11/10/2011 @ NSYSU
Topology and solid state physics
Ming-Che Chang
Department of Physics
1
A brief review of topology
extrinsic curvature K vs
intrinsic (Gaussian) curvature G
Positive and negative
Gaussian curvature
G>0
K≠0
G=0
G=0
G<0
K≠0
G≠0
2
The most beautiful theorem in differential topology
• Gauss-Bonnet theorem (for a 2-dim closed surface)

M
da G  2 ( M ),   2(1  g )
g 0
Euler
characteristic
g 1
g2
• Gauss-Bonnet theorem (for a surface with boundary)

M
da G   ds k g  2   M , M 
M
Marder, Phys Today, Feb 2007
3
A brief review of Berry phase
Adiabatic evolution of a quantum system
• Energy spectrum:
H (r , p;  )
• After a cyclic evolution
E(λ(t))
n+1
x
n
x
 (T )   (0)
 n, (T )
n-1
0
i
T
dt ' En (t ')

0
e
 n,  ( 0 )

dynamical
phase
λ(t)
• Phases of the snapshot states at different λ’s are
independent and can be arbitrarily assigned
 n , (t )  ei
n ( )
 n , ( t )
• Do we need to worry about this phase?
4
Fock, Z. Phys 1928
Schiff, Quantum Mechanics (3rd ed.) p.290
Pf :
No!
Consider the n-th level,
i n (  )
 (t )  e
H   (t )  i
 n  i  n ,
redefine the
phase,
i
t
dt ' En ( t ')

0
e
 n ,


  (t )
t
Stationary,
snapshot state
H n,  En n,

 n ,    0

≡An(λ)
 'n ,  ei
n ( )
An’(λ)  An(λ)
 n ,

n

Choose a  (λ) such
that,
An’(λ)=0
Thus removing the extra phase!
5
One problem:
  A( )
does not always have a
well-defined (global) solution
A
Vector flow
Vector flow
Contour of 
A
 is not
defined here
Contour of 
C
C
 (T )   (0) 

C
A d  0

C
A d  0
M. Berry, 1984 :
Parameter-dependent phase NOT always removable!
6
  (T )
T
dt ' E ( t ')

0
e e
  (0)
i C
i
Index n neglected
• Berry phase (path dependent, but gauge indep.)
C 

C
A( )  d 
• Berry connection
A( )  i      
3
• Berry curvature
 (t )
( )     A( )  i       
S
• Stokes theorem (3-dim here, can be higher)
C 

C
A  d      da
1
2
C
S
7
• Aharonov-Bohm (AB) phase (1959)
• Berry phase (1984)
Φ
solenoid
z
magnetic flux
y
B(t )
x
AB =2

0
Independent of precise path
S C
1
 C =  (C )
2
solid
angle
Independent of the speed
(as long as it’s slow)
8
• AB phase and persistent charge current (in a metal ring)
After one circle, the
electron gets a phase
Φ
kL  kL  2

0
AB phase
→ Persistent current
confirmed in 1990 (Levy et al, PRL)
• Berry phase and persistent spin current (Loss et al, PRL 1990)
B
textured
B field
S
e-
Ω(C)
S
After circling once, an electron gets
 z
kL  kL  2
 (C ) → persistent charge and spin current
0 2
9
Berry curvature and Bloch state

 p k

 2m


2

 U (r )  unk (r )   nk unk (r )  n (k )  i k unk  k unk


 For scalar Bloch state (non-degenerate band):
• Space inversion
symmetry
n (k )  n (k )
both symmetries
• Time reversal
symmetry
n (k )  n (k )
n (k )  0,  k
 When do we expect to have it?
• SI symmetry is broken
← electric polarization
• TR symmetry is broken
← QHE
• spinor Bloch state (degenerate band)
← SHE
Also, • band crossing
← monopole
10
Topology and Bloch state
=
Brillouin zone
1
2

