Rok Žitko

1. Topological insulators and superconductors Rok Žitko Ljubljana, 22. 7. 2011

2. States of matter • insulators • quantum Hall effect • Topological insulators (TI) • 2D TI and helical edge states • 3D TI and helical surface states • Proximity effect and topological superconductors • Majorana edge states • Detections schemes

3. States of matter • Characterized by • broken symmetries (long range correlations) • topological order • Quantified by • order parameter • topological quantum number • Described by • Landau theory of phase transitions • topological field theories

4. Solid-liquid phase transition Broken translation invariance Order parameter: FT of <r(r)r(0)>, Bragg peaks FLandau=ay2+by4 a=a0(T-Tc)

5. Insulators • Anderson insulatorsdisorder electrons become localized • Mott insulatorsCoulomb interaction(repulsion) between electrons motion suppressed • Band insulatorsabsence of conduction states at the Fermi level forbidden band

6. Band insulators • vacuum (“Dirac sea” model): Egap=2mc2=106 eV • atomic insulators (solid argon):Egap=10eV • covalent-bond semiconductors and insulators: Egap=1eV Bloch, 1928

7. 2D electron gas in strong magnetic field wc=eB/mc Landau levels Egap=ħwc

8. Quantum Hall Effect von Klitzing et al. (1980) Jy=sxyEx chiral edge states

9. Gauss-Bonnet theorem Gaussian curvature, K=1/R1R2 Sphere c=2 Torus c=0

10. Topological insulators • “Topological”: topological properties of the band structure in the reciprocal space • “Insulators”: well, not really. They have gap, but they are conducting (on edges)! • Quantum Hall effect: in high magnetic field, broken time-reversal symmetry (von Klitzing, 1980) • Time-reversal-invariant topological insulators (Kane, Mele, Fu, Zhang, Qi, Bernevig, Molenkamp, Hasan and others, from 2006 and still on-going)

11. Smooth transformations and topology Band structure: mapping from the Brillouin zone (k) to the Hilbert space (y): k|y(k) Bloch theorem: y(k)=eikr uk(r) uk(r)=uk(r+R) Smooth transformations: changes of the Hamiltonian such that the gap remains open at all times See Fig.

13. TKNN (Chern) invariant Thouless-Kohmoto-Nightingale-den Nijs, PRL 1982 Integernumber! Berry curvature Samenas insxy=ne2/h. An integer within an accuracy of at least10-9!New resistance standard: RK=h/e2=25812.807557(18) W

14. Spin-orbit coupling e- Nucleus Stronger effect forheavy elements (Pb, Bi, etc.) from the bottom of the periodic system

15. Reinterpretation:

16. Quantum Spin Hall effect (QSHE)(“2D topological insulators”) • Two copies of QHE, one for each spin, each seeing the opposite effective magnetic field induced by spin-orbit coupling. • Insulating in the bulk, conducting helical edge states. • Theoretically predicted (Bernevig, Hughes and Zhang, Science 2006) and experimentally observed (Koenig et al, Science 2007) in HgTe/CdTe quantum wells.

17. Edge states in 2D TIs Helical modes: on each edge one pair of 1D modes related by the TR symmetry. Propagate in opposite directions for opposite spin.

18. 3D topological insulators • Generalization of QSHE to 3D. • Insulating in the bulk, conducting helical surface states. • Theoretically predicted in 2006, experimentally discovered in BiSb alloys (Hsieh et al., Nature 2008) and in Bi2Se3 and similar layered materials (Xia et al., Nature Phys. 2009).

19. Surface states on 3D topological insulators • Conducting surface statesmust exist on the interface between two topologically different insulators, because the gap must close somewhere near the interface! • Single Dirac cone = ¼ of graphene. • In graphene, there is spin and valley degeneracy, i.e., fourfold degeneracy.

20. Experimental detection in Bi2Te3 Chen et al. Science (2009)

21. Spin-momentum locking Spin-resolved ARPES Hsieh, Science (2009)

22. Topological field theory q=0, topologically trivial, q=p, topological insulator Qi, Hughes, Zhang, PRB (2008), Wang, Qi, Zhang, NJP (2010).

23. Z2 invariants Fu, Kane, Mele (2007) Equivalence shown by Wang, Qi, Zhang (2010)

24. Time-reversal symmetry, t  -t -k k T • Time-reversal operator: T=K exp(ipsy) • Half-integer spin: rotation by 2p reverses the sign of the state. • Kramer’s theorem: T2=-1 degeneracy! • Spin-orbit coupling does not break TR. • Magnetic field breaks TR: Zeeman splitting! Time-derivatives (momenta) are reversed! -s s

25. Suppression of backreflection Kramers doublet: |k↑=T|-k↓ k↑|U|-k↓=0 for any time-reversal-invariant operator U Semiclassical picture: destructive interference. Quantum picture: spin-flip would break TRI.

