1. Topological insulators and superconductors
Rok Žitko
Ljubljana,

2. Topological insulators (TI)
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 Quantified by Described 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 insulators disorder  electrons become localized
Mott insulators Coulomb interaction (repulsion) between electrons  motion suppressed
Band insulators absence 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
If Zeeman splitting is included, each Landau level splits into a pair, one for spin up electrons and the other for spin down electrons. Then the occupation of each spin Landau level is just the ratio of fluxes D = Φ / Φ0.
Landau levels
Egap=ħwc

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

9. Gaussian curvature, K=1/R1R2
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
Integer number!
F: Berry’s curvature
A: Berry’s connection, measures the overlap between the wavefunctions.
Berry curvature
Same n as in sxy=ne2/h. An integer within an accuracy of at least 10-9! New resistance standard: RK=h/e2= (18) W

14. Spin-orbit coupling e-
Nucleus
Stronger effect for heavy 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 states must 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  -tk -k T
Time-derivatives (momenta) are reversed!
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! s -s
W/o spin-orbit, Kramer’s degeneracy is simply degeneracy between spin-up and spin-down.

25. Suppression of backreflection
Kramers doublet: |k↑=T|-k↓
k↑|U|-k↓=0 for any time-reversal-invariant operator U
Hermitian conjugate of an antiunitary operator does not exist.
Further properties: A(f1+f2)=Af1+Af2, A(cf)=c*Af.
Operator T is antiunitary: Ta|Tb=a|b*=b|a for spin-1/2 particles.
cf. Moore 2009
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, (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 edge K. G. Wilson, RMP (1975) H. R. Krisnamurthy et al., PRB (1980)
R. Žitko, Phys. Rev. B 81, (R) (2010)

29. Quantum anomalous Hall (QAH) state
= QHE without external magnetic field.
Proposal: magnetically doped HgTe quantum wells, Liu et al. (2008)
QSH effect = two copies of QAH effect (Liu et al. 2008)
Spin splitting induced by magnetization. The spin-down states penetrate in the bulk and disappear.
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 the induced 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.
spinless 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-magnet heterostructures
px+ipy chiral superconductor (i.e. Sr2RuO4)
Heterostructures: Fu, Kane PRL 103, (2008)
Magnet needed to obtain CHIRAL Majorana modes.
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 modelTK 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 Majorana edge mode
New realization of the 2CK model.
Source-drain linear conductance:
R. Žitko, Phys. Rev. B 83, (2011)

43. Impurity decoupled from one of the Majorana modes (a=0)
Standard Anderson impurity (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.