metallic topological defects inside a topological band insulator

1. Metallic Topological Defects inside a Topological Band Insulator Ref: Nature Physics 5, 289 (2009). arXiv:0810.5121

3. Band Topology Integer Quantum Hall States: Gapped in the bulk �The edge is the window into the bulk� � X.G. Wen Gapless Edge states that are chiral (propagate in one direction only) Leads to Quantized Hall effect. Integer classification. Also, superconducting analogs (px+ipy Sc)

4. Can one realize a quantum hall like insulator WITHOUT a magnetic field? Yes: Kane and Mele; Bernevig & Zhang (2005), Spin-orbit interaction � spin-dependent magnetic field

5. Two Dimensional Topological Insulator (Quantum Spin Hall Insulator)

6. Single Dirac node � impossible in 1D with time reversal symmetry Stable to (non-magetic) disorder: (no Anderson localization though 1D)

7. Topological insulator in D=3 One obvious route to topological insulator in D=3 : Stack 2D layers of quantum spin Hall insulators. Defined by the reciprocal vector of the stacking layers. �weak� topological insulators. Less obvious possibility (no quantum Hall analog) �strong� topological insulators in D=3. Characteristic feature � Surface state: single Dirac node.

8. Topological Indices in 3D Given a band structure, one can calculate the following topological indices:

9. 3D Strong T-I in Experiments Bi1-xSbx Surface modes ARPES confirms strong T-I. 5-crossings New materials: Bi2Se3, Bi2Te3 Eg. Y. Chen et al. (2009).

10. Broken Symmetry + Exotic Band Topology Superconducting order parameter: Vortex defects: Crystalline Solid � also broken symmetry phase. Analog in topological insulators? YES

11. Line Defects in a Crystal Dislocations: Defined by location R(s) and �strength� B (Burgers vector).

12. Visualizing Dislocations Volterra Process: Cut with an imaginary plane, that ends on the dislocation line R(s) Move all atoms on one side of the plane by the Burgers vector B Add/remove atoms if required.

13. Dislocations in Solids Always present Control mechanical properties eg. Plastic Flow

14. Dislocation in a Topological Insulator 1D Helical Metal occurs in a dislocation {R(s), B} embedded in a topological insulator { } iff:

15. Illustration � Diamond Lattice Top. Ins. Introduce a screw dislocation: B=(1,1,0). Easily introduced in tight binding. Momentum dependent phase factor for cut bonds.

16. Results: Screw Dislocation in Diamond Lattice Top. Ins. Insert a pair of screw dislocations (36x36x18 periodic BC). Momentum along the dislocations is a good quantum number. Two propagating modes per dislocation. `Helical metal�.

17. Stable to disorder: (TR symmetry no backscattering)

18. Proof for Weak Top. Ins. Weak Top.Ins. Adiabatically connected to a stack of decoupled 2D Top.Ins., stacking along Different proof for Strong TI

19. Anomaly argument for dislocation modes Consider defect terminating on a surface at a point. If seen by surface fermions as p flux.

20. Experimental Signatures Resistivity: dislocation contribution could dominate over surface conduction.

21. Experimental Signatures Stress dependent conductivity (strongly direction dependent - connected to lattice indices )

22. Experimental Signatures - STM Can determine atomic defect structure and enhanced Local Density of States (LDOS).

23. Conclusions 3D topological Band Insulator has protected helical mode in those dislocations that satisfy Weak topological insulator stable to disorder if dislocations do not proliferate. More stable than surface modes Another example of a `dressed� topological defect � intrinsically quantum phenomena.

24. Future Directions Effect of Interactions: Luttinger liquid behavior or spontaneous symmetry breaking of dislocation electron liquid. Topological quantum computing? Cold atom realizations? (Dislocations in optical lattices) Disorder and interaction effects in topological metals

26. Proof For General Top. Ins. 1 Screw dislocation � if surface Dirac node is at momentum then (-1) phase acquired on crossing the dislocation. In the weak surface connection limit => Dirac equation that changes mass term sign.

27. Proof For General Top. Ins. 2 Pair of zero modes at p1=0. Propagating 1D helical modes for general p1. Location of Surface Dirac Node � controlled by

28. Effect of Lattice Disorder Very Strong Disorder � dislocations proliferate; no meaning to Moderate disorder � dilute dislocation density. Can still characterize using gapless modes in dislocations and define . Weak insulators can be defined even with disorder. but Surface states localized. Moral: weak top. insulators can be defined even when surface states localized.

29. Topological Quantum Computing? Obtaining Majorana fermions in a Top.Ins.

30. Topological Quantum Computing? Dynamic Dislocations in optical lattices?

31. Electrical Transport in BiSb R=1�Om

32. HgTe quantum well 2D top. ins. in experiments

33. Topological Band Insulators Spin-orbit coupling+time reversal symmetry. 2-D Topological Insulator

34. Experimental Signatures - STM Diamond lattice strong Top.Ins. for demonstration. Resembles Bi0.9Sb0.1 in some ways.