Quantum Spin Hall Effect A New State of

Quantum Spin Hall Effect - A New State of Matter ? Aug. 1, 2006 @Banff Naoto Nagaosa Dept. Applied Phys. Univ. Tokyo Collaborators: M. Onoda (AIST), Y. Avishai (Ben-Grion)

magnetic field Voltage B Hall effect

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

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

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

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

Scaling Theory of Anderson Localization The change of the Thouless number Is determined only by the Thouless number Itself. In 3 D there is a metal-insulator transition In 1 D and 2 D all the states are localized for any finite disorder !!

Universality classes of Anderson Localization Orthogonal: Time-reversal symmetric system without the spin-orbit interaction Symplectic class with 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.

Chern number wave function

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

Chiral edge modes

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

Electrons with ”constraint” doubly degenerate positive energy states. Dirac electrons Projection onto positive energy state Spin-orbit interaction as SU(2) gauge connection 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

Distribution of momentum space “magnetic field” in momentum space of metallic ferromagnet with spin-orbit interaction. Gauge flux density Chern #'s : (-1, -2, 3, -4, 5 -1) Chern number = Integral of the gauge flux over the 1 st BZ. M. Onoda, N. N. J. P. S. P. 2002

Localization in Haldane model -- Quantized anomalous Hall effect M. Onoda-N. N. 2003

spin current time-reversal even y v -e -e -e E Electric field x Spin Hall Effect 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 S. Murakami-N. N. -S. C. Zhang J. Sinova-Q. Niu-A. Mac. Donald Ga. As

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

Experimental confirmation of spin Hall effect in Ga. As 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) Wunderlich et al. 　2004

Recent focus of theories Quantum spin Hall effect - A New State of Matter ?

Spin Hall Insulator with real Dissipationless spin current S. Murakami, N. N. , S. C. Zhang (2004) Bernevig-S. C. Zhang Kane-Mele Zero/narrow gap semiconductors Hg. Te, Hg. S, alpha-Sn Rocksalt structure: Pb. Te, Pb. S Finite spin Hall conductance but not quantized No edge modes for generic spin Hall insulator M. Onoda-NN (PRL 05) Quantum spin Hall Generic Spin Hall Insulator

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

Quantum Hall Problem TKNN Quantized Hall Conductance n ali TKNN m a ar p 2 - Topological Numbers Chern c. s Localization problem g La nd au Gauge invariance er Edge modes Conserved charge current and U(1) gauge invariance

Issues to be addressed Spin Hall Conductance Localization problem Sheng-Weng-Haldane Topological Numbers Spin Chern, Z 2 Kane-Mele Xu-Moore Wu-Bernevig-Zhang Qi-Wu-Zhang Edge modes No conserved spin current !!

Kane-Mele Model of quantum spin Hall system Lattice structure and/or inversion symmetry breaking Graphene, Hg. Te at interface, Bi surface 　　　　　　(Bernevig-S. C. Zhang) (Murakami) Pfaffian time-reversal operation Stability of edge modes Z 2 topological number = # of helical edge mode pairs Kane-Mele 2005

Two Dirac Fermions at K and K’ 8 components 1 st BZ K K’ K’ K K K’ SU(2) anomaly (Witten) ? helical edge modes Stability against the T-invariant disorder due to Kramer’s theorem Kane-Mele, Xu-Moore, Wu-Bernevig-Zhang

Sheng et al. 2006 Qi et al. 2006 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 Sheng-Weng-Haldane Topological Numbers Spin Chern, Z 2 ? Kane-Mele Xu-Moore Wu-Bernevig-Zhang Qi-Wu-Zhang Edge modes No conserved spin current !!

Generalized Kane-Mele Model Z 2 non-trivial Z 2 trivial Chern number =1, -1 Chern number =0 Two decoupled Haldane model (unitary)

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

(a-1) (a-2) (a-3) (b-1) (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 No quantized spin Hall conductance nor plateau Spin Hall Conductance Localization problem Topological Numbers Spin Chern, Z 2 Helical Edge modes No conserved spin current !!

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/Z 2 number Novel localization properties influenced by topology New universality class !? Graphene, Hg. Te, Bi (Murakami) Stability of the edge modes Spin Current physics Spin pumping and ME effect E E