1. Dirac’s inspiration in the search for topological insulators
Dirac Medal Award Ceremony, Trieste 2013 Shoucheng Zhang, Stanford University

2. As a student, I was deeply attracted to Dirac’s style of searching for physical laws of nature by looking for beautiful mathematical structures
Dirac equation
Dirac magnetic monopole

3. Mathematics and Physics
CN Yang advised me to look into condensed matter physics, to realize beautiful mathematical structures in real experiments

4. Chern-Simons-Landau-Ginzburg theory of the QHE
Zhang, Hansson, Kivelson, PRL (1989)

5. SO(5) theory of high Tc superconductivity
I tried to find a simple mathematical structure to explain a complex phenomenon

6. From 2D to 4D QHE
Haldane sphere construction based on the 1st Hopf map B
Our generalization to the 4-sphere, based
on the 2nd Hopf map
Zhang and Hu, Science 2001

7. A new topological class!
Time reversal symmetry:
The 2+1 dim QH described by the 2+1 dim CS theory breaks the T symmetry
The 4+1 dim QH described by the 4+1 dim CS theory preserves the T symmetry! T2=-1, spin-orbit coupling!
The 4+1 dim QH, or the 4D topological insulator is the grand father of 3D and 2D topological insulators discussed recently!

8. Trieste conference March 2005

9. The quantum spin Hall effect
Spin-orbit coupling:
Radial electric field:
Landau levels without B field!
Bernevig and Zhang, PRL 2006
Independent proposal of Kane and Mele in graphene

10. Where do we find QSH in nature?
Back to student days:
Eureka!
Bernevig, Hughes and Zhang, Science 2006

11. 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

12. 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)

13. Pz+, up, Pz-, up, Pz+, down, Pz-, down
Model for topological insulator Bi2Te3, (Zhang et al, 2009)
Pz+, up, Pz-, up, Pz+, down, Pz-, down
Single Dirac cone on the surface of Bi2Te3
Surface of Bi2Te3 = ¼ Graphene !

14. (a) Sb2Se3
(b) Sb2Te3
(c) Bi2Se3
(d) Bi2Te3

15. General theory of topological insulators
Topological field theory of topological insulators. Generally valid for interacting and disordered systems. Directly measurable physically. Quantized magneto-electric effect (Qi, Hughes and Zhang, PRB 2008))
For a periodic system, the system is time reversal symmetric only when
q=0 => trivial insulator
q=p => non-trivial insulator
Generalization of the Dirac quantization condition!

16. 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 and see also Haldane’s work on QH currents
Magnitude of B:

17. The quantum Hall trio

18. TI & TSC in relation to other branches of physics
Semiconductor physics
narrow gap semiconductors, dopant and defect physics, MBE
Magnetism: Spintronics, DMS, ferromagnetic layers on TI, half-metals
Superconductivity and superfluidity
Novel proximity effects, topological superconductors, vortex states, He3B is a topological superfluid! Oxide interface superconductivity
Quantum Hall effect
Quantum spin Hall effect, Quantum anomalous Hall effect, graphene
Heavy fermions: Mixed valence and the d-f band inversion
Cold atoms: Artificially engineered spin-orbit interaction
Exotic particles
Magnetic monopoles, axions, Majorana fermions
String theory
Modular invariance and anomaly determine topological stability
Standard model
Is the vacuum a topological insulator?

19. From traffic jam to info-superhighway
Traffic jam inside chips today
Info highways for the chips in the future

20. Dirac in condensed matter physics
Serendipitous discovery of new states matter => theoretical explanations
Superconductivity, quantum Hall effect, magnetism
Mathematical beauty => theoretical predictions => experimental observation
Topological insulators

21. Conclusion Topological insulator is a new state of quantum matter.
Material properties can be predicted and designed.
Frontiers of the field:
Improved material properties
New topological insulators
Magnetic monopoles and Majorana fermions
Topological superconductors
Quantized anomalous Hall effect
Strongly correlated topological insulators
Reviews:
Qi+Zhang, Physics Today
Moore, Nature
Hasan and Kane, RMP colloquium
Qi+Zhang, RMP, full article:
Rev. Mod. Phys. 83, 1057 (2011)