Topological delocalization of two-dimensional massless fermions

Transcription

1 - CMP Meets HEP at IPMU Kashiwa /10/010 - Topoloical delocalization of two-dimensional massless fermions Kentaro Nomura (Tohoku University) collaborators Shinsei Ryu (Berkeley) Mikito Koshino (Titech) Christopher Mudry (PSI) Akira Furusaki (RIKEN) references [1] Topoloical Delocalization of Two-Dimensional Massless Dirac Fermions KN, Mikito Koshino, Shinsei Ryu, PR 99, (007) [] Quantum Hall Effect of Massless Dirac Fermion in a Vanishin Manetic Field KN, S. Ryu, M. Koshino, C. Mudry, A. Furusaki, PR 100, (008)

2 - CMP Meets HEP at IPMU Kashiwa /10/010 - Topoloical delocalization of two-dimensional massless fermions Outline 1. Introduction: 1-1. Surface states of topoloical insulators 1-. Anderson localization and scalin theory. Topoloical delocalization of Dirac fermions 3. QHE of Dirac fermions in a vanishin manetic field 4. Summary

3 Spin Hall Effects (Ordinary) Spin Hall Effect Quantum Spin Hall Effect Murakami-Naaosa-Zhan (003) Sinova et al. (004) Kane-Mele (005) Bernevi-Zhan (006) Stron spin-orbit interaction Bulk : apless (metal) apped (topoloical insulator)

4 QSHE in D and 3D D topoloical insulator HTe Quantum Well, Thin Bi, Kane-Mele, Bernevi-Zhan, Murakami, E ky 3D topoloical insulator BiSb, BiSe, BiTe Moore-Balents, Roy, Fu-Kane-Mele, Dirac spectram ky kx

5 Stron and Weak Topoloical insulator (a) Stron topoloical insulators (STI) (b) Weak topoloical insulators (WTI) ky Odd # of Dirac cones on the surface ky Even # of Dirac cones on the surface Moore and Balents (006), Roy (006), Fu, Kane, Mele (007), Qi, Huhes, Zhan (008)

6 Is Surface of 3D STI robust? Question: Are these surface states robust aainst disorder (Anderson localization)???? impurities on the surface localized (insulator) delocalized (metal) Fraile or Robust

7 - CMP Meets HEP at IPMU Kashiwa /10/010 - Topoloical delocalization of two-dimensional massless fermions Outline 1. Introduction: 1-1. Surface states of topoloical insulators 1-. Anderson localization and scalin theory. Topoloical delocalization of Dirac fermions 3. QHE of Dirac fermions in a vanishin manetic field 4. Summary

8 Anderson ocalization Classical E >V(r) P. Drude (1900) Quantum y (r) P.W. Anderson (1958) Hy Ey

9 Scalin Theory of ocalization Abrahams, Anderson, icciardello, Ramakrishnan (1979) dimensionless conductance ( ) ( ) e / h d - ( ( + d) > ( ) + d) < ( ) metal b() = d() d insulator > 0 d : spatial dimension b() = d() d < 0

10 Scalin Theory of ocalization Abrahams, Anderson, icciardello, Ramakrishnan (1979) metal insulator d=3 d= ( + d) > ( ) metal b() = d() d > 0 ( + d) < ( ) insulator b() = d() d < 0

11 - CMP Meets HEP at IPMU Kashiwa /10/010 - Topoloical delocalization of two-dimensional massless fermions Outline 1. Introduction: 1-1. Surface states of topoloical insulators 1-. Anderson localization and scalin theory. Topoloical delocalization of Dirac fermions 3. QHE of Dirac fermions in a vanishin manetic field 4. Summary

12 Berry s phase in (kx,ky ) space E v F σ k k ( v k ) F k k y k x dk ik k C k Ando, Nakanishi, Saito (1998), Suzuura, Ando (00) Non-relativistic Relativistic -k k 0 -k k ( ) 0 -( ) ( ) 0 + ( )

13 b()=dln/dln Anti-localization Random SO model H p / m + V ( r) 1-loop correction + { ( r), p} σ / b( ) d d O 1 n + >>1 Hikami-arkin-Naaoka (1980) With SO couplin ( + d) > ( ) metal ( + d) < ( ) insulator

14 b()=dln/dln Anti-localization Random SO model H p / m + V ( r) 1-loop correction + { Massless Dirac model ( r), p} σ / b( ) d d O 1 n + H vσ p + V (r) >>1 Hikami-arkin-Naaoka (1980) Suzuura-Ando (00) same result With SO couplin ( + d) > ( ) metal ( + d) < ( ) insulator

