Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE) Shoucheng Zhang, Stanford University Les Houches, June 2006

References: Murakami, Nagaosa and Zhang, Science 301, 1348 (2003) Murakami, Nagaosa, Zhang, PRL 93, 156804 (2004) Bernevig and Zhang, PRL 95, 016801 (2005) Bernevig and Zhang, PRL 96, 106802 (2006); Qi, Wu, Zhang, condmat/0505308; Wu, Bernevig and Zhang, PRL 96, 106401 (2006); (Haldane, PRL 61, 2015 (1988)); Kane and Mele, PRL95 226801 (2005); Sheng et al, PRL 95, 136602 (2005); Xu and Moore cond-mat/0508291……

What about quantum spin Hall?

Key ingredients of the quantum Hall effect: Time reversal symmetry breaking. Bulk gap. Gapless chiral edge states. External magnetic field is not necessary! Quantized anomalous Hall effect: Time reversal symmetry breaking due to ferromagnetic moment. Topologically non-trivial bulk band gap. Gapless chiral edge states ensured by the index theorem.

Topological Quantization of the AHE (cond-mat/0505308) Magnetic semiconductor with SO coupling (no Landau levels): General 2×2 Hamiltonian Example Rashbar Spin-orbital Coupling

Topological Quantization of the AHE (cond-mat/0505308) Hall Conductivity Insulator Condition Quantization Rule The Example

Origin of Quantization: Skyrmion in momentum space Skyrmion number=1 Skyrmion in lattice momentum space (torus) Edge state due to monopole singularity

Band structure on stripe geometry and topological edge state

The intrinsic spin Hall effect Bulk GaAs Energy (eV) The intrinsic spin Hall effect Key advantage: electric field manipulation, rather than magnetic field. dissipationless response, since both spin current and the electric field are even under time reversal. Topological origin, due to Berry’s phase in momentum space similar to the QHE. Contrast between the spin current and the Ohm’s law:

Spin-Hall insulator: dissipationless spin transport without charge transport (PRL 93, 156804, 2004) In zero-gap semiconductors, such as HgTe, PbTe and a-Sn, the HH band is fully occupied while the LH band is completely empty. A bulk charge gap can be induced by quantum confinement in 2D or pressure. In this case, the spin Hall conductivity is maximal.

Spin-Orbit Coupling – Spin 3/2 Systems Luttinger Hamiltonian ( : spin-3/2 matrix) Symplectic symmetry structure

Spin-Orbit Coupling – Spin 3/2 Systems Natural structure SO(5) Vector Matrices SO(5) Tensor Matrices Inversion symmetric terms: d- wave Inversion asymmetric terms: p-wave Strain: Applied Rashba Field:

Luttinger Model for spin Hall insulator Bulk Material zero gap l=+1/2,-1/2 l=+3/2,-3/2 Symmetric Quantum Well, z-z mirror symmetry Decoupled between (-1/2, 3/2) and (1/2, -3/2)

Dirac Edge States Edge 1 y x Edge 2 L kx

From Dirac to Rashba Dirac at Beta=0 Rashba at Beta=1 0.0 0.02 1.0 0.2

From Luttinger to Rashba

Phase diagram Rashba Coupling 10^5 m/s 2.2 1.1 -1.1 -2.2

Topology in QHE: U(1) Chern Number and Edge States Relate more general many-body Chern number to edge states: “Goldstone theorem” for topological order. Generalized Twist boundary condition: Connection between periodical system and open boundary system Niu, Thouless and Wu, PRB Qi, Wu and Zhang, in progress

Topology in QHE: Chern Number and Edge States Non-vanishing Chern number Monopole in enlarged parameter space Gapless Edge States in the twisted Hamiltonian Monopole Gapless point boundary 3d parameter space

The Quantum Hall Effect with Landau Levels Spin – Orbit Coupling in varying external potential? for

Quantum Spin Hall 2D electron motion in increasing radial electric GaAs Inside a uniformly charged cylinder Electrons with large g-factor:

Quantum Spin Hall Hamiltonian for electrons: Spin - Spin - Tune to R=2 No inversion symm, shear strain ~ electric field (for SO coupling term)

Quantum Spin Hall Different strain configurations create the different “gauges” in the Landau level problem [110] Landau Gap and Strain Gradient

Helical Liquid at the Edge P,T-invariant system QSH characterized by number n of fermion PAIRS on ONE edge. Non-chiral edges => longitudinal charge conductance! Double Chern-Simons (Zhang, Hansson, Kivelson) (Michael Freedman, Chetan Nayak, Kirill Shtengel, Kevin Walker, Zhenghan Wang)

Quantum Spin Hall In Graphene (Kane and Mele) Graphene is a semimetal. Spin-orbit coupling opens a gap and forms non-trivial topological insulator with n=1 per edge (for certain gap val) Based on the Haldane model (PRL 1988) Quantized longitudinal conductance in the gap Experiment sees universal conductivity, SO gap too small Haldane, PRL 61, 2015 (1988) Kane and Mele, condmat/0411737 Bernevig and Zhang, condmat/0504147 Sheng et al, PRL 95, 136602 (2005) Kane and Mele PRL 95, 146802 (2005) Qi, Wu, Zhang, condmat/0505308 Wu, Bernevig and Zhang condmat/0508273 Xu and Moore cond-mat/0508291 …

Stability at the edge The edge states of the QSHE is the 1D helical liquid. Opposite spins have the opposite chirality at the same edge. It is different from the 1D chiral liquid (T breaking), and the 1D spinless fermions. T2=1 for spinless fermions and T2=-1 for helical liquids. Single particle backscattering is not possible for helical liquids!

Conclusions Quantum AHE in ferromagnetic insulators. Quantum SHE in “inverted band gap” insulators. Quantum SHE with Landau levels, caused by strain. New universality class of 1D liquid: helical liquid. QSHE is simpler to understand theoretically, well-classified by the global topology, easier to detect experimentally, purely intrinsic, can be engineered by band structure, enables spintronics without spin injection and spin detection.

Topological Quantization of Spin Hall Physical Understanding: Edge states In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states. Laughlin’s Gauge Argument: When turning on a flux threading a cylinder system, the edge states will transfer from one edge to another Energy spectrum on stripe geometry.

= + Topological Quantization of Spin Hall Physical Understanding: Edge states When an electric field is applied, n edge states with G12=+1(-1) transfer from left (right) to right (left). G12 accumulation  Spin accumulation Conserved Non-conserved = +

Topological Quantization of SHE Luttinger Hamiltonian rewritten as In the presence of mirror symmetry z->-z, <kz>=0d1=d2=0! In this case, the H becomes block-diagonal: LH HH SHE is topological quantized to be n/2p