the helical luttinger liquid and the edge of quantum spin hall systems
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1
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
2
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.
4
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.
5
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.
6
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
7
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.
8
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
9
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.
11
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.
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