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

• 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:

lkh

ewhereEJorRVI Fjj

22

/

)(6

,2

LF

HFspinkijkspin

ij kk

eEJ

Bulk GaAs

Ene

rgy

(eV

)

Spin-Hall insulator: dissipationless spin transport without charge transport (PRL 93, 156804, 2004)

• In zero-gap semiconductors, such as HgTe, PbTe and -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.

a

es 1.0

Spin-Orbit Coupling – Spin 3/2 Systems

• Symplectic symmetry structure

Luttinger Hamiltonian

( : spin-3/2 matrix)

• Natural structure5(4) (5)SU SO S

SO(5) VectorMatrices

SO(5) TensorMatrices

• Inversion symmetric terms: d- wave

• Inversion asymmetric terms: p-wave

Spin-Orbit Coupling – Spin 3/2 Systems

Strain:

Applied Rashba Field:

Luttinger Model for spin Hall insulator

Bulk Materialzero gap

Symmetric Quantum Well, z-z mirror

symmetryDecoupled between (-1/2,

3/2) and (1/2, -3/2)

Dirac Edge States

Edge 1

Edge 2

0L

xy

kx0

From Dirac to RashbaDirac at Beta=0Rashba at Beta=1

0.0

0.2 1.0

0.02

From Luttinger to Rashba

Phase diagramRashba Coupling

10^5 m/s

0

1.1

2.2

-1.1

-2.2

• 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

Topology in QHE: U(1) Chern Number and Edge States

Niu, Thouless and Wu, PRB

Qi, Wu and Zhang, in progress

Non-vanishing Chern number Monopole in enlarged parameter space

Topology in QHE: Chern Number and Edge States

Gapless Edge States in the twisted Hamiltonian

3d parameter space

MonopoleGapless point

boundary

The Quantum Hall Effect with Landau Levels

Spin – Orbit Coupling in varying external potential?

for

• 2D electron motion in increasing radial electric

raE charge

raE charge

GaAs

E• Inside a uniformly charged

cylinder

raE charge

• Electrons with large g-factor:

Quantum Spin Hall

• Hamiltonian for electrons:

• Tune to R=2

Quantum Spin Hall

• Spin - effectiveB• Spin - effectiveB

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

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

• Landau Gap and Strain Gradient

Quantum Spin Hall

• P,T-invariant system

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

Helical Liquid at the Edge

• 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.

yy SiSi eTeT 22 • T2=1 for spinless fermions and

T2=-1 for helical liquids.

)()( 1

11

RLLRRLLR

RLLR

TT

TTTT

• Single particle backscattering is not possible for helical liquids!

• 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.

Conclusions

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.

Energy spectrum on stripe geometry.

Laughlin’s Gauge Argument:

When turning on a flux threading a cylinder system, the edge states will transfer from one edge to another

Topological Quantization of Spin Hall • Physical Understanding: Edge states

When an electric field is applied, n edge states with transfer from left (right) to right (left).

accumulation Spin accumulation

Conserved Non-conserved

+=

Topological Quantization of SHE

LH

HH

SHE is topological quantized to be n/2

Luttinger Hamiltonian rewritten as

In the presence of mirror symmetry z->-z, <kz>=0d1=d2=0! In this case, the H becomes block-diagonal: