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## Rok Žitko

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**Topological insulators and superconductors**Rok Žitko Ljubljana, 22. 7. 2011**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**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=ay2+by4 a=a0(T-Tc)**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**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**2D electron gas in strong magnetic field**wc=eB/mc Landau levels Egap=ħwc**Quantum Hall Effect**von Klitzing et al. (1980) Jy=sxyEx chiral edge states**Gauss-Bonnet theorem**Gaussian curvature, K=1/R1R2 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 (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.**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**Spin-orbit coupling**e- Nucleus Stronger effect forheavy elements (Pb, Bi, etc.) from the bottom of the periodic system**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.**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.**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).**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.**Experimental detection in Bi2Te3**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).**Z2 invariants**Fu, Kane, Mele (2007) Equivalence shown by Wang, Qi, Zhang (2010)**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**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 edgeK. 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 HgTe 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 theinduced 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 y and y† defined as: y†0= 1, y1= 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.**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**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**Non-Abelian states of matter**• In 2D, excitations with unusual statistics, anyons (= particles which are neither fermions nor bosons):y1y2=eiqy2y1 with q0,p • Zero-energy Majorana modes degenerate ground state • Non-Abelian statistics: y1y2=y2y1U Wilczek, PRL 1982 unitary transformation within the ground state multiplet**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).**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**TK TD Can be solved by the numerical renormalization group (NRG), etc.**Bosonisation and refermionisation**One Majorana mode decouples! Emery, Kivelson (1992)**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)**Impurity decoupled from one of the Majorana modes (a=0)**Standard Andersonimpurity (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.