Topological Insulators and Superconductors

Transcription

1 Topological Insulators and Suprconductors Charls L. Kan, Univrsity of Pnnsylvania I. Topological Insulators and Band Thory Unifying thm: bulk boundary corrspondnc - Intgr Quantum Hall ffct - D Quantum Spin Hall Insulator - 3D Topological Insulator - Topological Suprconductivity, Majorana frmions II. Summary and Outlook - What w hav accomplishd - Challngs for th Futur Thanks to Gn Ml, Liang Fu, Jffry To

2 Th Insulating Stat.g. Silicon g ~ 1 V Th Intgr Quantum Hall Stat D Cyclotron Motion, Landau Lvls g c s xy = /h IQH with zro nt magntic fild Graphn with a priodic magntic fild B(r) (Haldan PRL 1988) B(r) = 0 Zro gap, Dirac point B(r) 0 nrgy gap s xy = /h g Band structur k

3 Topological Band Thory Th distinction btwn a convntional insulator and th quantum Hall stat is a topological proprty of th band structur H ( k ) : B lo c h H a m ilto n a n s B rillo u in zo n ( = to ru s T ) w ith n rg y g a p Th st of N occupid Bloch wavfunctions vctor bundl ovr th Brillouin zon torus. N u ( k ) i i 1 dfins a U(N) Classifid by th first Chrn numbr (or TKNN invariant) (Thoulss t al, 1984) Closly rlatd to thory of lctric polarization Brry connction Brry curvatur 1 st Chrn numbr Insulator : n = 0 IQH stat : s xy = n /h N A ( k ) - i u ( k ) u ( k ) i 1 F ( k ) A ( k ) k i 1 1 n A d k d C F k T k i -/a /a -/a Th TKNN invariant can only chang at a phas transition whr th nrgy gap gos to zro k y C k x /a

4 dg Stats Gaplss stats must xist at th intrfac btwn diffrnt topological phass IQH stat n=1 Vacuum n=0 dg stats ~ skipping orbits Gaplss Chiral Frmions : = v k K Haldan Modl gap k y Bulk Boundary Corrspondnc : K y x Dirac quation : M<0 n=1 n=0 Smooth transition : band invrsion ( ) H - iv s + s + M x s x x y y z Domain wall bound stat Jackiw, Rbbi (1976) Su, Schriffr, Hgr (1980) Dn = # Chiral dg Mods 0 ik y M>0 gap y x - M ( x ') d x '/ v

5 Tim Rvrsal Invariant Topological Insulator Tim Rvrsal Symmtry : Kramrs Thorm : - 1 H k H (- k ) is * - 1 All stats doubly dgnrat y topological invariant (n = 0,1) for D T-invariant band structurs Undrstand via Bulk-Boundary corrspondnc : dg Stats for 0<k</a n=0 : Convntional Insulator n=1 : Topological Insulator Kramrs dgnrat at tim rvrsal invariant momnta k* = -k* + G k*=0 k*=/a k*=0 k*=/a vn numbr of bands crossing Frmi nrgy Odd numbr of bands crossing Frmi nrgy

6 D Quantum Spin Hall Insulator I. Graphn Kan, Ml PRL 05 Intrinsic spin orbit intraction small (~10mK-1K) band gap S z consrvd : Haldan modl dg stats : G = /h 0 /a /a g II. HgCdT quantum wlls Thory: Brnvig, Hughs and Zhang, Scinc 06 xprimnt: Konig t al. Scinc 07 d Hg x Cd 1-x T Hg x Cd 1-x T HgT d < 6.3 nm : Normal band ordr G 6 ~ s k d > 6.3 nm : Invrtd band ordr G 8 ~ p k G ~ /h in QSHI Normal G 8 ~ p band invrsion G 6 ~ s Invrtd Convntional Insulator QSH Insulator ( ) + 1 n a ( ) - 1 n a

