Page 1
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
Page 2
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 -
Page 3
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
Page 4
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
Page 5
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)
Page 6
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
Page 7
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 -
Page 8
P. Drude (1900)
Anderson Localization
E >V(r)
y (r)
Hy Ey
P.W. Anderson (1958)
Classical
Quantum
Page 9
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) =
Page 10
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
Page 11
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 -
Page 12
)()( 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
Page 13
+
++
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
Page 14
+
++
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
Page 15
+
++
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
Page 16
+
++
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
Page 17
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)
Page 18
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
Page 19
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”
Page 20
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
Page 21
TRS breaking perturbations
)()()( xxaσxσ mVviH z+++-
QH transition point Ludwig et al.(1994)
V
Page 22
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 -
Page 23
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)
Page 24
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
Page 25
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)
Page 26
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)
Page 27
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)
Page 28
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)
Page 29
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
Page 30
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