Fiber bundles and topology for condensed matter systems

Transcription

1 Fber bundles and topology for condensed atter systes Hans-aner Trebn Insttut für Theoretsche und Angewandte Physk der Unverstät Stuttgart, Gerany Krakow, 4 Aprl 05 /3

2 . Tangent bundles and curvature Krakow, 4 Aprl 05 /3

3 The tangent bundle of the sphere e p T p S v e S Krakow, 4 Aprl 05 3/3

4 Parallel transport of vectors on S Coparson of dfferent tangent spaces by parallel transport along a path Lev Cvta connecton Krakow, 4 Aprl 05 4/3

5 Covarant dervatve Covarant dervatve: u u D D w : w ds dt t e 0 l t t t 0 w t 0 P t w t 0 w t w t 0 t 0 t s(t) : Connecton coeffcen ts Parallel transport : D w 0 Krakow, 4 Aprl 05 5/3

6 Curvature of S Curvature: transport along closed path rotates vector (holonoy) otaton angle: encrcled area K K d Gaussan curvature Curvature tensor two-densonal anfold:, Krakow, 4 Aprl 05 6/3

7 Prncpal bundle For descrpton of parallel transport and curvature: attach rotaton group SO() at each pont, yelds prncpal bundle (S,SO())=(S,S ) Change of orgn (unt operaton): gauge transforaton Krakow, 4 Aprl 05 7/3

8 . Fber bundles Krakow, 4 Aprl 05 8/3

9 Fber bundles Consst of a basc dfferental anfold M At each pont attached: fber F p whch s ether copy of a vector space ( vector bundle ) or of a (gauge) group ( prncple bundle ) Prescrptons for glueng the fbers together e.g. Moebus strp (S,) F p = M=S Krakow, 4 Aprl 05 9/3

10 Fber bundles Prescrpton for parallel transport : covarant dervatve and curvature D D Curvature tensor : j w : w e j w w, w w : Connecton coeffcen t atrx j j j j l, l j w w l l j F p = M=S Topologcal quantu nubers, e.g. for d tangent bundles: Euler characterstc: M d, g Krakow, 4 Aprl 05 0/3

11 3. Topologcal quantu nubers Krakow, 4 Aprl 05 /3

12 Euler characterstc χ E = χ E =0 χ E =0 χ E =- χ E =-4 χ E =-4 Krakow, 4 Aprl 05 /3

13 Euler characterstc E # faces # vertces # edges Krakow, 4 Aprl 05 3/3

14 4. Applcatons of fber bundles Krakow, 4 Aprl 05 4/3

15 Fber bundle n cosology TM 4 or 4 M,SO,3, M 4 spacete Krakow, 4 Aprl 05 5/3

16 Fber bundle n electrodynacs Classcal relatvstc physcs: Four oentu : In electroag netc feld : v p E c p, four velocty : v p qa, A p A Topology: Curvature equals the electroag netc feld tensor Quantu echancs: p D C,.e. secton through a Parallels? 3 0,,U v 0 prncple bundle q A q exp dx A phase change D U 3,C - bundle F F A ch 0 E E E x y z Topologcal quantu nuber A cp. 0 B E B y x z dx dx B E 0 d closed subanfold z B F y x B E B 0 x z y :Frst Chern nuber, Krakow, 4 Aprl 05 6/3

17 Magnetc onopole of strength γ Vector potental and electroag netc feld tensor ˆ A 0, 0, ˆ A Cp.Lev - Cvta connecton on the sphere n orthonora l bass 0 cos 0 Frst chern nuber : ch cos, F, ˆ d sn sn 0 d F actng on C : actng on : 3,U : Electroag nets 3,U SU : Electroweak nteracto n 3,SU 3 : Strong nteracto n Krakow, 4 Aprl 05 7/3

18 5. Topologcal quantu states Krakow, 4 Aprl 05 8/3

19 The Berry phase Quantu echancal ground states dependent on paraeter : M paraeter anfold Exaple : Spn - partcle n agnetc feld of fxed strength, but arbtrary orentato n : M S,, s deterned up to a phase factor e, syste s prncple bundle M, U Krakow, 4 Aprl 05 9/3

