The Helical Luttinger Liquid and the Edge of
Quantum Spin Hall Systems
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
B. Andrei Bernevig, and Shou-Cheng Zhang
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.
1
Introduction
• Spin Hall effect (SHE): Use electric field to generate transverse
spin current in spin-orbit (SO) coupling systems.
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).
• Quantum SHE systems: bulk is gapped; no charge current.
• Gapless edge modes support spin transport.
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., condmat/0505308; L. Sheng et al., PRL 95, 136602 (2005); D. N. Sheng et al, cond-mat/0603054.
• Stability of the gapless edge modes against impurity,
disorder under strong interactions.
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).
2
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).
upper edge
lower edge
• 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. Kane et al., PRL 2005, Wu et al., PRL 2006.
3
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.
E
EF
-p
• Double degeneracy occurs at
k=0 and p.
• Periodicity of the Brillouin zone.
p
• HLL with an odd number of
components can appear as the
edge states of a 2-D system.
H. B. Nielsen et al., Nucl Phys. B 185, 20 (1981); C. Wu et al., Phys. Rev. Lett. 96, 106401 (2006).
4
Instability: the single-particle back-scattering
• The non-interacting Hamiltonian.
H 0  v f  dx( Ri x R - Li x L )
• Kane and Mele : The non-interacting
helical systems with an odd number of
components remain gapless against
disorder and impurity scatterings.
not
allowed
• Single particle backscattering term breaks TR symmetry (T2=-1).
H bg   R L  L R
T -1H bgT  - H bg
• However, with strong interactions, HLL can indeed open the
gap from another mechanism.
5
Two-particle correlated back-scattering
• TR symmetry allows two-particle correlated back-scattering.
1
1
2
2
G2 (t ,0)   R1 (t ) R2 (t )  L2 (0) L1 (0)
connected
0
H um   s x (i ) s x ( j ) -s y (i ) s y ( j ),
ij
or  s x (i ) s y ( j ) s y (i ) s x ( j )
ij
• Microscopically, this Umklapp process can be generated
from anisotropic spin-spin interactions.
• Effective Hamiltonian:
H um  g u  dx e f  R ( x) R ( x   ) L ( x   ) L ( x)  h.c.
• U(1) rotation symmetry  Z2.
i 4k
sx  -sx , s y  -s y , sz  sz
6
Bosonization+Renormalization group
• Sine-Gordon theory at kf=p/2.
 R  ei
4p  R
,  L  e
- i 4p  L
;  ( )  R  L
gu
v 1
2
2
H 0   dx { ( x )  K ( x ) } 
cos 16p 
2
2 K
2(pa)
• If K<1/2 (strong repulsive interaction), the gap D opens.
Order parameters 2kf SDW orders Nx (gu<0) or Ny at (gu>0) .
D  a -1 ( gu )
1
4 (1/ 2- K )
N x  cos 4p  , N y  sin 4p 
• TR symmetry is spontaneously broken in the ground state.
• At D  T  0K , TR symmetry much be restored by thermal
fluctuations and the gap remains.
7
Random two-particle back-scattering
i ( x )
• Scattering amplitudes g u ( x)e
are quenched Gaussian
variables.
H int   dx
g u ( x)
cos( 16p    ( x))
2
2(pa)
g u ( x)e i ( x ) g u ( y )e -i ( y )  D ( x - y )
dD
 (3 - 8 K ) D
d ln t
Giamarchi, Quantum physics in one dimension, oxford press (2004).
• If K<3/8, gap D 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.
8
Single impurity scattering
• Boundary Sine-Gordon equation.
H int
gu
  dx
 ( x) cos( 16p  )
2
2(pa)
dg u
 (1 - 4 K ) g u
d ln t
C. Kane and M. P. A. Fisher, PRB 46, 15233 (1992).
• If K<1/4, gu term is relevant. 1D line is divided into two
segments.
• TR is restored by the instanton tunneling process.
N x, y

N x  cos 4p  , N y  sin 4p 
9
Kondo problem: magnetic impurity scattering
J //
H K   dx  ( x){ ( - R L    L R )  J z z ( R R - L L )}
2
• Poor man RG: critical coupling Jz is shifted by interactions.
• If K<1 (repulsive interaction), the Kondo singlet can form
with ferromagnetic couplings.
dJ z
 2 J //2 ,
d ln t
dJ z
 (1 - K  2 J z ) J //2
d ln t
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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.
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