1. Topology and solid state physics
NSYSU
Topology and solid state physics
Ming-Che Chang
Department of Physics

2. A brief review of topology
extrinsic curvature K vs intrinsic (Gaussian) curvature G
Positive and negative Gaussian curvature
G>0
G=0
G<0
K≠0
G=0
K≠0
G≠0

3. The most beautiful theorem in differential topology
Gauss-Bonnet theorem (for a 2-dim closed surface)
Euler characteristic
Gauss-Bonnet theorem (for a surface with boundary)
Marder, Phys Today, Feb 2007

4. x x A brief review of Berry phase n+1 n n-1
Adiabatic evolution of a quantum system
Energy spectrum:
After a cyclic evolution
E(λ(t))
n+1 x n x
n-1
dynamical phase
λ(t)
Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned
Do we need to worry about this phase?

5. Stationary, snapshot state
Fock, Z. Phys 1928
Schiff, Quantum Mechanics (3rd ed.) p.290
No!
Pf :
Consider the n-th level,
Stationary, snapshot state
≡An(λ)
redefine the phase,
An’(λ)
An(λ)
Choose a  (λ) such that,
An’(λ)=0
Thus removing the extra phase!

6. One problem:
does not always have a well-defined (global) solution
 is not defined here
Vector flow
Vector flow
Contour of 
Contour of  C
M. Berry, 1984 :
Parameter-dependent phase NOT always removable!

7. Stokes theorem (3-dim here, can be higher)
Index n neglected
Berry phase (path dependent, but gauge indep.)
Berry connection
Berry curvature C S
Stokes theorem (3-dim here, can be higher)

8. Aharonov-Bohm (AB) phase (1959)
Independent of precise path Φ
magnetic flux
solenoid
Berry phase (1984) C
Independent of the speed (as long as it’s slow)
solid angle

9. Φ e- AB phase and persistent charge current (in a metal ring)
After one circle, the electron gets a phase
AB phase
→ Persistent current
confirmed in 1990 (Levy et al, PRL)
Berry phase and persistent spin current (Loss et al, PRL 1990) B S e-
textured B field
Ω(C) S
After circling once, an electron gets
→ persistent charge and spin current

10. ← monopole Berry curvature and Bloch state
For scalar Bloch state (non-degenerate band):
Space inversion symmetry
both symmetries
Time reversal symmetry
When do we expect to have it?
SI symmetry is broken
TR symmetry is broken
spinor Bloch state (degenerate band)
band crossing
← electric polarization
← QHE
← SHE
← monopole
Also,

11. = → Quantization of Hall conductance (Thouless et al 1982)
Topology and Bloch state =
Brillouin zone
~ Gauss-Bonnet theorem
1st Chern ~ Euler characteristic number
→ Quantization of Hall conductance (Thouless et al 1982)
Remains quantized even with disorder, e-e interaction (Niu, Thouless, Wu, PRB, 1985)

12. 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.

13. Edge states in quantum Hall system
(Semiclassical picture)
Gapless excitations at the edge
LLs
number of edge modes = Hall conductance

14. The topology in “topological insulator”
First, a brief history of insulators
Band insulator (Wilson, Bloch)
Mott insulator
Anderson insulator
Quantum Hall insulator
Topological insulator
Peierls transition
Hubbard model
Scaling theory of localization
2D TI is also called QSHI

15. 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”

16. 2D TI characterized by a Z2 number (Fu and Kane 2006 )
~ Gauss-Bonnet theorem with edge
How can one get a TI?
: band inversion due to SO coupling
0-type
1-type

17. Bulk-edge correspondence in TI
TRIM
Dirac point
Fermi level
Bulk states
Edge states
(2-fold degeneracy at TRIM due to Kramer’s degeneracy)
helical edge states
robust
backscattering by non-magnetic impurity forbidden

18. A stack of 2D TI z 2D TI y x 3 TI indices fragile Helical SS
Helical edge state y x
3 TI indices

19. difference between two 2D TI indices
3 weak TI indices: (ν1, ν2, ν3 )
Eg., (x0, y0, z0 ) z+ x+ y+ z0 x0 y0
1 strong TI index: ν0
ν0 = z+-z0
(= y+-y0 = x+-x0 )
difference between two 2D TI indices
Fu, Kane, and Mele PRL Moore and Balents PRB 07 Roy, PRB 09

20. For systems with inversion symmetry, one can calculate the TI indices using parity eigenvalues of Bloch states. (Fu and Kane)
For example,

21. Topological insulators in real life
SO coupling only 10-3 meV
Graphene (Kane and Mele, PRLs 2005)
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)
thallium-based III-V-VI2 chalcogenides (TlBiSe2 …) (Lin, PRL 2010)
GenBi2mTe3m+n family (GeBi2Te4 …) 2D 3D
strong spin-orbit coupling
band inversion

22. Band inversion, parity change, and spin-momentum locking (helical Dirac cone)
S.Y. Xu et al Science 2011

23. Physics related to the Z2 invariant
Topological insulator
Helical surface state
graphene
Magneto-electric response
Quantum SHE
(Generalization of) Gauss-Bonnett theorem
Interplay with SC, magnetism
Spin pump
Majorana fermion in TI-SC interface QC
4D QHE …
Time reversal inv, spin-orbit coupling, band inversion

24. 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) …

25. Thank you!