26. Magnetic impurities can open gap Chen et al., Science (2010)

27. Kondo effect in helical electron liquids • Broken SU(2) symmetry for spin, but total angular momentum (orbital+spin) still conserved • Previous work: incomplete Kondo screening, residual degrees of freedom leading to anomalies in low-temperature thermodynamics • My little contribution: complete screening, no anomalous features R. Žitko, Phys. Rev. B 81, 241414(R) (2010)

28. The problem has time-reversal symmetry, so the persistance of Kondo screening seems likely. The Kramers symmetry, not the spin SU(2) symmetry, is essential for the Kondo effect. General approach: reduce the problem to a one-dimensional tight-binding Hamiltonian (Wilson chain Hamiltonian) with the impurity attached to one edgeK. G. Wilson, RMP (1975)H. R. Krisnamurthy et al., PRB (1980) R. Žitko, Phys. Rev. B 81, 241414(R) (2010)

29. Quantum anomalous Hall (QAH) state = QHE without external magnetic field. Proposal: magnetically doped HgTe quantum wells, Liu et al. (2008) See also Qi, Wu, Zhang, PRB (2006), Qi, Hughes, Zhang, PRB (2010)

30. Chiral topological superconductor = QAH + proximity induced superconductivity One has to tune both the magnetization, m, and theinduced superconducting gap, D. Qi, Hughes, Zhang, PRB (2010)

31. chiral Majorana mode Review: Qi, Zhang (2010), Hasan, Kane, RMP (2010)

32. Majorana fermions Two-state system: 0, 1 Complex “Dirac” fermionic operators y and y† defined as: y†0= 1, y1= 0, y 0=0, y†1=0 Canonical anticommutation relations: {y,y}=0, {y†,y†}=0, {y,y†}=1. We “decompose” complex operator  into its “real parts”: y=(h1+ih2)/2, y †=(h1-ih2)/2 Inverse transformation: h1=(y+y †)/2, h2=(y-y †)/(2i) Real operators: hi †=hi Canonical anticommutation relations: {h1,h1}=1, {h2,h2}=1, {h1,h2}=0. Thus hi2=1/2.

33. Is this merely a change of basis? • Not if a single Majorana mode is considered! (Or several spatially separated ones.) • Two separated Majorana fermions correspond to a two-state system (i.e., a qubit, cf. Kitaev 2001) where information is encoded non-locally. • Many-particle systems may have elementary excitations which behave as Majorana fermions. • Single Majorana fermion has half the degrees of freedom of a complex fermion → (1/2)ln2 entropy

34. Majorana excitations in superconductors • Solutions of the Bogoliubov-de Gennes equation come in pairs: y†(E) at energy E  y(E) at energy –E. • At E=0, a solution with y†=y is possible. •  Majorana fermion level at zero energy inside the vortex in a p-wave superconductor. Reed, Green, PRB (2000), Ivanov,PRL (2001), Volovik

35. Non-Abelian states of matter • In 2D, excitations with unusual statistics, anyons (= particles which are neither fermions nor bosons):y1y2=eiqy2y1 with q0,p • Zero-energy Majorana modes  degenerate ground state • Non-Abelian statistics: y1y2=y2y1U Wilczek, PRL 1982 unitary transformation within the ground state multiplet

36. Majorana fermions in condensed-matter systems • p-wave superconductors (Sr2RuO4, cold atom systems) • n=5/2 fractional quantum Hall state • topological superconductors • superconductor-topological insulator-magnetheterostructures Building blocks for topological quantum computers? For a review, see Nayak, Simon, Stern, Freedman, Das Sarma, RMP 80, 1083 (2008).

37. Detection of Majorana fermions • Problem: Majorana excitations in a superconductor have zero charge. • Proposals: • electrical transport measurements in interferometric setups (Akhmerov et al, 2009; Fu, Kane, 2009; Law, Lee, Ng 2009) • “teleportation” (Fu, 2010) • Josephson currents (Tanaka et al. 2009) • non-Fermi-liquid kind of the Kondo effect

38. Interferometric detection Electron can either be transmitted as an electron or as a hole (Andreev process), depending on the number of flux quanta enclosed. Akhmerov, Nilsson, Beenakker, PRL (2009); Fu, Kane, PRL (2009)

39. 2-ch Kondo effect – experimental detection in a quantum-dot system Potok, Rau, Shtrikman, Oreg, Goldhaber-Gordon (2007)

40. Two-channel Kondo model TK TD Can be solved by the numerical renormalization group (NRG), etc.

41. Bosonisation and refermionisation One Majorana mode decouples! Emery, Kivelson (1992)

42. Majorana detection via induced non-Fermi-liquid effects Chiral TSC:single Majoranaedge mode Source-drain linear conductance: R. Žitko, Phys. Rev. B 83,195137 (2011)

43. Impurity decoupled from one of the Majorana modes (a=0) Standard Andersonimpurity (a=45º) Parametrization:

45. Conclusion • Spin-orbit coupling leads to non-trivial topological properties of insulators containing heavy elements. • More surprises at the bottom of the periodic system? • Great news for surface physicists: the interesting things happen at the surface.