7 3D Topological Insulators Thr ar 4 surfac Dirac Points du to Kramrs dgnracy 4 k y 1 3 k x OR Moor & Balnts PRB 07 Roy, cond-mat 06 Fu, Kan & Ml PRL 07 Surfac Brillouin Zon D Dirac Point n 0 = 1 : Strong Topological Insulator Frmi circl ncloss odd numbr of Dirac points Topological Mtal : 1/4 graphn Robust to disordr: impossibl to localiz n 0 = 0 : Wak Topological Insulator Frmi circl ncloss vn numbr of Dirac points Rlatd to layrd D QSHI k= a k= b k= a k= b How do th Dirac points connct? Dtrmind by 4 bulk topological invariants n 0 ; (n 1 n n 3 ) F

8 Bi 1-x Sb x Thory: Prdict Bi 1-x Sb x is a topological insulator by xploiting invrsion symmtry of pur Bi, Sb (Fu,Kan PRL 07) xprimnt: ARPS (Hsih t al. Natur 08) Bi 1-x Sb x is a Strong Topological Insulator n 0 ;(n 1,n,n 3 ) = 1;(111) 5 surfac stat bands cross F btwn G and M Bi S 3 ARPS xprimnt : Y. Xia t al., Natur Phys. (009). Band Thory : H. Zhang t. al, Natur Phys. (009). n0;(n1,n,n3) = 1;(000) : Band invrsion at G nrgy gap: D ~.3 V : A room tmpratur topological insulator Control F on surfac by xposing to NO Simpl surfac stat structur : Similar to graphn, xcpt only a singl Dirac point F

9 Orbital QH : Surfac Quantum Hall ffct =0 Landau Lvl for Dirac frmions. Fractional IQH s xy n + h 1 s xy h B - s xy h n=1 chiral dg stat Anomalous QH : Induc a surfac gap by dpositing magntic matrial H F ( - iv s - + D s ) 0 M z Mass du to xchang fild s sg n ( D ) xy M h M M TI gap = D M Chiral dg Stat at Domain Wall : D M -D M + h - h

10 Topological Magntolctric ffct Considr a solid cylindr of TI with a magntically gappd surfac M J 1 J s n M xy + h Magntolctric Polarizability M Qi, Hughs, Zhang 08; ssin, Moor, Vandrbilt 09 n + h 1 topological q trm D L B q h Th fractional part of th magntolctric polarizability is dtrmind by th bulk, and indpndnt of th surfac (providd thr is a gap) Analogous to th lctric polarization, P, in 1D. T R s ym. : q 0 o r m o d DL formula uncrtainty quantum d=1 : Polarization P d=3 : Magntolctric poliarizability P B BZ Tr[ A ] [ ] Tr A d A + A A A BZ 4 h 3 / h (xtra nd lctron) (xtra surfac quantum Hall layr)

11 Topological Suprconductivity BCS man fild thory : * D 1 H H B d G k Bogoliubov d Gnns Hamiltonian H B d G H 0 D * D - H 0 Intrinsic anti-unitary particl hol symmtry B d G - 1 H - H B d G * x x Particl hol rdundancy - - k = sam stat k Bloch - BdG Hamiltonians satisfy Topological classification problm similar to tim rvrsal symmtry - 1 H ( k ) - H ( - k) B d G B d G

12 1D Topological Suprconductor : n = 0,1 Bulk-Boundary corrspondnc : Discrt nd stat spctrum ND (Kitav, 000) D 0 -D n=0 trivial n=1 topological - G G - G D 0 -D =0 Zro mod G G 0 0 Majorana frmion bound stat Majorana Frmion : Particl = Antiparticl Ral part of a Dirac frmion : Half a stat + ; + i i( - ) ; - i 1, i j ij i 1 Two Majorana frmions dfin a singl two lvl systm occupid H i 0 mpty

