d=3
Z2
p+ip SC
--
d+id SC
Z2
1 (p+ip)x(p-ip)
---SC
-Z
3He-B
Z
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
46
Periodic table of topological insulators/superconductors
period
d=2
period
d=8
A. Kitaev, AIP Conf. Proc. 1134, 22 (2009); arXiv:0901.2686
K-theory, Bott periodicity
Ryu, Schnyder, AF, Ludwig, NJP 12, 065010 (2010) massive Dirac Hamiltonian
Ryu, Takayanagi, PRD 82, 086914 (2010) Dp-brane & Dq-brane system
47
Summary and outlook
• Topological insulators/superconductors are new
states of matter!
• There are many such states to be discovered.
• Junctions: TI + SC, TI + Ferromagnets, ….
• Search for Majorana fermions
• So far, free fermions. What about interactions?
Outlook
• Effects of interactions among electrons
– Topological insulators of strongly correlated electrons??
– Fractional topological insulators ??
• Topological order
–
–
–
–
(no symmetry breaking)
Fractional QH states
Chern-Simons theory
Low-energy physics described by topological field theory
Fractionalization
Symmetry protected topological states
(e.g., Haldane spin chain in 1+1d)
X.-G. Wen
• Strongly correlated many-body systems
– have been (will remain to be) central problems
• High-Tc SC, heavy fermion SC, spin liquids, …
– but, very difficult to solve
• Theoretical approaches
– Analytical
• Application of new field theory techniques?
• ….
AdS/CMT?
– Numerical
• Quantum Monte Carlo (fermion sign problem)
• Density Matrix RG (only in 1+1 d)
• New algorithms: tensor-network RG, ….
Quantum information theory