Quantum anomalous Hall effect (QAHE) and the quantum spin Hall effect (QSHE)

1
Quantum anomalous Hall effect (QAHE) and the
quantum spin Hall effect (QSHE)
Shoucheng Zhang, Stanford University
Les Houches, June 2006
2
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

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What about quantum spin Hall?
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Key ingredients of the quantum Hall effect

• Time reversal symmetry breaking.
• Bulk gap.
• Gapless chiral edge states.

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Topological Quantization of the AHE
(cond-mat/0505308)
Magnetic semiconductor with SO coupling (no
Landau levels)
General 22 Hamiltonian
Example
Rashbar Spin-orbital Coupling
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Topological Quantization of the AHE
(cond-mat/0505308)
Hall Conductivity
Insulator Condition
Quantization Rule
The Example
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Origin of Quantization Skyrmion in momentum space
Skyrmion number1
Skyrmion in lattice momentum space (torus) Edge
state due to monopole singularity
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Band structure on stripe geometry and topological
edge state
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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 Berrys phase in
  momentum space similar to the QHE.
• Contrast between the spin current and the Ohms
  law

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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.

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Spin-Orbit Coupling Spin 3/2 Systems
Luttinger Hamiltonian
( spin-3/2 matrix)

• Symplectic symmetry structure

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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
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Luttinger Model for spin Hall insulator
Bulk Material zero gap
l1/2,-1/2
l3/2,-3/2
Symmetric Quantum Well, z?-z mirror
symmetry Decoupled between (-1/2, 3/2) and (1/2,
-3/2)
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Dirac Edge States
Edge 1
y
x
Edge 2
L
0
kx
0
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From Dirac to Rashba
Dirac at Beta0 Rashba at Beta1
0.0
0.02
1.0
0.2
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From Luttinger to Rashba
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Phase diagram
Rashba Coupling 105 m/s
2.2
1.1
0
-1.1
-2.2
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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
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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
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The Quantum Hall Effect with Landau Levels
Spin Orbit Coupling in varying external
potential?
for
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Quantum Spin Hall

• 2D electron motion in increasing radial electric

• Inside a uniformly charged cylinder

• Electrons with large g-factor

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Quantum Spin Hall

• Hamiltonian for electrons

• Tune to R2

• No inversion symm, shear strain electric field
  (for SO coupling term)

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Quantum Spin Hall

• Different strain configurations create the
  different gauges in the Landau level problem

• Landau Gap and Strain Gradient

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Helical Liquid at the Edge

• P,T-invariant system

• QSH characterized by number n of fermion PAIRS on
  ONE edge. Non-chiral edges gt longitudinal charge
  conductance!

• Double Chern-Simons

(Zhang, Hansson, Kivelson) (Michael Freedman,
Chetan Nayak, Kirill Shtengel, Kevin Walker,
Zhenghan Wang)
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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 n1 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

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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.

• T21 for spinless fermions and T2-1 for helical
  liquids.

• Single particle backscattering is not possible
  for helical liquids!

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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.

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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.
Laughlins 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.
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Topological Quantization of Spin Hall

• Physical Understanding Edge states

When an electric field is applied, n edge states
with G121(-1) transfer from left (right) to
right (left).
G12 accumulation ? Spin accumulation
Conserved
Non-conserved


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Topological Quantization of SHE
Luttinger Hamiltonian rewritten as
In the presence of mirror symmetry z-gt-z,
ltkzgt0?d1d20! In this case, the H becomes
block-diagonal
LH
HH
SHE is topological quantized to be n/2p