Topological delocalization

of two-dimensional massless fermions

- CMP Meets HEP at IPMU Kashiwa 2/10/2010 -

Kentaro Nomura (Tohoku University)

[1] “Topological Delocalization of Two-Dimensional Massless Dirac Fermions”

KN, Mikito Koshino, Shinsei Ryu, PRL 99, 146806 (2007)

[2] “Quantum Hall Effect of Massless Dirac Fermion in a Vanishing Magnetic Field”

KN, S. Ryu, M. Koshino, C. Mudry, A. Furusaki, PRL 100, 246806 (2008)

references

Shinsei Ryu (Berkeley)

Mikito Koshino (Titech)

Christopher Mudry (PSI)

Akira Furusaki (RIKEN)

collaborators

Outline

1. Introduction:

1-1. Surface states of topological insulators

1-2. Anderson localization and scaling theory

2. Topological delocalization of Dirac fermions

3. QHE of Dirac fermions in a vanishing magnetic field

4. Summary

Topological delocalization

of two-dimensional massless fermions

- CMP Meets HEP at IPMU Kashiwa 2/10/2010 -

Spin Hall Effects

(Ordinary) Spin Hall Effect Quantum Spin Hall Effect

Murakami-Nagaosa-Zhang (2003)

Sinova et al. (2004)

Kane-Mele (2005)

Bernevig-Zhang (2006)

Bulk : gapless (metal) gapped (topological insulator)

Strong spin-orbit

interaction

QSHE in 2D and 3D

2D topological insulator

3D topological insulator

HgTe Quantum Well, Thin Bi, …

Kane-Mele, Bernevig-Zhang, Murakami, …

BiSb, BiSe, BiTe

Moore-Balents, Roy, Fu-Kane-Mele, …

Dirac spectram

E

ky

ky

kx

Strong and Weak Topological insulator

(a) Strong topological insulators (STI) (b) Weak topological insulators (WTI)

Odd # of Dirac cones on the surface Even # of Dirac cones on the surface

ky ky

10 00

Moore and Balents (2006), Roy (2006), Fu, Kane, Mele (2007), Qi, Hughes, Zhang (2008)

Is Surface of 3D STI robust?

Question:

Are these surface states robust against disorder (Anderson localization)?

???

localized (insulator) delocalized (metal)

impurities

on the surface

Fragile or Robust

Outline

1. Introduction:

1-1. Surface states of topological insulators

1-2. Anderson localization and scaling theory

2. Topological delocalization of Dirac fermions

3. QHE of Dirac fermions in a vanishing magnetic field

4. Summary

Topological delocalization

of two-dimensional massless fermions

- CMP Meets HEP at IPMU Kashiwa 2/10/2010 -

P. Drude (1900)

Anderson Localization

E >V(r)

y (r)

Hy Ey

P.W. Anderson (1958)

Classical

Quantum

Scaling Theory of Localization

Abrahams, Anderson, Licciardello, Ramakrishnan (1979)

2

2 /

)()( - dL

he

LLg

dimensionless conductance

d : spatial dimension

metalLgdLLg >+ )()(

insulatorLgdLLg <+ )()(

dg(L)

d L> 0

L

gb(g) =

dg(L)

d L< 0

L

gb(g) =

Scaling Theory of Localization

Abrahams, Anderson, Licciardello, Ramakrishnan (1979)

metalLgdLLg >+ )()(

insulatorLgdLLg <+ )()(

dg(L)

d L> 0

L

gb(g) =

dg(L)

d L< 0

L

gb(g) =

d=3 d=2

metal insulator

Outline

1. Introduction:

1-1. Surface states of topological insulators

1-2. Anderson localization and scaling theory

2. Topological delocalization of Dirac fermions

3. QHE of Dirac fermions in a vanishing magnetic field

4. Summary

Topological delocalization

of two-dimensional massless fermions

- CMP Meets HEP at IPMU Kashiwa 2/10/2010 -

)()( 0 LggLg - )()( 0 LggLg +

kykx

E

Berry’s phase in (kx ,ky ) space

kkkkσ |)|( FF vv

kkk kidC

Ando, Nakanishi, Saito (1998), Suzuura, Ando (2002)

-k

“Non-relativistic” “Relativistic”

k 0 -k

k

+

++

ngO

gdL

dg

g

Lg

11)(b

Anti-localization

1-loop correction

g >>1

Hikami-Larkin-Nagaoka (1980)

b(g

)=dl

ng/d

lnL With SO coupling

metalLgdLLg >+ )()(

insulatorLgdLLg <+ )()(

Random SO model

2/}),({

)(2/2

σpr

rp

+

+

VmH

+

++

ngO

gdL

dg

g

Lg

11)(b

Anti-localization

1-loop correction

g >>1

Hikami-Larkin-Nagaoka (1980)

Suzuura-Ando (2002)same result

b(g

)=dl

ng/d

lnL With SO coupling

metalLgdLLg >+ )()(

insulatorLgdLLg <+ )()(

)(rpσ VvH +

Massless Dirac model

Random SO model

2/}),({

)(2/2

σpr

rp

+

+

VmH

+

++

ngO

gdL

dg

g

Lg

11)(b

Anti-localization

1-loop correction

g >>1

Hikami-Larkin-Nagaoka (1980)

