1. Effective Topological Field Theories in Condensed Matter Physics
Theoretical prediction: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experimental observation: Koenig et al, Science 318, 766 (2007) New Developments: Qi et al, Nature Physics 4, 273, 08’, Phy Rev B78, , 08’, Science 323, 1184, 09’

2. Quantum spin Hall effect and topological insulators
Theoretical prediction: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experimental observation: Koenig et al, Science 318, 766 (2007) New Developments: Nature Physics 4, 273, 08’, Phy Rev B78, , 08’, Science 323, 1184, 09’ Theoretical prediction: Zhang et al cond-mat/ Experimental observation: Chen et al, submitted

3. The search for new states of matter
The search for new elements led to a golden age of chemistry.
The search for new particles led to the golden age of particle physics.
In condensed matter physics, we ask what are the fundamental states of matter?
In the classical world we have solid, liquid and gas. The same H2O molecules can condense into ice, water or vapor.
In the quantum world we have metals, insulators, superconductors, magnets etc.
Most of these states are differentiated by the broken symmetry.
Magnet: Broken rotational symmetry
Superconductor: Broken gauge symmetry
Crystal: Broken translational symmetry

4. The quantum Hall state, a topologically non-trivial state of matter
TKNN integer=the first Chern number.
Topological states of matter are defined and described by topological field theory:
Physically measurable topological properties are all contained in the topological field theory, e.g. QHE, fractional charge, fractional statistics etc…

5. The Generalizations of the Hall Effect
Theoretical predictions of the spin Hall effect (Dyakonov, Murakami, Nagaosa and Zhang, Science 2003, Sinova et al PRL 2004)
The spin Hall effect has now been experimentally observed. (Kato et al, Science 2004, Wunderlich et al PRL 2004)
What about the quantum spin Hall effect?

6. Quantum Spin Hall Effect
The QSH state can be thought of as two copies of QH states, one for each spin component, each seeing the opposite magnetic field. (Bernevig and Zhang, PRL, 2006)
The QSH state does not break the time reversal symmetry, and can exist without any external magnetic field. x
Insulating gap in the bulk.
Helical edge states: Two states with opposite spins counter-propagate at a given edge.

7. Chiral (QHE) and helical (QSHE) liquids in D=1k kF
-kF k kF
-kF
The QHE state spatially separates the two
chiral states of a spinless 1D liquid
The QSHE state spatially separates the
four chiral states of a spinful 1D liquid x
2=1+1
4=2+2 x
No go theorems: chiral and helical states can never be constructed microscopically
from a purely 1D model. (Wu, Bernevig, Zhang, 2006) Helical liquid=1/2 of 1D fermi liquid!

8. Taking the square root in math and physics

9. Time reversal symmetry in quantum mechanics
Wave function of a particle with integer spin changes by 1 under 2p rotation.
Spin=1
Wave function of a half-integer spin changes by -1 under 2p rotation.
Kramers theorem, in a time reversal invariant system with half-integer spins, T2=-1, all states for degenerate doublets.
y=> y
Application in condensed matter physics: Anderson’s theorem. BCS pair=(k,up)+(-k,down). General pairing between Kramers doublets.
Spin=1/2
y=>-y

10. The topological distinction between a conventional insulator and a QSH insulator
Kane and Mele PRL, (2005); Wu, Bernevig and Zhang, PRL (2006); Xu and Moore, PRB (2006)
Band diagram of a conventional insulator, a conventional insulator with accidental surface states (with animation), a QSH insulator (with animation). Blue and red color code for up and down spins. e k
k=0 or p
Trivial
Trivial
Non-trivial

11. From topology to chemistry: the search for the QSH state
Graphene – spin-orbit coupling only about 10-3meV. Not realizable in experiments. (Kane and Mele, 2005, Yao et al, 2006, MacDonald group 2006)
Quantum spin Hall with Landau levels – spin-orbit coupling in GaAs too small. (Bernevig and Zhang, PRL, 2006)
Type III quantum wells work. HgTe has a negative band gap! (Bernevig, Hughes and Zhang, Science 2006)
Tuning the thickness of the HgTe/CdTe quantum well leads to a topological quantum phase transition into the QSH state.

12. Band Structure of HgTe S
P1/2
P3/2 S P S
P3/2
P1/2 S P

13. Quantum Well Sub-bands
Let us focus on E1, H1
bands close to
crossing point
HgTe
HgTe E1 H1
CdTe
CdTe
CdTe
CdTe H1 E1
normal
inverted

14. Effective tight-binding model
Square lattice with 4-orbitals per site:
Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k.
Relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! m<0 for d>dc.

