Quantum Spin Hall Effect

- A New State of Matter ? -

Naoto Nagaosa

Dept. Applied Phys. Univ. Tokyo

Collaborators:

M. Onoda (AIST), Y. Avishai (Ben-Grion)

Aug. 1, 2006 @Banff

Bmagnetic field

Voltage

Hall effect

(Integer) Quantum Hall Effect

Quantized Hall conductance in the unit of h

e2

Plateau as a function of magnetic field

(Integer) Quantum Hall Effect

Quantized Hall conductance in the unit of h

e2

Plateau as a function of magnetic field

pure case

Disorder effect and localization

pure case

Localized states do not contribute to xy

Extended states survive only at discrete energies

(Integer) Quantum Hall Effect

Anderson Localization of electronic wavefunctions

xx

ximpurity

Extended Bloch waveLocalized state

EELeGLg d /)/()( 22 Thouless number= Dimensionless conductance

E

L

E

Periodic boundary condition

Anti-periodic boundary condition

quantum interference betweenscattered waves.

Scaling Theory of Anderson Localization

)/),(()( LdLLgfdLLg The change of the Thouless numberIs determined only by the Thouless number Itself.

In 3D there is a metal-insulator transition

In 1D and 2D all the states are localized for any finite disorder !!

Symplectic class with Spin-orbit interaction

Universality classes of Anderson Localization

Orthogonal: Time-reversal symmetric system without the spin-orbit interaction

Symplectic: Time-reversal symmetric system with the spin-orbit interaction

Unitary: Time-reversal symmetry broken Under magnetic field or ferromagnets Chern number extended states

Universality of critical phenomena Spatial dimension, Symmetry, etc. determine the critical exponents.

xx k/

yy k/

0 2

2wave function

Chern number

cckk

kdi

Cheyx

xy .|4

)//( 22

Chern number is carried only by extended states.

Topology “protects” extended states.

Chiral edge modes

M

vy

x

-e

-e

-e

-eE

Anomalous Hall Effect

magnetization

Electricfield

Hall, Karplus-Luttinger, Smit,Berger, etc.

Berry phase

Electrons with ”constraint”

Projection onto positive energy stateSpin-orbit interaction

as SU(2) gauge connection

Dirac electrons

doublydegenerate

positive energy states.

E

k

Bloch electrons

Projection onto each bandBerry phase

of Bloch wavefunction

k

E

Berry Phase Curvature in k-space

Bloch wavefucntion )()( ruer nkikr

nk

nkknkn uuikA ||)( Berry phase connection in k-space

)()( kAikArx nknii i covariant derivative

)())()((],[ kiBkAkAiyx nznxknyk yx Curvature in k-space

y

VkB

m

k

y

Vyxi

m

kHxi

dt

tdxnz

xx

)(],[],[)(

xk yk

zk

Anomalous Velocity andAnomalous Hall Effect

New Quantum Mechanics !!Non-commutative Q.M.

knku| nku|

k

dt

tkdkB

k

k

dt

trdn

n )()(

)()(

dt

trdrB

r

rV

dt

tkd )()(

)()(

Duality between Real and Momentum Spaces

k- space curvature

r- space curvature

Gauge flux density

M.Onoda, N.N.J.P.S.P. 2002

Chern #'s : (-1, -2, 3, -4, 5 -1)

Chern number = Integral of the gauge fluxover the 1st BZ.

Distribution of momentum space “magnetic field” in momentum spaceof metallic ferromagnet with spin-orbit interaction.

M.Onoda-N.N. 2003

Localization in Haldane model -- Quantized anomalous Hall effect

vy

x

-e

-e

-e

E

Spin Hall Effect

Electric field

v-e

-e

-e

spin currenttime-reversal even

D’yakonov-Perel (1971)

Spin current induced by an electric field

x: current direction y: spin directionz: electric field

SU(2) analog of the QHE• topological origin• dissipationless • All occupied states in the valence ba

nd contribute.• Spin current is time-reversal even

zsLF

HF

zxy E

ekk

eEj

2

1

4 2

GaAs

E

x

y

z

S.Murakami-N.N.-S.C.ZhangJ.Sinova-Q.Niu-A.MacDonald

Let us extend the wave-packet formalism to the case with time-reversal symmetry.

Adiabatic transport = The wave-packet stays in the same band, but can transform inside the Kramers degeneracy.

