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……
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. Key ingredients of the quantum Hall effect: • Time reversal symmetry breaking. • Bulk gap. • Gapless chiral edge states.
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
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) Tensor Matrices SO(5) Vector 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 0 kx 0
From Dirac to Rashba Dirac at Beta=0 Rashba at Beta=1 0.0 0.02 1.0 0.2
Phase diagram Rashba Coupling 10^5 m/s 2.2 1.1 0 -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
GaAs Quantum Spin Hall • 2D electron motion in increasing radial electric • Inside a uniformly charged cylinder • Electrons with large g-factor:
Spin - • Spin - Quantum Spin Hall • Hamiltonian for electrons: • 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 • 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>=0d1=d2=0! In this case, the H becomes block-diagonal: LH HH SHE is topological quantized to be n/2p