15. Mass domain wall
Cutting the Hall bar along the y-direction we see a domain-wall structure in the
band structure mass term. This leads to states localized on the domain wall which still disperse along the x-direction. y y
m>0 x
m<0 m x
m>0 kx E
Bulk E

16. Experimental setup
High mobility samples of HgTe/CdTe quantum wells have been fabricated.
Because of the small band gap, about several meV, one can gate dope this system from n to p doped regimes.
Two tuning parameters, the thickness d of the quantum well, and the gate voltage.
(Koenig et al, Science 2007)

17. Experimental Predictionsk e k e

18. Smoking gun for the helical edge state: Magneto-Conductance
The crossing of the helical edge states is protected by the TR symmetry. TR breaking term such as the Zeeman magnetic field causes a singular perturbation and will open up a full insulating gap: e
B-Field k
Conductance now takes the activated form:

19. Experimental evidence for the QSH state in HgTe

20. Magnetic field dependence of the residual conductance

21. Nonlocal transport in the QSH regime
R14,14=3/4 h/e2 1 3 2 4
I: 1-4
V: 2-3
R14,23=1/4 h/e2 21

22. QSH state in InAs/GaSb type II quantum wells
HgTe is not a material that can be easily fabricated. We are searching for new semiconductor materials which can lead to QSH.
In HgTe, the band inversion occurs intrinsically in the material. However, in InAs/GaSb quantum wells, a similar inversion can occur, since the valance band edge of GaSb lies above the conduction band edge of InAs.
Our theoretical work show that the QSH can occur in InAs/Gab quantum wells. This material can be fabricated commercially in many places around the world.

23. mx x Fractional charge in the QSH state, E&M duality! G Vg E E G Vg
Since the mass is proportional to the magnetization, a magnetization domain wall leads to a mass domain wall on the edge. mx x
e/2 x
The fractional charge e/2 can be measured by a Coulomb blockade experiment, one at the time! Jackiw+Rebbie, Qi, Hughes & Zhang G Vg E
 V=e/C E G Vg

24. E B B E Electromagnetic response of an insulator
Electromagnetic response of an insulator is described by an effective action: E B
However, another quadratic term is also allowed:
4πP=(-1)E
4πM=(1-1/)B B E
Physically, this term describes the magneto-electric effect. Under time reversal:
4πP=a q/2p B
4πM=a q/2p E

25. q periodicity and time reversal
Consider an analog system of a period ring. The flux enters the partition function as:
Therefore, the physics is completely invariant under the shift of
Under time reversal, f=>-f, therefore, time reversal is recovered for two special values of f, f=0 and f=p.
The ME term is a total derivative, independent of the bulk values of the fields:
Integrated over a spatially and temporally periodic system,
Its contribution to the partition function is given by Therefore, the partition function is invariant under the shift:
Time reversal symmetry is recovered at

26. 3D insulators with a single Dirac cone on the surface
(b) z
(a) y x y x
Quintuple
layer
(c) C A t2 B t3 t1 C
Se2 Bi
Se1 A B C

27. Relevant orbitals of Bi2Se3 and the band inversion
0.6 Bi
E (eV)
0.2 Se c
-0.2
0.2
0.4 
(eV)
(I)
(II)
(III)

28. Bulk and surface states from first principle calculations
(a) Sb2Se3
(b) Sb2Te3
(c) Bi2Se3
(d) Bi2Te3

29. Pz+, up, Pz-, up, Pz+, down, Pz-, down
Effective model for Bi2Se3, Zhang et al
Pz+, up, Pz-, up, Pz+, down, Pz-, down
Minimal Dirac model on the surface of Bi2Se3, Zhang et al
Surface of Bi2Se3 = ¼ Graphene !

30. Doping evolution of the FS and band structure
Arpes experiment on Be2Te3 surface states, Shen group
Doping evolution of the FS and band structure
EF(undoped)
BCB bottom
Dirac point position
Undoped
Under-doped
Optimally-
doped
Over-doped
BVB bottom

31. General definition of a topological insulator
Z2 topological band invariant in momentum space based on single particle states.
(Fu, Kane and Mele, Moore and Balents, Roy)
Topological field theory term in the effective action. Generally valid for interacting and disordered systems. Directly measurable physically. Relates to axion physics! (Qi, Hughes and Zhang)
For a periodic system, the system is time reversal symmetric only when
q=0 => trivial insulator
q=p => non-trivial insulator
Arpes experiments (Hasan group)

32. E M j// q term with open boundaries
q=p implies QHE on the boundary with
For a sample with boundary, it is only insulating when a small T-breaking field is applied to the boundary. The surface theory is a CS term, describing the half QH.
Each Dirac cone contributes sxy=1/2e2/h to the QH. Therefore, q=p implies an odd number of Dirac cones on the surface!
T breaking E M
j//
Surface of a TI = ¼ graphene

