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

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

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

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 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=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 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

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

Gaussian curvature, K=1/R1R2 Gauss-Bonnet theorem Gaussian curvature, K=1/R1R2 http://mathworld.wolfram.com/Gauss-BonnetFormula.html http://en.wikipedia.org/wiki/Genus_%28mathematics%29 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 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=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) (“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 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 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-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.

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.

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 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)

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 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.

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. spinless 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 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

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, 237001 (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).

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 Majorana edge mode New realization of the 2CK model. 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.