Suzuura-Ando (2002)

b(g

)=dl

ng/d

lnL With SO coupling

)(rpσ VvH +

Massless Dirac model

Random SO model

same result

2/}),({

)(2/2

σpr

rp

+

+

VmH

+

++

ngO

gdL

dg

g

Lg

11)(b

Anti-localization

1-loop correction

g >>1

Hikami-Larkin-Nagaoka (1980)

Suzuura-Ando (2002)

)(rpσ VvH +

Massless Dirac model

Random SO model

same result

2/}),({

)(2/2

σpr

rp

+

+

VmH

)'(''|| ,' kkσkkk k'k -+ UH

2/}),({

)(2/2

σpr

rp

+

+

VmH

2

2 )(

d

Edg

)(2

0

+ n

Lkx

)(rpσ VvH +

Massless Dirac model

Random SO model

Spectral flow argument

)()( xeLx i yy +

)(nE

# even

# odd

KN, M. Koshino, S. Ryu, Phys. Rev. Lett. 99, 146806 (2007)

Z2 classification of band insulators

Z_2 class (bulk) _0 0 1

# crossing states even odd

Protected surface metal no yes

Weak topological

insulator (WTI)

Strong topological

insulator (STI)

momentum space (clean limit) experiments (ARPES)

Fu, Kane, Mele, PRL (2007) Hsieh et al. Nature (2007), Nat. Phys (2009)

WTI STI

NLM with Z_2 topological term

Z_2 topological term

cf. Ostrovsky et al

][tr8

1][ 2 QQgxdQS

dimensionless conductance

Scaling of “conductance” = RG flow of “coupling constant”

Z_2 topological term

cf. Ostrovsky et al

Open problem: derivation of the beta-function

][tr8

1][ 2 QQgxdQS

+

++

ngO

gLd

gdg

11

log

log)(b

NLM with Z_2 topological term

TRS breaking perturbations

)()()( xxaσxσ mVviH z+++-

QH transition point Ludwig et al.(1994)

V

Outline

1. Introduction:

1-1. Surface states of topological insulators

1-2. Anderson localization and scaling theory

2. Topological delocalization of Dirac fermions

3. QHE of Dirac fermions in a vanishing magnetic field

4. Summary

Topological delocalization

of two-dimensional massless fermions

- CMP Meets HEP at IPMU Kashiwa 2/10/2010 -

QHE of massless Dirac fermions

)()]([ rrAσ VeivH FK +--

Graphene ( half-integer x4 )

neB

h

Single Dirac fermions (half-integer)

B

Novoselov et al. Nature (2005)

QHE: Non-relativistic vs relativistic

“Non-relativistic” “Relativistic”

weak B field

(strong disorder)

strong B field

(weak disorder)

h

exy

2

2

0xy

?

3a

2 a

1

0

3/2

1/2

-1/2

-3/2

QHE in a vanishing B-field

QHE of Dirac fermions in Bg0h

exy

2

2

manifestation of parity anomaly

Phys. Rev. Lett. 100, 246806 (2008)

disorder

-½ 0 ½

x

Single Dirac fermion (surface of STI)

QHE in a vanishing B-field

QHE of Dirac fermions in Bg0h

exy

2

2

manifestation of parity anomaly

Phys. Rev. Lett. 100, 246806 (2008)

Ezj ˆ2

2

h

e

disorder

Bg0

Single Dirac fermion (surface of STI)

QHE in a vanishing B-field

QHE of Dirac fermions in Bg0h

exy

2

2

Ezj ˆ2

2

h

e

Qi, Li, Zang, Zhang (2009)

j

E

q , 0

Bg0

Single Dirac fermion (surface of STI)

QHE in a vanishing B-field

QHE of Dirac fermions in Bg0h

exy

2

2

Ezj ˆ2

2

h

e

j

E

q , 0

Bg0

“magnetic monopole”

image

Single Dirac fermion (surface of STI)

Qi, Li, Zang, Zhang (2009)

Conclusions

2D massless Dirac fermion on the surface of 3D Topological insulators

1. Robust against Time reversal perturbations

[topologically protected].

2. Half-integer QHEs survive in the B-> 0 limit.

[manifestation of parity anomaly and q-term]

Massless Dirac fernions emerge on the surface of STI

Thanks for your attention

Shinsei Ryu (Berkeley)

Mikito Koshino (Titech)

Christopher Mudry (PSI)

Akira Furusaki (RIKEN)

[1] “Topological Delocalization of Two-Dimensional Massless Dirac Fermions”

KN, Mikito Koshino, Shinsei Ryu, PRL 99, 146806 (2007)

[2] “Quantum Hall Effect of Massless Dirac Fermion in a Vanishing Magnetic Field”

KN, S. Ryu, M. Koshino, C. Mudry, A. Furusaki, PRL 100, 246806 (2008)

references