Topological insulators and superconductors Rok itko Ljubljana 22

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

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

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

Solid-liquid phase transition Broken translation invariance Order parameter: FT of <r(r)r(0)>, Bragg peaks FLandau=a 2+b 4 a=a 0(T-Tc)

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

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

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

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

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

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)

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

TKNN (Chern) invariant Thouless-Kohmoto-Nightingale-den Nijs, PRL 1982 Integer number! Berry curvature Same n as in sxy=ne 2/h. An integer within an accuracy of at least 10 -9! New resistance standard: RK=h/e 2=25812. 807557(18) W

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

Reinterpretation:

Quantum Spin Hall effect (QSHE) (“ 2 D 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 Hg. Te/Cd. Te quantum wells.

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

3 D topological insulators • Generalization of QSHE to 3 D. • Insulating in the bulk, conducting helical surface states. • Theoretically predicted in 2006, experimentally discovered in Bi. Sb alloys (Hsieh et al. , Nature 2008) and in Bi 2 Se 3 and similar layered materials (Xia et al. , Nature Phys. 2009).

Surface states on 3 D 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.

Experimental detection in Bi 2 Te 3 Chen et al. Science (2009)

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

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

Z 2 invariants Fu, Kane, Mele (2007) Equivalence shown by Wang, Qi, Zhang (2010)

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

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.

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

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)

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, 241414(R) (2010)

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

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)

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

Majorana fermions Two-state system: 0 , 1 Complex “Dirac” fermionic operators and † defined as: † 0 = 1 , 1 = 0 , 0 =0, † 1 =0 Canonical anticommutation relations: { , }=0, { †, †}=0, { , †}=1. We “decompose” complex operator into its “real parts”: =(h 1+ih 2)/ 2, † =(h 1 -ih 2)/ 2 Inverse transformation: h 1=( + † )/ 2, h 2=( - † )/( 2 i) Real operators: hi † =hi Canonical anticommutation relations: {h 1, h 1}=1, {h 2, h 2}=1, {h 1, h 2}=0. Thus hi 2=1/2.

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)ln 2 entropy

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

Non-Abelian states of matter • In 2 D, excitations with unusual statistics, anyons (= particles which are neither fermions nor bosons): 1 2=eiq 2 1 with q 0, p Wilczek, PRL 1982 • Zero-energy Majorana modes degenerate ground state • Non-Abelian statistics: 1 2= 2 1 U unitary transformation within the ground state multiplet

Majorana fermions in condensed-matter systems • p-wave superconductors (Sr 2 Ru. O 4, cold atom systems) • n=5/2 fractional quantum Hall state • topological superconductors • superconductor-topological insulator-magnet heterostructures Building blocks for topological quantum computers? For a review, see Nayak, Simon, Stern, Freedman, Das Sarma, RMP 80, 1083 (2008).

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

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)

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

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

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

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

Impurity decoupled from one of the Majorana modes (a=0) Standard Anderson impurity (a=45º) Parametrization:

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.