13 Priodic Tabl of Topological Insulators and Suprconductors Anti-Unitary Symmtris : - Tim Rvrsal : H - Particl - Hol : H - 1 ( k) + H (- k) ; 1-1 ( k) - H (- k) ; 1 Schnydr, Ryu, Furusaki, Ludwig 008 Kitav, 008 Unitary (chiral) symmtry : - 1 H ( k) - H ( k) ; Complx K-thory Altland- Zirnbaur Random Matrix Classs Ral K-thory Bott Priodicity

14 Furthr Rading: Hasan and Kan, Rv Mod Phys 8, 3045 (010). Moor, Natur 464, 194 (010). Qi and Zhang, Phys. Today 63, 33 (010). Ryu, Schnydr, Furusaki and Ludwig, Nw J. Phys. 1, (010). Qi and Zhang, Rv Mod Phys, to appar, arxiv: Moor and Hasan, Annual Rviw of Condnsd Mattr,, 44 (010).

15 Major accomplishmnts : Topological band thory of insulators and suprconductors is wll undrstood: - Topological Invariants and bulk-boundary corrspondnc - Robustnss to disordr and wak intractions - lctromagntic and/or gravitational rspons Rapid matrials progrss: - Svral matrials hav bn idntifid and charactrizd xprimntally. - vn mor matrials hav bn prdictd, basd on band structur calculations. - Dtaild charactrization of topological insulators via transport, optics and spctroscopy is dvloping.

16 Grand Challngs Prfct xisting and nw matrials Dsign and implmnt htrostructur dvics Find Majorana Classify and charactriz many body topological phass Find applications for tchnology

17 Prfct Nw and xisting Matrials Ral 3D topological insulator matrials ar not such grat insulators. lctrical conductanc is dominatd by th bulk. Challng for matrials thory in conjunction with xprimnts. Succss Story : Bi T S Xiong, t al (Princton) 11 lctrical rsistivity in Bi S 3 (Chcklsky t al 09)

18 Topological insulator dvics Rquirs control intrfacs btwn matrials. Challng for matrials thory and xprimnt Topological Insulator Trivial Insulator - protct th surfac stats - control th surfac stat Frmi nrgy (modulation doping) Topological Insulator Magntic Insulator - achiv magntically gappd surfac stats - anomalous quantum Hall ffct - topological magntolctric ffct Topological Insulator Suprconductor - achiv proximity inducd suprconductivity in th surfac stats. I. T.I. M. T.I. S.C. T.I.

19 Potntial Hosts : Find Majorana 1937 : Majorana publishs his modification of th Dirac quation that allows spin ½ particls to b thir own antiparticl : Majorana mystriously disappars at sa Obsrvation of a Majorana frmion is among th grat challngs of physics today Particl Physics : Nutrino (mayb) ttor Majorana ? - Allows nutrinolss doubl b-dcay. Condnsd mattr physics : Possibl du to pair condnsation - n=5/ Fractional quantum Hall ffct - Topological suprconductivity Topological Quantum Computation 0

20 What is th bst way to achiv topological suprconductivity? xotic suprconductors (suprfluids) - Surfac of 4 H - p+ip suprconductor (g Sr RuO 4 ) - Cu x Bi S 3? Ordinary suprconductor htrostructurs - suprconductor topological insulator SC TI Majorana bound stat at a vortx (0D) M TI SC 1D Chiral Majorana mod at a intrfac with a magntic matrial Majorana nd stat - suprconductor smiconductor (g InAs wir) SC InAs What ar th most fasibl xprimntal signaturs of Majorana mods?

21 Classify and Charactriz Intracting Topological Stats Topological Insulators ar lik th Intgr Quantum Hall ffct. Th singl particl nrgy gap is corrctly dscribd by non intracting band thory. Intracting systms xhibit a much richr collction of fractional quantum Hall stats. Undrstanding ths was on of th gratst triumphs of many body physics.

22 What is a fractional topological insulator? Classify possibl stats Charactriz quasiparticl xcitations and surfac stats. Nd to dvlop nw tchniqus: - Parton construction? - B-F thory? - ntanglmnt spctrum? What is th gnralization of th bulk boundary corrspondnc for intracting systms?