33. Topological stability of the surface states
No-go theorem: it is not possible to construct a 2D model with an odd number of Dirac cones, in a system with T2=-1 TR symmetry. Surface states of a TI with q=p is a holographic liquid! Wu, Bernevig & Zhang, Holographical principle
TI surface states can not rust away by surface chemistry.
For a sample with boundary, physics is not periodic in q. However, T-invariant perturbations, like disorder, can induce plateau transitions with Dsxy=1 e2/h, or Dq=2p. For TI with q=p, the surface QH can never disappear, no matter how strong the disorder! sxy=1/2 e2/h => sxy=-1/2 e2/h.
States related by interger plateau transition defines an equivalence class. There are only two classes!

34. q periodicity and time reversal
Consider an analog system of a period ring. The flux enters the partition function as:
Therefore, the physics is completely invariant under the shift of
Under time reversal, f=>-f, therefore, time reversal is recovered for two special values of f, f=0 and f=p.
The ME term is a total derivative, independent of the bulk values of the fields:
Integrated over a spatially and temporally periodic system T4,
Its contribution to the partition function is given by Therefore, the partition function is invariant under the shift:
Time reversal symmetry is recovered at

35. The Topological Magneto-Electric (TME) effect
Equations of axion electrodynamics predict the robust TME effect.
Wilzcek, axion electrodynamics
4πP=a q/2p B
4πM=a q/2p E
P3=q/2p is the electro-magnetic polarization, microscopically given by the CS term over the momentum space. Change of P3=2nd Chern number!

36. Low frequency Faraday/Kerr rotation
(Qi, Hughes and Zhang, PRB78, , 2008)
Adiabatic
Requirement:
(surface gap) Eg
Topological contribution
topo» 3.6x 10-3 rad
normal contribution

37. STM probe of the topological surface states
(Liu et al cond-mat/ )

38. Seeing the magnetic monopole thru the mirror of a TME insulator, (Qi et al, Science 323, 1184, 2009)
higher order feed back
(for =’, =’)
similar to Witten’s dyon effect
Magnitude of B:

39. An electron-monopole dyon becomes an anyon!

40. New topological states of quantum matter
QH insulator (U(1) integer), QSH insulator (Z2 number), chiral (U(1) integer) and helical (Z2 number) superconductors.
Chiral Majorana fermions
Chiral fermions
massless Majorana fermions
massless Dirac fermions

41. Taking the square root in math and physics

42. Topological superconductors and superfluids
The BCS-BdG Hamiltonian for equal spin pairing:
where p+=px+ipy. The edge Hamiltonian is given by:
forming a pair of Majorana fermions. Mass term breaks T symmetry=> topological protection! He3 B phase provides a physical realization!
See also
Roy
Schneider et al
Kitaev

43. Summary: the search for new states of matter
s-wave superconductor
Crystal
Magnet
Quantum Hall
Quantum Spin Hall

45. Recurrence of effective field theories
Dirac at MeV
Schroedinger at eV
Dirac at meV
Theta vacuum and axion of QCD
Topological insulators in CM
Monopoles in cosmology
table top experiments in CM
To see the world in a grain of sand,
To hold infinities in an hour!

46. Thanks! Summary Topological insulators Semiconductors
HgTe/CdTe, InAs/GaSb,
Bi1-xSbx, Bi2Se3,…
High energy physics
 vacuum, anomalies
axion, dyon
Magnetism
fractional charge,
spin charge separation
Topological
Magneto-electric effect
Topological
insulators
Thanks!
And what else?
Strongly correlated
electron systems
Topological Mott insulators,
Na2IrO3,…
Superconductivity
Nonabelian statistics,
Majorana fermion

47. Completing the table of Hall effects
1879
Anomalous Hall
1889
Spin Hall
2004
QHE
1980
QAHE
2008?
QSHE
2007

48. Momentum space topology of the tight-binding model
Critical points
Ferromagnetic
Skyrmion
Skyrmion
Ferromagnetic
(0, π)
(π, π) X X
(0, 0)
(π,0) X

49. Topological quantum phase transition
Meron in continuum picture:

50. Inversion symmetry breaking in zincblend lattices
Inversion breaking term comes in the form: ,
-spin 3/2 matrices
which couples E1+, H1- and E1-,H1+ states and is a constant in quasi-2d systems
E/t
Gap closes at nodes away from k=0, gap reopens at non-zero value of M/2B.
In the inverted regime, the helical edge state crossing is still robust.
Tight-binding model by X Dai, Z Fang, … k

51. Quantum control of the electron spin
The electron spin can be rotated by a pure AB flux, without any interaction with the electromagnetic field.