15 b()=dln/dln Anti-localization Random SO model H p / m + V ( r) 1-loop correction + { Massless Dirac model ( r), p} σ / b( ) d d O 1 n + H vσ p + V (r) >>1 Hikami-arkin-Naaoka (1980) Suzuura-Ando (00) same result With SO couplin

16 Anti-localization Random SO model H p / m + V ( r) 1-loop correction + { Massless Dirac model ( r), p} σ / b( ) d d O 1 n + H vσ p + V (r) >>1 Hikami-arkin-Naaoka (1980) Suzuura-Ando (00) same result k H k' ' kσ' k, k' + U( k -k')

17 Spectral flow arument Random SO model H p / m + V ( r) + { Massless Dirac model ( r), p} σ H vσ p + V (r) / KN, M. Koshino, S. Ryu, Phys. Rev. ett. 99, (007) ( x ) e i y + y ( x) E n () d E( ) d # even # odd k x ( n + ) 0

18 Z classification of band insulators Weak topoloical insulator (WTI) Stron topoloical insulator (STI) Z_ class (bulk) _0 0 1 # crossin states even odd Protected surface metal no yes momentum space (clean limit) experiments (ARPES) WTI STI Fu, Kane, Mele, PR (007) Hsieh et al. Nature (007), Nat. Phys (009)

19 NM with Z_ topoloical term cf. Ostrovsky et al S[ Q] d x 1 8 tr[ Q Q] Z_ topoloical term dimensionless conductance Scalin of conductance = RG flow of couplin constant

20 NM with Z_ topoloical term cf. Ostrovsky et al S[ Q] d x 1 8 tr[ Q Q] Z_ topoloical term Open problem: derivation of the beta-function b( ) d lo d lo O 1 n +

21 TRS breakin perturbations H -ivσ + V( x) + σa( x) + m( x) z V QH transition point udwi et al.(1994)

22 - CMP Meets HEP at IPMU Kashiwa /10/010 - Topoloical delocalization of two-dimensional massless fermions Outline 1. Introduction: 1-1. Surface states of topoloical insulators 1-. Anderson localization and scalin theory. Topoloical delocalization of Dirac fermions 3. QHE of Dirac fermions in a vanishin manetic field 4. Summary

23 QHE of massless Dirac fermions HK vfσ[ -i -ea( r)] + V( r) Graphene ( half-inteer x4 ) Sinle Dirac fermions (half-inteer) B Novoselov et al. Nature (005) h eb n

24 QHE: Non-relativistic vs relativistic Non-relativistic Relativistic weak B field (stron disorder) xy? e h xy a a stron B field (weak disorder) -3/ 1/ -1/ 3/

25 QHE in a vanishin B-field Sinle Dirac fermion (surface of STI) QHE of Dirac fermions in B0 xy e h disorder x -½ 0 ½ manifestation of parity anomaly Phys. Rev. ett. 100, (008)

26 QHE in a vanishin B-field Sinle Dirac fermion (surface of STI) QHE of Dirac fermions in B0 xy e h B0 j e h zˆ E disorder manifestation of parity anomaly Phys. Rev. ett. 100, (008)

27 QHE in a vanishin B-field Sinle Dirac fermion (surface of STI) QHE of Dirac fermions in B0 xy e h Qi, i, Zan, Zhan (009) B0 j e h zˆ E E j q, 0

28 QHE in a vanishin B-field Sinle Dirac fermion (surface of STI) QHE of Dirac fermions in B0 xy e h Qi, i, Zan, Zhan (009) B0 j e h zˆ E E j manetic monopole imae q, 0

29 Conclusions D massless Dirac fermion on the surface of 3D Topoloical insulators Massless Dirac fernions emere on the surface of STI 1. Robust aainst Time reversal perturbations [topoloically protected].. Half-inteer QHEs survive in the B-> 0 limit. [manifestation of parity anomaly and q-term]

30 Thanks for your attention Shinsei Ryu (Berkeley) Mikito Koshino (Titech) Christopher Mudry (PSI) Akira Furusaki (RIKEN) references [1] Topoloical Delocalization of Two-Dimensional Massless Dirac Fermions KN, Mikito Koshino, Shinsei Ryu, PR 99, (007) [] Quantum Hall Effect of Massless Dirac Fermion in a Vanishin Manetic Field KN, S. Ryu, M. Koshino, C. Mudry, A. Furusaki, PR 100, (008)