Topological Insulators and Superconductors. Tokyo 2010 Shoucheng Zhang, Stanford University

Transcription

1 Topological Insulators and Superconductors Tokyo 2010 Shoucheng Zhang, Stanford University

2 Colloborators Stanford group: Xiaoliang Qi, Andrei Bernevig, Congjun Wu, Chaoxing Liu, Taylor Hughes, Sri Raghu, Suk-bum Chung Stanford experimentalists: Yulin Chen, Ian Fisher, ZX Shen, Yi Cui, Aharon Kapitulnik, Wuerzburg colleagues: Laurens Molenkamp, Hartmut Buhmann, Markus Koenig Ewelina Hankiewicz, Bjoern Trauzettle IOP colleagues: Zhong Fang, Xi Dai, Haijun Zhang, Tsinghua colleagues: Qikun Xue, Jinfeng Jia, Xi Chen,

3 Outline Models and materials of topological insulators Topological field theory, experimental proposals Topological Mott insulators Topological superconductors and superfluids, experimental proposals

4 The search for new states of matter The search for new elements led to a golden age of chemistry. The search for new particles led to the golden age of particle physics. In condensed matter physics, we ask what are the fundamental states of matter? In the classical world we have solid, liquid and gas. The same H 2 O molecules can condense into ice, water or vapor. In the quantum world we have metals, insulators, superconductors, magnets etc. Most of these states are differentiated by the broken symmetry. Crystal: Broken translational symmetry Magnet: Broken rotational symmetry Superconductor: Broken gauge symmetry

5 The quantum Hall state, a topologically non-trivial state of matter 2 e xy n h TKNN integer=the first Chern number. 2 d k n F ( k) 2 (2 ) Topological states of matter are defined and described by topological field theory: S eff xy d 2 xdt 2 A A Physically measurable topological properties are all contained in the topological field theory, e.g. QHE, fractional charge, fractional statistics etc

6 Discovery of the 2D and 3D topological insulator HgTe Theory: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experiment: Koenig et al, Science 318, 766 (2007) BiSb Theory: Fu and Kane, PRB 76, (2007) Experiment: Hsieh et al, Nature 452, 907 (2008) Bi2Te3, Sb2Te3, Bi2Se3 Theory: Zhang et al, Nature Physics 5, 438 (2009) Theory and Experiment Bi2Se3: Xia et al, Nature Physics 5, 398 (2009), Experiment BieTe3: Chen et al Science 325, 178 (2009)

7 Topological Insulator is a New State of Quantum Matter

8 Chiral (QHE) and helical (QSHE) liquids in D=1 k k -k F k F -k F k F The QHE state spatially separates the two chiral states of a spinless 1D liquid The QSHE state spatially separates the four chiral states of a spinful 1D liquid 2=1+1 4=2+2 x x Kane and Mele (without Landau levels) Benervig and Zhang (with Landau levels)

9 Quantum protection by T 2 =-1 (Qi&Zhang, Phys Today, Jan, 2010) Transport experiments: Molenkamp group STM experiments: Yazdani group, Kapitulnik group, Xue group

10 From topology to chemistry: the search for the QSH state Type III quantum wells work. HgTe has a negative band gap! (Bernevig, Hughes and Zhang, Science 2006) Bandgap energy (ev) Tuning the thickness of the HgTe/CdTe quantum well leads to a topological quantum phase transition into the QSH state. Sign of the Dirac mass term determines the topological term in field theory Bandgap vs. lattice constant (at room temperature in zinc blende structure) lattice constant a [ 臸 0

11 Band Structure of HgTe S P P 3/2 S P 1/2 S P S P 3/2 P 1/2

12 Band inversion in HgTe leads to a topological quantum phase transition Let us focus on E1, H1 bands close to crossing point HgTe HgTe E1 H1 CdTe H1 CdTe CdTe E1 CdTe normal inverted

13 The model of the 2D topological insulator (BHZ, Science 2006) ), (,, (,,,, y x y x ip p ip p s s Square lattice with 4-orbitals per site: Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k. ) ( 0 0 ) ( ), ( k h k h k k H y x eff 2 2 ) ( ) ( ) ( ) ( ) sin (sin ) sin (sin ) ( ) ( Bk m ik k A ik k A Bk m k d k m k i k A k i k A k m k h y x y x a a y x y x Similar to relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! m/b<0 for d>d c.

14 Mass domain wall Cutting the Hall bar along the y-direction we see a domain-wall structure in the band structure mass term. This leads to states localized on the domain wall which still disperse along the x-direction. (similar to Jackiw-Rebbi solition). y y x x m/b<0 0 m E E Bulk k x 0 Bulk

15 Experimental observation of the QSH edge state (Konig et al, Science 2007) x x

16 Nonlocal transport in the QSH regime, (Roth et al Science 2009) R 14,14 =3/4 h/e 2 R (k) I: 1-4 V: R 14,23 =1/4 h/e V* (V)

17 No QSH in graphene Bond current model on honeycomb lattice (Haldane, PRL 1988). Spin Hall insulator (Murakami, Nagaosa and Zhang 2004) Spin-orbit coupling in graphene (Kane and Mele 2005). They took atomic spin-orbit coupling of 5meV to estimate the size of the gap. Yao et al, Min et al 2006 showed that the actual spin-orbit gap is given by 2 so / =10-3 mev. Similar to the seasaw mechanism of the neutrino mass in the Standard Model!

