Effective Topological Field Theories in Condensed Matter Physics

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, 195424, 08’, Science 323, 1184, 09’

2. 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. Superconductor: Broken gauge symmetry Magnet: Broken rotational symmetry Crystal: Broken translational symmetry

3. 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…

4. e k k kF kF -kF -kF x x Chiral (QHE) and helical (QSHE) liquids in D=1 e 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 2=1+1 4=2+2 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!

5. Wave function of a particle with integer spin changes by 1 under 2p rotation. Spin=1 Time reversal symmetry in quantum mechanics • 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

6. 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

7. 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.

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

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

10. 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.

11. E E Bulk kx 0 Bulk 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 0 x m>0

12. x x e e k k Experimental Predictions

13. Experimental evidence for the QSH state in HgTe

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

15. 3D insulators with a single Dirac cone on the surface (b) z y (a) y x x Quintuple layer (c) C A B t2 t3 t1 C Bi A Se1 B Se2 C

16. Relevant orbitals of Bi2Se3 and the band inversion (a) (b) 0.6 Bi 0.2 E (eV) Se c -0.2 0 0.2 0.4  (eV) (I) (II) (III)

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

18. 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 !

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

20. 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)

21. M E 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 • Surface of a TI = ¼ graphene

22. The Topological Magneto-Electric (TME) effect • Equations of axion electrodynamics predict the robust TME effect. Wilzcek, axion electrodynamics 4πP=aq/2p B 4πM=aq/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!

23. Low frequency Faraday/Kerr rotation (Qi, Hughes and Zhang, PRB78, 195424, 2008) Adiabatic Requirement: (surface gap) Eg Topological contribution topo» 3.6x 10-3 rad normal contribution

24. q TME insulator 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:

25. An electron-monopole dyon becomes an anyon!

26. 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 fermions Chiral Majorana fermions massless Dirac fermions massless Majorana fermions

27. Taking the square root in math and physics

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

29. 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!