Wave-packet formalism in systems with Kramers degeneracy

),(),,(),(),,(),()( 22113 LHntxqtqatxqtqaqdt cncnn

),(

),(1

),(

),(

2

1

22

212

1

tqa

tqa

aatqz

tqz

zAkiz

LHnzFzkk

Ex

Eek

n

nljj

l

n

l

,

Eq. of motion

Wunderlich et al. 　 2004

Experimental confirmation of spin Hall effect in GaAs D.D.Awschalom (n-type) 　　 UC Santa Barbara J.Wunderlich (p-type ) Hitachi Cambridge

Y.K.Kato,et.al.,Science,306,1910(2004)

n-type p-type

Recent focus of theories

Quantum spin Hall effect - A New State of Matter ?

Spin Hall Insulator with real Dissipationless spin current

Zero/narrow gap semiconductors

S.Murakami, N.N., S.C.Zhang (2004)

Rocksalt structure: PbTe, PbSe, PbSHgTe, HgSe, HgS, alpha-Sn

s

Bernevig-S.C.ZhangKane-Mele

rryxrryxr

rryrrxr

rrr

cccc

cccc

cMcH

H.c.3232

H.c.33

)(

52

33

52

33

52

4252

42

521

Quantum spin Hall GenericSpin Hall InsulatorM.Onoda-NN (PRL05)

0

Finite spin Hall conductance but not quantized

No edge modesfor generic spinHall insulator

Two sources of “conservation law”

Rotational symmetry Angular momentumGauge symmetry Conserved current

Topology winding number

Quantum Hall Problem

Quantized Hall Conductance

Localization problem

Topological Numbers

ChernEdge modes

TKNN

2-param. scalin

g

Gauge invariance

TKNN

Conserved charge current and U(1) gauge invariance

Landauer

Issues to be addressed

Spin Hall Conductance

Localization problem

Topological Numbers

Spin Chern, Z2Edge modes

No conserved spin current !!

Kane-MeleXu-MooreWu-Bernevig-ZhangQi-Wu-Zhang

Sheng-Weng-Haldane

Kane-Mele 2005

Kane-Mele Model of quantum spin Hall system

Stability of edge modes Z2 topological number = # of helical edge mode pairs

kk HH Lattice structureand/or inversion symmetry breakingGraphene, HgTe at interface, Bi surface　　　　　　 (Bernevig-S.C.Zhang) (Murakami)

Pfaffian

time-reversal operation

1st BZ

K

K

K

K’

K’

K’

Two Dirac Fermions at K and K’ 8 components

helical edge modes

SU(2) anomaly (Witten) ?

Stability against the T-invariant disorder due to Kramer’s theorem

Kane-Mele, Xu-Moore, Wu-Bernevig-Zhang

Sheng et al. 2006Qi et al. 2006

Chern Number Matrix

CC : spin Chern number

Generalized twisted boundary condition Qi-Wu-

Zhang(2006)

nn 4or 24 Spin Chern number

Issues to be addressed

Spin Hall Conductance

Localization problem

Topological Numbers

Spin Chern, Z2Edge modes

?

No conserved spin current !!

Kane-MeleXu-MooreWu-Bernevig-ZhangQi-Wu-Zhang

Sheng-Weng-Haldane

Two decoupled Haldane model(unitary)

Chern number =0

Chern number =1,-1

Z2 trivialZ2 non-trivial

xh

Generalized Kane-Mele Model

Numerical study of localization MacKinnon’s transfer matrix method and finite size scaling

M

L

Localization length ),( WM

/),1( LeLG

MWMWM /),(),(

(a-1)

(b-1)

(a-2) (a-3)

(b-2) (b-3)

(c-1) (c-2) (c-3)

2 copies of Haldane model

increasing disorder strength W

Two decoupled unitary modelwith Chern number +1,-1

Symplectic model

xh

Disappearance of the extended states in unitary model

hybridizes positive andnegative Chern number statesxh

xh

Disappearance of the extended states in trivial symplectic model

Scaling Analysis of the localization/delocalization transition

73.2symplectic 33.2unitary

Conjectures

Spin Hall Conductance

Localization problem

Topological Numbers

Spin Chern, Z2Helical Edge modes

No conserved spin current !!

No quantized spin Hall conductancenor plateau

Conclusions

Rich variety of Bloch wave functions in solids Symmetry classification Topological classification Anomalous velocity makes the insulator an active player.

Quantum spin Hall systems: No conserved spin current but Analogous to quantum Hall systems characterized by spin Chern number/Z2 number

Novel localization properties influenced by topology New universality class !? Graphene, HgTe, Bi (Murakami) Stability of the edge modes

Spin Current physics Spin pumping and ME effect

EE