d 2 k  z (k )  C1
~ Gauss-Bonnet theorem
BZ
1st Chern ~ Euler characteristic
number
→ Quantization of Hall conductance
(Thouless et al 1982)
e2
 H  C1
h
Remains quantized even with
disorder, e-e interaction
(Niu, Thouless, Wu, PRB, 1985)
11
Bulk-edge correspondence
Different topological classes
Semiclassical (adiabatic) picture:
energy levels must cross
(otherwise topology won’t change).
Eg
→ gapless states bound to the interface,
which are protected by topology.
12
Edge states in quantum Hall system
(Semiclassical picture)
Gapless excitations at the edge
LLs
number of edge modes = Hall conductance
13
The topology in “topological insulator”
First, a brief history of insulators
Band insulator
(Wilson, Bloch)
Mott
insulator
Anderson
insulator
Peierls
transition
1930
1940
1950
Hubbard
model
1960
1970
Quantum Hall
insulator
Topological
insulator
Scaling theory
of localization
1980
1990
2000
2010
2D TI is also called
QSHI
14
Lattice fermion with time reversal symmetry
2D Brillouin zone
• Without B field, Chern number C1= 0
• Bloch states at k, -k are not independent
• EBZ is a cylinder, not a closed torus.
∴ No obvious quantization.
Moore and Balents PRB 07
• C1 of closed surface may depend on caps
• C1 of the EBZ (mod 2) is independent of caps
(topological insulator, TI)
2 types of insulator, the “0-type”, and the “1-type”
15
• 2D TI characterized by a Z2 number (Fu and Kane 2006 )

1 
2
 mod 2
d
k


dk
A
  ( EBZ ) 
2  EBZ
~ Gauss-Bonnet theorem with edge
How can one get a TI?
: band inversion due to SO coupling
0-type
1-type
16
Bulk-edge correspondence in TI
Bulk states
Dirac
point
Edge states
Fermi
level
TRIM
(2-fold degeneracy at TRIM due to Kramer’s degeneracy)
helical edge states
robust
backscattering by
non-magnetic
impurity forbidden
17
A stack of 2D TI
2D TI
z
fragile SS
Helical
Helical edge state
y
x
3 TI indices
18
3 weak TI indices: (ν1, ν2, ν3 )
Eg., (x0, y0, z0 )
z+
z0
x0
x+
y0
y+
1 strong TI index: ν0
ν0 = z+-z0
(= y+-y0 = x+-x0 )
difference between
two 2D TI indices
Fu, Kane, and Mele PRL 07
Moore and Balents PRB 07
Roy, PRB 09
19
Topological insulators in real life
• Graphene (Kane and Mele, PRLs 2005)
2D
SO coupling only
10-3 meV
• HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 2006)
• Bi bilayer (Murakami, PRL 2006)
•…
• Bi1-xSbx, α-Sn ... (Fu, Kane, Mele, PRL, PRB 2007)
• Bi2Te3 (0.165 eV), Bi2Se3 (0.3 eV) … (Zhang, Nature Phys 2009)
• The half Heusler compounds (LuPtBi, YPtBi …) (Lin, Nature Material 2010)
3D
• thallium-based III-V-VI2 chalcogenides (TlBiSe2 …) (Lin, PRL 2010)
• GenBi2mTe3m+n family (GeBi2Te4 …)
•…
• strong spin-orbit coupling
• band inversion
21
Band inversion,
parity change,
and spin-momentum locking (helical Dirac cone)
S.Y. Xu et al Science 2011
22
Physics related to the Z2 invariant
Topological insulator
graphene
Quantum SHE
Spin pump
• (Generalization of)
Gauss-Bonnett theorem
4D QHE
Helical surface state
Magneto-electric
response
Interplay with SC,
magnetism
Majorana fermion
in TI-SC interface
QC
…
Time reversal inv,
spin-orbit coupling,
band inversion
23
Berry phase and topology in condensed matter physics
 1982 Quantized Hall conductance (Thouless et al)
 1983 Quantized charge transport (Thouless)
 1990 Persistent spin current in one-dimensional ring (Loss et al)
 1992 Quantum tunneling in magnetic cluster (Loss et al)
 1993 Modern theory of electric polarization (King-Smith et al)
 1996 Semiclassical dynamics in Bloch band (Chang et al)
 2001 Anomalous Hall effect (Taguchi et al)
 2003 Spin Hall effect (Murakami et al)
 2006 Topological insulator (Kane et al)
…
24
Thank you!
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