Quantum Spin Hall Effect - A New State of Matter ? -

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Quantum Spin Hall Effect - A New State of Matter
? -
Aug. 1, 2006 _at_Banff

• Naoto Nagaosa
• Dept. Applied Phys. Univ. Tokyo
• Collaborators
• M. Onoda (AIST), Y. Avishai (Ben-Grion)

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magnetic field
B
Voltage
Hall effect
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(Integer) Quantum Hall Effect
Quantized Hall conductance in the unit of
Plateau as a function of magnetic field

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(Integer) Quantum Hall Effect
pure case
Quantized Hall conductance in the unit of
Plateau as a function of magnetic field
Disorder effect and localization

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(Integer) Quantum Hall Effect
pure case

Localized states do not contribute to
Extended states survive only at
discrete energies
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Anderson Localization of electronic wavefunctions
x
x
Extended Bloch wave
x
impurity
Localized state
quantum interference between scattered waves.
Thouless number Dimensionless conductance
Periodic boundary condition
Anti-periodic boundary condition
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Scaling Theory of Anderson Localization
The change of the Thouless number Is 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 !!
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Universality classes of Anderson Localization
Symplectic class with Spin-orbit interaction
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.
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Chern number
wave function
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Chern number is carried only by extended
states. Topology protects extended states.

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Chiral edge modes

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M
magnetization
y
v
Electric field
E
x
Anomalous Hall Effect
Hall, Karplus-Luttinger, Smit, Berger, etc.
Berry phase
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Electrons with constraint
doublydegeneratepositive energy states.
Dirac electrons
Bloch electrons

• Projection onto positive energy state
• Spin-orbit interaction
• as SU(2) gauge connection

Projection onto each band Berry phase
of Bloch wavefunction
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Berry Phase Curvature in k-space
Bloch wavefucntion
Berry phase connection in k-space
covariant derivative

Curvature in k-space
Anomalous Velocity and Anomalous Hall Effect
New Quantum Mechanics !! Non-commutative Q.M.
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Duality between Real and Momentum
Spaces
k- space curvature
r- space curvature
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Gauge flux density
Distribution of momentum space magnetic field
in momentum space of metallic ferromagnet with
spin-orbit interaction.
Chern 's (-1, -2, 3, -4, 5 -1)
Chern number Integral of the gauge flux over
the 1st BZ.
M.Onoda, N.N. J.P.S.P. 2002
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Localization in Haldane model -- Quantized
anomalous Hall effect
M.Onoda-N.N. 2003
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spin current time-reversal even
y
v
v
E
Electric field
x
Spin Hall Effect
Dyakonov-Perel (1971)
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Spin current induced by an electric field
x current direction y spin direction z
electric field

• SU(2) analog of the QHE
• topological origin
• dissipationless
• All occupied states in the valence band
  contribute.
• Spin current is time-reversal even

S.Murakami-N.N.-S.C.Zhang J.Sinova-Q.Niu-A.MacDona
ld
GaAs
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Wave-packet formalism in systems with Kramers
degeneracy
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.
Eq. of motion
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Experimental confirmation of spin Hall effect in
GaAs D.D.Awschalom (n-type) UC Santa Barbara
J.Wunderlich (p-type ) Hitachi Cambridge
n-type
p-type
Y.K.Kato,et.al.,Science,306,1910(2004)
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Recent focus of
theories Quantum spin Hall effect - A New
State of Matter ?
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Spin Hall Insulator with real Dissipationless
spin current
S.Murakami, N.N., S.C.Zhang (2004)
Bernevig-S.C.Zhang Kane-Mele
HgTe, HgSe, HgS, alpha-Sn
Zero/narrow gap semiconductors
Rocksalt structure PbTe, PbSe, PbS
Finite spin Hall conductance but not quantized
No edge modes for generic spin Hall insulator
Quantum spin Hall
Generic Spin Hall Insulator
M.Onoda-NN (PRL05)
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Two sources of conservation law
Rotational symmetry ? Angular
momentum Gauge symmetry ? Conserved
current
Topology ? winding number
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Quantum Hall Problem
Quantized Hall Conductance
Localization problem
TKNN
2-param. scaling
TKNN
Landauer
Topological Numbers Chern
Edge modes
Gauge invariance
Conserved charge current and U(1) gauge invariance
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Issues to be addressed
Spin Hall Conductance
Localization problem
Sheng-Weng-Haldane
Topological Numbers Spin Chern, Z2
Edge modes
Kane-Mele Xu-Moore Wu-Bernevig-Zhang Qi-Wu-Zhang
No conserved spin current !!
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Kane-Mele Model of quantum spin Hall system
Lattice structure and/or inversion symmetry
breaking Graphene, HgTe at interface, Bi
surface (Bernevig-S.C.Zhang) (Murakami)
Pfaffian
time-reversal operation
Stability of edge modes Z2 topological number
of helical edge mode pairs
Kane-Mele 2005
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1st BZ
Two Dirac Fermions at K and K ? 8 components
K
K
K
K
K
K
SU(2) anomaly (Witten) ?
helical edge modes
Stability against the T-invariant disorder due to
Kramers theorem
Kane-Mele, Xu-Moore, Wu-Bernevig-Zhang
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Sheng et al. 2006 Qi et al. 2006
Chern Number Matrix
spin Chern number
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Generalized twisted boundary condition
Qi-Wu-Zhang(2006)
Spin Chern number
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Issues to be addressed
Spin Hall Conductance
Localization problem
?
Sheng-Weng-Haldane
Topological Numbers Spin Chern, Z2
Edge modes
Kane-Mele Xu-Moore Wu-Bernevig-Zhang Qi-Wu-Zhang
No conserved spin current !!
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Generalized Kane-Mele Model
Z2 non-trivial
Z2 trivial
Chern number 1,-1
Chern number 0
Two decoupled Haldane model (unitary)
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Numerical study of
localization MacKinnons transfer matrix method
and finite size scaling
M
L
Localization length
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(a-2)
(a-3)
(a-1)
2 copies of Haldane model
(b-1)
(b-2)
(b-3)
(c-1)
(c-2)
(c-3)
increasing disorder strength W
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Two decoupled unitary model with Chern number
1,-1
Symplectic model
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Disappearance of the extended states in unitary
model
hybridizes positive and negative Chern
number states
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Disappearance of the extended states in trivial
symplectic model
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Scaling Analysis of the localization/delocalizatio
n transition
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Conjectures
No quantized spin Hall conductance nor plateau
Spin Hall Conductance
Localization problem
Topological Numbers Spin Chern, Z2
Helical Edge modes
No conserved spin current !!
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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
E
E