20 The Berry phase Adabatc oton and energy gap : partcles rean n nstantane ous state Crcular oton : phase change, holonoy Covarant dervatve D A wth Berry connecton A Berry curvature F Frst chern nuber for d paraeter ch M M dx dx F space Exaple spn - - partcle : - A 0, - A cos, F sn, ch Krakow, 4 Aprl 05 0/3

21 The Quantu Hall Effect (von Kltzng 980) B z B z j x d crystal n agnetc feld E y yx j x j x xy E y yx h n e xy e n h E y xy xy B z B z B z Krakow, 4 Aprl 05 /3

22 The Quantu Hall Effect Wave functons labelled by wavevector k Wavevector space s also perodc : k Prncple bundle T,U k,k : u k od K,K T Hasan MZ, Kane CL 00 MP 8, 3045 Kubo transport forula for Hall conductvty : K k T xy e h occuped bands d Torus k u u u u Berry curvature F K xy e h ch occuped bands n T,U Krakow, 4 Aprl 05 /3

23 6. Bascs for general topologcal classfcaton Krakow, 4 Aprl 05 3/3

24 Search for topologcal nsulators Insulator : descrbed by a appng T k d H k? Set of Bloch - Haltona ns wth gap. Set T d, of topologca lly dfferent appngs? Procedure : eplace by an equvalent atheatc ally known standard space such that T d, T d, Krakow, 4 Aprl 05 4/3

25 Topologcal classfcaton of nsulators Frst step : dagonalz e Bloch - Halton atrces : H k k r H k H k s j r, j r H k egenstate s of H k s s j U H k U k k k n k U H k U k U U n k U Krakow, 4 Aprl 05 5/3

26 Topologcal classfcaton of nsulators Second step : defor bandstructure k, k j K k U U wthout closng gap UD n U + s "Orbt" of D Fxpont group of D : U U n n S U U n U S D U S s soorphc to the "Grassannan" coset space G Topologcal quantu nubers for solators : T n, d,g n under the acton of the group U UD C U n /U U n n, n C Z for d,3 n U n - Krakow, 4 Aprl 05 6/3

27 The spn quantu Hall effect estrcto n on the base anfold by te reversal syetry New base anfold New classfca ton by HgTe/CdTe quantu well structures n d Spn polarzed edge currents B -x T Sb x d /,B,G S 3,B n, T C Te 3 d / Z 0,, d,3 n 3d, surface currents : k -k K k k T K Hasan MZ, Kane CL 00 MP 8, 3045 Krakow, 4 Aprl 05 7/3

28 7. Suary Krakow, 4 Aprl 05 8/3

29 Suary Up to 980: Quantu nubers based on syetry Easy to break, lft of degeneraces Snce 980: Topologcal quantu nubers New states of quantu atter obust Krakow, 4 Aprl 05 9/3

30 Lterature Hasan MZ, Kane CL 00 ev.mod.phys. 8, 3045 Q X-L, Chang S-Ch 0 ev.mod.phys. 83, 057 Budch JC, Trauzettel B 03 phys.stat.sol.(l) 7, 09 v. Kltzng K, Dorda G, Pepper M 980 PL 45, 494 Thouless DJ, Kohoto M, Nghtngale MP, den Njs M 98 PL 49, 405 Kane CL, Mele EJ 005 PL 95, 4680 Bernevg BA, Zhang S-Ch 006 PL 96, 0680 Bernevg BA, Hughes DL, Zhang S-Ch 006 Scence 34, 757 Köng M, Wedann S, Brüne Ch, oth A, Buhann H, Molenkap W, Q X-L, Zhang S-Ch 007 Scence 38, 766 Hseh D, Qan D, Wray L, Xa Y, Hor YS, Cava J, Hasan MZ 008 Nature 45, 970 Krakow, 4 Aprl 05 30/3

31 The End Krakow, 4 Aprl 05 3/3