Transcript Slide 1

Quantum anomalous Hall effect
(QAHE) and the quantum spin
Hall effect (QSHE)
Shoucheng Zhang, Stanford University
Les Houches, June 2006
References:
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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);
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(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 Spinorbital Coupling
Topological Quantization of the AHE
Hall Conductivity
Insulator Condition
Quantization
Rule
The Example
(cond-mat/0505308)
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
e
6
2
Bulk GaAs
(k F  k F )
H
L
• 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:
Energy (eV)
J j   spin  ijk Ek ,  spin 
i
e2 2
I  V / R or J j   E j where   k F l
h
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.
e
 s  0.1
a
Spin-Orbit Coupling – Spin 3/2 Systems
Luttinger Hamiltonian
(
• Symplectic symmetry structure
: spin-3/2 matrix)
Spin-Orbit Coupling – Spin 3/2 Systems
• Natural SU (4)  SO(5)  S 5 structure
SO(5) Vector
Matrices
• Inversion symmetric terms: d- wave
• Inversion asymmetric terms: p-wave
Strain:
Applied Rashba Field:
SO(5) Tensor
Matrices
Luttinger Model for spin Hall insulator
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)
Bulk Material
zero gap
Dirac Edge States
Edge 1
y
x
Edge 2
0
L
0
kx
From Dirac to Rashba
Dirac at Beta=0
Rashba at Beta=1
0.0
0.2
0.02
1.0
From Luttinger to Rashba
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
3d parameter space
boundary
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
charge
E
• Inside a uniformly charged cylinder
charge
• Electrons with large g-factor:
E  ar
GaAs
charge
E  ar
E  ar
Quantum Spin Hall
• Hamiltonian for electrons:
• Spin
-
B
Beffective
effective
• 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
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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.
iS y
T e
T e
2
2iS y
• T2=1 for spinless fermions and
T2=-1 for helical liquids.
T RT
1
  L T LT
1
  R 
T ( + R  L +  + L  R )T 1  ( + R  L +  + L  R )
• 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
=
Nonconserved
+
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/2