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,

B magnetic field Voltage Hall effect

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

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

pure case Localized states do not contribute to Extended states survive only at discrete energies (Integer) Quantum Hall Effect

Anderson Localization of electronic wavefunctions x x x impurity Extended Bloch wave Localized state Thouless number = Dimensionless conductance Periodic boundary condition Anti-periodic boundary condition quantum interference between scattered waves.

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 !!

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.

wave function Chern number

Chern number is carried only by extended states. Topology “protects” extended states.

Chiral edge modes

M v y x -e E Anomalous Hall Effect magnetization Electric field Hall, Karplus-Luttinger, Smit, Berger, etc. Berry phase

Electrons with ”constraint” Projection onto positive energy state Spin-orbit interaction as SU(2) gauge connection Dirac electrons doubly degenerate positive energy states. Bloch electrons Projection onto each band Berry phase of Bloch wavefunction

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.

Duality between Real and Momentum Spaces k- space curvature r- space curvature

Gauge flux density M.Onoda, N.N. J.P.S.P Chern #'s : (-1, -2, 3, -4, 5 -1) Chern number = Integral of the gauge flux over the 1 st BZ. Distribution of momentum space “magnetic field” in momentum space of metallic ferromagnet with spin-orbit interaction.

M.Onoda-N.N Localization in Haldane model -- Quantized anomalous Hall effect

v y x -e E Spin Hall Effect Electric field v -e spin current time-reversal even D’yakonov-Perel (1971)

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 GaAs S.Murakami-N.N.-S.C.Zhang J.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 Eq. of motion

Wunderlich et al 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, PbS HgTe, HgSe, HgS, alpha-Sn Bernevig-S.C.Zhang Kane-Mele Quantum spin Hall Generic Spin Hall Insulator M.Onoda-NN (PRL05) Finite spin Hall conductance but not quantized No edge modes for generic spin Hall insulator

Two sources of “conservation law” Rotational symmetry  Angular momentum Gauge symmetry  Conserved current Topology  winding number

Quantum Hall Problem Quantized Hall Conductance Localization problem Topological Numbers Chern Edge modes TKNN 2-param. scaling 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, Z2 Edge modes No conserved spin current !! Kane-Mele Xu-Moore Wu-Bernevig-Zhang Qi-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 Lattice structure and/or inversion symmetry breaking Graphene, HgTe at interface, Bi surface (Bernevig-S.C.Zhang) (Murakami) Pfaffian time-reversal operation

1st BZ 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 Qi et al Chern Number Matrix : spin Chern number

Generalized twisted boundary condition Qi-Wu- Zhang(2006) Spin Chern number

Issues to be addressed Spin Hall Conductance Localization problem Topological Numbers Spin Chern, Z2 Edge modes ? No conserved spin current !! Kane-Mele Xu-Moore Wu-Bernevig-Zhang Qi-Wu-Zhang Sheng-Weng-Haldane

Two decoupled Haldane model (unitary) Chern number =0 Chern number =1,-1 Z2 trivial Z2 non-trivial Generalized Kane-Mele Model

Numerical study of localization MacKinnon’s transfer matrix method and finite size scaling M L Localization length

(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 model with Chern number +1,-1 Symplectic model

Disappearance of the extended states in unitary model hybridizes positive and negative Chern number states

Disappearance of the extended states in trivial symplectic model

Scaling Analysis of the localization/delocalization transition

Conjectures Spin Hall Conductance Localization problem Topological Numbers Spin Chern, Z2 Helical Edge modes No conserved spin current !! No quantized spin Hall conductance nor 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 E E