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The Helical Luttinger Liquid and the Edge of Quantum Spin Hall Systems

Congjun Wu

Physics Department, Stanford University

Ref: C. Wu, B. A. Bernevig, and S. C. Zhang, Phys. Rev. Lett. 96, 106401 (2006).

March Meeting, 03/15/2006, 3:42pm.

B. Andrei Bernevig, and Shou-Cheng Zhang

Kavli Institute for Theoretical Physics, UCSB

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Introduction

• Quantum SHE systems: bulk is gapped; no charge current.

• Gapless edge modes support spin transport.

• Spin Hall effect (SHE): Use electric field to generate transverse spin current in spin-orbit (SO) coupling systems.

• Stability of the gapless edge modes against impurity, disorder under strong interactions.

F. D. M. Haldane, PRL 61, 2015 (1988); Kane et al., cond-mat/0411737, Phys. Rev. Lett. 95, 146802 (2005); B. A. Bernevig et al., cond-mat/0504147, to appear in PRL; X. L Qi et al., cond-mat/0505308; L. Sheng et al., PRL 95, 136602 (2005); D. N. Sheng et al, cond-mat/0603054.

S. Murakami et al., Science 301 (2003); J. Sinova et al. PRL 92, 126603 (2004).

Y. Kato et al., PRL 93, 176001 (2004); J. Wunderlich et al. PRL 94, 47204 (2005).

C. Wu, B. A. Bernevig, and S. C. Zhang, cond-mat/0508273, to appear in Phys. Rev. Lett;

C. Xu and J. Moore, PRB, 45322 (2006).

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QSHE edges: Helical Luttinger liquids (HLL)

• Edge modes are characterized by helicity.

• Right-movers with spin up, and left-movers with spin down:

• n-component HLL: n-branches of time-reversal pairs (T2=-1).

• HLL with an odd number of components are special.

chiral Luttinger liquids in quantum Hall edges break TR symmetry;

spinless non-chiral Luttinger liquids: T2=1;

non-chiral spinful Luttinger liquids have an even number of branches of TR pairs.

upper edge

lower edge

Kane et al., PRL 2005, Wu et al., PRL 2006.

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EF

The no-go theorem for helical Luttinger liquids

• 1D HLL with an odd number of components can NOT be constructed in a purely 1D lattice system.

H. B. Nielsen et al., Nucl Phys. B 185, 20 (1981); C. Wu et al., Phys. Rev. Lett. 96, 106401 (2006).

• Double degeneracy occurs at k=0 and .

• Periodicity of the Brillouin zone.

E

• HLL with an odd number of components can appear as the edge states of a 2-D system.

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Instability: the single-particle back-scattering

bgbgRLLRbg HTHTH

1

• Single particle backscattering term breaks TR symmetry (T2=-1).

• Kane and Mele : The non-interacting helical systems with an odd number of components remain gapless against disorder and impurity scatterings.

• The non-interacting Hamiltonian.

)(0

LxLRxRf iidxvH

not allowed

• However, with strong interactions, HLL can indeed open the gap from another mechanism.

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Two-particle correlated back-scattering

0)0()0()()()0,( 12212

connectedLLRR tttG

• TR symmetry allows two-particle correlated back-scattering.

• Effective Hamiltonian:

1

2

1

2

)()()()(or

),()()()(

jsisjsis

jsisjsisH

xyij

yx

yyij

xxum

• Microscopically, this Umklapp process can be generated from anisotropic spin-spin interactions.

zzyyxx ssssss ,,• U(1) rotation symmetry Z2.

..)()()()(4 chxxxxedxgH LLRR

kiuum

f

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Bosonization+Renormalization group

• Sine-Gordon theory at kf=/2.

16cos)(2

})()(1{2 2

220 a

gK

K

vdxH u

xx

LR

i

L

i

RLR ee )(;, 44

• If K<1/2 (strong repulsive interaction), the gap opens. Order parameters 2kf SDW orders Nx (gu<0) or Ny at (gu>0) .

4sin,4cos yx NN)2/1(4

1

1 )( Kuga

• At , TR symmetry much be restored by thermal fluctuations and the gap remains.

K0 T

• TR symmetry is spontaneously broken in the ground state.

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Random two-particle back-scattering

• Scattering amplitudes are quenched Gaussian variables.

))(16cos()(2

)(2int x

a

xgdxH u

• If K<3/8, gap opens. SDW order is spatially disordered but static in the time domain.

• TR symmetry is spontaneously broken.

• At small but finite temperatures, gap remains but TR is restored by thermal fluctuations.

DKtd

dDyxDeygexg yi

uxi

u )83(ln

)()()( )()(

Giamarchi, Quantum physics in one dimension, oxford press (2004).

)()( xiu exg

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Single impurity scattering

)16cos()()(2 2int

x

a

gdxH u

• If K<1/4, gu term is relevant. 1D line is divided into two segments.

• TR is restored by the instanton tunneling process.

C. Kane and M. P. A. Fisher, PRB 46, 15233 (1992).

• Boundary Sine-Gordon equation.

uu gKtd

dg)41(

ln

yxN ,

4sin,4cos yx NN

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Kondo problem: magnetic impurity scattering

)}()(2

){( //

LLRRzzRLLRK J

JxdxH

2//

2//

)21(ln

,2ln

JJKtd

dJ

Jtd

dJ

zz

z

• Poor man RG: critical coupling Jz is shifted by interactions.

• If K<1 (repulsive interaction), the Kondo singlet can form with ferromagnetic couplings.

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Summary

• Helical Luttinger liquid (HLL) as edge states of QSHE systems.• No-go theorem: HLL with odd number of components can not be constructed in a purely 1D lattice system.

• Instability problem: Two-particle correlated back-scattering is allowed by TR symmetry, and becomes relevant at:

Kc<1/2 for Umklapp scattering at commensurate fillings.

Kc<3/8 for random disorder scattering.

Kc<1/4 for a single impurity scattering.• Critical Kondo coupling Jz is shifted by interaction effects.