18 3D insulators with a single Dirac cone on the surface (a) z y (b) x y Quintuple layer (c) C x t 1 t 2 t 3 Bi Se1 Se2 A B C A B C

19 Relevant orbitals of Bi2Se3 and the band inversion (a) (b) 0.6 Bi Se E (ev) 0.2 c (ev) (I) (II) (III)

20 Bulk and surface states from first principle calculations (a) Sb 2 Se 3 (b) Sb 2 Te 3 (c) Bi 2 Se 3 (d) Bi 2 Te 3

21 Model for topological insulator Bi2Te3, (Zhang et al, 2009) Pz+, up, Pz-, up, Pz+, down, Pz-, down Single Dirac cone on the surface of Bi2Te3 Surface of Bi2Te3 = ¼ Graphene!

22 Arpes experiment on Bi2Te3 surface states, Shen group Doping evolution of the FS and band structure Undoped Under doped Optimallydoped Over doped E F (undoped) BCB bottom BVB bottom Dirac point position

23 Arpes experiment on Bi2Se3 surface states, Hasan group

24 General definition of a topological insulator Topological field theory term in the effective action. Generally valid for interacting and disordered systems. Directly measurable physically. Relates to axion physics! (Qi, Hughes and Zhang) For a periodic system, the system is time reversal symmetric only when =0 => trivial insulator = => non-trivial insulator Z2 topological band invariant in momentum space based on single particle states. (Fu, Kane and Mele, Moore and Balents, Roy)

25 term with open boundaries = implies QHE on the boundary with 2 1 e xy 2 h For a sample with boundary, it is only insulating when a small T-breaking field is applied to the boundary. The surface theory is a CS term, describing the half QH. Each Dirac cone contributes xy =1/2e 2 /h to the QH. Therefore, = implies an odd number of Dirac cones on the surface! T breaking M E j // Surface of a TI = ¼ graphene

26 Generalization of the QH topology state in d=2 to time reversal invariant topological state in d>2, in Science 2001

27 Time Reversal Invariant Topological Insulators Zhang & Hu 2001: TRI topological insulator in D=4. => Root state of all TRI topological insulators. Murakami, Nagaosa & Zhang 2004: Spin Hall insulator with spin-orbit coupled band structure. Kane and Mele, Bernevig and Zhang 2005: Quantum spin Hall insulator with and without Landau levels. Fu, Kane & Mele, Moore and Balents, Roy 2007: Topological band theory based on Z2 Qi, Hughes and Zhang 2008: Topological field theory based on F F dual. TRB Chern-Simons term in D=2: A 0 =even, A i =odd S D d k da( k) d x A( x) da( x) TRI Chern-Simons term in D=4: A 0 =even, A i =odd S D d k da( k) da( k) d x A( x) da( x) da( x)

28 Dimensional reduction From 4D QHE to the 3D topological insulator Zhang & Hu, Qi, Hughes & Zhang S 4DQH S d 3D 3 xdt( d d 4 xdt 3 dx 5 A 5 A ( x, t)) xdt ( x, t) F F F F F F x 5 A 5 (x, y, z) S From 3D axion action to the 2D QSH 3D S d 2D 2 e 3 J2D d xdt A A 2 2 xdt e ( dzaz ( x, t)) A J1 D 2 2 d xdt A Goldstone & Wilzcek

29 Topological field theory and the family tree Topological field theory of the QHE: (Thouless et al, Zhang, Hansson and Kivelson) S d 2 k da( k) x A( x) da( x) Topological field theory of the TI: (Qi, Hughes and Zhang, 2008) d 3 S d 3 k( a( k) da( k)..) d 4 x da( x) da( x) More extensive and general classification soon followed (Kitaev, Ludwig et al)

30 Equivalence between the integral and the discrete topological invariants (Wang, Qi and Zhang, ) LHS=QHZ definition of TI, RHS=FKM definition of TI

31 Low frequency Faraday/Kerr rotation (Qi, Hughes and Zhang, PRB78, , 2008) Adiabatic E g Requirement: (surface gap) normal contribution Topological contribution topo»3.6x10-3 rad

32 Seeing the magnetic monopole thru the mirror of a TME insulator, (Qi et al, Science 323, 1184, 2009) q TME insulator (for =, = ) similar to Witten s dyon effect higher order feed back Magnitude of B:

33 Dynamic axions in topological magnetic insulators (Li et al, Nature Physics 2010) Hubbard interactions leads to anti-ferromagnetic order Effective action for dynamical axion

34 Spin-plasmon collective mode (Raghu et al, ) General operator identity: Density-spin coupling:

35 Topological Mott insulators Dynamic generation of spin-orbit coupling can give rise to TMI (Raghu et al, PRL 2008). Interplay between spin-orbit coupling and Mott physics in 5d transition metal Ir oxides, Nagaosa, SCZ et al PRL 2009, Balents et al, Franz et al. Topological Kondo insulators (Coleman et al)

36 Topological insulators and superconductors Full pairing gap in the bulk, gapless Majorana edge and surface states Chiral Majorana fermions Chiral fermions massless Majorana fermions massless Dirac fermions Qi, Hughes, Raghu and Zhang, PRL, 2009

37 Probing He3B as a topological superfluid (Chung and Zhang, 2009) The BCS-BdG model for He3B Model of the 3D TI by Zhang et al Surface Majorana state: Qi, Hughes, Raghu and Zhang, PRL, 2009 Schnyder et al, PRB, 2008 Kitaev Roy

38 General reading For a video introduction of topological insulators and superconductors, see com/watch?v=qg8yu -Ju3Vw

39 Summary: discovery of new states of matter Crystal Magnet s-wave superconductor Quantum Spin Hall Topological insulators